An integral equation formulation for predicting radiation patterns of a space shuttle annular slot antenna

NASA Technical Reports Server (NTRS)

Jones, J. E.; Richmond, J. H.

1974-01-01

An integral equation formulation is applied to predict pitch- and roll-plane radiation patterns of a thin VHF/UHF (very high frequency/ultra high frequency) annular slot communications antenna operating at several locations in the nose region of the space shuttle orbiter. Digital computer programs used to compute radiation patterns are given and the use of the programs is illustrated. Experimental verification of computed patterns is given from measurements made on 1/35-scale models of the orbiter.

Numerical solution of boundary-integral equations for molecular electrostatics.

PubMed

Bardhan, Jaydeep P

2009-03-07

Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.

An integrable family of Monge-Ampère equations and their multi-Hamiltonian structure

NASA Astrophysics Data System (ADS)

Nutku, Y.; Sarioǧlu, Ö.

1993-02-01

We have identified a completely integrable family of Monge-Ampère equations through an examination of their Hamiltonian structure. Starting with a variational formulation of the Monge-Ampère equations we have constructed the first Hamiltonian operator through an application of Dirac's theory of constraints. The completely integrable class of Monge-Ampère equations are then obtained by solving the Jacobi identities for a sufficiently general form of the second Hamiltonian operator that is compatible with the first.

Computation of Sound Propagation by Boundary Element Method

NASA Technical Reports Server (NTRS)

Guo, Yueping

2005-01-01

This report documents the development of a Boundary Element Method (BEM) code for the computation of sound propagation in uniform mean flows. The basic formulation and implementation follow the standard BEM methodology; the convective wave equation and the boundary conditions on the surfaces of the bodies in the flow are formulated into an integral equation and the method of collocation is used to discretize this equation into a matrix equation to be solved numerically. New features discussed here include the formulation of the additional terms due to the effects of the mean flow and the treatment of the numerical singularities in the implementation by the method of collocation. The effects of mean flows introduce terms in the integral equation that contain the gradients of the unknown, which is undesirable if the gradients are treated as additional unknowns, greatly increasing the sizes of the matrix equation, or if numerical differentiation is used to approximate the gradients, introducing numerical error in the computation. It is shown that these terms can be reformulated in terms of the unknown itself, making the integral equation very similar to the case without mean flows and simple for numerical implementation. To avoid asymptotic analysis in the treatment of numerical singularities in the method of collocation, as is conventionally done, we perform the surface integrations in the integral equation by using sub-triangles so that the field point never coincide with the evaluation points on the surfaces. This simplifies the formulation and greatly facilitates the implementation. To validate the method and the code, three canonic problems are studied. They are respectively the sound scattering by a sphere, the sound reflection by a plate in uniform mean flows and the sound propagation over a hump of irregular shape in uniform flows. The first two have analytical solutions and the third is solved by the method of Computational Aeroacoustics (CAA), all of which are used to compare the BEM solutions. The comparisons show very good agreements and validate the accuracy of the BEM approach implemented here.

Equations of motion of slung load systems with results for dual lift

NASA Technical Reports Server (NTRS)

Cicolani, Luigi S.; Kanning, Gerd

1990-01-01

General simulation equations are derived for the rigid body motion of slung load systems. These systems are viewed as consisting of several rigid bodies connected by straight-line cables or links. The suspension can be assumed to be elastic or inelastic, both cases being of interest in simulation and control studies. Equations for the general system are obtained via D'Alembert's principle and the introduction of generalized velocity coordinates. Three forms are obtained. Two of these generalize previous case-specific results for single helicopter systems with elastic or inelastic suspensions. The third is a new formulation for inelastic suspensions. It is derived from the elastic suspension equations by choosing the generalized coordinates so as to separate motion due to cable stretching from motion with invariant cable lengths. The result is computationally more efficient than the conventional formulation, and is readily integrated with the elastic suspension formulation and readily applied to the complex dual lift and multilift systems. Equations are derived for dual lift systems. Three proposed suspension arrangements can be integrated in a single equation set. The equations are given in terms of the natural vectors and matrices of three-dimensional rigid body mechanics and are tractable for both analysis and programming.

A boundary integral approach in primitive variables for free surface flows

NASA Astrophysics Data System (ADS)

Casciola, C.; Piva, R.

The boundary integral formulation, very efficient for free surface potential flows, was considered for its possible extension to rotational flows either inviscid or viscous. We first analyze a general formulation for unsteady Navier-Stokes equations in primitive variables, which reduces to a representation for the Euler equations in the limiting case of Reynolds infinity. A first simplified model for rotational flows, obtained by decoupling kinematics and dynamics, reduces the integral equations to a known kinematical form whose mathematical and numerical properties have been studied. The dynamics equations to complete the model are obtained for the free surface and the wake. A simple and efficient scheme for the study of the non linear evolution of the wave system and its interaction with the body wake is presented. A steady state version for the calculation of the wave resistance is also reported. A second model was proposed for the simulation of rotational separated regions, by coupling the integral equations in velocity with an integral equation for the vorticity at the body boundary. The same procedure may be extended to include the diffusion of the vorticity in the flowfield. The vortex shedding from a cylindrical body in unsteady motion is discussed, as a first application of the model.

A momentum-space formulation without partial wave decomposition for scattering of two spin-half particles

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fachruddin, Imam, E-mail: imam.fachruddin@sci.ui.ac.id; Salam, Agus

2016-03-11

A new momentum-space formulation for scattering of two spin-half particles, both either identical or unidentical, is formulated. As basis states the free linear-momentum states are not expanded into the angular-momentum states, the system’s spin states are described by the product of the spin states of the two particles, and the system’s isospin states by the total isospin states of the two particles. We evaluate the Lippmann-Schwinger equations for the T-matrix elements in these basis states. The azimuthal behavior of the potential and of the T-matrix elements leads to a set of coupled integral equations for the T-matrix elements in twomore » variables only, which are the magnitude of the relative momentum and the scattering angle. Some symmetry relations for the potential and the T-matrix elements reduce the number of the integral equations to be solved. A set of six spin operators to express any interaction of two spin-half particles is introduced. We show the spin-averaged differential cross section as being calculated in terms of the solution of the set of the integral equations.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Manjunath, Naren; Samajdar, Rhine; Jain, Sudhir R., E-mail: srjain@barc.gov.in

Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in Samajdar and Jain (2014). The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.

Variational estimate method for solving autonomous ordinary differential equations

NASA Astrophysics Data System (ADS)

Mungkasi, Sudi

2018-04-01

In this paper, we propose a method for solving first-order autonomous ordinary differential equation problems using a variational estimate formulation. The variational estimate is constructed with a Lagrange multiplier which is chosen optimally, so that the formulation leads to an accurate solution to the problem. The variational estimate is an integral form, which can be computed using a computer software. As the variational estimate is an explicit formula, the solution is easy to compute. This is a great advantage of the variational estimate formulation.

The unified acoustic and aerodynamic prediction theory of advanced propellers in the time domain

NASA Technical Reports Server (NTRS)

Farassat, F.

1984-01-01

This paper presents some numerical results for the noise of an advanced supersonic propeller based on a formulation published last year. This formulation was derived to overcome some of the practical numerical difficulties associated with other acoustic formulations. The approach is based on the Ffowcs Williams-Hawkings equation and time domain analysis is used. To illustrate the method of solution, a model problem in three dimensions and based on the Laplace equation is solved. A brief sketch of derivation of the acoustic formula is then given. Another model problem is used to verify validity of the acoustic formulation. A recent singular integral equation for aerodynamic applications derived from the acoustic formula is also presented here.

Advanced development of BEM for elastic and inelastic dynamic analysis of solids

NASA Technical Reports Server (NTRS)

Banerjee, P. K.; Ahmad, S.; Wang, H. C.

1989-01-01

Direct Boundary Element formulations and their numerical implementation for periodic and transient elastic as well as inelastic transient dynamic analyses of two-dimensional, axisymmetric and three-dimensional solids are presented. The inelastic formulation is based on an initial stress approach and is the first of its kind in the field of Boundary Element Methods. This formulation employs the Navier-Cauchy equation of motion, Graffi's dynamic reciprocal theorem, Stokes' fundamental solution, and the divergence theorem, together with kinematical and constitutive equations to obtain the pertinent integral equations of the problem in the time domain within the context of the small displacement theory of elastoplasticity. The dynamic (periodic, transient as well as nonlinear transient) formulations have been applied to a range of problems. The numerical formulations presented here are included in the BEST3D and GPBEST systems.

A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method.

PubMed

Chen, I L; Chen, J T; Kuo, S R; Liang, M T

2001-03-01

Integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems (radiation and scattering). It was recently found that the real-part boundary element method (BEM) for the interior problem results in spurious eigensolutions if the singular (UT) or the hypersingular (LM) equation is used alone. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation (UT method) results in fictitious solutions for the exterior problem. To solve this problem, a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) is proposed. Based on the CHEEF method, the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully. Finally, two examples for the eigensolutions of circular and rectangular cavities are considered. The optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied. Also, numerical experiments were designed to verify the analytical results. It is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry, while the nodal line of the rectangular is on a fixed position.

The Transmission Line as a Simple Example for Introducing Integral Equations to Undergraduates

ERIC Educational Resources Information Center

Rothwell, E. J.

2009-01-01

Integral equations are becoming a common means for describing problems in electromagnetics, and so it is important to expose students to methods for their solution. Typically this is done using examples in antennas, scattering, or electrostatics. Unfortunately, many difficult issues arise in the formulation and solution of the associated…

Modeling erosion and sedimentation coupled with hydrological and overland flow processes at the watershed scale

NASA Astrophysics Data System (ADS)

Kim, Jongho; Ivanov, Valeriy Y.; Katopodes, Nikolaos D.

2013-09-01

A novel two-dimensional, physically based model of soil erosion and sediment transport coupled to models of hydrological and overland flow processes has been developed. The Hairsine-Rose formulation of erosion and deposition processes is used to account for size-selective sediment transport and differentiate bed material into original and deposited soil layers. The formulation is integrated within the framework of the hydrologic and hydrodynamic model tRIBS-OFM, Triangulated irregular network-based, Real-time Integrated Basin Simulator-Overland Flow Model. The integrated model explicitly couples the hydrodynamic formulation with the advection-dominated transport equations for sediment of multiple particle sizes. To solve the system of equations including both the Saint-Venant and the Hairsine-Rose equations, the finite volume method is employed based on Roe's approximate Riemann solver on an unstructured grid. The formulation yields space-time dynamics of flow, erosion, and sediment transport at fine scale. The integrated model has been successfully verified with analytical solutions and empirical data for two benchmark cases. Sensitivity tests to grid resolution and the number of used particle sizes have been carried out. The model has been validated at the catchment scale for the Lucky Hills watershed located in southeastern Arizona, USA, using 10 events for which catchment-scale streamflow and sediment yield data were available. Since the model is based on physical laws and explicitly uses multiple types of watershed information, satisfactory results were obtained. The spatial output has been analyzed and the driving role of topography in erosion processes has been discussed. It is expected that the integrated formulation of the model has the promise to reduce uncertainties associated with typical parameterizations of flow and erosion processes. A potential for more credible modeling of earth-surface processes is thus anticipated.

Finite element techniques for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation

NASA Technical Reports Server (NTRS)

Glaisner, F.; Tezduyar, T. E.

1987-01-01

Finite element procedures for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation have been implemented. For both formulations, streamline-upwind/Petrov-Galerkin techniques are used for the discretization of the transport equations. The main problem associated with the vorticity stream-function formulation is the lack of boundary conditions for vorticity at solid surfaces. Here an implicit treatment of the vorticity at no-slip boundaries is incorporated in a predictor-multicorrector time integration scheme. For the primitive variable formulation, mixed finite-element approximations are used. A nine-node element and a four-node + bubble element have been implemented. The latter is shown to exhibit a checkerboard pressure mode and a numerical treatment for this spurious pressure mode is proposed. The two methods are compared from the points of view of simulating internal and external flows and the possibilities of extensions to three dimensions.

Integrated force method versus displacement method for finite element analysis

NASA Technical Reports Server (NTRS)

Patnaik, S. N.; Berke, L.; Gallagher, R. H.

1991-01-01

A novel formulation termed the integrated force method (IFM) has been developed in recent years for analyzing structures. In this method all the internal forces are taken as independent variables, and the system equilibrium equations (EEs) are integrated with the global compatibility conditions (CCs) to form the governing set of equations. In IFM the CCs are obtained from the strain formulation of St. Venant, and no choices of redundant load systems have to be made, in constrast to the standard force method (SFM). This property of IFM allows the generation of the governing equation to be automated straightforwardly, as it is in the popular stiffness method (SM). In this report IFM and SM are compared relative to the structure of their respective equations, their conditioning, required solution methods, overall computational requirements, and convergence properties as these factors influence the accuracy of the results. Overall, this new version of the force method produces more accurate results than the stiffness method for comparable computational cost.

Integrated force method versus displacement method for finite element analysis

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Berke, Laszlo; Gallagher, Richard H.

1990-01-01

A novel formulation termed the integrated force method (IFM) has been developed in recent years for analyzing structures. In this method all the internal forces are taken as independent variables, and the system equilibrium equations (EE's) are integrated with the global compatibility conditions (CC's) to form the governing set of equations. In IFM the CC's are obtained from the strain formulation of St. Venant, and no choices of redundant load systems have to be made, in constrast to the standard force method (SFM). This property of IFM allows the generation of the governing equation to be automated straightforwardly, as it is in the popular stiffness method (SM). In this report IFM and SM are compared relative to the structure of their respective equations, their conditioning, required solution methods, overall computational requirements, and convergence properties as these factors influence the accuracy of the results. Overall, this new version of the force method produces more accurate results than the stiffness method for comparable computational cost.

Energy, momentum, and angular momentum of sound pulses.

PubMed

Lekner, John

2017-12-01

Pulse solutions of the wave equation can be expressed as superpositions of scalar monochromatic beam wavefunctions (solutions of the Helmholtz equation). This formulation leads to causal (unidirectional) propagation, in contrast to all currently known closed-form solutions of the wave equation. Application is made to the evaluation of the energy, momentum, and angular momentum of acoustic pulses, as integrals over the beam and pulse weight functions. Equivalence is established between integration over space of the energy, momentum, and angular momentum densities, and integration over the wavevector weight function. The inequality linking the total energy and the total momentum is made explicit in terms of the weight function formulation. It is shown that a general pulse can be viewed as a superposition of phonons, each with energy ℏck, z component of momentum ℏq, and z component of angular momentum ℏm. A closed-form solution of the wave equation is found, which is localized and causal, and its energy and momentum are evaluated explicitly.

Dual boundary element formulation for elastoplastic fracture mechanics

NASA Astrophysics Data System (ADS)

Leitao, V.; Aliabadi, M. H.; Rooke, D. P.

1995-01-01

In this paper the extension of the dual boundary element method (DBEM) to the analysis of elastoplastic fracture mechanics (EPFM) problems is presented. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. In order to avoid collocation at crack tips, crack kinks and crack-edge corners, both crack surfaces are discretized with discontinuous quadratic boundary elements. The elasto-plastic behavior is modelled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield. This region is discretized with internal quadratic, quadrilateral and/or triangular cells. This formulation was implemented for two-dimensional domains only, although there is no theoretical or numerical limitation to its application to three-dimensional ones. A center-cracked plate and a slant edge-cracked plate subjected to tensile load are analysed and the results are compared with others available in the literature. J-type integrals are calculated.

An integral equation formulation for rigid bodies in Stokes flow in three dimensions

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Greengard, Leslie; Rachh, Manas; Veerapaneni, Shravan

2017-03-01

We present a new derivation of a boundary integral equation (BIE) for simulating the three-dimensional dynamics of arbitrarily-shaped rigid particles of genus zero immersed in a Stokes fluid, on which are prescribed forces and torques. Our method is based on a single-layer representation and leads to a simple second-kind integral equation. It avoids the use of auxiliary sources within each particle that play a role in some classical formulations. We use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials, so that only a small number of spatial discretization points per particle are required. The resulting discrete sums are computed in O (n) time, where n denotes the number of particles, using the fast multipole method (FMM). The particle positions and orientations are updated by a high-order time-stepping scheme. We illustrate the accuracy, conditioning and scaling of our solvers with several numerical examples.

Normal compression wave scattering by a permeable crack in a fluid-saturated poroelastic solid

NASA Astrophysics Data System (ADS)

Song, Yongjia; Hu, Hengshan; Rudnicki, John W.

2017-04-01

A mathematical formulation is presented for the dynamic stress intensity factor (mode I) of a finite permeable crack subjected to a time-harmonic propagating longitudinal wave in an infinite poroelastic solid. In particular, the effect of the wave-induced fluid flow due to the presence of a liquid-saturated crack on the dynamic stress intensity factor is analyzed. Fourier sine and cosine integral transforms in conjunction with Helmholtz potential theory are used to formulate the mixed boundary-value problem as dual integral equations in the frequency domain. The dual integral equations are reduced to a Fredholm integral equation of the second kind. It is found that the stress intensity factor monotonically decreases with increasing frequency, decreasing the fastest when the crack width and the slow wave wavelength are of the same order. The characteristic frequency at which the stress intensity factor decays the fastest shifts to higher frequency values when the crack width decreases.

Equations of motion of slung-load systems, including multilift systems

NASA Technical Reports Server (NTRS)

Cicolani, Luigi S.; Kanning, Gerd

1992-01-01

General simulation equations are derived for the rigid body motion of slung-load systems. This work is motivated by an interest in trajectory control for slung loads carried by two or more helicopters. An approximation of these systems consists of several rigid bodies connected by straight-line cables or links. The suspension can be assumed elastic or inelastic. Equations for the general system are obtained from the Newton-Euler rigid-body equations with the introduction of generalized velocity coordinates. Three forms are obtained: two generalize previous case-specific results for single-helicopter systems with elastic and inelastic suspensions, respectively; and the third is a new formulation for inelastic suspensions. The latter is derived from the elastic suspension equations by choosing the generalized coordinates so that motion induced by cable stretching is separated from motion with invariant cable lengths, and by then nulling the stretching coordinates to get a relation for the suspension forces. The result is computationally more efficient than the conventional formulation, is readily integrated with the elastic suspension formulation, and is easily applied to the complex dual-lift and multilift systems. Results are given for two-helicopter systems; three configurations are included and these can be integrated in a single simulation. Equations are also given for some single-helicopter systems, for comparison with the previous literature, and for a multilift system. Equations for degenerate-body approximations (point masses, rigid rods) are also formulated and results are given for dual-lift and multilift systems. Finally, linearlized equations of motion are given for general slung-load systems are presented along with results for the two-helicopter system with a spreader bar.

Spinning particle and gauge theories as integrability conditions

NASA Astrophysics Data System (ADS)

Eisenberg, Yeshayahu

1992-02-01

Starting from a new four dimensional spinning point particle we obtain new representations of the standard four dimensional gauge field equations in terms of a generalized space (Minkowski + light cone). In terms of this new formulation we define linear systems whose integrability conditions imply the massive Dirac-Maxwell and the Yang-Mills equations. Research supported by the Rothschild Fellowship.

Efficient Solution of Three-Dimensional Problems of Acoustic and Electromagnetic Scattering by Open Surfaces

NASA Technical Reports Server (NTRS)

Turc, Catalin; Anand, Akash; Bruno, Oscar; Chaubell, Julian

2011-01-01

We present a computational methodology (a novel Nystrom approach based on use of a non-overlapping patch technique and Chebyshev discretizations) for efficient solution of problems of acoustic and electromagnetic scattering by open surfaces. Our integral equation formulations (1) Incorporate, as ansatz, the singular nature of open-surface integral-equation solutions, and (2) For the Electric Field Integral Equation (EFIE), use analytical regularizes that effectively reduce the number of iterations required by iterative linear-algebra solution based on Krylov-subspace iterative solvers.

Oblique scattering from radially inhomogeneous dielectric cylinders: An exact Volterra integral equation formulation

NASA Astrophysics Data System (ADS)

Tsalamengas, John L.

2018-07-01

We study plane-wave electromagnetic scattering by radially and strongly inhomogeneous dielectric cylinders at oblique incidence. The method of analysis relies on an exact reformulation of the underlying field equations as a first-order 4 × 4 system of differential equations and on the ability to restate the associated initial-value problem in the form of a system of coupled linear Volterra integral equations of the second kind. The integral equations so derived are discretized via a sophisticated variant of the Nyström method. The proposed method yields results accurate up to machine precision without relying on approximations. Numerical results and case studies ably demonstrate the efficiency and high accuracy of the algorithms.

An efficient algorithm for the generalized Foldy-Lax formulation

NASA Astrophysics Data System (ADS)

Huang, Kai; Li, Peijun; Zhao, Hongkai

2013-02-01

Consider the scattering of a time-harmonic plane wave incident on a two-scale heterogeneous medium, which consists of scatterers that are much smaller than the wavelength and extended scatterers that are comparable to the wavelength. In this work we treat those small scatterers as isotropic point scatterers and use a generalized Foldy-Lax formulation to model wave propagation and capture multiple scattering among point scatterers and extended scatterers. Our formulation is given as a coupled system, which combines the original Foldy-Lax formulation for the point scatterers and the regular boundary integral equation for the extended obstacle scatterers. The existence and uniqueness of the solution for the formulation is established in terms of physical parameters such as the scattering coefficient and the separation distances. Computationally, an efficient physically motivated Gauss-Seidel iterative method is proposed to solve the coupled system, where only a linear system of algebraic equations for point scatterers or a boundary integral equation for a single extended obstacle scatterer is required to solve at each step of iteration. The convergence of the iterative method is also characterized in terms of physical parameters. Numerical tests for the far-field patterns of scattered fields arising from uniformly or randomly distributed point scatterers and single or multiple extended obstacle scatterers are presented.

Application of boundary integral equations to elastoplastic problems

NASA Technical Reports Server (NTRS)

Mendelson, A.; Albers, L. U.

1975-01-01

The application of boundary integral equations to elastoplastic problems is reviewed. Details of the analysis as applied to torsion problems and to plane problems is discussed. Results are presented for the elastoplastic torsion of a square cross section bar and for the plane problem of notched beams. A comparison of different formulations as well as comparisons with experimental results are presented.

A new formulation of electromagnetic wave scattering using an on-surface radiation boundary condition approach

NASA Technical Reports Server (NTRS)

Kriegsmann, Gregory A.; Taflove, Allen; Umashankar, Koradar R.

1987-01-01

A new formulation of electromagnetic wave scattering by convex, two-dimensional conducting bodies is reported. This formulation, called the on-surface radiation condition (OSRC) approach, is based upon an expansion of the radiation condition applied directly on the surface of a scatterer. It is now shown that application of a suitable radiation condition directly on the surface of a convex conducting scatterer can lead to substantial simplification of the frequency-domain integral equation for the scattered field, which is reduced to just a line integral. For the transverse magnetic case, the integrand is known explicitly. For the transverse electric case, the integrand can be easily constructed by solving an ordinary differential equation around the scatterer surface contour. Examples are provided which show that OSRC yields computed near and far fields which approach the exact results for canonical shapes such as the circular cylinder, square cylinder, and strip. Electrical sizes for the examples are ka = 5 and ka = 10. The new OSRC formulation of scattering may present a useful alternative to present integral equation and uniform high-frequency approaches for convex cylinders larger than ka = 1. Structures with edges or corners can also be analyzed, although more work is needed to incorporate the physics of singular currents at these discontinuities. Convex dielectric structures can also be treated using OSRC.

Methods of Attenuation Correction for Dual-Wavelength and Dual-Polarization Weather Radar Data

NASA Technical Reports Server (NTRS)

Meneghini, R.; Liao, L.

2007-01-01

In writing the integral equations for the median mass diameter and number concentration, or comparable parameters of the raindrop size distribution, it is apparent that the forms of the equations for dual-polarization and dual-wavelength radar data are identical when attenuation effects are included. The differential backscattering and extinction coefficients appear in both sets of equations: for the dual-polarization equations, the differences are taken with respect to polarization at a fixed frequency while for the dual-wavelength equations, the differences are taken with respect to frequency at a fixed polarization. An alternative to the integral equation formulation is that based on the k-Z (attenuation coefficient-radar reflectivity factor) parameterization. This-technique was originally developed for attenuating single-wavelength radars, a variation of which has been applied to the TRMM Precipitation Radar data (PR). Extensions of this method have also been applied to dual-polarization data. In fact, it is not difficult to show that nearly identical equations are applicable as well to dualwavelength radar data. In this case, the equations for median mass diameter and number concentration take the form of coupled, but non-integral equations. Differences between this and the integral equation formulation are a consequence of the different ways in which attenuation correction is performed under the two formulations. For both techniques, the equations can be solved either forward from the radar outward or backward from the final range gate toward the radar. Although the forward-going solutions tend to be unstable as the attenuation out to the range of interest becomes large in some sense, an independent estimate of path attenuation is not required. This is analogous to the case of an attenuating single-wavelength radar where the forward solution to the Hitschfeld-Bordan equation becomes unstable as the attenuation increases. To circumvent this problem, the equations can be expressed in the form of a final-value problem so that the recursion begins at the far range gate and proceeds inward towards the radar. Solving the problem in this way traditionally requires estimates of path attenuation to the final gate: in the case of orthogonal linear polarizations, the attenuations at horizontal and vertical polarizations (same frequency) are required while in the dual-wavelength case, attenuations at the two frequencies (same polarization) are required.

Alternative bi-Hamiltonian structures for WDVV equations of associativity

NASA Astrophysics Data System (ADS)

Kalayci, J.; Nutku, Y.

1998-01-01

The WDVV equations of associativity in two-dimensional topological field theory are completely integrable third-order Monge-Ampère equations which admit bi-Hamiltonian structure. The time variable plays a distinguished role in the discussion of Hamiltonian structure, whereas in the theory of WDVV equations none of the independent variables merits such a distinction. WDVV equations admit very different alternative Hamiltonian structures under different possible choices of the time variable, but all these various Hamiltonian formulations can be brought together in the framework of the covariant theory of symplectic structure. They can be identified as different components of the covariant Witten-Zuckerman symplectic 2-form current density where a variational formulation of the WDVV equation that leads to the Hamiltonian operator through the Dirac bracket is available.

Identification of Spurious Signals from Permeable Ffowcs Williams and Hawkings Surfaces

NASA Technical Reports Server (NTRS)

Lopes, Leonard V.; Boyd, David D., Jr.; Nark, Douglas M.; Wiedemann, Karl E.

2017-01-01

Integral forms of the permeable surface formulation of the Ffowcs Williams and Hawkings (FW-H) equation often require an input in the form of a near field Computational Fluid Dynamics (CFD) solution to predict noise in the near or far field from various types of geometries. The FW-H equation involves three source terms; two surface terms (monopole and dipole) and a volume term (quadrupole). Many solutions to the FW-H equation, such as several of Farassat's formulations, neglect the quadrupole term. Neglecting the quadrupole term in permeable surface formulations leads to inaccuracies called spurious signals. This paper explores the concept of spurious signals, explains how they are generated by specifying the acoustic and hydrodynamic surface properties individually, and provides methods to determine their presence, regardless of whether a correction algorithm is employed. A potential approach based on the equivalent sources method (ESM) and the sensitivity of Formulation 1A (Formulation S1A) is also discussed for the removal of spurious signals.

Evaluation of the path integral for flow through random porous media

NASA Astrophysics Data System (ADS)

Westbroek, Marise J. E.; Coche, Gil-Arnaud; King, Peter R.; Vvedensky, Dimitri D.

2018-04-01

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multivariate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

Gaussian-windowed frame based method of moments formulation of surface-integral-equation for extended apertures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shlivinski, A., E-mail: amirshli@ee.bgu.ac.il; Lomakin, V., E-mail: vlomakin@eng.ucsd.edu

2016-03-01

Scattering or coupling of electromagnetic beam-field at a surface discontinuity separating two homogeneous or inhomogeneous media with different propagation characteristics is formulated using surface integral equation, which are solved by the Method of Moments with the aid of the Gabor-based Gaussian window frame set of basis and testing functions. The application of the Gaussian window frame provides (i) a mathematically exact and robust tool for spatial-spectral phase-space formulation and analysis of the problem; (ii) a system of linear equations in a transmission-line like form relating mode-like wave objects of one medium with mode-like wave objects of the second medium; (iii)more » furthermore, an appropriate setting of the frame parameters yields mode-like wave objects that blend plane wave properties (as if solving in the spectral domain) with Green's function properties (as if solving in the spatial domain); and (iv) a representation of the scattered field with Gaussian-beam propagators that may be used in many large (in terms of wavelengths) systems.« less

On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation.

PubMed

Khusnutdinova, K R; Klein, C; Matveev, V B; Smirnov, A O

2013-03-01

There exist two versions of the Kadomtsev-Petviashvili (KP) equation, related to the Cartesian and cylindrical geometries of the waves. In this paper, we derive and study a new version, related to the elliptic cylindrical geometry. The derivation is given in the context of surface waves, but the derived equation is a universal integrable model applicable to generic weakly nonlinear weakly dispersive waves. We also show that there exist nontrivial transformations between all three versions of the KP equation associated with the physical problem formulation, and use them to obtain new classes of approximate solutions for water waves.

The dual boundary element formulation for elastoplastic fracture mechanics

NASA Astrophysics Data System (ADS)

Leitao, V.; Aliabadi, M. H.; Rooke, D. P.

1993-08-01

The extension of the dual boundary element method (DBEM) to the analysis of elastoplastic fracture mechanics (EPFM) problems is presented. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied to one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. In order to avoid collocation at crack tips, crack kinks, and crack-edge corners, both crack surfaces are discretized with discontinuous quadratic boundary elements. The elastoplastic behavior is modeled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield. This region is discretized with internal quadratic, quadrilateral, and/or triangular cells. A center-cracked plate and a slant edge-cracked plate subjected to tensile load are analyzed and the results are compared with others available in the literature. J-type integrals are calculated.

A new aerodynamic integral equation based on an acoustic formula in the time domain

NASA Technical Reports Server (NTRS)

Farassat, F.

1984-01-01

An aerodynamic integral equation for bodies moving at transonic and supersonic speeds is presented. Based on a time-dependent acoustic formula for calculating the noise emanating from the outer portion of a propeller blade travelling at high speed (the Ffowcs Williams-Hawking formulation), the loading terms and a conventional thickness source terms are retained. Two surface and three line integrals are employed to solve an equation for the loading noise. The near-field term is regularized using the collapsing sphere approach to obtain semiconvergence on the blade surface. A singular integral equation is thereby derived for the unknown surface pressure, and is amenable to numerical solutions using Galerkin or collocation methods. The technique is useful for studying the nonuniform inflow to the propeller.

Large-Deformation Displacement Transfer Functions for Shape Predictions of Highly Flexible Slender Aerospace Structures

NASA Technical Reports Server (NTRS)

Ko, William L.; Fleischer, Van Tran

2013-01-01

Large deformation displacement transfer functions were formulated for deformed shape predictions of highly flexible slender structures like aircraft wings. In the formulation, the embedded beam (depth wise cross section of structure along the surface strain sensing line) was first evenly discretized into multiple small domains, with surface strain sensing stations located at the domain junctures. Thus, the surface strain (bending strains) variation within each domain could be expressed with linear of nonlinear function. Such piecewise approach enabled piecewise integrations of the embedded beam curvature equations [classical (Eulerian), physical (Lagrangian), and shifted curvature equations] to yield closed form slope and deflection equations in recursive forms.

The compressible aerodynamics of rotating blades based on an acoustic formulation

NASA Technical Reports Server (NTRS)

Long, L. N.

1983-01-01

An acoustic formula derived for the calculation of the noise of moving bodies is applied to aerodynamic problems. The acoustic formulation is a time domain result suitable for slender wings and bodies moving at subsonic speeds. A singular integral equation is derived in terms of the surface pressure which must then be solved numerically for aerodynamic purposes. However, as the 'observer' is moved onto the body surface, the divergent integrals in the acoustic formulation are semiconvergent. The procedure for regularization (or taking principal values of divergent integrals) is explained, and some numerical examples for ellipsoids, wings, and lifting rotors are presented. The numerical results show good agreement with available measured surface pressure data.

On the Solution of Elliptic Partial Differential Equations on Regions with Corners

DTIC Science & Technology

2015-07-09

In this report we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations . We observe...that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of...efficient numerical algorithms. The results are illustrated by a number of numerical examples. On the solution of elliptic partial differential equations on

Extremely Fast Numerical Integration of Ocean Surface Wave Dynamics: Building Blocks for a Higher Order Method

DTIC Science & Technology

2006-09-30

equation known as the Kadomtsev - Petviashvili (KP) equation ): (ηt + coηx +αηηx + βη )x +γηyy = 0 (4) where γ = co / 2 . The KdV equation ...using the spectral formulation of the Kadomtsev - Petviashvili equation , a standard equation for nonlinear, shallow water wave dynamics that is a... Petviashvili and nonlinear Schroedinger equations and higher order corrections have been developed as prerequisites to coding the Boussinesq and Euler

Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras

NASA Astrophysics Data System (ADS)

Grahovski, Georgi G.; Mikhailov, Alexander V.

2013-12-01

Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.

Interface equation and viscosity contrast in Hele-Shaw flow

DOE Office of Scientific and Technical Information (OSTI.GOV)

Casademunt, J.; Jasnow, D.; Hernandez-Machado, A.

1992-05-20

In this paper, the authors derive an integro-differential equation for the evolution of the interface separating two immiscible viscous fluids in a Hele-Shaw cell with a channel geometry, for arbitrary viscosity contrast. The authors' equation differs from a previous one obtained by a vortex-sheet formulation of the problem, in that the normal component of the interface velocity is formally decoupled from the gauge-dependent tangential part. The result is thus a closed integral equation for the normal velocity. The authors briefly comment on the advantages of such a formulation and implement an alternative computational algorithm based on it. Preliminary numerical resultsmore » confirm a highly inefficient finger competition in the zero viscosity contrast limit.« less

Computation of solar perturbations with Poisson series

NASA Technical Reports Server (NTRS)

Broucke, R.

1974-01-01

Description of a project for computing first-order perturbations of natural or artificial satellites by integrating the equations of motion on a computer with automatic Poisson series expansions. A basic feature of the method of solution is that the classical variation-of-parameters formulation is used rather than rectangular coordinates. However, the variation-of-parameters formulation uses the three rectangular components of the disturbing force rather than the classical disturbing function, so that there is no problem in expanding the disturbing function in series. Another characteristic of the variation-of-parameters formulation employed is that six rather unusual variables are used in order to avoid singularities at the zero eccentricity and zero (or 90 deg) inclination. The integration process starts by assuming that all the orbit elements present on the right-hand sides of the equations of motion are constants. These right-hand sides are then simple Poisson series which can be obtained with the use of the Bessel expansions of the two-body problem in conjunction with certain interation methods. These Poisson series can then be integrated term by term, and a first-order solution is obtained.

Solution of steady and unsteady transonic-vortex flows using Euler and full-potential equations

NASA Technical Reports Server (NTRS)

Kandil, Osama A.; Chuang, Andrew H.; Hu, Hong

1989-01-01

Two methods are presented for inviscid transonic flows: unsteady Euler equations in a rotating frame of reference for transonic-vortex flows and integral solution of full-potential equation with and without embedded Euler domains for transonic airfoil flows. The computational results covered: steady and unsteady conical vortex flows; 3-D steady transonic vortex flow; and transonic airfoil flows. The results are in good agreement with other computational results and experimental data. The rotating frame of reference solution is potentially efficient as compared with the space fixed reference formulation with dynamic gridding. The integral equation solution with embedded Euler domain is computationally efficient and as accurate as the Euler equations.

The problem of a finite strip compressed between two rough rigid stamps

NASA Technical Reports Server (NTRS)

Gupta, G. D.

1975-01-01

A finite strip compressed between two rough rigid stamps is considered. The elastostatic problem is formulated in terms of a singular integral equation from which the proper stress singularities at the corners are determined. The singular integral equation is solved numerically to determine the stresses along the fixed ends of the strip. The effect of material properties and strip geometry on the stress-intensity factor is presented graphically.

A symplectic integration method for elastic filaments

NASA Astrophysics Data System (ADS)

Ladd, Tony; Misra, Gaurav

2009-03-01

Elastic rods are a ubiquitous coarse-grained model of semi-flexible biopolymers such as DNA, actin, and microtubules. The Worm-Like Chain (WLC) is the standard numerical model for semi-flexible polymers, but it is only a linearized approximation to the dynamics of an elastic rod, valid for small deflections; typically the torsional motion is neglected as well. In the standard finite-difference and finite-element formulations of an elastic rod, the continuum equations of motion are discretized in space and time, but it is then difficult to ensure that the Hamiltonian structure of the exact equations is preserved. Here we discretize the Hamiltonian itself, expressed as a line integral over the contour of the filament. This discrete representation of the continuum filament can then be integrated by one of the explicit symplectic integrators frequently used in molecular dynamics. The model systematically approximates the continuum partial differential equations, but has the same level of computational complexity as molecular dynamics and is constraint free. Numerical tests show that the algorithm is much more stable than a finite-difference formulation and can be used for high aspect ratio filaments, such as actin. We present numerical results for the deterministic and stochastic motion of single filaments.

Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation

NASA Astrophysics Data System (ADS)

Zhu, Ying; Herbert, John M.

2018-01-01

The "real time" formulation of time-dependent density functional theory (TDDFT) involves integration of the time-dependent Kohn-Sham (TDKS) equation in order to describe the time evolution of the electron density following a perturbation. This approach, which is complementary to the more traditional linear-response formulation of TDDFT, is more efficient for computation of broad-band spectra (including core-excited states) and for systems where the density of states is large. Integration of the TDKS equation is complicated by the time-dependent nature of the effective Hamiltonian, and we introduce several predictor/corrector algorithms to propagate the density matrix, one of which can be viewed as a self-consistent extension of the widely used modified-midpoint algorithm. The predictor/corrector algorithms facilitate larger time steps and are shown to be more efficient despite requiring more than one Fock build per time step, and furthermore can be used to detect a divergent simulation on-the-fly, which can then be halted or else the time step modified.

Efficient Meshfree Large Deformation Simulation of Rainfall Induced Soil Slope Failure

NASA Astrophysics Data System (ADS)

Wang, Dongdong; Li, Ling

2010-05-01

An efficient Lagrangian Galerkin meshfree framework is presented for large deformation simulation of rainfall-induced soil slope failure. Detailed coupled soil-rainfall seepage equations are given for the proposed formulation. This nonlinear meshfree formulation is featured by the Lagrangian stabilized conforming nodal integration method where the low cost nature of nodal integration approach is kept and at the same time the numerical stability is maintained. The initiation and evolution of progressive failure in the soil slope is modeled by the coupled constitutive equations of isotropic damage and Drucker-Prager pressure-dependent plasticity. The gradient smoothing in the stabilized conforming integration also serves as a non-local regularization of material instability and consequently the present method is capable of effectively capture the shear band failure. The efficacy of the present method is demonstrated by simulating the rainfall-induced failure of two typical soil slopes.

An accurate and efficient acoustic eigensolver based on a fast multipole BEM and a contour integral method

NASA Astrophysics Data System (ADS)

Zheng, Chang-Jun; Gao, Hai-Feng; Du, Lei; Chen, Hai-Bo; Zhang, Chuanzeng

2016-01-01

An accurate numerical solver is developed in this paper for eigenproblems governed by the Helmholtz equation and formulated through the boundary element method. A contour integral method is used to convert the nonlinear eigenproblem into an ordinary eigenproblem, so that eigenvalues can be extracted accurately by solving a set of standard boundary element systems of equations. In order to accelerate the solution procedure, the parameters affecting the accuracy and efficiency of the method are studied and two contour paths are compared. Moreover, a wideband fast multipole method is implemented with a block IDR (s) solver to reduce the overall solution cost of the boundary element systems of equations with multiple right-hand sides. The Burton-Miller formulation is employed to identify the fictitious eigenfrequencies of the interior acoustic problems with multiply connected domains. The actual effect of the Burton-Miller formulation on tackling the fictitious eigenfrequency problem is investigated and the optimal choice of the coupling parameter as α = i / k is confirmed through exterior sphere examples. Furthermore, the numerical eigenvalues obtained by the developed method are compared with the results obtained by the finite element method to show the accuracy and efficiency of the developed method.

Variational formulation for dissipative continua and an incremental J-integral

NASA Astrophysics Data System (ADS)

Rahaman, Md. Masiur; Dhas, Bensingh; Roy, D.; Reddy, J. N.

2018-01-01

Our aim is to rationally formulate a proper variational principle for dissipative (viscoplastic) solids in the presence of inertia forces. As a first step, a consistent linearization of the governing nonlinear partial differential equations (PDEs) is carried out. An additional set of complementary (adjoint) equations is then formed to recover an underlying variational structure for the augmented system of linearized balance laws. This makes it possible to introduce an incremental Lagrangian such that the linearized PDEs, including the complementary equations, become the Euler-Lagrange equations. Continuous groups of symmetries of the linearized PDEs are computed and an analysis is undertaken to identify the variational groups of symmetries of the linearized dissipative system. Application of Noether's theorem leads to the conservation laws (conserved currents) of motion corresponding to the variational symmetries. As a specific outcome, we exploit translational symmetries of the functional in the material space and recover, via Noether's theorem, an incremental J-integral for viscoplastic solids in the presence of inertia forces. Numerical demonstrations are provided through a two-dimensional plane strain numerical simulation of a compact tension specimen of annealed mild steel under dynamic loading.

On integrable boundaries in the 2 dimensional O(N) σ-models

NASA Astrophysics Data System (ADS)

Aniceto, Inês; Bajnok, Zoltán; Gombor, Tamás; Kim, Minkyoo; Palla, László

2017-09-01

We make an attempt to map the integrable boundary conditions for 2 dimensional non-linear O(N) σ-models. We do it at various levels: classically, by demanding the existence of infinitely many conserved local charges and also by constructing the double row transfer matrix from the Lax connection, which leads to the spectral curve formulation of the problem; at the quantum level, we describe the solutions of the boundary Yang-Baxter equation and derive the Bethe-Yang equations. We then show how to connect the thermodynamic limit of the boundary Bethe-Yang equations to the spectral curve.

Quantum kinetic expansion in the spin-boson model: Matrix formulation and system-bath factorized initial state.

PubMed

Gong, Zhihao; Tang, Zhoufei; Wang, Haobin; Wu, Jianlan

2017-12-28

Within the framework of the hierarchy equation of motion (HEOM), the quantum kinetic expansion (QKE) method of the spin-boson model is reformulated in the matrix representation. The equivalence between the two formulations (HEOM matrices and quantum operators) is numerically verified from the calculation of the time-integrated QKE rates. The matrix formulation of the QKE is extended to the system-bath factorized initial state. Following a one-to-one mapping between HEOM matrices and quantum operators, a quantum kinetic equation is rederived. The rate kernel is modified by an extra term following a systematic expansion over the site-site coupling. This modified QKE is numerically tested for its reliability by calculating the time-integrated rate and non-Markovian population kinetics. For an intermediate-to-strong dissipation strength and a large site-site coupling, the population transfer is found to be significantly different when the initial condition is changed from the local equilibrium to system-bath factorized state.

A finite element-boundary integral method for cavities in a circular cylinder

NASA Technical Reports Server (NTRS)

Kempel, Leo C.; Volakis, John L.

1992-01-01

Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. However, due to a lack of rigorous mathematical models for conformal antenna arrays, antenna designers resort to measurement and planar antenna concepts for designing non-planar conformal antennas. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. We extend this formulation to conformal arrays on large metallic cylinders. In this report, we develop the mathematical formulation. In particular, we discuss the shape functions, the resulting finite elements and the boundary integral equations, and the solution of the conformal finite element-boundary integral system. Some validation results are presented and we further show how this formulation can be applied with minimal computational and memory resources.

Investigation of Volumetric Sources in Airframe Noise Simulations

NASA Technical Reports Server (NTRS)

Casper, Jay H.; Lockard, David P.; Khorrami, Mehdi R.; Streett, Craig L.

2004-01-01

Hybrid methods for the prediction of airframe noise involve a simulation of the near field flow that is used as input to an acoustic propagation formula. The acoustic formulations discussed herein are those based on the Ffowcs Williams and Hawkings equation. Some questions have arisen in the published literature in regard to an apparently significant dependence of radiated noise predictions on the location of the integration surface used in the solution of the Ffowcs Williams and Hawkings equation. These differences in radiated noise levels are most pronounced between solid-body surface integrals and off-body, permeable surface integrals. Such differences suggest that either a non-negligible volumetric source is contributing to the total radiation or the input flow simulation is suspect. The focus of the current work is the issue of internal consistency of the flow calculations that are currently used as input to airframe noise predictions. The case study for this research is a computer simulation for a three-element, high-lift wing profile during landing conditions. The noise radiated from this flow is predicted by a two-dimensional, frequency-domain formulation of the Ffowcs Williams and Hawkings equation. Radiated sound from volumetric sources is assessed by comparison of a permeable surface integration with the sum of a solid-body surface integral and a volume integral. The separate noise predictions are found in good agreement.

The Improvement of Efficiency in the Numerical Computation of Orbit Trajectories

NASA Technical Reports Server (NTRS)

Dyer, J.; Danchick, R.; Pierce, S.; Haney, R.

1972-01-01

An analysis, system design, programming, and evaluation of results are described for numerical computation of orbit trajectories. Evaluation of generalized methods, interaction of different formulations for satellite motion, transformation of equations of motion and integrator loads, and development of efficient integrators are also considered.

A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures

NASA Technical Reports Server (NTRS)

Jin, Jian-Ming; Volakis, John L.

1990-01-01

A numerical technique is proposed for the electromagnetic characterization of the scattering by a three-dimensional cavity-backed aperture in an infinite ground plane. The technique combines the finite element and boundary integral methods to formulate a system of equations for the solution of the aperture fields and those inside the cavity. Specifically, the finite element method is employed to formulate the fields in the cavity region and the boundary integral approach is used in conjunction with the equivalence principle to represent the fields above the ground plane. Unlike traditional approaches, the proposed technique does not require knowledge of the cavity's Green's function and is, therefore, applicable to arbitrary shape depressions and material fillings. Furthermore, the proposed formulation leads to a system having a partly full and partly sparse as well as symmetric and banded matrix which can be solved efficiently using special algorithms.

On the Formulation of Weakly Singular Displacement/Traction Integral Equations; and Their Solution by the MLPG Method

NASA Technical Reports Server (NTRS)

Atluri, Satya N.; Shen, Shengping

2002-01-01

In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.

The influence of a wind tunnel on helicopter rotational noise: Formulation of analysis

NASA Technical Reports Server (NTRS)

Mosher, M.

1984-01-01

An analytical model is discussed that can be used to examine the effects of wind tunnel walls on helicopter rotational noise. A complete physical model of an acoustic source in a wind tunnel is described and a simplified version is then developed. This simplified model retains the important physical processes involved, yet it is more amenable to analysis. The simplified physical model is then modeled as a mathematical problem. An inhomogeneous partial differential equation with mixed boundary conditions is set up and then transformed into an integral equation. Details of generating a suitable Green's function and integral equation are included and the equation is discussed and also given for a two-dimensional case.

An integral formulation for wave propagation on weakly non-uniform potential flows

NASA Astrophysics Data System (ADS)

Mancini, Simone; Astley, R. Jeremy; Sinayoko, Samuel; Gabard, Gwénaël; Tournour, Michel

2016-12-01

An integral formulation for acoustic radiation in moving flows is presented. It is based on a potential formulation for acoustic radiation on weakly non-uniform subsonic mean flows. This work is motivated by the absence of suitable kernels for wave propagation on non-uniform flow. The integral solution is formulated using a Green's function obtained by combining the Taylor and Lorentz transformations. Although most conventional approaches based on either transform solve the Helmholtz problem in a transformed domain, the current Green's function and associated integral equation are derived in the physical space. A dimensional error analysis is developed to identify the limitations of the current formulation. Numerical applications are performed to assess the accuracy of the integral solution. It is tested as a means of extrapolating a numerical solution available on the outer boundary of a domain to the far field, and as a means of solving scattering problems by rigid surfaces in non-uniform flows. The results show that the error associated with the physical model deteriorates with increasing frequency and mean flow Mach number. However, the error is generated only in the domain where mean flow non-uniformities are significant and is constant in regions where the flow is uniform.

A Global Interpolation Function (GIF) boundary element code for viscous flows

NASA Technical Reports Server (NTRS)

Reddy, D. R.; Lafe, O.; Cheng, A. H-D.

1995-01-01

Using global interpolation functions (GIF's), boundary element solutions are obtained for two- and three-dimensional viscous flows. The solution is obtained in the form of a boundary integral plus a series of global basis functions. The unknown coefficients of the GIF's are determined to ensure the satisfaction of the governing equations at selected collocation points. The values of the coefficients involved in the boundary integral equations are determined by enforcing the boundary conditions. Both primitive variable and vorticity-velocity formulations are examined.

Application of Two-Parameter Stabilizing Functions in Solving a Convolution-Type Integral Equation by Regularization Method

NASA Astrophysics Data System (ADS)

Maslakov, M. L.

2018-04-01

This paper examines the solution of convolution-type integral equations of the first kind by applying the Tikhonov regularization method with two-parameter stabilizing functions. The class of stabilizing functions is expanded in order to improve the accuracy of the resulting solution. The features of the problem formulation for identification and adaptive signal correction are described. A method for choosing regularization parameters in problems of identification and adaptive signal correction is suggested.

Robust multiscale field-only formulation of electromagnetic scattering

NASA Astrophysics Data System (ADS)

Sun, Qiang; Klaseboer, Evert; Chan, Derek Y. C.

2017-01-01

We present a boundary integral formulation of electromagnetic scattering by homogeneous bodies that are characterized by linear constitutive equations in the frequency domain. By working with the Cartesian components of the electric E and magnetic H fields and with the scalar functions (r .E ) and (r .H ) where r is a position vector, the problem can be cast as having to solve a set of scalar Helmholtz equations for the field components that are coupled by the usual electromagnetic boundary conditions at material boundaries. This facilitates a direct solution for the surface values of E and H rather than having to work with surface currents or surface charge densities as intermediate quantities in existing methods. Consequently, our formulation is free of the well-known numerical instability that occurs in the zero-frequency or long-wavelength limit in traditional surface integral solutions of Maxwell's equations and our numerical results converge uniformly to the static results in the long-wavelength limit. Furthermore, we use a formulation of the scalar Helmholtz equation that is expressed as classically convergent integrals and does not require the evaluation of principal value integrals or any knowledge of the solid angle. Therefore, standard quadrature and higher order surface elements can readily be used to improve numerical precision for the same number of degrees of freedom. In addition, near and far field values can be calculated with equal precision, and multiscale problems in which the scatterers possess characteristic length scales that are both large and small relative to the wavelength can be easily accommodated. From this we obtain results for the scattering and transmission of electromagnetic waves at dielectric boundaries that are valid for any ratio of the local surface curvature to the wave number. This is a generalization of the familiar Fresnel formula and Snell's law, valid at planar dielectric boundaries, for the scattering and transmission of electromagnetic waves at surfaces of arbitrary curvature. Implementation details are illustrated with scattering by multiple perfect electric conductors as well as dielectric bodies with complex geometries and composition.

Divergent conservation laws in hyperbolic thermoelasticity

NASA Astrophysics Data System (ADS)

Murashkin, E. V.; Radayev, Y. N.

2018-05-01

The present study is devoted to the problem of formulation of conservation laws in divergent form for hyperbolic thermoelastic continua. The field formalism is applied to study the problem. A natural density of thermoelastic action and the corresponding variational least action principle are formulated. A special form of the first variation of the action is employed to obtain 4-covariant divergent conservation laws. Differential field equations and constitutive laws are derived from a special form of the first variation of the action integral. The objectivity of constitutive equations is provided by the rotationally invariant forms of the Lagrangian employed.

Analytic Formulation and Numerical Implementation of an Acoustic Pressure Gradient Prediction

NASA Technical Reports Server (NTRS)

Lee, Seongkyu; Brentner, Kenneth S.; Farassat, F.; Morris, Philip J.

2008-01-01

Two new analytical formulations of the acoustic pressure gradient have been developed and implemented in the PSU-WOPWOP rotor noise prediction code. The pressure gradient can be used to solve the boundary condition for scattering problems and it is a key aspect to solve acoustic scattering problems. The first formulation is derived from the gradient of the Ffowcs Williams-Hawkings (FW-H) equation. This formulation has a form involving the observer time differentiation outside the integrals. In the second formulation, the time differentiation is taken inside the integrals analytically. This formulation avoids the numerical time differentiation with respect to the observer time, which is computationally more efficient. The acoustic pressure gradient predicted by these new formulations is validated through comparison with available exact solutions for a stationary and moving monopole sources. The agreement between the predictions and exact solutions is excellent. The formulations are applied to the rotor noise problems for two model rotors. A purely numerical approach is compared with the analytical formulations. The agreement between the analytical formulations and the numerical method is excellent for both stationary and moving observer cases.

Completed Beltrami-Michell Formulation for Analyzing Radially Symmetrical Bodies

NASA Technical Reports Server (NTRS)

Kaljevic, Igor; Saigal, Sunil; Hopkins, Dale A.; Patnaik, Surya N.

1994-01-01

A force method formulation, the completed Beltrami-Michell formulation (CBMF), has been developed for analyzing boundary value problems in elastic continua. The CBMF is obtained by augmenting the classical Beltrami-Michell formulation with novel boundary compatibility conditions. It can analyze general elastic continua with stress, displacement, or mixed boundary conditions. The CBMF alleviates the limitations of the classical formulation, which can solve stress boundary value problems only. In this report, the CBMF is specialized for plates and shells. All equations of the CBMF, including the boundary compatibility conditions, are derived from the variational formulation of the integrated force method (IFM). These equations are defined only in terms of stresses. Their solution for kinematically stable elastic continua provides stress fields without any reference to displacements. In addition, a stress function formulation for plates and shells is developed by augmenting the classical Airy's formulation with boundary compatibility conditions expressed in terms of the stress function. The versatility of the CBMF and the augmented stress function formulation is demonstrated through analytical solutions of several mixed boundary value problems. The example problems include a composite circular plate and a composite circular cylindrical shell under the simultaneous actions of mechanical and thermal loads.

A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ciraolo, Giulio, E-mail: g.ciraolo@math.unipa.it; Gargano, Francesco, E-mail: gargano@math.unipa.it; Sciacca, Vincenzo, E-mail: sciacca@math.unipa.it

2013-08-01

We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition. The index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations. We prove analytical results on the convergence of the approximate solution. Numerical examples for different shapes of the artificial boundary and for non-constant indexes of refraction will be presented.

FAST TRACK COMMUNICATION Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples

NASA Astrophysics Data System (ADS)

Holm, D. D.; Ivanov, R. I.

2010-12-01

The Lax pair formulation of the two-component Camassa-Holm equation (CH2) is generalized to produce an integrable multi-component family, CH(n, k), of equations with n components and 1 <= |k| <= n velocities. All of the members of the CH(n, k) family show fluid-dynamics properties with coherent solitons following particle characteristics. We determine their Lie-Poisson Hamiltonian structures and give numerical examples of their soliton solution behaviour. We concentrate on the CH(2, k) family with one or two velocities, including the CH(2, -1) equation in the Dym position of the CH2 hierarchy. A brief discussion of the CH(3, 1) system reveals the underlying graded Lie-algebraic structure of the Hamiltonian formulation for CH(n, k) when n >= 3. Fondly recalling our late friend Jerry Marsden.

The prediction of the noise of supersonic propellers in time domain - New theoretical results

NASA Technical Reports Server (NTRS)

Farassat, F.

1983-01-01

In this paper, a new formula for the prediction of the noise of supersonic propellers is derived in the time domain which is superior to the previous formulations in several respects. The governing equation is based on the Ffowcs Williams-Hawkings (FW-H) equation with the thickness source term replaced by an equivalent loading source term derived by Isom (1975). Using some results of generalized function theory and simple four-dimensional space-time geometry, the formal solution of the governing equation is manipulated to a form requiring only the knowledge of blade surface pressure data and geometry. The final form of the main result of this paper consists of some surface and line integrals. The surface integrals depend on the surface pressure, time rate of change of surface pressure, and surface pressure gradient. These integrals also involve blade surface curvatures. The line integrals which depend on local surface pressure are along the trailing edge, the shock traces on the blade, and the perimeter of the airfoil section at the inner radius of the blade. The new formulation is for the full blade surface and does not involve any numerical observer time differentiation. The method of implementation on a computer for numerical work is also discussed.

Stability of nonuniform rotor blades in hover using a mixed formulation

NASA Technical Reports Server (NTRS)

Stephens, W. B.; Hodges, D. H.; Avila, J. H.; Kung, R. M.

1980-01-01

A mixed formulation for calculating static equilibrium and stability eigenvalues of nonuniform rotor blades in hover is presented. The static equilibrium equations are nonlinear and are solved by an accurate and efficient collocation method. The linearized perturbation equations are solved by a one step, second order integration scheme. The numerical results correlate very well with published results from a nearly identical stability analysis based on a displacement formulation. Slight differences in the results are traced to terms in the equations that relate moments to derivatives of rotations. With the present ordering scheme, in which terms of the order of squares of rotations are neglected with respect to unity, it is not possible to achieve completely equivalent models based on mixed and displacement formulations. The one step methods reveal that a second order Taylor expansion is necessary to achieve good convergence for nonuniform rotating blades. Numerical results for a hypothetical nonuniform blade, including the nonlinear static equilibrium solution, were obtained with no more effort or computer time than that required for a uniform blade.

Simulation of a steady-state integrated human thermal system.

NASA Technical Reports Server (NTRS)

Hsu, F. T.; Fan, L. T.; Hwang, C. L.

1972-01-01

The mathematical model of an integrated human thermal system is formulated. The system consists of an external thermal regulation device on the human body. The purpose of the device (a network of cooling tubes held in contact with the surface of the skin) is to maintain the human body in a state of thermoneutrality. The device is controlled by varying the inlet coolant temperature and coolant mass flow rate. The differential equations of the model are approximated by a set of algebraic equations which result from the application of the explicit forward finite difference method to the differential equations. The integrated human thermal system is simulated for a variety of combinations of the inlet coolant temperature, coolant mass flow rate, and metabolic rates.

Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship.

PubMed

Yan, Zai You; Hung, Kin Chew; Zheng, Hui

2003-05-01

Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.

A boundary element method for steady incompressible thermoviscous flow

NASA Technical Reports Server (NTRS)

Dargush, G. F.; Banerjee, P. K.

1991-01-01

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modeling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.

Nonlinear mechanical behavior of thermoplastic matrix materials for advanced composites

NASA Technical Reports Server (NTRS)

Arenz, R. J.; Landel, R. F.

1989-01-01

Two recent theories of nonlinear mechanical response are quantitatively compared and related to experimental data. Computer techniques are formulated to handle the numerical integration and iterative procedures needed to solve the associated sets of coupled nonlinear differential equations. Problems encountered during these formulations are discussed and some open questions described. Bearing in mind these cautions, the consequences of changing parameters that appear in the formulations on the resulting engineering properties are discussed. Hence, engineering approaches to the analysis of thermoplastic matrix material can be suggested.

A new numerical approach for uniquely solvable exterior Riemann-Hilbert problem on region with corners

NASA Astrophysics Data System (ADS)

Zamzamir, Zamzana; Murid, Ali H. M.; Ismail, Munira

2014-06-01

Numerical solution for uniquely solvable exterior Riemann-Hilbert problem on region with corners at offcorner points has been explored by discretizing the related integral equation using Picard iteration method without any modifications to the left-hand side (LHS) and right-hand side (RHS) of the integral equation. Numerical errors for all iterations are converge to the required solution. However, for certain problems, it gives lower accuracy. Hence, this paper presents a new numerical approach for the problem by treating the generalized Neumann kernel at LHS and the function at RHS of the integral equation. Due to the existence of the corner points, Gaussian quadrature is employed which avoids the corner points during numerical integration. Numerical example on a test region is presented to demonstrate the effectiveness of this formulation.

Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method

NASA Astrophysics Data System (ADS)

Ansari, R.; Torabi, J.; Norouzzadeh, A.

2018-04-01

Due to the capability of Eringen's nonlocal elasticity theory to capture the small length scale effect, it is widely used to study the mechanical behaviors of nanostructures. Previous studies have indicated that in some cases, the differential form of this theory cannot correctly predict the behavior of structure, and the integral form should be employed to avoid obtaining inconsistent results. The present study deals with the bending analysis of nanoplates resting on elastic foundation based on the integral formulation of Eringen's nonlocal theory. Since the formulation is presented in a general form, arbitrary kernel functions can be used. The first order shear deformation plate theory is considered to model the nanoplates, and the governing equations for both integral and differential forms are presented. Finally, the finite element method is applied to solve the problem. Selected results are given to investigate the effects of elastic foundation and to compare the predictions of integral nonlocal model with those of its differential nonlocal and local counterparts. It is found that by the use of proposed integral formulation of Eringen's nonlocal model, the paradox observed for the cantilever nanoplate is resolved.

Hybrid state vector methods for structural dynamic and aeroelastic boundary value problems

NASA Technical Reports Server (NTRS)

Lehman, L. L.

1982-01-01

A computational technique is developed that is suitable for performing preliminary design aeroelastic and structural dynamic analyses of large aspect ratio lifting surfaces. The method proves to be quite general and can be adapted to solving various two point boundary value problems. The solution method, which is applicable to both fixed and rotating wing configurations, is based upon a formulation of the structural equilibrium equations in terms of a hybrid state vector containing generalized force and displacement variables. A mixed variational formulation is presented that conveniently yields a useful form for these state vector differential equations. Solutions to these equations are obtained by employing an integrating matrix method. The application of an integrating matrix provides a discretization of the differential equations that only requires solutions of standard linear matrix systems. It is demonstrated that matrix partitioning can be used to reduce the order of the required solutions. Results are presented for several example problems in structural dynamics and aeroelasticity to verify the technique and to demonstrate its use. These problems examine various types of loading and boundary conditions and include aeroelastic analyses of lifting surfaces constructed from anisotropic composite materials.

On the solution of the Helmholtz equation on regions with corners.

PubMed

Serkh, Kirill; Rokhlin, Vladimir

2016-08-16

In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

On the solution of the Helmholtz equation on regions with corners

PubMed Central

Serkh, Kirill; Rokhlin, Vladimir

2016-01-01

In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples. PMID:27482110

A new solution procedure for a nonlinear infinite beam equation of motion

NASA Astrophysics Data System (ADS)

Jang, T. S.

2016-10-01

Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.

A space-time tensor formulation for continuum mechanics in general curvilinear, moving, and deforming coordinate systems

NASA Technical Reports Server (NTRS)

Avis, L. M.

1976-01-01

Tensor methods are used to express the continuum equations of motion in general curvilinear, moving, and deforming coordinate systems. The space-time tensor formulation is applicable to situations in which, for example, the boundaries move and deform. Placing a coordinate surface on such a boundary simplifies the boundary condition treatment. The space-time tensor formulation is also applicable to coordinate systems with coordinate surfaces defined as surfaces of constant pressure, density, temperature, or any other scalar continuum field function. The vanishing of the function gradient components along the coordinate surfaces may simplify the set of governing equations. In numerical integration of the equations of motion, the freedom of motion of the coordinate surfaces provides a potential for enhanced resolution of the continuum field function. An example problem of an incompressible, inviscid fluid with a top free surface is considered, where the surfaces of constant pressure (including the top free surface) are coordinate surfaces.

Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lu Benzhuo; Holst, Michael J.; Center for Theoretical Biological Physics, University of California San Diego, La Jolla, CA 92093

2010-09-20

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for simulating electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised formore » time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.« less

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions

PubMed Central

Lu, Benzhuo; Holst, Michael J.; McCammon, J. Andrew; Zhou, Y. C.

2010-01-01

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems. PMID:21709855

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions.

PubMed

Lu, Benzhuo; Holst, Michael J; McCammon, J Andrew; Zhou, Y C

2010-09-20

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.

Boundary element modelling of dynamic behavior of piecewise homogeneous anisotropic elastic solids

NASA Astrophysics Data System (ADS)

Igumnov, L. A.; Markov, I. P.; Litvinchuk, S. Yu

2018-04-01

A traditional direct boundary integral equations method is applied to solve three-dimensional dynamic problems of piecewise homogeneous linear elastic solids. The materials of homogeneous parts are considered to be generally anisotropic. The technique used to solve the boundary integral equations is based on the boundary element method applied together with the Radau IIA convolution quadrature method. A numerical example of suddenly loaded 3D prismatic rod consisting of two subdomains with different anisotropic elastic properties is presented to verify the accuracy of the proposed formulation.

Integrability: mathematical methods for studying solitary waves theory

NASA Astrophysics Data System (ADS)

Wazwaz, Abdul-Majid

2014-03-01

In recent decades, substantial experimental research efforts have been devoted to linear and nonlinear physical phenomena. In particular, studies of integrable nonlinear equations in solitary waves theory have attracted intensive interest from mathematicians, with the principal goal of fostering the development of new methods, and physicists, who are seeking solutions that represent physical phenomena and to form a bridge between mathematical results and scientific structures. The aim for both groups is to build up our current understanding and facilitate future developments, develop more creative results and create new trends in the rapidly developing field of solitary waves. The notion of the integrability of certain partial differential equations occupies an important role in current and future trends, but a unified rigorous definition of the integrability of differential equations still does not exist. For example, an integrable model in the Painlevé sense may not be integrable in the Lax sense. The Painlevé sense indicates that the solution can be represented as a Laurent series in powers of some function that vanishes on an arbitrary surface with the possibility of truncating the Laurent series at finite powers of this function. The concept of Lax pairs introduces another meaning of the notion of integrability. The Lax pair formulates the integrability of nonlinear equation as the compatibility condition of two linear equations. However, it was shown by many researchers that the necessary integrability conditions are the existence of an infinite series of generalized symmetries or conservation laws for the given equation. The existence of multiple soliton solutions often indicates the integrability of the equation but other tests, such as the Painlevé test or the Lax pair, are necessary to confirm the integrability for any equation. In the context of completely integrable equations, studies are flourishing because these equations are able to describe the real features in a variety of vital areas in science, technology and engineering. In recognition of the importance of solitary waves theory and the underlying concept of integrable equations, a variety of powerful methods have been developed to carry out the required analysis. Examples of such methods which have been advanced are the inverse scattering method, the Hirota bilinear method, the simplified Hirota method, the Bäcklund transformation method, the Darboux transformation, the Pfaffian technique, the Painlevé analysis, the generalized symmetry method, the subsidiary ordinary differential equation method, the coupled amplitude-phase formulation, the sine-cosine method, the sech-tanh method, the mapping and deformation approach and many new other methods. The inverse scattering method, viewed as a nonlinear analogue of the Fourier transform method, is a powerful approach that demonstrates the existence of soliton solutions through intensive computations. At the center of the theory of integrable equations lies the bilinear forms and Hirota's direct method, which can be used to obtain soliton solutions by using exponentials. The Bäcklund transformation method is a useful invariant transformation that transforms one solution into another of a differential equation. The Darboux transformation method is a well known tool in the theory of integrable systems. It is believed that there is a connection between the Bäcklund transformation and the Darboux transformation, but it is as yet not known. Archetypes of integrable equations are the Korteweg-de Vries (KdV) equation, the modified KdV equation, the sine-Gordon equation, the Schrödinger equation, the Vakhnenko equation, the KdV6 equation, the Burgers equation, the fifth-order Lax equation and many others. These equations yield soliton solutions, multiple soliton solutions, breather solutions, quasi-periodic solutions, kink solutions, homo-clinic solutions and other solutions as well. The couplings of linear and nonlinear equations were recently discovered and subsequently received considerable attention. The concept of couplings forms a new direction for developing innovative construction methods. The recently obtained results in solitary waves theory highlight new approaches for additional creative ideas, promising further achievements and increased progress in this field. We are grateful to all of the authors who accepted our invitation to contribute to this comment section.

Calculating corner singularities by boundary integral equations.

PubMed

Shi, Hualiang; Lu, Ya Yan; Du, Qiang

2017-06-01

Accurate numerical solutions for electromagnetic fields near sharp corners and edges are important for nanophotonics applications that rely on strong near fields to enhance light-matter interactions. For cylindrical structures, the singularity exponents of electromagnetic fields near sharp edges can be solved analytically, but in general the actual fields can only be calculated numerically. In this paper, we use a boundary integral equation method to compute electromagnetic fields near sharp edges, and construct the leading terms in asymptotic expansions based on numerical solutions. Our integral equations are formulated for rescaled unknown functions to avoid unbounded field components, and are discretized with a graded mesh and properly chosen quadrature schemes. The numerically found singularity exponents agree well with the exact values in all the test cases presented here, indicating that the numerical solutions are accurate.

An interative solution of an integral equation for radiative transfer by using variational technique

NASA Technical Reports Server (NTRS)

Yoshikawa, K. K.

1973-01-01

An effective iterative technique is introduced to solve a nonlinear integral equation frequently associated with radiative transfer problems. The problem is formulated in such a way that each step of an iterative sequence requires the solution of a linear integral equation. The advantage of a previously introduced variational technique which utilizes a stepwise constant trial function is exploited to cope with the nonlinear problem. The method is simple and straightforward. Rapid convergence is obtained by employing a linear interpolation of the iterative solutions. Using absorption coefficients of the Milne-Eddington type, which are applicable to some planetary atmospheric radiation problems. Solutions are found in terms of temperature and radiative flux. These solutions are presented numerically and show excellent agreement with other numerical solutions.

Numerical calculations of velocity and pressure distribution around oscillating airfoils

NASA Technical Reports Server (NTRS)

Bratanow, T.; Ecer, A.; Kobiske, M.

1974-01-01

An analytical procedure based on the Navier-Stokes equations was developed for analyzing and representing properties of unsteady viscous flow around oscillating obstacles. A variational formulation of the vorticity transport equation was discretized in finite element form and integrated numerically. At each time step of the numerical integration, the velocity field around the obstacle was determined for the instantaneous vorticity distribution from the finite element solution of Poisson's equation. The time-dependent boundary conditions around the oscillating obstacle were introduced as external constraints, using the Lagrangian Multiplier Technique, at each time step of the numerical integration. The procedure was then applied for determining pressures around obstacles oscillating in unsteady flow. The obtained results for a cylinder and an airfoil were illustrated in the form of streamlines and vorticity and pressure distributions.

Advantages of formulating an evolution equation directly for elastic distortional deformation in finite deformation plasticity

NASA Astrophysics Data System (ADS)

Rubin, M. B.; Cardiff, P.

2017-11-01

Simo (Comput Methods Appl Mech Eng 66:199-219, 1988) proposed an evolution equation for elastic deformation together with a constitutive equation for inelastic deformation rate in plasticity. The numerical algorithm (Simo in Comput Methods Appl Mech Eng 68:1-31, 1988) for determining elastic distortional deformation was simple. However, the proposed inelastic deformation rate caused plastic compaction. The corrected formulation (Simo in Comput Methods Appl Mech Eng 99:61-112, 1992) preserves isochoric plasticity but the numerical integration algorithm is complicated and needs special methods for calculation of the exponential map of a tensor. Alternatively, an evolution equation for elastic distortional deformation can be proposed directly with a simplified constitutive equation for inelastic distortional deformation rate. This has the advantage that the physics of inelastic distortional deformation is separated from that of dilatation. The example of finite deformation J2 plasticity with linear isotropic hardening is used to demonstrate the simplicity of the numerical algorithm.

Symmetric and arbitrarily high-order Birkhoff-Hermite time integrators and their long-time behaviour for solving nonlinear Klein-Gordon equations

NASA Astrophysics Data System (ADS)

Liu, Changying; Iserles, Arieh; Wu, Xinyuan

2018-03-01

The Klein-Gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering. Its computation represents a major challenge. The main theme of this paper is the construction of symmetric and arbitrarily high-order time integrators for the nonlinear Klein-Gordon equation by integrating Birkhoff-Hermite interpolation polynomials. To this end, under the assumption of periodic boundary conditions, we begin with the formulation of the nonlinear Klein-Gordon equation as an abstract second-order ordinary differential equation (ODE) and its operator-variation-of-constants formula. We then derive a symmetric and arbitrarily high-order Birkhoff-Hermite time integration formula for the nonlinear abstract ODE. Accordingly, the stability, convergence and long-time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix, subject to suitable temporal and spatial smoothness. A remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for PDEs in the literature. Numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches.

Compliance matrices for cracked bodies

NASA Technical Reports Server (NTRS)

Ballarini, R.

1986-01-01

An algorithm is developed to construct the compliance matrix for a cracked solid in the integral-equation formulation of two-dimensional linear-elastic fracture mechanics. The integral equation is reduced to a system of algebraic equations for unknown values of the dislocation-density function at discrete points on the interval from -1 to 1, using the numerical procedure described by Gerasoulis (1982). Sample numerical results are presented, and it is suggested that the algorithm is especially useful in cases where iterative solutions are required; e.g., models of fiber-reinforced concrete, rocks, or ceramics where microcracking, fiber bridging, and other nonlinear effects are treated as nonlinear springs along the crack surfaces (Ballarini et al., 1984).

Dynamic analysis of a system of hinge-connected rigid bodies with nonrigid appendages. [equations of motion

NASA Technical Reports Server (NTRS)

Likins, P. W.

1974-01-01

Equations of motion are derived for use in simulating a spacecraft or other complex electromechanical system amenable to idealization as a set of hinge-connected rigid bodies of tree topology, with rigid axisymmetric rotors and nonrigid appendages attached to each rigid body in the set. In conjunction with a previously published report on finite-element appendage vibration equations, this report provides a complete minimum-dimension formulation suitable for generic programming for digital computer numerical integration.

Implicit Incompressible SPH.

PubMed

Ihmsen, Markus; Cornelis, Jens; Solenthaler, Barbara; Horvath, Christopher; Teschner, Matthias

2013-07-25

We propose a novel formulation of the projection method for Smoothed Particle Hydrodynamics (SPH). We combine a symmetric SPH pressure force and an SPH discretization of the continuity equation to obtain a discretized form of the pressure Poisson equation (PPE). In contrast to previous projection schemes, our system does consider the actual computation of the pressure force. This incorporation improves the convergence rate of the solver. Furthermore, we propose to compute the density deviation based on velocities instead of positions as this formulation improves the robustness of the time-integration scheme. We show that our novel formulation outperforms previous projection schemes and state-of-the-art SPH methods. Large time steps and small density deviations of down to 0.01% can be handled in typical scenarios. The practical relevance of the approach is illustrated by scenarios with up to 40 million SPH particles.

Implicit incompressible SPH.

PubMed

Ihmsen, Markus; Cornelis, Jens; Solenthaler, Barbara; Horvath, Christopher; Teschner, Matthias

2014-03-01

We propose a novel formulation of the projection method for Smoothed Particle Hydrodynamics (SPH). We combine a symmetric SPH pressure force and an SPH discretization of the continuity equation to obtain a discretized form of the pressure Poisson equation (PPE). In contrast to previous projection schemes, our system does consider the actual computation of the pressure force. This incorporation improves the convergence rate of the solver. Furthermore, we propose to compute the density deviation based on velocities instead of positions as this formulation improves the robustness of the time-integration scheme. We show that our novel formulation outperforms previous projection schemes and state-of-the-art SPH methods. Large time steps and small density deviations of down to 0.01 percent can be handled in typical scenarios. The practical relevance of the approach is illustrated by scenarios with up to 40 million SPH particles.

A finite element-boundary integral method for conformal antenna arrays on a circular cylinder

NASA Technical Reports Server (NTRS)

Kempel, Leo C.; Volakis, John L.; Woo, Alex C.; Yu, C. Long

1992-01-01

Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. In the past, antenna designers have had to resort to expensive measurements in order to develop a conformal array design. This is due to the lack of rigorous mathematical models for conformal antenna arrays, and as a result the design of conformal arrays is primarily based on planar antenna design concepts. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. Herewith we shall extend this formulation for conformal arrays on large metallic cylinders. In this we develop the mathematical formulation. In particular we discuss the finite element equations, the shape elements, and the boundary integral evaluation, and it is shown how this formulation can be applied with minimal computation and memory requirements. The implementation shall be discussed in a later report.

A finite element-boundary integral method for conformal antenna arrays on a circular cylinder

NASA Technical Reports Server (NTRS)

Kempel, Leo C.; Volakis, John L.

1992-01-01

Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. In the past, antenna designers have had to resort to expensive measurements in order to develop a conformal array design. This was due to the lack of rigorous mathematical models for conformal antenna arrays. As a result, the design of conformal arrays was primarily based on planar antenna design concepts. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. We are extending this formulation to conformal arrays on large metallic cylinders. In doing so, we will develop a mathematical formulation. In particular, we discuss the finite element equations, the shape elements, and the boundary integral evaluation. It is shown how this formulation can be applied with minimal computation and memory requirements.

A novel formulation for unsteady counterflow flames using a thermal-conductivity-weighted coordinate

NASA Astrophysics Data System (ADS)

Weiss, Adam D.; Vera, Marcos; Liñán, Amable; Sánchez, Antonio L.; Williams, Forman A.

2018-01-01

A general formulation is given for the description of reacting mixing layers in stagnation-type flows subject to both time-varying strain and pressure. The salient feature of the formulation is the introduction of a thermal-conductivity-weighted transverse coordinate that leads to a compact transport operator that facilitates numerical integration and theoretical analysis. For steady counterflow mixing layers, the associated transverse mass flux is shown to be effectively linear in terms of the new coordinate, so that the conservation equations for energy and chemical species uncouple from the mass and momentum conservation equations, thereby greatly simplifying the solution. Comparisons are shown with computations of diffusion flames with infinitely fast reaction using both the classic Howarth-Dorodnitzyn density-weighted coordinate and the new thermal-conductivity-weighted coordinate, illustrating the advantages of the latter. Also, as an illustrative application of the formulation to the computation of unsteady counterflows, the flame response to harmonically varying strain is examined in the linear limit.

Deformed coset models from gauged WZW actions

NASA Astrophysics Data System (ADS)

Park, Q.-Han

1994-06-01

A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term Tr(gTg -1overlineT) , where algebra elements T, overlineT belong to the center of the algebra h associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equation, we give a systematic way to construct infinitely many conserved currents as well as soliton solutions. In the case of the parafermionic SU(2)/U(1)-coset model, we derive n-solitons and conserved currents explicitly.

Sub-optimal control of unsteady boundary layer separation and optimal control of Saltzman-Lorenz model

NASA Astrophysics Data System (ADS)

Sardesai, Chetan R.

The primary objective of this research is to explore the application of optimal control theory in nonlinear, unsteady, fluid dynamical settings. Two problems are considered: (1) control of unsteady boundary-layer separation, and (2) control of the Saltzman-Lorenz model. The unsteady boundary-layer equations are nonlinear partial differential equations that govern the eruptive events that arise when an adverse pressure gradient acts on a boundary layer at high Reynolds numbers. The Saltzman-Lorenz model consists of a coupled set of three nonlinear ordinary differential equations that govern the time-dependent coefficients in truncated Fourier expansions of Rayleigh-Renard convection and exhibit deterministic chaos. Variational methods are used to derive the nonlinear optimal control formulations based on cost functionals that define the control objective through a performance measure and a penalty function that penalizes the cost of control. The resulting formulation consists of the nonlinear state equations, which must be integrated forward in time, and the nonlinear control (adjoint) equations, which are integrated backward in time. Such coupled forward-backward time integrations are computationally demanding; therefore, the full optimal control problem for the Saltzman-Lorenz model is carried out, while the more complex unsteady boundary-layer case is solved using a sub-optimal approach. The latter is a quasi-steady technique in which the unsteady boundary-layer equations are integrated forward in time, and the steady control equation is solved at each time step. Both sub-optimal control of the unsteady boundary-layer equations and optimal control of the Saltzman-Lorenz model are found to be successful in meeting the control objectives for each problem. In the case of boundary-layer separation, the control results indicate that it is necessary to eliminate the recirculation region that is a precursor to the unsteady boundary-layer eruptions. In the case of the Saltzman-Lorenz model, it is possible to control the system about either of the two unstable equilibrium points representing clockwise and counterclockwise rotation of the convection roles in a parameter regime for which the uncontrolled solution would exhibit deterministic chaos.

The evolution of methods for noise prediction of high speed rotors and propellers in the time domain

NASA Technical Reports Server (NTRS)

Farassat, F.

1986-01-01

Linear wave equation models which have been used over the years at NASA Langley for describing noise emissions from high speed rotating blades are summarized. The noise sources are assumed to lie on a moving surface, and analysis of the situation has been based on the Ffowcs Williams-Hawkings (FW-H) equation. Although the equation accounts for two surface and one volume source, the NASA analyses have considered only the surface terms. Several variations on the FW-H model are delineated for various types of applications, noting the computational benefits of removing the frequency dependence of the calculations. Formulations are also provided for compact and noncompact sources, and features of Long's subsonic integral equation and Farassat's high speed integral equation are discussed. The selection of subsonic or high speed models is dependent on the Mach number of the blade surface where the source is located.

A vectorized algorithm for 3D dynamics of a tethered satellite

NASA Technical Reports Server (NTRS)

Wilson, Howard B.

1989-01-01

Equations of motion characterizing the three dimensional motion of a tethered satellite during the retrieval phase are studied. The mathematical model involves an arbitrary number of point masses connected by weightless cords. Motion occurs in a gravity gradient field. The formulation presented accounts for general functions describing support point motion, rate of tether retrieval, and arbitrary forces applied to the point masses. The matrix oriented program language MATLAB is used to produce an efficient vectorized formulation for computing natural frequencies and mode shapes for small oscillations about the static equilibrium configuration; and for integrating the nonlinear differential equations governing large amplitude motions. An example of time response pertaining to the skip rope effect is investigated.

Multigrid for hypersonic viscous two- and three-dimensional flows

NASA Technical Reports Server (NTRS)

Turkel, E.; Swanson, R. C.; Vatsa, V. N.; White, J. A.

1991-01-01

The use of a multigrid method with central differencing to solve the Navier-Stokes equations for hypersonic flows is considered. The time dependent form of the equations is integrated with an explicit Runge-Kutta scheme accelerated by local time stepping and implicit residual smoothing. Variable coefficients are developed for the implicit process that removes the diffusion limit on the time step, producing significant improvement in convergence. A numerical dissipation formulation that provides good shock capturing capability for hypersonic flows is presented. This formulation is shown to be a crucial aspect of the multigrid method. Solutions are given for two-dimensional viscous flow over a NACA 0012 airfoil and three-dimensional flow over a blunt biconic.

Wigner distribution functions for complex dynamical systems: the emergence of the Wigner-Boltzmann equation.

PubMed

Sels, Dries; Brosens, Fons

2013-10-01

The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.

Evolution equations for connected and disconnected sea parton distributions

NASA Astrophysics Data System (ADS)

Liu, Keh-Fei

2017-08-01

It has been revealed from the path-integral formulation of the hadronic tensor that there are connected sea and disconnected sea partons. The former is responsible for the Gottfried sum rule violation primarily and evolves the same way as the valence. Therefore, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations can be extended to accommodate them separately. We discuss its consequences and implications vis-á-vis lattice calculations.

Formulation of an explicit-multiple-time-step time integration method for use in a global primitive equation grid model

NASA Technical Reports Server (NTRS)

Chao, W. C.

1982-01-01

With appropriate modifications, a recently proposed explicit-multiple-time-step scheme (EMTSS) is incorporated into the UCLA model. In this scheme, the linearized terms in the governing equations that generate the gravity waves are split into different vertical modes. Each mode is integrated with an optimal time step, and at periodic intervals these modes are recombined. The other terms are integrated with a time step dictated by the CFL condition for low-frequency waves. This large time step requires a special modification of the advective terms in the polar region to maintain stability. Test runs for 72 h show that EMTSS is a stable, efficient and accurate scheme.

Theoretical analysis of linearized acoustics and aerodynamics of advanced supersonic propellers

NASA Technical Reports Server (NTRS)

Farassat, F.

1985-01-01

The derivation of a formula for prediction of the noise of supersonic propellers using time domain analysis is presented. This formula is a solution of the Ffowcs Williams-Hawkings equation and does not have the Doppler singularity of some other formulations. The result presented involves some surface integrals over the blade and line integrals over the leading and trailing edges. The blade geometry, motion and surface pressure are needed for noise calculation. To obtain the blade surface pressure, the observer is moved onto the blade surface and a linear singular integral equation is derived which can be solved numerically. Two examples of acoustic calculations using a computer program are currently under development.

Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system

NASA Astrophysics Data System (ADS)

Liao, Feng; Zhang, Luming; Wang, Shanshan

2018-02-01

In this article, we formulate an efficient and accurate numerical method for approximations of the coupled Schrödinger-Boussinesq (SBq) system. The main features of our method are based on: (i) the applications of a time-splitting Fourier spectral method for Schrödinger-like equation in SBq system, (ii) the utilizations of exponential wave integrator Fourier pseudospectral for spatial derivatives in the Boussinesq-like equation. The scheme is fully explicit and efficient due to fast Fourier transform. The numerical examples are presented to show the efficiency and accuracy of our method.

A numerical study of electromagnetic scattering from ocean like surfaces

NASA Technical Reports Server (NTRS)

Lentz, R. R.

1972-01-01

The integral equations describing electromagnetic scattering from one dimensional conducting surfaces are formulated and numerical results are presented. The results are compared with those obtained using approximate methods such as physical optics, geometrical optics, and perturbation theory. The integral equation solutions show that the surface radius of curvature must be greater than 2.5 wavelengths for either the physical optics or geometric optics to give satisfactory results. It has also been shown that perturbation theory agrees with the exact fields as long as the root mean square surface roughness is less than one-tenth of a wavelength.

Error behavior of multistep methods applied to unstable differential systems

NASA Technical Reports Server (NTRS)

Brown, R. L.

1977-01-01

The problem of modeling a dynamic system described by a system of ordinary differential equations which has unstable components for limited periods of time is discussed. It is shown that the global error in a multistep numerical method is the solution to a difference equation initial value problem, and the approximate solution is given for several popular multistep integration formulas. Inspection of the solution leads to the formulation of four criteria for integrators appropriate to unstable problems. A sample problem is solved numerically using three popular formulas and two different stepsizes to illustrate the appropriateness of the criteria.

Determination of welding residual stresses by inverse approach with eigenstrain formulations of boundary integral equation

NASA Astrophysics Data System (ADS)

Ma, Hang; Wang, Ying; Qin, Qing-Hua

2011-04-01

Based on the concept of eigenstrain, a straightforward computational model of the inverse approach is proposed for determining the residual stress field induced by welding using the eigenstrain formulations of boundary integral equations. The eigenstrains are approximately expressed in terms of low-order polynomials in the local area around welded zones. The domain integrals with polynomial eigenstrains are transformed into the boundary integrals to preserve the favourable features of the boundary-only discretization in the process of numerical solutions. The sensitivity matrices in the inverse approach for evaluating the eigenstrain fields are constructed by either the measured deformations (displacements) on the boundary or the measured stresses in the domain after welding over a number of selected measuring points, or by both the measured information. It shows from the numerical examples that the results of residual stresses from deformation measurements are always better than those from stress measurements but they are sensitive to the noises from experiments. The results from stress measurements can be improved by introducing a few deformation measuring points while reducing the number of points for stress measuring to reduce the cost since the measurement of deformation is easier than that of stresses in practice.

Calculation of precision satellite orbits with nonsingular elements /VOP formulation/

NASA Technical Reports Server (NTRS)

Velez, C. E.; Cefola, P. J.; Long, A. C.; Nimitz, K. S.

1974-01-01

Review of some results obtained in an effort to develop efficient, high-precision trajectory computation processes for artificial satellites by optimum selection of the form of the equations of motion of the satellite and the numerical integration method. In particular, the matching of a Gaussian variation-of-parameter (VOP) formulation is considered which is expressed in terms of equinoctial orbital elements and partially decouples the motion of the orbital frame from motion within the orbital frame. The performance of the resulting orbit generators is then compared with the popular classical Cowell/Gauss-Jackson formulation/integrator pair for two distinctly different orbit types - namely, the orbit of the ATS satellite at near-geosynchronous conditions and the near-circular orbit of the GEOS-C satellite at 1000 km.

Scale-dependent behavior of scale equations.

PubMed

Kim, Pilwon

2009-09-01

We propose a new mathematical framework to formulate scale structures of general systems. Stack equations characterize a system in terms of accumulative scales. Their behavior at each scale level is determined independently without referring to other levels. Most standard geometries in mathematics can be reformulated in such stack equations. By involving interaction between scales, we generalize stack equations into scale equations. Scale equations are capable to accommodate various behaviors at different scale levels into one integrated solution. On contrary to standard geometries, such solutions often reveal eccentric scale-dependent figures, providing a clue to understand multiscale nature of the real world. Especially, it is suggested that the Gaussian noise stems from nonlinear scale interactions.

Integrable discrete PT symmetric model.

PubMed

Ablowitz, Mark J; Musslimani, Ziad H

2014-09-01

An exactly solvable discrete PT invariant nonlinear Schrödinger-like model is introduced. It is an integrable Hamiltonian system that exhibits a nontrivial nonlinear PT symmetry. A discrete one-soliton solution is constructed using a left-right Riemann-Hilbert formulation. It is shown that this pure soliton exhibits unique features such as power oscillations and singularity formation. The proposed model can be viewed as a discretization of a recently obtained integrable nonlocal nonlinear Schrödinger equation.

The influence of wind-tunnel walls on discrete frequency noise

NASA Technical Reports Server (NTRS)

Mosher, M.

1984-01-01

This paper describes an analytical model that can be used to examine the effects of wind-tunnel walls on discrete frequency noise. First, a complete physical model of an acoustic source in a wind tunnel is described, and a simplified version is then developed. This simplified model retains the important physical processes involved, yet it is more amenable to analysis. Second, the simplified physical model is formulated as a mathematical problem. An inhomogeneous partial differential equation with mixed boundary conditions is set up and then transformed into an integral equation. The integral equation has been solved with a panel program on a computer. Preliminary results from a simple model problem will be shown and compared with the approximate analytic solution.

Full Wave Parallel Code for Modeling RF Fields in Hot Plasmas

NASA Astrophysics Data System (ADS)

Spencer, Joseph; Svidzinski, Vladimir; Evstatiev, Evstati; Galkin, Sergei; Kim, Jin-Soo

2015-11-01

FAR-TECH, Inc. is developing a suite of full wave RF codes in hot plasmas. It is based on a formulation in configuration space with grid adaptation capability. The conductivity kernel (which includes a nonlocal dielectric response) is calculated by integrating the linearized Vlasov equation along unperturbed test particle orbits. For Tokamak applications a 2-D version of the code is being developed. Progress of this work will be reported. This suite of codes has the following advantages over existing spectral codes: 1) It utilizes the localized nature of plasma dielectric response to the RF field and calculates this response numerically without approximations. 2) It uses an adaptive grid to better resolve resonances in plasma and antenna structures. 3) It uses an efficient sparse matrix solver to solve the formulated linear equations. The linear wave equation is formulated using two approaches: for cold plasmas the local cold plasma dielectric tensor is used (resolving resonances by particle collisions), while for hot plasmas the conductivity kernel is calculated. Work is supported by the U.S. DOE SBIR program.

A Riemann-Hilbert formulation for the finite temperature Hubbard model

NASA Astrophysics Data System (ADS)

Cavaglià, Andrea; Cornagliotto, Martina; Mattelliano, Massimo; Tateo, Roberto

2015-06-01

Inspired by recent results in the context of AdS/CFT integrability, we reconsider the Thermodynamic Bethe Ansatz equations describing the 1D fermionic Hubbard model at finite temperature. We prove that the infinite set of TBA equations are equivalent to a simple nonlinear Riemann-Hilbert problem for a finite number of unknown functions. The latter can be transformed into a set of three coupled nonlinear integral equations defined over a finite support, which can be easily solved numerically. We discuss the emergence of an exact Bethe Ansatz and the link between the TBA approach and the results by Jüttner, Klümper and Suzuki based on the Quantum Transfer Matrix method. We also comment on the analytic continuation mechanism leading to excited states and on the mirror equations describing the finite-size Hubbard model with twisted boundary conditions.

Integral equation methods for vesicle electrohydrodynamics in three dimensions

NASA Astrophysics Data System (ADS)

Veerapaneni, Shravan

2016-12-01

In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.

A global multilevel atmospheric model using a vector semi-Lagrangian finite-difference scheme. I - Adiabatic formulation

NASA Technical Reports Server (NTRS)

Bates, J. R.; Moorthi, S.; Higgins, R. W.

1993-01-01

An adiabatic global multilevel primitive equation model using a two time-level, semi-Lagrangian semi-implicit finite-difference integration scheme is presented. A Lorenz grid is used for vertical discretization and a C grid for the horizontal discretization. The momentum equation is discretized in vector form, thus avoiding problems near the poles. The 3D model equations are reduced by a linear transformation to a set of 2D elliptic equations, whose solution is found by means of an efficient direct solver. The model (with minimal physics) is integrated for 10 days starting from an initialized state derived from real data. A resolution of 16 levels in the vertical is used, with various horizontal resolutions. The model is found to be stable and efficient, and to give realistic output fields. Integrations with time steps of 10 min, 30 min, and 1 h are compared, and the differences are found to be acceptable.

Nonlocal Reformulations of Water and Internal Waves and Asymptotic Reductions

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.

2009-09-01

Nonlocal reformulations of the classical equations of water waves and two ideal fluids separated by a free interface, bounded above by either a rigid lid or a free surface, are obtained. The kinematic equations may be written in terms of integral equations with a free parameter. By expressing the pressure, or Bernoulli, equation in terms of the surface/interface variables, a closed system is obtained. An advantage of this formulation, referred to as the nonlocal spectral (NSP) formulation, is that the vertical component is eliminated, thus reducing the dimensionality and fixing the domain in which the equations are posed. The NSP equations and the Dirichlet-Neumann operators associated with the water wave or two-fluid equations can be related to each other and the Dirichlet-Neumann series can be obtained from the NSP equations. Important asymptotic reductions obtained from the two-fluid nonlocal system include the generalizations of the Benney-Luke and Kadomtsev-Petviashvili (KP) equations, referred to as intermediate-long wave (ILW) generalizations. These 2+1 dimensional equations possess lump type solutions. In the water wave problem high-order asymptotic series are obtained for two and three dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known hyperbolic secant squared solution of the KdV equation; in three dimensions, the first term is the rational lump solution of the KP equation.

Completed Beltrami-Michell Formulation in Polar Coordinates

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Hopkins, Dale A.

2005-01-01

A set of conditions had not been formulated on the boundary of an elastic continuum since the time of Saint-Venant. This limitation prevented the formulation of a direct stress calculation method in elasticity for a continuum with a displacement boundary condition. The missed condition, referred to as the boundary compatibility condition, is now formulated in polar coordinates. The augmentation of the new condition completes the Beltrami-Michell formulation in polar coordinates. The completed formulation that includes equilibrium equations and a compatibility condition in the field as well as the traction and boundary compatibility condition is derived from the stationary condition of the variational functional of the integrated force method. The new method is illustrated by solving an example of a mixed boundary value problem for mechanical as well as thermal loads.

Functional integral for non-Lagrangian systems

NASA Astrophysics Data System (ADS)

Kochan, Denis

2010-02-01

A functional integral formulation of quantum mechanics for non-Lagrangian systems is presented. The approach, which we call “stringy quantization,” is based solely on classical equations of motion and is free of any ambiguity arising from Lagrangian and/or Hamiltonian formulation of the theory. The functionality of the proposed method is demonstrated on several examples. Special attention is paid to the stringy quantization of systems with a general A-power friction force -κq˙A. Results for A=1 are compared with those obtained in the approaches by Caldirola-Kanai, Bateman, and Kostin. Relations to the Caldeira-Leggett model and to the Feynman-Vernon approach are discussed as well.

Many-body perturbation theory using the density-functional concept: beyond the GW approximation.

PubMed

Bruneval, Fabien; Sottile, Francesco; Olevano, Valerio; Del Sole, Rodolfo; Reining, Lucia

2005-05-13

We propose an alternative formulation of many-body perturbation theory that uses the density-functional concept. Instead of the usual four-point integral equation for the polarizability, we obtain a two-point one, which leads to excellent optical absorption and energy-loss spectra. The corresponding three-point vertex function and self-energy are then simply calculated via an integration, for any level of approximation. Moreover, we show the direct impact of this formulation on the time-dependent density-functional theory. Numerical results for the band gap of bulk silicon and solid argon illustrate corrections beyond the GW approximation for the self-energy.

A plane wave model for direct simulation of reflection and transmission by discretely inhomogeneous plane parallel media

NASA Astrophysics Data System (ADS)

Mackowski, Daniel; Ramezanpour, Bahareh

2018-07-01

A formulation is developed for numerically solving the frequency domain Maxwell's equations in plane parallel layers of inhomogeneous media. As was done in a recent work [1], the plane parallel layer is modeled as an infinite square lattice of W × W × H unit cells, with W being a sample width of the layer and H the layer thickness. As opposed to the 3D volume integral/discrete dipole formulation, the derivation begins with a Fourier expansion of the electric field amplitude in the lateral plane, and leads to a coupled system of 1D ordinary differential equations in the depth direction of the layer. A 1D dyadic Green's function is derived for this system and used to construct a set of coupled 1D integral equations for the field expansion coefficients. The resulting mathematical formulation is considerably simpler and more compact than that derived, for the same system, using the discrete dipole approximation applied to the periodic plane lattice. Furthermore, the fundamental property variable appearing in the formulation is the Fourier transformed complex permittivity distribution in the unit cell, and the method obviates any need to define or calculate a dipole polarizability. Although designed primarily for random media calculations, the method is also capable of predicting the single scattering properties of individual particles; comparisons are presented to demonstrate that the method can accurately reproduce, at scattering angles not too close to 90°, the polarimetric scattering properties of single and multiple spheres. The derivation of the dyadic Green's function allows for an analytical preconditioning of the equations, and it is shown that this can result in significantly accelerated solution times when applied to densely-packed systems of particles. Calculation results demonstrate that the method, when applied to inhomogeneous media, can predict coherent backscattering and polarization opposition effects.

Integrability and Poisson Structures of Three Dimensional Dynamical Systems and Equations of Hydrodynamic Type

NASA Astrophysics Data System (ADS)

Gumral, Hasan

Poisson structure of completely integrable 3 dimensional dynamical systems can be defined in terms of an integrable 1-form. We take advantage of this fact and use the theory of foliations in discussing the geometrical structure underlying complete and partial integrability. We show that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a non-trivial Godbillon-Vey class. On the other hand, for the Euler top and a special case of 3-species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the sl_2 structure is a quadratic unfolding of an integrable 1-form in 3 + 1 dimensions. We complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler's equation and a continuum limit of Toda lattice. We present further infinite sequences of conserved quantities for shallow water equations and show that their generalizations by Kodama admit bi-Hamiltonian structure. We present a simple way of constructing the second Hamiltonian operators for N-component equations admitting some scaling properties. The Kodama reduction of the dispersionless-Boussinesq equations and the Lax reduction of the Benney moment equations are shown to be equivalent by a symmetry transformation. They can be cast into the form of a triplet of conservation laws which enable us to recognize a non-trivial scaling symmetry. The resulting bi-Hamiltonian structure generates three infinite sequences of conserved densities.

Analysis of Large Quasistatic Deformations of Inelastic Solids by a New Stress Based Finite Element Method. Ph.D. Thesis Final Report

NASA Technical Reports Server (NTRS)

Reed, Kenneth W.

1992-01-01

A new hybrid stress finite element algorithm suitable for analyses of large quasistatic deformation of inelastic solids is presented. Principal variables in the formulation are the nominal stress rate and spin. The finite element equations which result are discrete versions of the equations of compatibility and angular momentum balance. Consistent reformulation of the constitutive equation and accurate and stable time integration of the stress are discussed at length. Examples which bring out the feasibility and performance of the algorithm conclude the work.

Degenerate variational integrators for magnetic field line flow and guiding center trajectories

NASA Astrophysics Data System (ADS)

Ellison, C. L.; Finn, J. M.; Burby, J. W.; Kraus, M.; Qin, H.; Tang, W. M.

2018-05-01

Symplectic integrators offer many benefits for numerically approximating solutions to Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two important Hamiltonian systems encountered in plasma physics—the flow of magnetic field lines and the guiding center motion of magnetized charged particles—resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates. New algorithms were recently developed using the variational integration formalism; however, those integrators were found to admit parasitic mode instabilities due to their multistep character. This work eliminates the multistep character, and therefore the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem "degenerate variational integration." Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that the resultant Euler-Lagrange equations are systems of first-order ordinary differential equations. We show that retaining the same degree of degeneracy when constructing discrete Lagrangians yields one-step variational integrators preserving a non-canonical symplectic structure. Numerical examples demonstrate the benefits of the new algorithms, including superior stability relative to the existing variational integrators for these systems and superior qualitative behavior relative to non-conservative algorithms.

Scaling and scale invariance of conservation laws in Reynolds transport theorem framework

NASA Astrophysics Data System (ADS)

Haltas, Ismail; Ulusoy, Suleyman

2015-07-01

Scale invariance is the case where the solution of a physical process at a specified time-space scale can be linearly related to the solution of the processes at another time-space scale. Recent studies investigated the scale invariance conditions of hydrodynamic processes by applying the one-parameter Lie scaling transformations to the governing equations of the processes. Scale invariance of a physical process is usually achieved under certain conditions on the scaling ratios of the variables and parameters involved in the process. The foundational axioms of hydrodynamics are the conservation laws, namely, conservation of mass, conservation of linear momentum, and conservation of energy from continuum mechanics. They are formulated using the Reynolds transport theorem. Conventionally, Reynolds transport theorem formulates the conservation equations in integral form. Yet, differential form of the conservation equations can also be derived for an infinitesimal control volume. In the formulation of the governing equation of a process, one or more than one of the conservation laws and, some times, a constitutive relation are combined together. Differential forms of the conservation equations are used in the governing partial differential equation of the processes. Therefore, differential conservation equations constitute the fundamentals of the governing equations of the hydrodynamic processes. Applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework instead of applying to the governing partial differential equations may lead to more fundamental conclusions on the scaling and scale invariance of the hydrodynamic processes. This study will investigate the scaling behavior and scale invariance conditions of the hydrodynamic processes by applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework.

Interaction between a circular inclusion and an arbitrarily oriented crack

NASA Technical Reports Server (NTRS)

Erdogan, F.; Gupta, G. D.; Ratwani, M.

1975-01-01

The plane interaction problem for a circular elastic inclusion embedded in an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations as Green's functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equations with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress-intensity factors representing the strength of the stress singularities are obtained in terms of the asymptotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress-intensity factors are given.

Higher Order Bases in a 2D Hybrid BEM/FEM Formulation

NASA Technical Reports Server (NTRS)

Fink, Patrick W.; Wilton, Donald R.

2002-01-01

The advantages of using higher order, interpolatory basis functions are examined in the analysis of transverse electric (TE) plane wave scattering by homogeneous, dielectric cylinders. A boundary-element/finite-element (BEM/FEM) hybrid formulation is employed in which the interior dielectric region is modeled with the vector Helmholtz equation, and a radiation boundary condition is supplied by an Electric Field Integral Equation (EFIE). An efficient method of handling the singular self-term arising in the EFIE is presented. The iterative solution of the partially dense system of equations is obtained using the Quasi-Minimal Residual (QMR) algorithm with an Incomplete LU Threshold (ILUT) preconditioner. Numerical results are shown for the case of an incident wave impinging upon a square dielectric cylinder. The convergence of the solution is shown versus the number of unknowns as a function of the completeness order of the basis functions.

Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: A tutorial

NASA Astrophysics Data System (ADS)

Mishchenko, Michael I.; Yurkin, Maxim A.

2018-07-01

Although free space cannot generate electromagnetic waves, the majority of existing accounts of frequency-domain electromagnetic scattering by particles and particle groups are based on the postulate of existence of an impressed incident field, usually in the form of a plane wave. In this tutorial we discuss how to account for the actual existence of impressed source currents rather than impressed incident fields. Specifically, we outline a self-consistent theoretical formalism describing electromagnetic scattering by an arbitrary finite object in the presence of arbitrarily distributed impressed currents, some of which can be far removed from the object and some can reside in its vicinity, including inside the object. To make the resulting formalism applicable to a wide range of scattering-object morphologies, we use the framework of the volume integral equation formulation of electromagnetic scattering, couple it with the notion of the transition operator, and exploit the fundamental symmetry property of this operator. Among novel results, this tutorial includes a streamlined proof of fundamental symmetry (reciprocity) relations, a simplified derivation of the Foldy equations, and an explicit analytical expression for the transition operator of a multi-component scattering object.

Finite element formulation of fluctuating hydrodynamics for fluids filled with rigid particles using boundary fitted meshes

DOE Office of Scientific and Technical Information (OSTI.GOV)

De Corato, M., E-mail: marco.decorato@unina.it; Slot, J.J.M., E-mail: j.j.m.slot@tue.nl; Hütter, M., E-mail: m.huetter@tue.nl

In this paper, we present a finite element implementation of fluctuating hydrodynamics with a moving boundary fitted mesh for treating the suspended particles. The thermal fluctuations are incorporated into the continuum equations using the Landau and Lifshitz approach [1]. The proposed implementation fulfills the fluctuation–dissipation theorem exactly at the discrete level. Since we restrict the equations to the creeping flow case, this takes the form of a relation between the diffusion coefficient matrix and friction matrix both at the particle and nodal level of the finite elements. Brownian motion of arbitrarily shaped particles in complex confinements can be considered withinmore » the present formulation. A multi-step time integration scheme is developed to correctly capture the drift term required in the stochastic differential equation (SDE) describing the evolution of the positions of the particles. The proposed approach is validated by simulating the Brownian motion of a sphere between two parallel plates and the motion of a spherical particle in a cylindrical cavity. The time integration algorithm and the fluctuating hydrodynamics implementation are then applied to study the diffusion and the equilibrium probability distribution of a confined circle under an external harmonic potential.« less

A Tensor-Train accelerated solver for integral equations in complex geometries

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Rahimian, Abtin; Zorin, Denis

2017-04-01

We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O (log ⁡ N) and once the inverse is computed, it can be applied in O (Nlog ⁡ N) . We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

Iso-chemical potential trajectories in the P-T plane for He II

NASA Technical Reports Server (NTRS)

Maytal, B.; Nissen, J. A.; Van Sciver, S. W.

1990-01-01

Trajectories of constant chemical potential in the P-T plane serve as an integral formulation of London's equation. The trajectories are useful for analysis and synthesis of fountain effect pump performance. A family of trajectories is generated from available numerical codes.

Root finding in the complex plane for seismo-acoustic propagation scenarios with Green's function solutions.

PubMed

McCollom, Brittany A; Collis, Jon M

2014-09-01

A normal mode solution to the ocean acoustic problem of the Pekeris waveguide with an elastic bottom using a Green's function formulation for a compressional wave point source is considered. Analytic solutions to these types of waveguide propagation problems are strongly dependent on the eigenvalues of the problem; these eigenvalues represent horizontal wavenumbers, corresponding to propagating modes of energy. The eigenvalues arise as singularities in the inverse Hankel transform integral and are specified by roots to a characteristic equation. These roots manifest themselves as poles in the inverse transform integral and can be both subtle and difficult to determine. Following methods previously developed [S. Ivansson et al., J. Sound Vib. 161 (1993)], a root finding routine has been implemented using the argument principle. Using the roots to the characteristic equation in the Green's function formulation, full-field solutions are calculated for scenarios where an acoustic source lies in either the water column or elastic half space. Solutions are benchmarked against laboratory data and existing numerical solutions.

A review on the systematic formulation of 3-D multiparameter full waveform inversion in viscoelastic medium

NASA Astrophysics Data System (ADS)

Yang, Pengliang; Brossier, Romain; Métivier, Ludovic; Virieux, Jean

2016-10-01

In this paper, we study 3-D multiparameter full waveform inversion (FWI) in viscoelastic media based on the generalized Maxwell/Zener body including arbitrary number of attenuation mechanisms. We present a frequency-domain energy analysis to establish the stability condition of a full anisotropic viscoelastic system, according to zero-valued boundary condition and the elastic-viscoelastic correspondence principle: the real-valued stiffness matrix becomes a complex-valued one in Fourier domain when seismic attenuation is taken into account. We develop a least-squares optimization approach to linearly relate the quality factor with the anelastic coefficients by estimating a set of constants which are independent of the spatial coordinates, which supplies an explicit incorporation of the parameter Q in the general viscoelastic wave equation. By introducing the Lagrangian multipliers into the matrix expression of the wave equation with implicit time integration, we build a systematic formulation of multiparameter FWI for full anisotropic viscoelastic wave equation, while the equivalent form of the state and adjoint equation with explicit time integration is available to be resolved efficiently. In particular, this formulation lays the foundation for the inversion of the parameter Q in the time domain with full anisotropic viscoelastic properties. In the 3-D isotropic viscoelastic settings, the anelastic coefficients and the quality factors using bulk and shear moduli parametrization can be related to the counterparts using P and S velocity. Gradients with respect to any other parameter of interest can be found by chain rule. Pioneering numerical validations as well as the real applications of this most generic framework will be carried out to disclose the potential of viscoelastic FWI when adequate high-performance computing resources and the field data are available.

Stability and Interaction of Coherent Structure in Supersonic Reactive Wakes

NASA Technical Reports Server (NTRS)

Menon, Suresh

1983-01-01

A theoretical formulation and analysis is presented for a study of the stability and interaction of coherent structure in reacting free shear layers. The physical problem under investigation is a premixed hydrogen-oxygen reacting shear layer in the wake of a thin flat plate. The coherent structure is modeled as a periodic disturbance and its stability is determined by the application of linearized hydrodynamic stability theory which results in a generalized eigenvalue problem for reactive flows. Detailed stability analysis of the reactive wake for neutral, symmetrical and antisymmetrical disturbance is presented. Reactive stability criteria is shown to be quite different from classical non-reactive stability. The interaction between the mean flow, coherent structure and fine-scale turbulence is theoretically formulated using the von-Kaman integral technique. Both time-averaging and conditional phase averaging are necessary to separate the three types of motion. The resulting integro-differential equations can then be solved subject to initial conditions with appropriate shape functions. In the laminar flow transition region of interest, the spatial interaction between the mean motion and coherent structure is calculated for both non-reactive and reactive conditions and compared with experimental data wherever available. The fine-scale turbulent motion determined by the application of integral analysis to the fluctuation equations. Since at present this turbulence model is still untested, turbulence is modeled in the interaction problem by a simple algebraic eddy viscosity model. The applicability of the integral turbulence model formulated here is studied parametrically by integrating these equations for the simple case of self-similar mean motion with assumed shape functions. The effect of the motion of the coherent structure is studied and very good agreement is obtained with previous experimental and theoretical works for non-reactive flow. For the reactive case, lack of experimental data made direct comparison difficult. It was determined that the growth rate of the disturbance amplitude is lower for reactive case. The results indicate that the reactive flow stability is in qualitative agreement with experimental observation.

Modeling of thin-walled structures interacting with acoustic media as constrained two-dimensional continua

NASA Astrophysics Data System (ADS)

Rabinskiy, L. N.; Zhavoronok, S. I.

2018-04-01

The transient interaction of acoustic media and elastic shells is considered on the basis of the transition function approach. The three-dimensional hyperbolic initial boundary-value problem is reduced to a two-dimensional problem of shell theory with integral operators approximating the acoustic medium effect on the shell dynamics. The kernels of these integral operators are determined by the elementary solution of the problem of acoustic waves diffraction at a rigid obstacle with the same boundary shape as the wetted shell surface. The closed-form elementary solution for arbitrary convex obstacles can be obtained at the initial interaction stages on the background of the so-called “thin layer hypothesis”. Thus, the shell–wave interaction model defined by integro-differential dynamic equations with analytically determined kernels of integral operators becomes hence two-dimensional but nonlocal in time. On the other hand, the initial interaction stage results in localized dynamic loadings and consequently in complex strain and stress states that require higher-order shell theories. Here the modified theory of I.N.Vekua–A.A.Amosov-type is formulated in terms of analytical continuum dynamics. The shell model is constructed on a two-dimensional manifold within a set of field variables, Lagrangian density, and constraint equations following from the boundary conditions “shifted” from the shell faces to its base surface. Such an approach allows one to construct consistent low-order shell models within a unified formal hierarchy. The equations of the N th-order shell theory are singularly perturbed and contain second-order partial derivatives with respect to time and surface coordinates whereas the numerical integration of systems of first-order equations is more efficient. Such systems can be obtained as Hamilton–de Donder–Weyl-type equations for the Lagrangian dynamical system. The Hamiltonian formulation of the elementary N th-order shell theory is here briefly described.

Similarity solution of the Boussinesq equation

NASA Astrophysics Data System (ADS)

Lockington, D. A.; Parlange, J.-Y.; Parlange, M. B.; Selker, J.

Similarity transforms of the Boussinesq equation in a semi-infinite medium are available when the boundary conditions are a power of time. The Boussinesq equation is reduced from a partial differential equation to a boundary-value problem. Chen et al. [Trans Porous Media 1995;18:15-36] use a hodograph method to derive an integral equation formulation of the new differential equation which they solve by numerical iteration. In the present paper, the convergence of their scheme is improved such that numerical iteration can be avoided for all practical purposes. However, a simpler analytical approach is also presented which is based on Shampine's transformation of the boundary value problem to an initial value problem. This analytical approximation is remarkably simple and yet more accurate than the analytical hodograph approximations.

A theoretical analysis of fluid flow and energy transport in hydrothermal systems

USGS Publications Warehouse

Faust, Charles R.; Mercer, James W.

1977-01-01

A mathematical derivation for fluid flow and energy transport in hydrothermal systems is presented. Specifically, the mathematical model describes the three-dimensional flow of both single- and two-phase, single-component water and the transport of heat in porous media. The derivation begins with the point balance equations for mass, momentum, and energy. These equations are then averaged over a finite volume to obtain the macroscopic balance equations for a porous medium. The macroscopic equations are combined by appropriate constitutive relationships to form two similified partial differential equations posed in terms of fluid pressure and enthalpy. A two-dimensional formulation of the simplified equations is also derived by partial integration in the vertical dimension. (Woodard-USGS)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nakatsuji, Hiroshi, E-mail: h.nakatsuji@qcri.or.jp; Nakashima, Hiroyuki

The free-complement (FC) method is a general method for solving the Schrödinger equation (SE): The produced wave function has the potentially exact structure as the solution of the Schrödinger equation. The variables included are determined either by using the variational principle (FC-VP) or by imposing the local Schrödinger equations (FC-LSE) at the chosen set of the sampling points. The latter method, referred to as the local Schrödinger equation (LSE) method, is integral-free and therefore applicable to any atom and molecule. The purpose of this paper is to formulate the basic theories of the LSE method and explain their basic features.more » First, we formulate three variants of the LSE method, the AB, HS, and H{sup T}Q methods, and explain their properties. Then, the natures of the LSE methods are clarified in some detail using the simple examples of the hydrogen atom and the Hooke’s atom. Finally, the ideas obtained in this study are applied to solving the SE of the helium atom highly accurately with the FC-LSE method. The results are very encouraging: we could get the world’s most accurate energy of the helium atom within the sampling-type methodologies, which is comparable to those obtained with the FC-VP method. Thus, the FC-LSE method is an easy and yet a powerful integral-free method for solving the Schrödinger equation of general atoms and molecules.« less

On some control problems of dynamic of reactor

NASA Astrophysics Data System (ADS)

Baskakov, A. V.; Volkov, N. P.

2017-12-01

The paper analyzes controllability of the transient processes in some problems of nuclear reactor dynamics. In this case, the mathematical model of nuclear reactor dynamics is described by a system of integro-differential equations consisting of the non-stationary anisotropic multi-velocity kinetic equation of neutron transport and the balance equation of delayed neutrons. The paper defines the formulation of the linear problem on control of transient processes in nuclear reactors with application of spatially distributed actions on internal neutron sources, and the formulation of the nonlinear problems on control of transient processes with application of spatially distributed actions on the neutron absorption coefficient and the neutron scattering indicatrix. The required control actions depend on the spatial and velocity coordinates. The theorems on existence and uniqueness of these control actions are proved in the paper. To do this, the control problems mentioned above are reduced to equivalent systems of integral equations. Existence and uniqueness of the solution for this system of integral equations is proved by the method of successive approximations, which makes it possible to construct an iterative scheme for numerical analyses of transient processes in a given nuclear reactor with application of the developed mathematical model. Sufficient conditions for controllability of transient processes are also obtained. In conclusion, a connection is made between the control problems and the observation problems, which, by to the given information, allow us to reconstruct either the function of internal neutron sources, or the neutron absorption coefficient, or the neutron scattering indicatrix....

Coupled NASTRAN/boundary element formulation for acoustic scattering

NASA Technical Reports Server (NTRS)

Everstine, Gordon C.; Henderson, Francis M.; Schuetz, Luise S.

1987-01-01

A coupled finite element/boundary element capability is described for calculating the sound pressure field scattered by an arbitrary submerged 3-D elastic structure. Structural and fluid impedances are calculated with no approximation other than discretization. The surface fluid pressures and normal velocities are first calculated by coupling a NASTRAN finite element model of the structure with a discretized form of the Helmholtz surface integral equation for the exterior field. Far field pressures are then evaluated from the surface solution using the Helmholtz exterior integral equation. The overall approach is illustrated and validated using a known analytic solution for scattering from submerged spherical shells.

Torsion analysis of cracked circular bars actuated by a piezoelectric coating

NASA Astrophysics Data System (ADS)

Hassani, A. R.; Faal, R. T.

2016-12-01

This study presents a formulation for a bar with circular cross-section, coated by a piezoelectric layer and subjected to Saint-Venant torsion loading. The bar is weakened by a Volterra-type screw dislocation. First, with aid of the finite Fourier transform, the stress fields in the circular bar and the piezoelectric layer are obtained. The problem is then reduced to a set of singular integral equations with a Cauchy-type singularity. Unknown dislocation density is achieved by numerical solution of these integral equations. Numerical results are discussed, to reveal the effect of the piezoelectric layer on the reduction of the mechanical stress intensity factor in the bar.

A generalized nonlocal vector calculus

NASA Astrophysics Data System (ADS)

Alali, Bacim; Liu, Kuo; Gunzburger, Max

2015-10-01

A nonlocal vector calculus was introduced in Du et al. (Math Model Meth Appl Sci 23:493-540, 2013) that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A formulation is developed that provides a more general setting for the nonlocal vector calculus that is independent of particular nonlocal models. It is shown that general nonlocal calculus operators are integral operators with specific integral kernels. General nonlocal calculus properties are developed, including nonlocal integration by parts formula and Green's identities. The nonlocal vector calculus introduced in Du et al. (Math Model Meth Appl Sci 23:493-540, 2013) is shown to be recoverable from the general formulation as a special example. This special nonlocal vector calculus is used to reformulate the peridynamics equation of motion in terms of the nonlocal gradient operator and its adjoint. A new example of nonlocal vector calculus operators is introduced, which shows the potential use of the general formulation for general nonlocal models.

Fidelity of the Integrated Force Method Solution

NASA Technical Reports Server (NTRS)

Hopkins, Dale; Halford, Gary; Coroneos, Rula; Patnaik, Surya

2002-01-01

The theory of strain compatibility of the solid mechanics discipline was incomplete since St. Venant's 'strain formulation' in 1876. We have addressed the compatibility condition both in the continuum and the discrete system. This has lead to the formulation of the Integrated Force Method. A dual Integrated Force Method with displacement as the primal variable has also been formulated. A modest finite element code (IFM/Analyzers) based on the IFM theory has been developed. For a set of standard test problems the IFM results were compared with the stiffness method solutions and the MSC/Nastran code. For the problems IFM outperformed the existing methods. Superior IFM performance is attributed to simultaneous compliance of equilibrium equation and compatibility condition. MSC/Nastran organization expressed reluctance to accept the high fidelity IFM solutions. This report discusses the solutions to the examples. No inaccuracy was detected in the IFM solutions. A stiffness method code with a small programming effort can be improved to reap the many IFM benefits when implemented with the IFMD elements. Dr. Halford conducted a peer-review on the Integrated Force Method. Reviewers' response is included.

Generalized cable equation model for myelinated nerve fiber.

PubMed

Einziger, Pinchas D; Livshitz, Leonid M; Mizrahi, Joseph

2005-10-01

Herein, the well-known cable equation for nonmyelinated axon model is extended analytically for myelinated axon formulation. The myelinated membrane conductivity is represented via the Fourier series expansion. The classical cable equation is thereby modified into a linear second order ordinary differential equation with periodic coefficients, known as Hill's equation. The general internal source response, expressed via repeated convolutions, uniformly converges provided that the entire periodic membrane is passive. The solution can be interpreted as an extended source response in an equivalent nonmyelinated axon (i.e., the response is governed by the classical cable equation). The extended source consists of the original source and a novel activation function, replacing the periodic membrane in the myelinated axon model. Hill's equation is explicitly integrated for the specific choice of piecewise constant membrane conductivity profile, thereby resulting in an explicit closed form expression for the transmembrane potential in terms of trigonometric functions. The Floquet's modes are recognized as the nerve fiber activation modes, which are conventionally associated with the nonlinear Hodgkin-Huxley formulation. They can also be incorporated in our linear model, provided that the periodic membrane point-wise passivity constraint is properly modified. Indeed, the modified condition, enforcing the periodic membrane passivity constraint on the average conductivity only leads, for the first time, to the inclusion of the nerve fiber activation modes in our novel model. The validity of the generalized transmission-line and cable equation models for a myelinated nerve fiber, is verified herein through a rigorous Green's function formulation and numerical simulations for transmembrane potential induced in three-dimensional myelinated cylindrical cell. It is shown that the dominant pole contribution of the exact modal expansion is the transmembrane potential solution of our generalized model.

Bonded half planes containing an arbitrarily oriented crack

NASA Technical Reports Server (NTRS)

Erdogan, F.; Aksogan, O.

1973-01-01

The plane elastostatic problem for two bonded half planes containing an arbitrarily oriented crack in the neighborhood of the interface is considered. Using Mellin transforms, the problem is formulated as a system of singular integral equations. The equations are solved for various crack orientations, material combinations, and external loads. The numerical results given include the stress intensity factors, tHe strain energy release rates, and tHe probable cleavage angles giving the direction of crack propagation.

Direct integration of the inverse Radon equation for X-ray computed tomography.

PubMed

Libin, E E; Chakhlov, S V; Trinca, D

2016-11-22

A new mathematical appoach using the inverse Radon equation for restoration of images in problems of linear two-dimensional x-ray tomography is formulated. In this approach, Fourier transformation is not used, and it gives the chance to create the practical computing algorithms having more reliable mathematical substantiation. Results of software implementation show that for especially for low number of projections, the described approach performs better than standard X-ray tomographic reconstruction algorithms.

A boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions.

PubMed

Yang, S A

2002-10-01

This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.

Numerical study of hydrogen-air supersonic combustion by using elliptic and parabolized equations

NASA Technical Reports Server (NTRS)

Chitsomboon, T.; Tiwari, S. N.

1986-01-01

The two-dimensional Navier-Stokes and species continuity equations are used to investigate supersonic chemically reacting flow problems which are related to scramjet-engine configurations. A global two-step finite-rate chemistry model is employed to represent the hydrogen-air combustion in the flow. An algebraic turbulent model is adopted for turbulent flow calculations. The explicit unsplit MacCormack finite-difference algorithm is used to develop a computer program suitable for a vector processing computer. The computer program developed is then used to integrate the system of the governing equations in time until convergence is attained. The chemistry source terms in the species continuity equations are evaluated implicitly to alleviate stiffness associated with fast chemical reactions. The problems solved by the elliptic code are re-investigated by using a set of two-dimensional parabolized Navier-Stokes and species equations. A linearized fully-coupled fully-implicit finite difference algorithm is used to develop a second computer code which solves the governing equations by marching in spce rather than time, resulting in a considerable saving in computer resources. Results obtained by using the parabolized formulation are compared with the results obtained by using the fully-elliptic equations. The comparisons indicate fairly good agreement of the results of the two formulations.

A parallel time integrator for noisy nonlinear oscillatory systems

NASA Astrophysics Data System (ADS)

Subber, Waad; Sarkar, Abhijit

2018-06-01

In this paper, we adapt a parallel time integration scheme to track the trajectories of noisy non-linear dynamical systems. Specifically, we formulate a parallel algorithm to generate the sample path of nonlinear oscillator defined by stochastic differential equations (SDEs) using the so-called parareal method for ordinary differential equations (ODEs). The presence of Wiener process in SDEs causes difficulties in the direct application of any numerical integration techniques of ODEs including the parareal algorithm. The parallel implementation of the algorithm involves two SDEs solvers, namely a fine-level scheme to integrate the system in parallel and a coarse-level scheme to generate and correct the required initial conditions to start the fine-level integrators. For the numerical illustration, a randomly excited Duffing oscillator is investigated in order to study the performance of the stochastic parallel algorithm with respect to a range of system parameters. The distributed implementation of the algorithm exploits Massage Passing Interface (MPI).

Integrable Time-Dependent Quantum Hamiltonians

NASA Astrophysics Data System (ADS)

Sinitsyn, Nikolai A.; Yuzbashyan, Emil A.; Chernyak, Vladimir Y.; Patra, Aniket; Sun, Chen

2018-05-01

We formulate a set of conditions under which the nonstationary Schrödinger equation with a time-dependent Hamiltonian is exactly solvable analytically. The main requirement is the existence of a non-Abelian gauge field with zero curvature in the space of system parameters. Known solvable multistate Landau-Zener models satisfy these conditions. Our method provides a strategy to incorporate time dependence into various quantum integrable models while maintaining their integrability. We also validate some prior conjectures, including the solution of the driven generalized Tavis-Cummings model.

On the theory of oscillating airfoils of finite span in subsonic compressible flow

NASA Technical Reports Server (NTRS)

Reissner, Eric

1950-01-01

The problem of oscillating lifting surface of finite span in subsonic compressible flow is reduced to an integral equation. The kernel of the integral equation is approximated by a simpler expression, on the basis of the assumption of sufficiently large aspect ratio. With this approximation the double integral occurring in the formulation of the problem is reduced to two single integrals, one of which is taken over the chord and the other over the span of the lifting surface. On the basis of this reduction the three-dimensional problem appears separated into two two-dimensional problems, one of them being effectively the problem of two-dimensional flow and the other being the problem of spanwise circulation distribution. Earlier results concerning the oscillating lifting surface of finite span in incompressible flow are contained in the present more general results.

Boundary integral equation analysis for suspension of spheres in Stokes flow

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Veerapaneni, Shravan

2018-06-01

We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.

Transient difference solutions of the inhomogeneous wave equation - Simulation of the Green's function

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1983-01-01

A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.

Transient difference solutions of the inhomogeneous wave equation: Simulation of the Green's function

NASA Technical Reports Server (NTRS)

Baumeiste, K. J.

1983-01-01

A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.

Modified equations, rational solutions, and the Painleve property for the Kadomtsev--Petviashvili and Hirota--Satsuma equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Weiss, J.

1985-09-01

We propose a method for finding the Lax pairs and rational solutions of integrable partial differential equations. That is, when an equation possesses the Painleve property, a Baecklund transformation is defined in terms of an expansion about the singular manifold. This Baecklund transformation obtains (1) a type of modified equation that is formulated in terms of Schwarzian derivatives and (2) a Miura transformation from the modified to the original equation. By linearizing the (Ricati-type) Miura transformation the Lax pair is found. On the other hand, consideration of the (distinct) Baecklund transformations of the modified equations provides a method for themore » iterative construction of rational solutions. This also obtains the Lax pairs for the modified equations. In this paper we apply this method to the Kadomtsev--Petviashvili equation and the Hirota--Satsuma equations.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Garcia-Fernandez, Ignacio; Pla-Castells, Marta; Martinez-Dura, Rafael J.

A model of a cable and pulleys is presented that can be used in Real Time Computer Graphics applications. The model is formulated by the coupling of a damped spring and a variable coefficient wave equation, and can be integrated in more complex mechanical models of lift systems, such as cranes, elevators, etc. with a high degree of interactivity.

Dispersion analysis of leaky guided waves in fluid-loaded waveguides of generic shape.

PubMed

Mazzotti, M; Marzani, A; Bartoli, I

2014-01-01

A fully coupled 2.5D formulation is proposed to compute the dispersive parameters of waveguides with arbitrary cross-section immersed in infinite inviscid fluids. The discretization of the waveguide is performed by means of a Semi-Analytical Finite Element (SAFE) approach, whereas a 2.5D BEM formulation is used to model the impedance of the surrounding infinite fluid. The kernels of the boundary integrals contain the fundamental solutions of the space Fourier-transformed Helmholtz equation, which governs the wave propagation process in the fluid domain. Numerical difficulties related to the evaluation of singular integrals are avoided by using a regularization procedure. To improve the numerical stability of the discretized boundary integral equations for the external Helmholtz problem, the so called CHIEF method is used. The discrete wave equation results in a nonlinear eigenvalue problem in the complex axial wavenumbers that is solved at the frequencies of interest by means of a contour integral algorithm. In order to separate physical from non-physical solutions and to fulfill the requirement of holomorphicity of the dynamic stiffness matrix inside the complex wavenumber contour, the phase of the radial bulk wavenumber is uniquely defined by enforcing the Snell-Descartes law at the fluid-waveguide interface. Three numerical applications are presented. The computed dispersion curves for a circular bar immersed in oil are in agreement with those extracted using the Global Matrix Method. Novel results are presented for viscoelastic steel bars of square and L-shaped cross-section immersed in water. Copyright © 2013 Elsevier B.V. All rights reserved.

Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3

NASA Technical Reports Server (NTRS)

Chakravarthy, Sukumar R.

1990-01-01

An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach.

Radiative transfer in a sphere illuminated by a parallel beam - An integral equation approach. [in planetary atmosphere

NASA Technical Reports Server (NTRS)

Shia, R.-L.; Yung, Y. L.

1986-01-01

The problem of multiple scattering of nonpolarized light in a planetary body of arbitrary shape illuminated by a parallel beam is formulated using the integral equation approach. There exists a simple functional whose stationarity condition is equivalent to solving the equation of radiative transfer and whose value at the stationary point is proportional to the differential cross section. The analysis reveals a direct relation between the microscopic symmetry of the phase function for each scattering event and the macroscopic symmetry of the differential cross section for the entire planetary body, and the interconnection of these symmetry relations and the variational principle. The case of a homogeneous sphere containing isotropic scatterers is investigated in detail. It is shown that the solution can be expanded in a multipole series such that the general spherical problem is reduced to solving a set of decoupled integral equations in one dimension. Computations have been performed for a range of parameters of interest, and illustrative examples of applications to planetary problems as provided.

A New and General Formulation of the Parametric HFGMC Micromechanical Method for Three-Dimensional Multi-Phase Composites

NASA Technical Reports Server (NTRS)

Haj-Ali, Rami; Aboudi, Jacob

2012-01-01

The recent two-dimensional (2-D) parametric formulation of the high fidelity generalized method of cells (HFGMC) reported by the authors is generalized for the micromechanical analysis of three-dimensional (3-D) multiphase composites with periodic microstructure. Arbitrary hexahedral subcell geometry is developed to discretize a triply periodic repeating unit-cell (RUC). Linear parametric-geometric mapping is employed to transform the arbitrary hexahedral subcell shapes from the physical space to an auxiliary orthogonal shape, where a complete quadratic displacement expansion is performed. Previously in the 2-D case, additional three equations are needed in the form of average moments of equilibrium as a result of the inclusion of the bilinear terms. However, the present 3-D parametric HFGMC formulation eliminates the need for such additional equations. This is achieved by expressing the coefficients of the full quadratic polynomial expansion of the subcell in terms of the side or face average-displacement vectors. The 2-D parametric and orthogonal HFGMC are special cases of the present 3-D formulation. The continuity of displacements and tractions, as well as the equilibrium equations, are imposed in the average (integral) sense as in the original HFGMC formulation. Each of the six sides (faces) of a subcell has an independent average displacement micro-variable vector which forms an energy-conjugate pair with the transformed average-traction vector. This allows generating symmetric stiffness matrices along with internal resisting vectors for the subcells which enhances the computational efficiency. The established new parametric 3-D HFGMC equations are formulated and solution implementations are addressed. Several applications for triply periodic 3-D composites are presented to demonstrate the general capability and varsity of the present parametric HFGMC method for refined micromechanical analysis by generating the spatial distributions of local stress fields. These applications include triply periodic composites with inclusions in the form of a cavity, spherical inclusion, ellipsoidal inclusion, discontinuous aligned short fiber. A 3-D repeating unit-cell for foam material composite is simulated.

A New Time Domain Formulation for Broadband Noise Predictions

NASA Technical Reports Server (NTRS)

Casper, J.; Farassat, F.

2002-01-01

A new analytic result in acoustics called "Formulation 1B," proposed by Farassat, is used to compute the loading noise from an unsteady surface pressure distribution on a thin airfoil in the time domain. This formulation is a new solution of the Ffowcs Williams-Hawkings equation with the loading source term. The formulation contains a far field surface integral that depends on the time derivative and the surface gradient of the pressure on the airfoil, as well as a contour integral on the boundary of the airfoil surface. As a first test case, the new formulation is used to compute the noise radiated from a flat plate, moving through a sinusoidal gust of constant frequency. The unsteady surface pressure for this test case is analytically specified from a result based on linear airfoil theory. This test case is used to examine the velocity scaling properties of Formulation 1B and to demonstrate its equivalence to Formulation 1A of Farassat. The new acoustic formulation, again with an analytic surface pressure, is then used to predict broadband noise radiated from an airfoil immersed in homogeneous, isotropic turbulence. The results are compared with experimental data previously reported by Paterson and Amiet. Good agreement between predictions and measurements is obtained. Finally, an alternative form of Formulation 1B is described for statistical analysis of broadband noise.

A New Time Domain Formulation for Broadband Noise Predictions

NASA Technical Reports Server (NTRS)

Casper, Jay H.; Farassat, Fereidoun

2002-01-01

A new analytic result in acoustics called "Formulation 1B," proposed by Farassat, is used to compute the loading noise from an unsteady surface pressure distribution on a thin airfoil in the time domain. This formulation is a new solution of the Ffowcs Williams-Hawkings equation with the loading source term. The formulation contains a far field surface integral that depends on the time derivative and the surface gradient of the pressure on the airfoil, as well as a contour integral on the boundary of the airfoil surface. As a first test case, the new formulation is used to compute the noise radiated from a flat plate, moving through a sinusoidal gust of constant frequency. The unsteady surface pressure for this test case is analytically specied from a result based on linear airfoil theory. This test case is used to examine the velocity scaling properties of Formulation 1B and to demonstrate its equivalence to Formulation 1A of Farassat. The new acoustic formulation, again with an analytic surface pressure, is then used to predict broadband noise radiated from an airfoil immersed in homogeneous, isotropic turbulence. The results are compared with experimental data previously reported by Paterson and Amiet. Good agreement between predictions and measurements is obtained. Finally, an alternative form of Formulation 1B is described for statistical analysis of broadband noise.

Explicit formulation of second and third order optical nonlinearity in the FDTD framework

NASA Astrophysics Data System (ADS)

Varin, Charles; Emms, Rhys; Bart, Graeme; Fennel, Thomas; Brabec, Thomas

2018-01-01

The finite-difference time-domain (FDTD) method is a flexible and powerful technique for rigorously solving Maxwell's equations. However, three-dimensional optical nonlinearity in current commercial and research FDTD softwares requires solving iteratively an implicit form of Maxwell's equations over the entire numerical space and at each time step. Reaching numerical convergence demands significant computational resources and practical implementation often requires major modifications to the core FDTD engine. In this paper, we present an explicit method to include second and third order optical nonlinearity in the FDTD framework based on a nonlinear generalization of the Lorentz dispersion model. A formal derivation of the nonlinear Lorentz dispersion equation is equally provided, starting from the quantum mechanical equations describing nonlinear optics in the two-level approximation. With the proposed approach, numerical integration of optical nonlinearity and dispersion in FDTD is intuitive, transparent, and fully explicit. A strong-field formulation is also proposed, which opens an interesting avenue for FDTD-based modelling of the extreme nonlinear optics phenomena involved in laser filamentation and femtosecond micromachining of dielectrics.

Stress intensity factor in a tapered specimen

NASA Technical Reports Server (NTRS)

Xue-Hui, L.; Erdogan, F.

1985-01-01

The general problem of a tapered specimen containing an edge crack is formulated in terms of a system of singular integral equations. The equations are solved and the stress intensity factor is calculated for a compact and for a slender tapered specimen, the latter simulating the double cantilever beam. The results are obtained primarily for a pair of concentrated forces and for crack surface wedge forces. The stress intensity factors are also obtained for a long strip under uniform tension which contains inclined edge cracks.

u-w formulation for dynamic problems in large deformation regime solved through an implicit meshfree scheme

NASA Astrophysics Data System (ADS)

Navas, Pedro; Sanavia, Lorenzo; López-Querol, Susana; Yu, Rena C.

2017-12-01

Solving dynamic problems for fluid saturated porous media at large deformation regime is an interesting but complex issue. An implicit time integration scheme is herein developed within the framework of the u-w (solid displacement-relative fluid displacement) formulation for the Biot's equations. In particular, liquid water saturated porous media is considered and the linearization of the linear momentum equations taking into account all the inertia terms for both solid and fluid phases is for the first time presented. The spatial discretization is carried out through a meshfree method, in which the shape functions are based on the principle of local maximum entropy LME. The current methodology is firstly validated with the dynamic consolidation of a soil column and the plastic shear band formulation of a square domain loaded by a rigid footing. The feasibility of this new numerical approach for solving large deformation dynamic problems is finally demonstrated through the application to an embankment problem subjected to an earthquake.

Opening of an interface flaw in a layered elastic half-plane under compressive loading

NASA Technical Reports Server (NTRS)

Kennedy, J. M.; Fichter, W. B.; Goree, J. G.

1984-01-01

A static analysis is given of the problem of an elastic layer perfectly bonded, except for a frictionless interface crack, to a dissimilar elastic half-plane. The free surface of the layer is loaded by a finite pressure distribution directly over the crack. The problem is formulated using the two dimensional linear elasticity equations. Using Fourier transforms, the governing equations are converted to a pair of coupled singular integral equations. The integral equations are reduced to a set of simultaneous algebraic equations by expanding the unknown functions in a series of Jacobi polynomials and then evaluating the singular Cauchy-type integrals. The resulting equations are found to be ill-conditioned and, consequently, are solved in the least-squares sense. Results from the analysis show that, under a normal pressure distribution on the free surface of the layer and depending on the combination of geometric and material parameters, the ends of the crack can open. The resulting stresses at the crack-tips are singular, implying that crack growth is possible. The extent of the opening and the crack-top stress intensity factors depend on the width of the pressure distribution zone, the layer thickness, and the relative material properties of the layer and half-plane.

Generalized Case ``Van Kampen theory for electromagnetic oscillations in a magnetized plasma

NASA Astrophysics Data System (ADS)

Bairaktaris, F.; Hizanidis, K.; Ram, A. K.

2017-10-01

The Case-Van Kampen theory is set up to describe electrostatic oscillations in an unmagnetized plasma. Our generalization to electromagnetic oscillations in magnetized plasma is formulated in the relativistic position-momentum phase space of the particles. The relativistic Vlasov equation includes the ambient, homogeneous, magnetic field, and space-time dependent electromagnetic fields that satisfy Maxwell's equations. The standard linearization technique leads to an equation for the perturbed distribution function in terms of the electromagnetic fields. The eigenvalues and eigenfunctions are obtained from three integrals `` each integral being over two different components of the momentum vector. Results connecting phase velocity, frequency, and wave vector will be presented. Supported in part by the Hellenic National Programme on Controlled Thermonuclear Fusion associated with the EUROfusion Consortium, and by DoE Grant DE-FG02-91ER-54109.

Modeling RF Fields in Hot Plasmas with Parallel Full Wave Code

NASA Astrophysics Data System (ADS)

Spencer, Andrew; Svidzinski, Vladimir; Zhao, Liangji; Galkin, Sergei; Kim, Jin-Soo

2016-10-01

FAR-TECH, Inc. is developing a suite of full wave RF plasma codes. It is based on a meshless formulation in configuration space with adapted cloud of computational points (CCP) capability and using the hot plasma conductivity kernel to model the nonlocal plasma dielectric response. The conductivity kernel is calculated by numerically integrating the linearized Vlasov equation along unperturbed particle trajectories. Work has been done on the following calculations: 1) the conductivity kernel in hot plasmas, 2) a monitor function based on analytic solutions of the cold-plasma dispersion relation, 3) an adaptive CCP based on the monitor function, 4) stencils to approximate the wave equations on the CCP, 5) the solution to the full wave equations in the cold-plasma model in tokamak geometry for ECRH and ICRH range of frequencies, and 6) the solution to the wave equations using the calculated hot plasma conductivity kernel. We will present results on using a meshless formulation on adaptive CCP to solve the wave equations and on implementing the non-local hot plasma dielectric response to the wave equations. The presentation will include numerical results of wave propagation and absorption in the cold and hot tokamak plasma RF models, using DIII-D geometry and plasma parameters. Work is supported by the U.S. DOE SBIR program.

Plates and shells containing a surface crack under general loading conditions

NASA Technical Reports Server (NTRS)

Joseph, Paul F.; Erdogan, Fazil

1986-01-01

The severity of the underlying assumptions of the line-spring model (LSM) are such that verification with three-dimensional solutions is necessary. Such comparisons show that the model is quite accurate, and therefore, its use in extensive parameter studies is justified. Investigations into the endpoint behavior of the line-spring model have led to important conclusions about the ability of the model to predict stresses in front of the crack tip. An important application of the LSM was to solve the contact plate bending problem. Here the flexibility of the model to allow for any crack shape is exploited. The use of displacement quantities as unknowns in the formulation of the problem leads to strongly singular integral equations, rather than singular integral equations which result from using displacement derivatives. The collocation method of solving the integral equations was found to be better and more convenient than the quadrature technique. Orthogonal polynomials should be used as fitting functions when using the LSM as opposed to simpler functions such as power series.

A time-domain Kirchhoff formula for the convective acoustic wave equation

NASA Astrophysics Data System (ADS)

Ghorbaniasl, Ghader; Siozos-Rousoulis, Leonidas; Lacor, Chris

2016-03-01

Kirchhoff's integral method allows propagated sound to be predicted, based on the pressure and its derivatives in time and space obtained on a data surface located in the linear flow region. Kirchhoff's formula for noise prediction from high-speed rotors and propellers suffers from the limitation of the observer located in uniform flow, thus requiring an extension to arbitrarily moving media. This paper presents a Kirchhoff formulation for moving surfaces in a uniform moving medium of arbitrary configuration. First, the convective wave equation is derived in a moving frame, based on the generalized functions theory. The Kirchhoff formula is then obtained for moving surfaces in the time domain. The formula has a similar form to the Kirchhoff formulation for moving surfaces of Farassat and Myers, with the presence of additional terms owing to the moving medium effect. The equation explicitly accounts for the influence of mean flow and angle of attack on the radiated noise. The formula is verified by analytical cases of a monopole source located in a moving medium.

Stress and Fracture Analyses Under Elastic-plastic and Creep Conditions: Some Basic Developments and Computational Approaches

NASA Technical Reports Server (NTRS)

Reed, K. W.; Stonesifer, R. B.; Atluri, S. N.

1983-01-01

A new hybrid-stress finite element algorith, suitable for analyses of large quasi-static deformations of inelastic solids, is presented. Principal variables in the formulation are the nominal stress-rate and spin. A such, a consistent reformulation of the constitutive equation is necessary, and is discussed. The finite element equations give rise to an initial value problem. Time integration has been accomplished by Euler and Runge-Kutta schemes and the superior accuracy of the higher order schemes is noted. In the course of integration of stress in time, it has been demonstrated that classical schemes such as Euler's and Runge-Kutta may lead to strong frame-dependence. As a remedy, modified integration schemes are proposed and the potential of the new schemes for suppressing frame dependence of numerically integrated stress is demonstrated. The topic of the development of valid creep fracture criteria is also addressed.

Designing ROW Methods

NASA Technical Reports Server (NTRS)

Freed, Alan D.

1996-01-01

There are many aspects to consider when designing a Rosenbrock-Wanner-Wolfbrandt (ROW) method for the numerical integration of ordinary differential equations (ODE's) solving initial value problems (IVP's). The process can be simplified by constructing ROW methods around good Runge-Kutta (RK) methods. The formulation of a new, simple, embedded, third-order, ROW method demonstrates this design approach.

Prediction of submarine scattered noise by the acoustic analogy

NASA Astrophysics Data System (ADS)

Testa, C.; Greco, L.

2018-07-01

The prediction of the noise scattered by a submarine subject to the propeller tonal noise is here addressed through a non-standard frequency-domain formulation that extends the use of the acoustic analogy to scattering problems. A boundary element method yields the scattered pressure upon the hull surface by the solution of a boundary integral equation, whereas the noise radiated in the fluid domain is evaluated by the corresponding boundary integral representation. Propeller-induced incident pressure field on the scatterer is detected by combining an unsteady three-dimensional panel method with the Bernoulli equation. For each frequency of interest, numerical results concern with sound pressure levels upon the hull and in the flowfield. The validity of the results is established by a comparison with a time-marching hydrodynamic panel method that solves propeller and hull jointly. Within the framework of potential-flow hydrodynamics, it is found out that the scattering formulation herein proposed is appropriate to successfully capture noise magnitude and directivity both on the hull surface and in the flowfield, yielding a computationally efficient solution procedure that may be useful in preliminary design/multidisciplinary optimization applications.

Approach to atmospheric laser-propagation theory based on the extended Huygens-Fresnel principle and a self-consistency concept.

PubMed

Bochove, Erik J; Rao Gudimetla, V S

2017-01-01

We propose a self-consistency condition based on the extended Huygens-Fresnel principle, which we apply to the propagation kernel of the mutual coherence function of a partially coherent laser beam propagating through a turbulent atmosphere. The assumption of statistical independence of turbulence in neighboring propagation segments leads to an integral equation in the propagation kernel. This integral equation is satisfied by a Gaussian function, with dependence on the transverse coordinates that is identical to the previous Gaussian formulation by Yura [Appl. Opt.11, 1399 (1972)APOPAI0003-693510.1364/AO.11.001399], but differs in the transverse coherence length's dependence on propagation distance, so that this established version violates our self-consistency principle. Our formulation has one free parameter, which in the context of Kolmogorov's theory is independent of turbulence strength and propagation distance. We determined its value by numerical fitting to the rigorous beam propagation theory of Yura and Hanson [J. Opt. Soc. Am. A6, 564 (1989)JOAOD60740-323210.1364/JOSAA.6.000564], demonstrating in addition a significant improvement over other Gaussian models.

Vortex conception of rotor and mutual effect of screw/propellers

NASA Technical Reports Server (NTRS)

Lepilkin, A. M.

1986-01-01

A vortex theory of screw/propellers with variable circulation according to the blade and its azimuth is proposed, the problem is formulated and circulation is expanded in a Fourier series. Equations are given for inductive velocities in space for crews, including those with an infinitely large number of blades and expansion of the inductive velocity by blade azimuth of a second screw. Multiparameter improper integrals are given as a combination of elliptical integrals and elementary functions, and it is shown how to reduce elliptical integrals of the third kind with a complex parameter to integrals with a real parameter.

Some examples of exact and approximate solutions in small particle scattering - A progress report

NASA Technical Reports Server (NTRS)

Greenberg, J. M.

1974-01-01

The formulation of basic equations from which the scattering of radiation by a localized variation in a medium is discussed. These equations are developed in both the differential and the integral form. Primary interest is in the scattering of electromagnetic waves for which the solution of the vector wave equation with appropriate boundary conditions must be considered. Scalar scattering by an infinite homogeneous isotropic circular cylinder, and scattering of electromagnetic waves by infinite circular cylinders are treated, and the case of the finite circular cylinder is considered. A procedure is given for obtaining angular scattering distributions from spheroids.

New formulations for tsunami runup estimation

NASA Astrophysics Data System (ADS)

Kanoglu, U.; Aydin, B.; Ceylan, N.

2017-12-01

We evaluate shoreline motion and maximum runup in two folds: One, we use linear shallow water-wave equations over a sloping beach and solve as initial-boundary value problem similar to the nonlinear solution of Aydın and Kanoglu (2017, Pure Appl. Geophys., https://doi.org/10.1007/s00024-017-1508-z). Methodology we present here is simple; it involves eigenfunction expansion and, hence, avoids integral transform techniques. We then use several different types of initial wave profiles with and without initial velocity, estimate shoreline properties and confirm classical runup invariance between linear and nonlinear theories. Two, we use the nonlinear shallow water-wave solution of Kanoglu (2004, J. Fluid Mech. 513, 363-372) to estimate maximum runup. Kanoglu (2004) presented a simple integral solution for the nonlinear shallow water-wave equations using the classical Carrier and Greenspan transformation, and further extended shoreline position and velocity to a simpler integral formulation. In addition, Tinti and Tonini (2005, J. Fluid Mech. 535, 33-64) defined initial condition in a very convenient form for near-shore events. We use Tinti and Tonini (2005) type initial condition in Kanoglu's (2004) shoreline integral solution, which leads further simplified estimates for shoreline position and velocity, i.e. algebraic relation. We then use this algebraic runup estimate to investigate effect of earthquake source parameters on maximum runup and present results similar to Sepulveda and Liu (2016, Coast. Eng. 112, 57-68).

A unique set of micromechanics equations for high temperature metal matrix composites

NASA Technical Reports Server (NTRS)

Hopkins, D. A.; Chamis, C. C.

1985-01-01

A unique set of micromechanic equations is presented for high temperature metal matrix composites. The set includes expressions to predict mechanical properties, thermal properties and constituent microstresses for the unidirectional fiber reinforced ply. The equations are derived based on a mechanics of materials formulation assuming a square array unit cell model of a single fiber, surrounding matrix and an interphase to account for the chemical reaction which commonly occurs between fiber and matrix. A three-dimensional finite element analysis was used to perform a preliminary validation of the equations. Excellent agreement between properties predicted using the micromechanics equations and properties simulated by the finite element analyses are demonstrated. Implementation of the micromechanics equations as part of an integrated computational capability for nonlinear structural analysis of high temperature multilayered fiber composites is illustrated.

Theoretical study on the sound absorption of electrolytic solutions. I. Theoretical formulation.

PubMed

Yamaguchi, T; Matsuoka, T; Koda, S

2007-04-14

A theory is formulated that describes the sound absorption of electrolytic solutions due to the relative motion of ions, including the formation of ion pairs. The theory is based on the Kubo-Green formula for the bulk viscosity. The time correlation function of the pressure is projected onto the bilinear product of the density modes of ions. The time development of the product of density modes is described by the diffusive limit of the generalized Langevin equation, and approximate expressions for the three- and four-body correlation functions required are given with the hypernetted-chain integral equation theory. Calculations on the aqueous solutions of model electrolytes are performed. It is demonstrated that the theory describes both the activated barrier crossing between contact and solvent-separated ion pairs and the Coulombic correlation between ions.

Theoretical study on the sound absorption of electrolytic solutions. I. Theoretical formulation

NASA Astrophysics Data System (ADS)

Yamaguchi, T.; Matsuoka, T.; Koda, S.

2007-04-01

A theory is formulated that describes the sound absorption of electrolytic solutions due to the relative motion of ions, including the formation of ion pairs. The theory is based on the Kubo-Green formula for the bulk viscosity. The time correlation function of the pressure is projected onto the bilinear product of the density modes of ions. The time development of the product of density modes is described by the diffusive limit of the generalized Langevin equation, and approximate expressions for the three- and four-body correlation functions required are given with the hypernetted-chain integral equation theory. Calculations on the aqueous solutions of model electrolytes are performed. It is demonstrated that the theory describes both the activated barrier crossing between contact and solvent-separated ion pairs and the Coulombic correlation between ions.

Analysis of crack propagation in roller bearings using the boundary integral equation method - A mixed-mode loading problem

NASA Technical Reports Server (NTRS)

Ghosn, L. J.

1988-01-01

Crack propagation in a rotating inner raceway of a high-speed roller bearing is analyzed using the boundary integral method. The model consists of an edge plate under plane strain condition upon which varying Hertzian stress fields are superimposed. A multidomain boundary integral equation using quadratic elements was written to determine the stress intensity factors KI and KII at the crack tip for various roller positions. The multidomain formulation allows the two faces of the crack to be modeled in two different subregions, making it possible to analyze crack closure when the roller is positioned on or close to the crack line. KI and KII stress intensity factors along any direction were computed. These calculations permit determination of crack growth direction along which the average KI times the alternating KI is maximum.

A finite element formulation for supersonic flows around complex configurations

NASA Technical Reports Server (NTRS)

Morino, L.

1974-01-01

The problem of small perturbation potential supersonic flow around complex configurations is considered. This problem requires the solution of an integral equation relating the values of the potential on the surface of the body to the values of the normal derivative, which is known from the small perturbation boundary conditions. The surface of the body is divided into small (hyperboloidal quadrilateral) surface elements which are described in terms of the Cartesian components of the four corner points. The values of the potential (and its normal derivative) within each element are assumed to be constant and equal to its value at the centroid of the element. This yields a set of linear algebraic equations whose coefficients are given by source and doublet integrals over the surface elements. Closed form evaluations of the integrals are presented.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liao, Haitao, E-mail: liaoht@cae.ac.cn

The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results inmore » an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method.« less

Efficient sensitivity analysis method for chaotic dynamical systems

NASA Astrophysics Data System (ADS)

Liao, Haitao

2016-05-01

The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method.

Broadband Noise Predictions Based on a New Aeroacoustic Formulation

NASA Technical Reports Server (NTRS)

Casper, J.; Farassat, F.

2002-01-01

A new analytic result in acoustics called 'Formulation 1B,' proposed by Farassat, is used to compute the loading noise from an unsteady surface pressure distribution on a thin airfoil in the time domain. This formulation is a new solution of the Ffowcs Williams-Hawkings equation with the loading source term. The formulation contains a far-field surface integral that depends on the time derivative and the surface gradient of the pressure on the airfoil, as well as a contour integral on the boundary of the airfoil surface. As a first test case, the new formulation is used to compute the noise radiated from a flat plate, moving through a sinusoidal gust of constant frequency. The unsteady surface pressure for this test case is specified analytically from a result that is based on linear airfoil theory. This test case is used to examine the velocity scaling properties of Formulation 1B, and to demonstrate its equivalence to Formulation 1A, of Farassat. The new acoustic formulation, again with an analytic surface pressure, is then used to predict broadband noise radiated from an airfoil immersed in homogeneous turbulence. The results are compared with experimental data previously reported by Paterson and Amiet. Good agreement between predictions and measurements is obtained. The predicted results also agree very well with those of Paterson and Amiet, who used a frequency-domain approach. Finally, an alternative form of Formulation 1B is described for statistical analysis of broadband noise.

A finite-element analysis for steady and oscillatory supersonic flows around complex configurations

NASA Technical Reports Server (NTRS)

Morino, L.; Chen, L. T.

1974-01-01

The problem of small perturbation potential supersonic flow around complex configurations is considered. This problem requires the solution of an integral equation relating the values of the potential on the surface of the body to the values of the normal derivative, which is known from the small perturbation boundary conditions. The surface of the body is divided into small (hyperboloidal quadrilateral) surface elements, sigma sub i, which are described in terms of the Cartesian components of the four corner points. The values of the potential (and its normal derivative) within each element is assumed to be constant and equal to its value at the centroid of the element, and this yields a set of linear algebraic equations. The coefficients of the equation are given by source and doublet integrals over the surface elements, sigma sub i. The results obtained using the above formulation are compared with existing analytical and experimental results.

Multirate Particle-in-Cell Time Integration Techniques of Vlasov-Maxwell Equations for Collisionless Kinetic Plasma Simulations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, Guangye; Chacon, Luis; Knoll, Dana Alan

2015-07-31

A multi-rate PIC formulation was developed that employs large timesteps for slow field evolution, and small (adaptive) timesteps for particle orbit integrations. Implementation is based on a JFNK solver with nonlinear elimination and moment preconditioning. The approach is free of numerical instabilities (ω peΔt >>1, and Δx >> λ D), and requires many fewer dofs (vs. explicit PIC) for comparable accuracy in challenging problems. Significant gains (vs. conventional explicit PIC) may be possible for large scale simulations. The paper is organized as follows: Vlasov-Maxwell Particle-in-cell (PIC) methods for plasmas; Explicit, semi-implicit, and implicit time integrations; Implicit PIC formulation (Jacobian-Free Newton-Krylovmore » (JFNK) with nonlinear elimination allows different treatments of disparate scales, discrete conservation properties (energy, charge, canonical momentum, etc.)); Some numerical examples; and Summary.« less

On the energy integral for first post-Newtonian approximation

NASA Astrophysics Data System (ADS)

O'Leary, Joseph; Hill, James M.; Bennett, James C.

2018-07-01

The post-Newtonian approximation for general relativity is widely adopted by the geodesy and astronomy communities. It has been successfully exploited for the inclusion of relativistic effects in practically all geodetic applications and techniques such as satellite/lunar laser ranging and very long baseline interferometry. Presently, the levels of accuracy required in geodetic techniques require that reference frames, planetary and satellite orbits and signal propagation be treated within the post-Newtonian regime. For arbitrary scalar W and vector gravitational potentials W^j (j=1,2,3), we present a novel derivation of the energy associated with a test particle in the post-Newtonian regime. The integral so obtained appears not to have been given previously in the literature and is deduced through algebraic manipulation on seeking a Jacobi-like integral associated with the standard post-Newtonian equations of motion. The new integral is independently verified through a variational formulation using the post-Newtonian metric components and is subsequently verified by numerical integration of the post-Newtonian equations of motion.

Gauge and integrable theories in loop spaces

NASA Astrophysics Data System (ADS)

Ferreira, L. A.; Luchini, G.

2012-05-01

We propose an integral formulation of the equations of motion of a large class of field theories which leads in a quite natural and direct way to the construction of conservation laws. The approach is based on generalized non-abelian Stokes theorems for p-form connections, and its appropriate mathematical language is that of loop spaces. The equations of motion are written as the equality of a hyper-volume ordered integral to a hyper-surface ordered integral on the border of that hyper-volume. The approach applies to integrable field theories in (1+1) dimensions, Chern-Simons theories in (2+1) dimensions, and non-abelian gauge theories in (2+1) and (3+1) dimensions. The results presented in this paper are relevant for the understanding of global properties of those theories. As a special byproduct we solve a long standing problem in (3+1)-dimensional Yang-Mills theory, namely the construction of conserved charges, valid for any solution, which are invariant under arbitrary gauge transformations.

WARP3D-Release 10.8: Dynamic Nonlinear Analysis of Solids using a Preconditioned Conjugate Gradient Software Architecture

NASA Technical Reports Server (NTRS)

Koppenhoefer, Kyle C.; Gullerud, Arne S.; Ruggieri, Claudio; Dodds, Robert H., Jr.; Healy, Brian E.

1998-01-01

This report describes theoretical background material and commands necessary to use the WARP3D finite element code. WARP3D is under continuing development as a research code for the solution of very large-scale, 3-D solid models subjected to static and dynamic loads. Specific features in the code oriented toward the investigation of ductile fracture in metals include a robust finite strain formulation, a general J-integral computation facility (with inertia, face loading), an element extinction facility to model crack growth, nonlinear material models including viscoplastic effects, and the Gurson-Tver-gaard dilatant plasticity model for void growth. The nonlinear, dynamic equilibrium equations are solved using an incremental-iterative, implicit formulation with full Newton iterations to eliminate residual nodal forces. The history integration of the nonlinear equations of motion is accomplished with Newmarks Beta method. A central feature of WARP3D involves the use of a linear-preconditioned conjugate gradient (LPCG) solver implemented in an element-by-element format to replace a conventional direct linear equation solver. This software architecture dramatically reduces both the memory requirements and CPU time for very large, nonlinear solid models since formation of the assembled (dynamic) stiffness matrix is avoided. Analyses thus exhibit the numerical stability for large time (load) steps provided by the implicit formulation coupled with the low memory requirements characteristic of an explicit code. In addition to the much lower memory requirements of the LPCG solver, the CPU time required for solution of the linear equations during each Newton iteration is generally one-half or less of the CPU time required for a traditional direct solver. All other computational aspects of the code (element stiffnesses, element strains, stress updating, element internal forces) are implemented in the element-by- element, blocked architecture. This greatly improves vectorization of the code on uni-processor hardware and enables straightforward parallel-vector processing of element blocks on multi-processor hardware.

Integral Equation Method for Electromagnetic Wave Propagation in Stratified Anisotropic Dielectric-Magnetic Materials

NASA Astrophysics Data System (ADS)

Shu, Wei-Xing; Fu, Na; Lü, Xiao-Fang; Luo, Hai-Lu; Wen, Shuang-Chun; Fan, Dian-Yuan

2010-11-01

We investigate the propagation of electromagnetic waves in stratified anisotropic dielectric-magnetic materials using the integral equation method (IEM). Based on the superposition principle, we use Hertz vector formulations of radiated fields to study the interaction of wave with matter. We derive in a new way the dispersion relation, Snell's law and reflection/transmission coefficients by self-consistent analyses. Moreover, we find two new forms of the generalized extinction theorem. Applying the IEM, we investigate the wave propagation through a slab and disclose the underlying physics, which are further verified by numerical simulations. The results lead to a unified framework of the IEM for the propagation of wave incident either from a medium or vacuum in stratified dielectric-magnetic materials.

Analysis of scattering by a linear chain of spherical inclusions in an optical fiber

NASA Astrophysics Data System (ADS)

Chremmos, Ioannis D.; Uzunoglu, Nikolaos K.

2006-12-01

The scattering by a linear chain of spherical dielectric inclusions, embedded along the axis of an optical fiber, is analyzed using a rigorous integral equation formulation, based on the dyadic Green's function theory. The coupled electric field integral equations are solved by applying the Galerkin technique with Mie-type expansion of the field inside the spheres in terms of spherical waves. The analysis extends the previously studied case of a single spherical inhomogeneity inside a fiber to the multisphere-scattering case, by utilizing the classic translational addition theorems for spherical waves in order to analytically extract the direct-intersphere-coupling coefficients. Results for the transmitted and reflected power, on incidence of the fundamental HE11 mode, are presented for several cases.

Domain identification in impedance computed tomography by spline collocation method

NASA Technical Reports Server (NTRS)

Kojima, Fumio

1990-01-01

A method for estimating an unknown domain in elliptic boundary value problems is considered. The problem is formulated as an inverse problem of integral equations of the second kind. A computational method is developed using a splice collocation scheme. The results can be applied to the inverse problem of impedance computed tomography (ICT) for image reconstruction.

Computational cost of two alternative formulations of Cahn-Hilliard equations

NASA Astrophysics Data System (ADS)

Paszyński, Maciej; Gurgul, Grzegorz; Łoś, Marcin; Szeliga, Danuta

2018-05-01

In this paper we propose two formulations of Cahn-Hilliard equations, which have several applications in cancer growth modeling and material science phase-field simulations. The first formulation uses one C4 partial differential equations (PDEs) the second one uses two C2 PDEs. Finally, we compare the computational costs of direct solvers for both formulations, using the refined isogeometric analysis (rIGA) approach.

Local-in-Time Adjoint-Based Method for Optimal Control/Design Optimization of Unsteady Compressible Flows

NASA Technical Reports Server (NTRS)

Yamaleev, N. K.; Diskin, B.; Nielsen, E. J.

2009-01-01

.We study local-in-time adjoint-based methods for minimization of ow matching functionals subject to the 2-D unsteady compressible Euler equations. The key idea of the local-in-time method is to construct a very accurate approximation of the global-in-time adjoint equations and the corresponding sensitivity derivative by using only local information available on each time subinterval. In contrast to conventional time-dependent adjoint-based optimization methods which require backward-in-time integration of the adjoint equations over the entire time interval, the local-in-time method solves local adjoint equations sequentially over each time subinterval. Since each subinterval contains relatively few time steps, the storage cost of the local-in-time method is much lower than that of the global adjoint formulation, thus making the time-dependent optimization feasible for practical applications. The paper presents a detailed comparison of the local- and global-in-time adjoint-based methods for minimization of a tracking functional governed by the Euler equations describing the ow around a circular bump. Our numerical results show that the local-in-time method converges to the same optimal solution obtained with the global counterpart, while drastically reducing the memory cost as compared to the global-in-time adjoint formulation.

Iterative Methods to Solve Linear RF Fields in Hot Plasma

NASA Astrophysics Data System (ADS)

Spencer, Joseph; Svidzinski, Vladimir; Evstatiev, Evstati; Galkin, Sergei; Kim, Jin-Soo

2014-10-01

Most magnetic plasma confinement devices use radio frequency (RF) waves for current drive and/or heating. Numerical modeling of RF fields is an important part of performance analysis of such devices and a predictive tool aiding design and development of future devices. Prior attempts at this modeling have mostly used direct solvers to solve the formulated linear equations. Full wave modeling of RF fields in hot plasma with 3D nonuniformities is mostly prohibited, with memory demands of a direct solver placing a significant limitation on spatial resolution. Iterative methods can significantly increase spatial resolution. We explore the feasibility of using iterative methods in 3D full wave modeling. The linear wave equation is formulated using two approaches: for cold plasmas the local cold plasma dielectric tensor is used (resolving resonances by particle collisions), while for hot plasmas the conductivity kernel (which includes a nonlocal dielectric response) is calculated by integrating along test particle orbits. The wave equation is discretized using a finite difference approach. The initial guess is important in iterative methods, and we examine different initial guesses including the solution to the cold plasma wave equation. Work is supported by the U.S. DOE SBIR program.

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

NASA Astrophysics Data System (ADS)

Galucio, A. C.; Deü, J.-F.; Ohayon, R.

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler-Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.

The MHOST finite element program: 3-D inelastic analysis methods for hot section components. Volume 1: Theoretical manual

NASA Technical Reports Server (NTRS)

Nakazawa, Shohei

1991-01-01

Formulations and algorithms implemented in the MHOST finite element program are discussed. The code uses a novel concept of the mixed iterative solution technique for the efficient 3-D computations of turbine engine hot section components. The general framework of variational formulation and solution algorithms are discussed which were derived from the mixed three field Hu-Washizu principle. This formulation enables the use of nodal interpolation for coordinates, displacements, strains, and stresses. Algorithmic description of the mixed iterative method includes variations for the quasi static, transient dynamic and buckling analyses. The global-local analysis procedure referred to as the subelement refinement is developed in the framework of the mixed iterative solution, of which the detail is presented. The numerically integrated isoparametric elements implemented in the framework is discussed. Methods to filter certain parts of strain and project the element discontinuous quantities to the nodes are developed for a family of linear elements. Integration algorithms are described for linear and nonlinear equations included in MHOST program.

Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves

NASA Astrophysics Data System (ADS)

Yi, Taishan; Chen, Yuming; Wu, Jianhong

Reaction diffusion equations with delayed nonlinear reaction terms are used as prototypes to motivate an appropriate abstract formulation of dynamical systems with unimodal nonlinearity. For such non-monotone dynamical systems, we develop a general comparison principle and show how this general comparison principle, coupled with some existing results for monotone dynamical systems, can be used to establish results on the asymptotic speeds of spread and travelling waves. We illustrate our main results by an integral equation which includes a nonlocal delayed reaction diffusion equation and a nonlocal delayed lattice differential system in an unbounded domain, with the non-monotone nonlinearities including the Ricker birth function and the Mackey-Glass hematopoiesis feedback.

Three-dimensional finite element analysis for high velocity impact. [of projectiles from space debris

NASA Technical Reports Server (NTRS)

Chan, S. T. K.; Lee, C. H.; Brashears, M. R.

1975-01-01

A finite element algorithm for solving unsteady, three-dimensional high velocity impact problems is presented. A computer program was developed based on the Eulerian hydroelasto-viscoplastic formulation and the utilization of the theorem of weak solutions. The equations solved consist of conservation of mass, momentum, and energy, equation of state, and appropriate constitutive equations. The solution technique is a time-dependent finite element analysis utilizing three-dimensional isoparametric elements, in conjunction with a generalized two-step time integration scheme. The developed code was demonstrated by solving one-dimensional as well as three-dimensional impact problems for both the inviscid hydrodynamic model and the hydroelasto-viscoplastic model.

A finite-element analysis for steady and oscillatory subsonic flow around complex configurations

NASA Technical Reports Server (NTRS)

Chen, L. T.; Suciu, E. O.; Morino, L.

1974-01-01

The problem of potential subsonic flow around complex configurations is considered. The solution is given of an integral equation relating the values of the potential on the surface of the body to the values of the normal derivative, which is known from the boundary conditions. The surface of the body is divided into small (hyperboloidal quadrilateral) surface elements, which are described in terms of the Cartesian components of the four corner points. The values of the potential (and its normal derivative) within each element is assumed to be constant and equal to its value at the centroid of the element. The coefficients of the equation are given by source and doublet integrals over the surface elements. Closed form evaluations of the integrals are presented. The results obtained with the above formulation are compared with existing analytical and experimental results.

Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Morozov, Oleg I.

2018-06-01

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDE s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.

Adhesive contact between a rigid spherical indenter and an elastic multi-layer coated substrate

PubMed Central

Stan, Gheorghe; Adams, George G.

2016-01-01

In this work the frictionless, adhesive contact between a rigid spherical indenter and an elastic multi-layer coated half-space was investigated by means of an integral transform formulation. The indented multi-layer coats were considered as made of isotropic layers that are perfectly bonded to each other and to an isotropic substrate. The adhesive interaction between indenter and contacting surface was treated as Maugis-type adhesion to provide general applicability within the entire range of adhesive interactions. By using a transfer matrix method, the stress-strain equations of the system were reduced to two coupled integral equations for the stress distribution under the indenter and the ratio between the adhesion radius and the contact radius, respectively. These resulting integral equations were solved through a numerical collocation technique, with solutions for the load dependencies of the contact radius and indentation depth for various values of the adhesion parameter and layer composition. The method developed here can be used to calculate the force-distance response of adhesive contacts on various inhomogeneous half-spaces that can be modeled as multi-layer coated half-spaces. PMID:27574338

Simulation of a steady-state integrated human thermal system.

NASA Technical Reports Server (NTRS)

Hsu, F. T.; Fan, L. T.; Hwang, C. L.

1972-01-01

The mathematical model of an integrated human thermal system is formulated. The system consists of an external thermal regulation device on the human body. The purpose of the device (a network of cooling tubes held in contact with the surface of the skin) is to maintain the human body in a state of thermoneutrality. The device is controlled by varying the inlet coolant temperature and coolant mass flow rate. The differential equations of the model are approximated by a set of algebraic equations which result from the application of the explicit forward finite difference method to the differential equations. The integrated human thermal system is simulated for a variety of combinations of the inlet coolant temperature, coolant mass flow rate, and metabolic rates. Two specific cases are considered: (1) the external thermal regulation device is placed only on the head and (2) the devices are placed on the head and the torso. The results of the simulation indicate that when the human body is exposed to hot environment, thermoneutrality can be attained by localized cooling if the operating variables of the external regulation device(s) are properly controlled.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wang, Y. B.; Zhu, X. W., E-mail: xiaowuzhu1026@znufe.edu.cn; Dai, H. H.

Though widely used in modelling nano- and micro- structures, Eringen’s differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings aremore » considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.« less

Modeling of Inverted Annular Film Boiling using an integral method

NASA Astrophysics Data System (ADS)

Sridharan, Arunkumar

In modeling Inverted Annular Film Boiling (IAFB), several important phenomena such as interaction between the liquid and the vapor phases and irregular nature of the interface, which greatly influence the momentum and heat transfer at the interface, need to be accounted for. However, due to the complexity of these phenomena, they were not modeled in previous studies. Since two-phase heat transfer equations and relationships rely heavily on experimental data, many closure relationships that were used in previous studies to solve the problem are empirical in nature. Also, in deriving the relationships, the experimental data were often extrapolated beyond the intended range of conditions, causing errors in predictions. In some cases, empirical correlations that were derived from situations other than IAFB, and whose applicability to IAFB was questionable, were used. Moreover, arbitrary constants were introduced in the model developed in previous studies to provide good fit to the experimental data. These constants have no physical basis, thereby leading to questionable accuracy in the model predictions. In the present work, modeling of Inverted Annular Film Boiling (IAFB) is done using Integral Method. Two-dimensional formulation of IAFB is presented. Separate equations for the conservation of mass, momentum and energy are derived from first principles, for the vapor film and the liquid core. Turbulence is incorporated in the formulation. The system of second-order partial differential equations is integrated over the radial direction to obtain a system of integral differential equations. In order to solve the system of equations, second order polynomial profiles are used to describe the nondimensional velocity and temperatures. The unknown coefficients in the profiles are functions of the axial direction alone. Using the boundary conditions that govern the physical problem, equations for the unknown coefficients are derived in terms of the primary dependent variables: wall shear stress, interfacial shear stress, film thickness, pressure, wall temperature and the mass transfer rate due to evaporation. A system of non-linear first order coupled ordinary differential equations is obtained. Due to the inherent mathematical complexity of the system of equations, simplifying assumptions are made to obtain a numerical solution. The system of equations is solved numerically to obtain values of the unknown quantities at each subsequent axial location. Derived quantities like void fraction and heat transfer coefficient are calculated at each axial location. The calculation is terminated when the void fraction reaches a value of 0.6, the upper limit of IAFB. The results obtained agree with the experimental trends observed. Void fraction increases along the heated length, while the heat transfer coefficient drops due to the increased resistance of the vapor film as expected.

Solution of Volterra and Fredholm Classes of Equations via Triangular Orthogonal Function (A Combination of Right Hand Triangular Function and Left Hand Triangular Function) and Hybrid Orthogonal Function (A Combination of Sample Hold Function and Right Hand Triangular Function)

NASA Astrophysics Data System (ADS)

Mukhopadhyay, Anirban; Ganguly, Anindita; Chatterjee, Saumya Deep

2018-04-01

In this paper the authors have dealt with seven kinds of non-linear Volterra and Fredholm classes of equations. The authors have formulated an algorithm for solving the aforementioned equation types via Hybrid Function (HF) and Triangular Function (TF) piecewise-linear orthogonal approach. In this approach the authors have reduced integral equation or integro-differential equation into equivalent system of simultaneous non-linear equation and have employed either Newton's method or Broyden's method to solve the simultaneous non-linear equations. The authors have calculated the L2-norm error and the max-norm error for both HF and TF method for each kind of equations. Through the illustrated examples, the authors have shown that the HF based algorithm produces stable result, on the contrary TF-computational method yields either stable, anomalous or unstable results.

A time-domain Kirchhoff formula for the convective acoustic wave equation

PubMed Central

Ghorbaniasl, Ghader; Siozos-Rousoulis, Leonidas; Lacor, Chris

2016-01-01

Kirchhoff’s integral method allows propagated sound to be predicted, based on the pressure and its derivatives in time and space obtained on a data surface located in the linear flow region. Kirchhoff’s formula for noise prediction from high-speed rotors and propellers suffers from the limitation of the observer located in uniform flow, thus requiring an extension to arbitrarily moving media. This paper presents a Kirchhoff formulation for moving surfaces in a uniform moving medium of arbitrary configuration. First, the convective wave equation is derived in a moving frame, based on the generalized functions theory. The Kirchhoff formula is then obtained for moving surfaces in the time domain. The formula has a similar form to the Kirchhoff formulation for moving surfaces of Farassat and Myers, with the presence of additional terms owing to the moving medium effect. The equation explicitly accounts for the influence of mean flow and angle of attack on the radiated noise. The formula is verified by analytical cases of a monopole source located in a moving medium. PMID:27118912

Hamiltonian structure of real Monge - Ampère equations

NASA Astrophysics Data System (ADS)

Nutku, Y.

1996-06-01

The variational principle for the real homogeneous Monge - Ampère equation in two dimensions is shown to contain three arbitrary functions of four variables. There exist two different specializations of this variational principle where the Lagrangian is degenerate and furthermore contains an arbitrary function of two variables. The Hamiltonian formulation of these degenerate Lagrangian systems requires the use of Dirac's theory of constraints. As in the case of most completely integrable systems the constraints are second class and Dirac brackets directly yield the Hamiltonian operators. Thus the real homogeneous Monge - Ampère equation in two dimensions admits two classes of infinitely many Hamiltonian operators, namely a family of local, as well as another family non-local Hamiltonian operators and symplectic 2-forms which depend on arbitrary functions of two variables. The simplest non-local Hamiltonian operator corresponds to the Kac - Moody algebra of vector fields and functions on the unit circle. Hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair. In the case of real Monge - Ampère equations with constant right-hand side this compatible pair is the only pair of Hamiltonian operators that survives. Then the complete integrability of all these real Monge - Ampère equations follows by Magri's theorem. Some of the remarkable properties we have obtained for the Hamiltonian structure of the real homogeneous Monge - Ampère equation in two dimensions turn out to be generic to the real homogeneous Monge - Ampère equation and the geodesic flow for the complex homogeneous Monge - Ampère equation in arbitrary number of dimensions. Hence among all integrable nonlinear evolution equations in one space and one time dimension, the real homogeneous Monge - Ampère equation is distinguished as one that retains its character as an integrable system in multiple dimensions.

Electromagnetic pulses, localized and causal

NASA Astrophysics Data System (ADS)

Lekner, John

2018-01-01

We show that pulse solutions of the wave equation can be expressed as time Fourier superpositions of scalar monochromatic beam wave functions (solutions of the Helmholtz equation). This formulation is shown to be equivalent to Bateman's integral expression for solutions of the wave equation, for axially symmetric solutions. A closed-form one-parameter solution of the wave equation, containing no backward-propagating parts, is constructed from a beam which is the tight-focus limit of two families of beams. Application is made to transverse electric and transverse magnetic pulses, with evaluation of the energy, momentum and angular momentum for a pulse based on the general localized and causal form. Such pulses can be represented as superpositions of photons. Explicit total energy and total momentum values are given for the one-parameter closed-form pulse.

Discovery of a general method of solving the Schrödinger and dirac equations that opens a way to accurately predictive quantum chemistry.

PubMed

Nakatsuji, Hiroshi

2012-09-18

Just as Newtonian law governs classical physics, the Schrödinger equation (SE) and the relativistic Dirac equation (DE) rule the world of chemistry. So, if we can solve these equations accurately, we can use computation to predict chemistry precisely. However, for approximately 80 years after the discovery of these equations, chemists believed that they could not solve SE and DE for atoms and molecules that included many electrons. This Account reviews ideas developed over the past decade to further the goal of predictive quantum chemistry. Between 2000 and 2005, I discovered a general method of solving the SE and DE accurately. As a first inspiration, I formulated the structure of the exact wave function of the SE in a compact mathematical form. The explicit inclusion of the exact wave function's structure within the variational space allows for the calculation of the exact wave function as a solution of the variational method. Although this process sounds almost impossible, it is indeed possible, and I have published several formulations and applied them to solve the full configuration interaction (CI) with a very small number of variables. However, when I examined analytical solutions for atoms and molecules, the Hamiltonian integrals in their secular equations diverged. This singularity problem occurred in all atoms and molecules because it originates from the singularity of the Coulomb potential in their Hamiltonians. To overcome this problem, I first introduced the inverse SE and then the scaled SE. The latter simpler idea led to immediate and surprisingly accurate solution for the SEs of the hydrogen atom, helium atom, and hydrogen molecule. The free complement (FC) method, also called the free iterative CI (free ICI) method, was efficient for solving the SEs. In the FC method, the basis functions that span the exact wave function are produced by the Hamiltonian of the system and the zeroth-order wave function. These basis functions are called complement functions because they are the elements of the complete functions for the system under consideration. We extended this idea to solve the relativistic DE and applied it to the hydrogen and helium atoms, without observing any problems such as variational collapse. Thereafter, we obtained very accurate solutions of the SE for the ground and excited states of the Born-Oppenheimer (BO) and non-BO states of very small systems like He, H(2)(+), H(2), and their analogues. For larger systems, however, the overlap and Hamiltonian integrals over the complement functions are not always known mathematically (integration difficulty); therefore we formulated the local SE (LSE) method as an integral-free method. Without any integration, the LSE method gave fairly accurate energies and wave functions for small atoms and molecules. We also calculated continuous potential curves of the ground and excited states of small diatomic molecules by introducing the transferable local sampling method. Although the FC-LSE method is simple, the achievement of chemical accuracy in the absolute energy of larger systems remains time-consuming. The development of more efficient methods for the calculations of ordinary molecules would allow researchers to make these calculations more easily.

Kinetic and dynamic probability-density-function descriptions of disperse turbulent two-phase flows

NASA Astrophysics Data System (ADS)

Minier, Jean-Pierre; Profeta, Christophe

2015-11-01

This article analyzes the status of two classical one-particle probability density function (PDF) descriptions of the dynamics of discrete particles dispersed in turbulent flows. The first PDF formulation considers only the process made up by particle position and velocity Zp=(xp,Up) and is represented by its PDF p (t ;yp,Vp) which is the solution of a kinetic PDF equation obtained through a flux closure based on the Furutsu-Novikov theorem. The second PDF formulation includes fluid variables into the particle state vector, for example, the fluid velocity seen by particles Zp=(xp,Up,Us) , and, consequently, handles an extended PDF p (t ;yp,Vp,Vs) which is the solution of a dynamic PDF equation. For high-Reynolds-number fluid flows, a typical formulation of the latter category relies on a Langevin model for the trajectories of the fluid seen or, conversely, on a Fokker-Planck equation for the extended PDF. In the present work, a new derivation of the kinetic PDF equation is worked out and new physical expressions of the dispersion tensors entering the kinetic PDF equation are obtained by starting from the extended PDF and integrating over the fluid seen. This demonstrates that, under the same assumption of a Gaussian colored noise and irrespective of the specific stochastic model chosen for the fluid seen, the kinetic PDF description is the marginal of a dynamic PDF one. However, a detailed analysis reveals that kinetic PDF models of particle dynamics in turbulent flows described by statistical correlations constitute incomplete stand-alone PDF descriptions and, moreover, that present kinetic-PDF equations are mathematically ill posed. This is shown to be the consequence of the non-Markovian characteristic of the stochastic process retained to describe the system and the use of an external colored noise. Furthermore, developments bring out that well-posed PDF descriptions are essentially due to a proper choice of the variables selected to describe physical systems and guidelines are formulated to emphasize the key role played by the notion of slow and fast variables.

High-order accurate finite-volume formulations for the pressure gradient force in layered ocean models

NASA Astrophysics Data System (ADS)

Engwirda, Darren; Kelley, Maxwell; Marshall, John

2017-08-01

Discretisation of the horizontal pressure gradient force in layered ocean models is a challenging task, with non-trivial interactions between the thermodynamics of the fluid and the geometry of the layers often leading to numerical difficulties. We present two new finite-volume schemes for the pressure gradient operator designed to address these issues. In each case, the horizontal acceleration is computed as an integration of the contact pressure force that acts along the perimeter of an associated momentum control-volume. A pair of new schemes are developed by exploring different control-volume geometries. Non-linearities in the underlying equation-of-state definitions and thermodynamic profiles are treated using a high-order accurate numerical integration framework, designed to preserve hydrostatic balance in a non-linear manner. Numerical experiments show that the new methods achieve high levels of consistency, maintaining hydrostatic and thermobaric equilibrium in the presence of strongly-sloping layer geometries, non-linear equations-of-state and non-uniform vertical stratification profiles. These results suggest that the new pressure gradient formulations may be appropriate for general circulation models that employ hybrid vertical coordinates and/or terrain-following representations.

Compatibility Condition in Theory of Solid Mechanics (Elasticity, Structures, and Design Optimization)

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Pai, Shantaram S.; Hopkins, Dale A.

2007-01-01

The strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. This shortcoming prevented the formulation of a direct method to calculate stress. We have researched and understood the compatibility condition for linear problems in elasticity and in finite element analysis. This has lead to the completion of the method of force with stress (or stress resultant) as the primary unknown. The method in elasticity is referred to as the completed Beltrami-Michell formulation (CBMF), and it is the integrated force method (IFM) in structures. The dual integrated force method (IFMD) with displacement as the primary unknown has been formulated. IFM and IFMD produce identical responses. The variational derivation of the CBMF yielded the new boundary compatibility conditions. The CBMF can be used to solve stress, displacement, and mixed boundary value problems. The IFM in structures produced high-fidelity response even with a modest finite element model. The IFM has influenced structural design considerably. A fully utilized design method for strength and stiffness limitation has been developed. The singularity condition in optimization has been identified. The CBMF and IFM tensorial approaches are robust formulations because of simultaneous emphasis on the equilibrium equation and the compatibility condition.

A Galleria Boundary Element Method for two-dimensional nonlinear magnetostatics

NASA Astrophysics Data System (ADS)

Brovont, Aaron D.

The Boundary Element Method (BEM) is a numerical technique for solving partial differential equations that is used broadly among the engineering disciplines. The main advantage of this method is that one needs only to mesh the boundary of a solution domain. A key drawback is the myriad of integrals that must be evaluated to populate the full system matrix. To this day these integrals have been evaluated using numerical quadrature. In this research, a Galerkin formulation of the BEM is derived and implemented to solve two-dimensional magnetostatic problems with a focus on accurate, rapid computation. To this end, exact, closed-form solutions have been derived for all the integrals comprising the system matrix as well as those required to compute fields in post-processing; the need for numerical integration has been eliminated. It is shown that calculation of the system matrix elements using analytical solutions is 15-20 times faster than with numerical integration of similar accuracy. Furthermore, through the example analysis of a c-core inductor, it is demonstrated that the present BEM formulation is a competitive alternative to the Finite Element Method (FEM) for linear magnetostatic analysis. Finally, the BEM formulation is extended to analyze nonlinear magnetostatic problems via the Dual Reciprocity Method (DRBEM). It is shown that a coarse, meshless analysis using the DRBEM is able to achieve RMS error of 3-6% compared to a commercial FEM package in lightly saturated conditions.

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

NASA Astrophysics Data System (ADS)

Lin, Zeng; Wang, Dongdong

2017-10-01

Due to the nonlocal property of the fractional derivative, the finite element analysis of fractional diffusion equation often leads to a dense and non-symmetric stiffness matrix, in contrast to the conventional finite element formulation with a particularly desirable symmetric and banded stiffness matrix structure for the typical diffusion equation. This work first proposes a finite element formulation that preserves the symmetry and banded stiffness matrix characteristics for the fractional diffusion equation. The key point of the proposed formulation is the symmetric weak form construction through introducing a fractional weight function. It turns out that the stiffness part of the present formulation is identical to its counterpart of the finite element method for the conventional diffusion equation and thus the stiffness matrix formulation becomes trivial. Meanwhile, the fractional derivative effect in the discrete formulation is completely transferred to the force vector, which is obviously much easier and efficient to compute than the dense fractional derivative stiffness matrix. Subsequently, it is further shown that for the general fractional advection-diffusion-reaction equation, the symmetric and banded structure can also be maintained for the diffusion stiffness matrix, although the total stiffness matrix is not symmetric in this case. More importantly, it is demonstrated that under certain conditions this symmetric diffusion stiffness matrix formulation is capable of producing very favorable numerical solutions in comparison with the conventional non-symmetric diffusion stiffness matrix finite element formulation. The effectiveness of the proposed methodology is illustrated through a series of numerical examples.

A numerical technique for linear elliptic partial differential equations in polygonal domains.

PubMed

Hashemzadeh, P; Fokas, A S; Smitheman, S A

2015-03-08

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform ( or the Fokas transform ) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map . The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

The Davey-Stewartson Equation on the Half-Plane

NASA Astrophysics Data System (ADS)

Fokas, A. S.

2009-08-01

The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Scott, James R.; Chang, Sin-Chung

1993-01-01

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number.

G-DYN Multibody Dynamics Engine

NASA Technical Reports Server (NTRS)

Acikmese, Behcet; Blackmore, James C.; Broderick, Daniel

2011-01-01

G-DYN is a multi-body dynamic simulation software engine that automatically assembles and integrates equations of motion for arbitrarily connected multibody dynamic systems. The algorithm behind G-DYN is based on a primal-dual formulation of the dynamics that captures the position and velocity vectors (primal variables) of each body and the interaction forces (dual variables) between bodies, which are particularly useful for control and estimation analysis and synthesis. It also takes full advantage of the spare matrix structure resulting from the system dynamics to numerically integrate the equations of motion efficiently. Furthermore, the dynamic model for each body can easily be replaced without re-deriving the overall equations of motion, and the assembly of the equations of motion is done automatically. G-DYN proved an essential software tool in the simulation of spacecraft systems used for small celestial body surface sampling, specifically in simulating touch-and-go (TAG) maneuvers of a robotic sampling system from a comet and asteroid. It is used extensively in validating mission concepts for small body sample return, such as Comet Odyssey and Galahad New Frontiers proposals.

Kinematic validation of a quasi-geostrophic model for the fast dynamics in the Earth's outer core

NASA Astrophysics Data System (ADS)

Maffei, S.; Jackson, A.

2017-09-01

We derive a quasi-geostrophic (QG) system of equations suitable for the description of the Earth's core dynamics on interannual to decadal timescales. Over these timescales, rotation is assumed to be the dominant force and fluid motions are strongly invariant along the direction parallel to the rotation axis. The diffusion-free, QG system derived here is similar to the one derived in Canet et al. but the projection of the governing equations on the equatorial disc is handled via vertical integration and mass conservation is applied to the velocity field. Here we carefully analyse the properties of the resulting equations and we validate them neglecting the action of the Lorentz force in the momentum equation. We derive a novel analytical solution describing the evolution of the magnetic field under these assumptions in the presence of a purely azimuthal flow and an alternative formulation that allows us to numerically solve the evolution equations with a finite element method. The excellent agreement we found with the analytical solution proves that numerical integration of the QG system is possible and that it preserves important physical properties of the magnetic field. Implementation of magnetic diffusion is also briefly considered.

Systematic generation of multibody equations of motion suitable for recursive and parallel manipulation

NASA Technical Reports Server (NTRS)

Nikravesh, Parviz E.; Gim, Gwanghum; Arabyan, Ara; Rein, Udo

1989-01-01

The formulation of a method known as the joint coordinate method for automatic generation of the equations of motion for multibody systems is summarized. For systems containing open or closed kinematic loops, the equations of motion can be reduced systematically to a minimum number of second order differential equations. The application of recursive and nonrecursive algorithms to this formulation, computational considerations and the feasibility of implementing this formulation on multiprocessor computers are discussed.

The Boltzmann equation in the difference formulation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Szoke, Abraham; Brooks III, Eugene D.

2015-05-06

First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.

Properties of the Residual Stress of the Temporally Filtered Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Pruett, C. D.; Gatski, T. B.; Grosch, C. E.; Thacker, W. D.

2002-01-01

The development of a unifying framework among direct numerical simulations, large-eddy simulations, and statistically averaged formulations of the Navier-Stokes equations, is of current interest. Toward that goal, the properties of the residual (subgrid-scale) stress of the temporally filtered Navier-Stokes equations are carefully examined. Causal time-domain filters, parameterized by a temporal filter width 0 less than Delta less than infinity, are considered. For several reasons, the differential forms of such filters are preferred to their corresponding integral forms; among these, storage requirements for differential forms are typically much less than for integral forms and, for some filters, are independent of Delta. The behavior of the residual stress in the limits of both vanishing and in infinite filter widths is examined. It is shown analytically that, in the limit Delta to 0, the residual stress vanishes, in which case the Navier-Stokes equations are recovered from the temporally filtered equations. Alternately, in the limit Delta to infinity, the residual stress is equivalent to the long-time averaged stress, and the Reynolds-averaged Navier-Stokes equations are recovered from the temporally filtered equations. The predicted behavior at the asymptotic limits of filter width is further validated by numerical simulations of the temporally filtered forced, viscous Burger's equation. Finally, finite filter widths are also considered, and a priori analyses of temporal similarity and temporal approximate deconvolution models of the residual stress are conducted.

On the use of co-ordinate stretching in the numeral computation of high frequency scattering. [of jet engine noise by fuselage

NASA Technical Reports Server (NTRS)

Bayliss, A.

1978-01-01

The scattering of the sound of a jet engine by an airplane fuselage is modeled by solving the axially symmetric Helmholtz equation exterior to a long thin ellipsoid. The integral equation method based on the single layer potential formulation is used. A family of coordinate systems on the body is introduced and an algorithm is presented to determine the optimal coordinate system. Numerical results verify that the optimal choice enables the solution to be computed with a grid that is coarse relative to the wavelength.

Landau ghost pole problem in quantum field theory: From 50th of last century to the present day

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jafarov, Rauf G., E-mail: rauf-jafarov@hotmail.com; Mutallimov, Mutallim M.

2016-03-25

In this paper we present our results of the investigation of asymptotical behavior of amplitude at short distances in four-dimensional scalar field theory with ϕ{sup 4} interaction. To formulate of our calculating model – two-particle approximation of the mean-field expansion we have used an Rochev’s iteration scheme of solution of the Schwinger-Dyson equations with the fermion bilocal source. We have considered the nonlinear integral equations in deep-inelastic region of momenta. As result we have a non-trivial behavior of amplitude at large momenta.

Lectures on the scattering of light. [by dielectric sphere

NASA Technical Reports Server (NTRS)

Saxon, D. S.

1974-01-01

The exact (Mie) theory for the scattering of a plane wave by a dielectric sphere is presented. Since this infinite series solution is computationally impractical for large spheres, another formulation is given in terms of an integral equation valid for a bounded, but otherwise general array of scatterers. This equation is applied to the scattering by a single sphere, and several methods are suggested for approximating the scattering cross section in closed form. A tensor scattering matrix is introduced, in terms of which some general scattering theorems are derived. The application of the formalism to multiple scattering is briefly considered.

Correcting magnetic probe perturbations on current density measurements of current carrying plasmas.

PubMed

Knoblauch, P; Raspa, V; Di Lorenzo, F; Lazarte, A; Clausse, A; Moreno, C

2010-09-01

A method to infer the current density distribution in the current sheath of a plasma focus discharge from a magnetic probe is formulated and then applied to experimental data obtained in a 1.1 kJ device. Distortions on the magnetic probe signal caused by current redistribution and by a time-dependent total discharge current are considered simultaneously, leading to an integral equation for the current density. Two distinct, easy to implement, numerical procedures are given to solve such equation. Experimental results show the coexistence of at least two maxima in the current density structure of a nitrogen sheath.

Generalized vector calculus on convex domain

NASA Astrophysics Data System (ADS)

Agrawal, Om P.; Xu, Yufeng

2015-06-01

In this paper, we apply recently proposed generalized integral and differential operators to develop generalized vector calculus and generalized variational calculus for problems defined over a convex domain. In particular, we present some generalization of Green's and Gauss divergence theorems involving some new operators, and apply these theorems to generalized variational calculus. For fractional power kernels, the formulation leads to fractional vector calculus and fractional variational calculus for problems defined over a convex domain. In special cases, when certain parameters take integer values, we obtain formulations for integer order problems. Two examples are presented to demonstrate applications of the generalized variational calculus which utilize the generalized vector calculus developed in the paper. The first example leads to a generalized partial differential equation and the second example leads to a generalized eigenvalue problem, both in two dimensional convex domains. We solve the generalized partial differential equation by using polynomial approximation. A special case of the second example is a generalized isoperimetric problem. We find an approximate solution to this problem. Many physical problems containing integer order integrals and derivatives are defined over arbitrary domains. We speculate that future problems containing fractional and generalized integrals and derivatives in fractional mechanics will be defined over arbitrary domains, and therefore, a general variational calculus incorporating a general vector calculus will be needed for these problems. This research is our first attempt in that direction.

An unstructured-mesh finite-volume MPDATA for compressible atmospheric dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kühnlein, Christian, E-mail: christian.kuehnlein@ecmwf.int; Smolarkiewicz, Piotr K., E-mail: piotr.smolarkiewicz@ecmwf.int

An advancement of the unstructured-mesh finite-volume MPDATA (Multidimensional Positive Definite Advection Transport Algorithm) is presented that formulates the error-compensative pseudo-velocity of the scheme to rely only on face-normal advective fluxes to the dual cells, in contrast to the full vector employed in previous implementations. This is essentially achieved by expressing the temporal truncation error underlying the pseudo-velocity in a form consistent with the flux-divergence of the governing conservation law. The development is especially important for integrating fluid dynamics equations on non-rectilinear meshes whenever face-normal advective mass fluxes are employed for transport compatible with mass continuity—the latter being essential for flux-formmore » schemes. In particular, the proposed formulation enables large-time-step semi-implicit finite-volume integration of the compressible Euler equations using MPDATA on arbitrary hybrid computational meshes. Furthermore, it facilitates multiple error-compensative iterations of the finite-volume MPDATA and improved overall accuracy. The advancement combines straightforwardly with earlier developments, such as the nonoscillatory option, the infinite-gauge variant, and moving curvilinear meshes. A comprehensive description of the scheme is provided for a hybrid horizontally-unstructured vertically-structured computational mesh for efficient global atmospheric flow modelling. The proposed finite-volume MPDATA is verified using selected 3D global atmospheric benchmark simulations, representative of hydrostatic and non-hydrostatic flow regimes. Besides the added capabilities, the scheme retains fully the efficacy of established finite-volume MPDATA formulations.« less

Formulating viscous hydrodynamics for large velocity gradients

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pratt, Scott

2008-02-15

Viscous corrections to relativistic hydrodynamics, which are usually formulated for small velocity gradients, have recently been extended from Navier-Stokes formulations to a class of treatments based on Israel-Stewart equations. Israel-Stewart treatments, which treat the spatial components of the stress-energy tensor {tau}{sub ij} as dynamical objects, introduce new parameters, such as the relaxation times describing nonequilibrium behavior of the elements {tau}{sub ij}. By considering linear response theory and entropy constraints, we show how the additional parameters are related to fluctuations of {tau}{sub ij}. Furthermore, the Israel-Stewart parameters are analyzed for their ability to provide stable and physical solutions for sound waves.more » Finally, it is shown how these parameters, which are naturally described by correlation functions in real time, might be constrained by lattice calculations, which are based on path-integral formulations in imaginary time.« less

Feynman-Kac formula for stochastic hybrid systems.

PubMed

Bressloff, Paul C

2017-01-01

We derive a Feynman-Kac formula for functionals of a stochastic hybrid system evolving according to a piecewise deterministic Markov process. We first derive a stochastic Liouville equation for the moment generator of the stochastic functional, given a particular realization of the underlying discrete Markov process; the latter generates transitions between different dynamical equations for the continuous process. We then analyze the stochastic Liouville equation using methods recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment generating function, averaged with respect to realizations of the discrete Markov process. The resulting Feynman-Kac formula takes the form of a differential Chapman-Kolmogorov equation. We illustrate the theory by calculating the occupation time for a one-dimensional velocity jump process on the infinite or semi-infinite real line. Finally, we present an alternative derivation of the Feynman-Kac formula based on a recent path-integral formulation of stochastic hybrid systems.

A transformed path integral approach for solution of the Fokker-Planck equation

NASA Astrophysics Data System (ADS)

Subramaniam, Gnana M.; Vedula, Prakash

2017-10-01

A novel path integral (PI) based method for solution of the Fokker-Planck equation is presented. The proposed method, termed the transformed path integral (TPI) method, utilizes a new formulation for the underlying short-time propagator to perform the evolution of the probability density function (PDF) in a transformed computational domain where a more accurate representation of the PDF can be ensured. The new formulation, based on a dynamic transformation of the original state space with the statistics of the PDF as parameters, preserves the non-negativity of the PDF and incorporates short-time properties of the underlying stochastic process. New update equations for the state PDF in a transformed space and the parameters of the transformation (including mean and covariance) that better accommodate nonlinearities in drift and non-Gaussian behavior in distributions are proposed (based on properties of the SDE). Owing to the choice of transformation considered, the proposed method maps a fixed grid in transformed space to a dynamically adaptive grid in the original state space. The TPI method, in contrast to conventional methods such as Monte Carlo simulations and fixed grid approaches, is able to better represent the distributions (especially the tail information) and better address challenges in processes with large diffusion, large drift and large concentration of PDF. Additionally, in the proposed TPI method, error bounds on the probability in the computational domain can be obtained using the Chebyshev's inequality. The benefits of the TPI method over conventional methods are illustrated through simulations of linear and nonlinear drift processes in one-dimensional and multidimensional state spaces. The effects of spatial and temporal grid resolutions as well as that of the diffusion coefficient on the error in the PDF are also characterized.

Cable Connected Spinning Spacecraft, 1. the Canonical Equations, 2. Urban Mass Transportation, 3

NASA Technical Reports Server (NTRS)

Sitchin, A.

1972-01-01

Work on the dynamics of cable-connected spinning spacecraft was completed by formulating the equations of motion by both the canonical equations and Lagrange's equations and programming them for numerical solution on a digital computer. These energy-based formulations will permit future addition of the effect of cable mass. Comparative runs indicate that the canonical formulation requires less computer time. Available literature on urban mass transportation was surveyed. Areas of the private rapid transit concept of urban transportation are also studied.

A wideband FMBEM for 2D acoustic design sensitivity analysis based on direct differentiation method

NASA Astrophysics Data System (ADS)

Chen, Leilei; Zheng, Changjun; Chen, Haibo

2013-09-01

This paper presents a wideband fast multipole boundary element method (FMBEM) for two dimensional acoustic design sensitivity analysis based on the direct differentiation method. The wideband fast multipole method (FMM) formed by combining the original FMM and the diagonal form FMM is used to accelerate the matrix-vector products in the boundary element analysis. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The strongly singular and hypersingular integrals in the sensitivity equations can be evaluated explicitly and directly by using the piecewise constant discretization. The iterative solver GMRES is applied to accelerate the solution of the linear system of equations. A set of optimal parameters for the wideband FMBEM design sensitivity analysis are obtained by observing the performances of the wideband FMM algorithm in terms of computing time and memory usage. Numerical examples are presented to demonstrate the efficiency and validity of the proposed algorithm.

Integrability from point symmetries in a family of cosmological Horndeski Lagrangians

NASA Astrophysics Data System (ADS)

Dimakis, N.; Giacomini, Alex; Paliathanasis, Andronikos

2017-07-01

For a family of Horndeski theories, formulated in terms of a generalized Galileon model, we study the integrability of the field equations in a Friedmann-Lemaître-Robertson-Walker space-time. We are interested in point transformations which leave invariant the field equations. Noether's theorem is applied to determine the conservation laws for a family of models that belong to the same general class. The cosmological scenarios with or without an extra perfect fluid with constant equation of state parameter are the two important cases of our study. The de Sitter universe and ideal gas solutions are derived by using the invariant functions of the symmetry generators as a demonstration of our result. Furthermore, we discuss the connection of the different models under conformal transformations while we show that when the Horndeski theory reduces to a canonical field the same holds for the conformal equivalent theory. Finally, we discuss how singular solutions provides nonsingular universes in a different frame and vice versa.

On the Milankovitch orbital elements for perturbed Keplerian motion

NASA Astrophysics Data System (ADS)

Rosengren, Aaron J.; Scheeres, Daniel J.

2014-03-01

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace-Runge-Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

NASA Astrophysics Data System (ADS)

Peshkov, Ilya; Pavelka, Michal; Romenski, Evgeniy; Grmela, Miroslav

2018-01-01

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

A combined finite element-boundary integral formulation for solution of two-dimensional scattering problems via CGFFT. [Conjugate Gradient Fast Fourier Transformation

NASA Technical Reports Server (NTRS)

Collins, Jeffery D.; Volakis, John L.; Jin, Jian-Ming

1990-01-01

A new technique is presented for computing the scattering by 2-D structures of arbitrary composition. The proposed solution approach combines the usual finite element method with the boundary-integral equation to formulate a discrete system. This is subsequently solved via the conjugate gradient (CG) algorithm. A particular characteristic of the method is the use of rectangular boundaries to enclose the scatterer. Several of the resulting boundary integrals are therefore convolutions and may be evaluated via the fast Fourier transform (FFT) in the implementation of the CG algorithm. The solution approach offers the principal advantage of having O(N) memory demand and employs a 1-D FFT versus a 2-D FFT as required with a traditional implementation of the CGFFT algorithm. The speed of the proposed solution method is compared with that of the traditional CGFFT algorithm, and results for rectangular bodies are given and shown to be in excellent agreement with the moment method.

A combined finite element and boundary integral formulation for solution via CGFFT of 2-dimensional scattering problems

NASA Technical Reports Server (NTRS)

Collins, Jeffery D.; Volakis, John L.

1989-01-01

A new technique is presented for computing the scattering by 2-D structures of arbitrary composition. The proposed solution approach combines the usual finite element method with the boundary integral equation to formulate a discrete system. This is subsequently solved via the conjugate gradient (CG) algorithm. A particular characteristic of the method is the use of rectangular boundaries to enclose the scatterer. Several of the resulting boundary integrals are therefore convolutions and may be evaluated via the fast Fourier transform (FFT) in the implementation of the CG algorithm. The solution approach offers the principle advantage of having O(N) memory demand and employs a 1-D FFT versus a 2-D FFT as required with a traditional implementation of the CGFFT algorithm. The speed of the proposed solution method is compared with that of the traditional CGFFT algorithm, and results for rectangular bodies are given and shown to be in excellent agreement with the moment method.

Time-local equation for exact time-dependent optimized effective potential in time-dependent density functional theory

NASA Astrophysics Data System (ADS)

Liao, Sheng-Lun; Ho, Tak-San; Rabitz, Herschel; Chu, Shih-I.

2017-04-01

Solving and analyzing the exact time-dependent optimized effective potential (TDOEP) integral equation has been a longstanding challenge due to its highly nonlinear and nonlocal nature. To meet the challenge, we derive an exact time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham orbitals and effective memory orbitals. For illustration, the dipole evolution dynamics of a one-dimension-model chain of hydrogen atoms is numerically evaluated and examined to demonstrate the utility of the proposed time-local formulation. Importantly, it is shown that the zero-force theorem, violated by the time-dependent Krieger-Li-Iafrate approximation, is fulfilled in the current TDOEP framework. This work was partially supported by DOE.

Strong nonlinear rupture theory of thin free liquid films

NASA Astrophysics Data System (ADS)

Chi-Chuan, Hwang; Jun-Liang, Chen; Li-Fu, Shen; Cheng-I, Weng

1996-02-01

A simplified governing equation with high-order effects is formulated after a procedure of evaluating the order of magnitude. Furthermore, the nonlinear evolution equations are derived by the Kármán-Polhausen integral method with a specified velocity profile. Particularly, the effects of surface tension, van der Waals potential, inertia and high-order viscous dissipation are taken into consideration in these equation. The numerical results reveal that the rupture time of free film is much shorter than that of a film on a flat plate. It is shown that because of a more complete high-order viscous dissipation effect discussed in the present study, the rupture process of present model is slower than is predicted by the high-order long wave theory.

A Potential Function Derivation of a Constitutive Equation for Inelastic Material Response

NASA Technical Reports Server (NTRS)

Stouffer, D. C.; Elfoutouh, N. A.

1983-01-01

Physical and thermodynamic concepts are used to develop a potential function for application to high temperature polycrystalline material response. Inherent in the formulation is a differential relationship between the potential function and constitutive equation in terms of the state variables. Integration of the differential relationship produces a state variable evolution equation that requires specification of the initial value of the state variable and its time derivative. It is shown that the initial loading rate, which is directly related to the initial hardening rate, can significantly influence subsequent material response. This effect is consistent with observed material behavior on the macroscopic and microscopic levels, and may explain the wide scatter in response often found in creep testing.

A Mixed Model for Real-Time, Interactive Simulation of a Cable Passing Through Several Pulleys

NASA Astrophysics Data System (ADS)

García-Fernández, Ignacio; Pla-Castells, Marta; Martínez-Durá, Rafael J.

2007-09-01

A model of a cable and pulleys is presented that can be used in Real Time Computer Graphics applications. The model is formulated by the coupling of a damped spring and a variable coefficient wave equation, and can be integrated in more complex mechanical models of lift systems, such as cranes, elevators, etc. with a high degree of interactivity.

Evolution of magnetic field and atmospheric response. I - Three-dimensional formulation by the method of projected characteristics. II - Formulation of proper boundary equations. [stellar magnetohydrodynamics

NASA Technical Reports Server (NTRS)

Nakagawa, Y.

1981-01-01

The method described as the method of nearcharacteristics by Nakagawa (1980) is renamed the method of projected characteristics. Making full use of properties of the projected characteristics, a new and simpler formulation is developed. As a result, the formulation for the examination of the general three-dimensional problems is presented. It is noted that since in practice numerical solutions must be obtained, the final formulation is given in the form of difference equations. The possibility of including effects of viscous and ohmic dissipations in the formulation is considered, and the physical interpretation is discussed. A systematic manner is then presented for deriving physically self-consistent, time-dependent boundary equations for MHD initial boundary problems. It is demonstrated that the full use of the compatibility equations (differential equations relating variations at two spatial locations and times) is required in determining the time-dependent boundary conditions. In order to provide a clear physical picture as an example, the evolution of axisymmetric global magnetic field by photospheric differential rotation is considered.

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Meade, Andrew James, Jr.

1989-01-01

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.

Exploring the propagation of relativistic quantum wavepackets in the trajectory-based formulation

NASA Astrophysics Data System (ADS)

Tsai, Hung-Ming; Poirier, Bill

2016-03-01

In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [Phys. Rev. X, 4, 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a “many interacting worlds” interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.

A new multi-domain method based on an analytical control surface for linear and second-order mean drift wave loads on floating bodies

NASA Astrophysics Data System (ADS)

Liang, Hui; Chen, Xiaobo

2017-10-01

A novel multi-domain method based on an analytical control surface is proposed by combining the use of free-surface Green function and Rankine source function. A cylindrical control surface is introduced to subdivide the fluid domain into external and internal domains. Unlike the traditional domain decomposition strategy or multi-block method, the control surface here is not panelized, on which the velocity potential and normal velocity components are analytically expressed as a series of base functions composed of Laguerre function in vertical coordinate and Fourier series in the circumference. Free-surface Green function is applied in the external domain, and the boundary integral equation is constructed on the control surface in the sense of Galerkin collocation via integrating test functions orthogonal to base functions over the control surface. The external solution gives rise to the so-called Dirichlet-to-Neumann [DN2] and Neumann-to-Dirichlet [ND2] relations on the control surface. Irregular frequencies, which are only dependent on the radius of the control surface, are present in the external solution, and they are removed by extending the boundary integral equation to the interior free surface (circular disc) on which the null normal derivative of potential is imposed, and the dipole distribution is expressed as Fourier-Bessel expansion on the disc. In the internal domain, where the Rankine source function is adopted, new boundary integral equations are formulated. The point collocation is imposed over the body surface and free surface, while the collocation of the Galerkin type is applied on the control surface. The present method is valid in the computation of both linear and second-order mean drift wave loads. Furthermore, the second-order mean drift force based on the middle-field formulation can be calculated analytically by using the coefficients of the Fourier-Laguerre expansion.

Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

NASA Astrophysics Data System (ADS)

Scholle, M.; Gaskell, P. H.; Marner, F.

2018-04-01

An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell's theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems, two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries.

The crack problem for a nonhomogeneous plane

NASA Technical Reports Server (NTRS)

Delale, F.; Erdogan, F.

1982-01-01

The plane elasticity problem for a nonhomogeneous medium containing a crack is considered. It is assumed that the Poisson's ratio of the medium is constant and the Young's modulus E varies exponentially with the coordinate parallel to the crack. First the half plane problem is formulated and the solution is given for arbitrary tractions along the boundary. Then the integral equation for the crack problem is derived. It is shown that the integral equation having the derivative of the crack surface displacement as the density function has a simple Cauchy type kernel. Hence, its solution and the stresses around the crack tips have the conventional square root singularity. The solution is given for various loading conditions. The results show that the effect of the Poisson's ratio and consequently that of the thickness constraint on the stress intensity factors are rather negligible.

Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations

PubMed Central

Feischl, Michael; Gantner, Gregor; Praetorius, Dirk

2015-01-01

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence. PMID:26085698

On the background independence of two-dimensional topological gravity

NASA Astrophysics Data System (ADS)

Imbimbo, Camillo

1995-04-01

We formulate two-dimensional topological gravity in a background covariant Lagrangian framework. We derive the Ward identities which characterize the dependence of physical correlators on the background world-sheet metric defining the gauge-slice. We point out the existence of an "anomaly" in Ward identitites involving correlators of observables with higher ghost number. This "anomaly" represents an obstruction for physical correlators to be globally defined forms on moduli space which could be integrated in a background independent way. Starting from the anomalous Ward identities, we derive "descent" equations whose solutions are cocycles of the Lie algebra of the diffeomorphism group with values in the space of local forms on the moduli space. We solve the descent equations and provide explicit formulas for the cocycles, which allow for the definition of background independent integrals of physical correlators on the moduli space.

The crack problem for a nonhomogeneous plane

NASA Technical Reports Server (NTRS)

Delale, F.; Erdogan, F.

1983-01-01

The plane elasticity problem for a nonhomogeneous medium containing a crack is considered. It is assumed that the Poisson's ratio of the medium is constant and the Young's modulus E varies exponentially with the coordinate parallel to the crack. First the half plane problem is formulated and the solution is given for arbitrary tractions along the boundary. Then the integral equation for the crack problem is derived. It is shown that the integral equation having the derivative of the crack surface displacement as the density function has a simple Cauchy type kernel. Hence, its solution and the stresses around the crack tips have the conventional square root singularity. The solution is given for various loading conditions. The results show that the effect of the Poisson's ratio and consequently that of the thickness constraint on the stress intensity factors are rather negligible.

Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method.

PubMed

Guizal, Brahim; Barchiesi, Dominique; Felbacq, Didier

2003-12-01

We have developed a new formulation of the coupled-wave method (CWM) to handle aperiodic lamellar structures, and it will be referred to as the aperiodic coupled-wave method (ACWM). The space is still divided into three regions, but the fields are written by use of their Fourier integrals instead of the Fourier series. In the modulated region the relative permittivity is represented by its Fourier transform, and then a set of integro-differential equations is derived. Discretizing the last system leads to a set of ordinary differential equations that is reduced to an eigenvalue problem, as is usually done in the CWM. To assess the method, we compare our results with three independent formalisms: the Rayleigh perturbation method for small samples, the volume integral method, and the finite-element method.

Integral equation and discontinuous Galerkin methods for the analysis of light-matter interaction

NASA Astrophysics Data System (ADS)

Baczewski, Andrew David

Light-matter interaction is among the most enduring interests of the physical sciences. The understanding and control of this physics is of paramount importance to the design of myriad technologies ranging from stained glass, to molecular sensing and characterization techniques, to quantum computers. The development of complex engineered systems that exploit this physics is predicated at least partially upon in silico design and optimization that properly capture the light-matter coupling. In this thesis, the details of computational frameworks that enable this type of analysis, based upon both Integral Equation and Discontinuous Galerkin formulations will be explored. There will be a primary focus on the development of efficient and accurate software, with results corroborating both. The secondary focus will be on the use of these tools in the analysis of a number of exemplary systems.

Topics in structural dynamics: Nonlinear unsteady transonic flows and Monte Carlo methods in acoustics

NASA Technical Reports Server (NTRS)

Haviland, J. K.

1974-01-01

The results are reported of two unrelated studies. The first was an investigation of the formulation of the equations for non-uniform unsteady flows, by perturbation of an irrotational flow to obtain the linear Green's equation. The resulting integral equation was found to contain a kernel which could be expressed as the solution of the adjoint flow equation, a linear equation for small perturbations, but with non-constant coefficients determined by the steady flow conditions. It is believed that the non-uniform flow effects may prove important in transonic flutter, and that in such cases, the use of doublet type solutions of the wave equation would then prove to be erroneous. The second task covered an initial investigation into the use of the Monte Carlo method for solution of acoustical field problems. Computed results are given for a rectangular room problem, and for a problem involving a circular duct with a source located at the closed end.

Non-autonomous equations with unpredictable solutions

NASA Astrophysics Data System (ADS)

Akhmet, Marat; Fen, Mehmet Onur

2018-06-01

To make research of chaos more amenable to investigating differential and discrete equations, we introduce the concepts of an unpredictable function and sequence. The topology of uniform convergence on compact sets is applied to define unpredictable functions [1,2]. The unpredictable sequence is defined as a specific unpredictable function on the set of integers. The definitions are convenient to be verified as solutions of differential and discrete equations. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a delay differential equation are proved as well as for quasilinear discrete systems. As a corollary of the theorem, a similar assertion for a quasilinear ordinary differential equation is formulated. The results are demonstrated numerically, and an application to Hopfield neural networks is provided. In particular, Poincaré chaos near periodic orbits is observed. The completed research contributes to the theory of chaos as well as to the theory of differential and discrete equations, considering unpredictable solutions.

Newton-Euler Dynamic Equations of Motion for a Multi-body Spacecraft

NASA Technical Reports Server (NTRS)

Stoneking, Eric

2007-01-01

The Magnetospheric MultiScale (MMS) mission employs a formation of spinning spacecraft with several flexible appendages and thruster-based control. To understand the complex dynamic interaction of thruster actuation, appendage motion, and spin dynamics, each spacecraft is modeled as a tree of rigid bodies connected by spherical or gimballed joints. The method presented facilitates assembling by inspection the exact, nonlinear dynamic equations of motion for a multibody spacecraft suitable for solution by numerical integration. The building block equations are derived by applying Newton's and Euler's equations of motion to an "element" consisting of two bodies and one joint (spherical and gimballed joints are considered separately). Patterns in the "mass" and L'force" matrices guide assembly by inspection of a general N-body tree-topology system. Straightforward linear algebra operations are employed to eliminate extraneous constraint equations, resulting in a minimum-dimension system of equations to solve. This method thus combines a straightforward, easily-extendable, easily-mechanized formulation with an efficient computer implementation.

Stable Algorithm For Estimating Airdata From Flush Surface Pressure Measurements

NASA Technical Reports Server (NTRS)

Whitmore, Stephen, A. (Inventor); Cobleigh, Brent R. (Inventor); Haering, Edward A., Jr. (Inventor)

2001-01-01

An airdata estimation and evaluation system and method, including a stable algorithm for estimating airdata from nonintrusive surface pressure measurements. The airdata estimation and evaluation system is preferably implemented in a flush airdata sensing (FADS) system. The system and method of the present invention take a flow model equation and transform it into a triples formulation equation. The triples formulation equation eliminates the pressure related states from the flow model equation by strategically taking the differences of three surface pressures, known as triples. This triples formulation equation is then used to accurately estimate and compute vital airdata from nonintrusive surface pressure measurements.

Free vibration analysis of elastic structures submerged in an infinite or semi-infinite fluid domain by means of a coupled FE-BE solver

NASA Astrophysics Data System (ADS)

Zheng, Chang-Jun; Bi, Chuan-Xing; Zhang, Chuanzeng; Gao, Hai-Feng; Chen, Hai-Bo

2018-04-01

The vibration behavior of thin elastic structures can be noticeably influenced by the surrounding water, which represents a kind of heavy fluid. Since the feedback of the acoustic pressure onto the structure cannot be neglected in this case, a strong coupled scheme between the structural and fluid domains is usually required. In this work, a coupled finite element and boundary element (FE-BE) solver is developed for the free vibration analysis of structures submerged in an infinite fluid domain or a semi-infinite fluid domain with a free water surface. The structure is modeled by the finite element method (FEM). The compressibility of the fluid is taken into account, and hence the Helmholtz equation serves as the governing equation of the fluid domain. The boundary element method (BEM) is employed to model the fluid domain, and a boundary integral formulation with a half-space fundamental solution is used to satisfy the Dirichlet boundary condition on the free water surface exactly. The resulting nonlinear eigenvalue problem (NEVP) is converted into a small linear one by using a contour integral method. Adequate modifications are suggested to improve the efficiency of the contour integral method and avoid missing the eigenfrequencies of interest. The Burton-Miller method is used to filter out the fictitious eigenfrequencies of the boundary integral formulations. Numerical examples are given to demonstrate the accuracy and applicability of the developed eigensolver, and also show that the fluid-loading effect strongly depends on both the water depth and the mode shapes.

Exact solutions for the static bending of Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model

NASA Astrophysics Data System (ADS)

Wang, Y. B.; Zhu, X. W.; Dai, H. H.

2016-08-01

Though widely used in modelling nano- and micro- structures, Eringen's differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.

Ring polymer dynamics in curved spaces

NASA Astrophysics Data System (ADS)

Wolf, S.; Curotto, E.

2012-07-01

We formulate an extension of the ring polymer dynamics approach to curved spaces using stereographic projection coordinates. We test the theory by simulating the particle in a ring, {T}^1, mapped by a stereographic projection using three potentials. Two of these are quadratic, and one is a nonconfining sinusoidal model. We propose a new class of algorithms for the integration of the ring polymer Hamilton equations in curved spaces. These are designed to improve the energy conservation of symplectic integrators based on the split operator approach. For manifolds, the position-position autocorrelation function can be formulated in numerous ways. We find that the position-position autocorrelation function computed from configurations in the Euclidean space {R}^2 that contains {T}^1 as a submanifold has the best statistical properties. The agreement with exact results obtained with vector space methods is excellent for all three potentials, for all values of time in the interval simulated, and for a relatively broad range of temperatures.

A combined finite element-boundary element formulation for solution of axially symmetric bodies

NASA Technical Reports Server (NTRS)

Collins, Jeffrey D.; Volakis, John L.

1991-01-01

A new method is presented for the computation of electromagnetic scattering from axially symmetric bodies. To allow the simulation of inhomogeneous cross sections, the method combines the finite element and boundary element techniques. Interior to a fictitious surface enclosing the scattering body, the finite element method is used which results in a sparce submatrix, whereas along the enclosure the Stratton-Chu integral equation is enforced. By choosing the fictitious enclosure to be a right circular cylinder, most of the resulting boundary integrals are convolutional and may therefore be evaluated via the FFT with which the system is iteratively solved. In view of the sparce matrix associated with the interior fields, this reduces the storage requirement of the entire system to O(N) making the method attractive for large scale computations. The details of the corresponding formulation and its numerical implementation are described.

Hamiltonian formulation of systems with balanced loss-gain and exactly solvable models

NASA Astrophysics Data System (ADS)

Ghosh, Pijush K.; Sinha, Debdeep

2018-01-01

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occurs in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion are constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered.

Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface

NASA Astrophysics Data System (ADS)

Shao, Songdong; Lo, Edmond Y. M.

An incompressible smoothed particle hydrodynamics (SPH) method is presented to simulate Newtonian and non-Newtonian flows with free surfaces. The basic equations solved are the incompressible mass conservation and Navier-Stokes equations. The method uses prediction-correction fractional steps with the temporal velocity field integrated forward in time without enforcing incompressibility in the prediction step. The resulting deviation of particle density is then implicitly projected onto a divergence-free space to satisfy incompressibility through a pressure Poisson equation derived from an approximate pressure projection. Various SPH formulations are employed in the discretization of the relevant gradient, divergence and Laplacian terms. Free surfaces are identified by the particles whose density is below a set point. Wall boundaries are represented by particles whose positions are fixed. The SPH formulation is also extended to non-Newtonian flows and demonstrated using the Cross rheological model. The incompressible SPH method is tested by typical 2-D dam-break problems in which both water and fluid mud are considered. The computations are in good agreement with available experimental data. The different flow features between Newtonian and non-Newtonian flows after the dam-break are discussed.

One-dimensional Lagrangian implicit hydrodynamic algorithm for Inertial Confinement Fusion applications

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ramis, Rafael, E-mail: rafael.ramis@upm.es

A new one-dimensional hydrodynamic algorithm, specifically developed for Inertial Confinement Fusion (ICF) applications, is presented. The scheme uses a fully conservative Lagrangian formulation in planar, cylindrical, and spherically symmetric geometries, and supports arbitrary equations of state with separate ion and electron components. Fluid equations are discretized on a staggered grid and stabilized by means of an artificial viscosity formulation. The space discretized equations are advanced in time using an implicit algorithm. The method includes several numerical parameters that can be adjusted locally. In regions with low Courant–Friedrichs–Lewy (CFL) number, where stability is not an issue, they can be adjusted tomore » optimize the accuracy. In typical problems, the truncation error can be reduced by a factor between 2 to 10 in comparison with conventional explicit algorithms. On the other hand, in regions with high CFL numbers, the parameters can be set to guarantee unconditional stability. The method can be integrated into complex ICF codes. This is demonstrated through several examples covering a wide range of situations: from thermonuclear ignition physics, where alpha particles are managed as an additional species, to low intensity laser–matter interaction, where liquid–vapor phase transitions occur.« less

Boundary-Layer Receptivity and Integrated Transition Prediction

NASA Technical Reports Server (NTRS)

Chang, Chau-Lyan; Choudhari, Meelan

2005-01-01

The adjoint parabold stability equations (PSE) formulation is used to calculate the boundary layer receptivity to localized surface roughness and suction for compressible boundary layers. Receptivity efficiency functions predicted by the adjoint PSE approach agree well with results based on other nonparallel methods including linearized Navier-Stokes equations for both Tollmien-Schlichting waves and crossflow instability in swept wing boundary layers. The receptivity efficiency function can be regarded as the Green's function to the disturbance amplitude evolution in a nonparallel (growing) boundary layer. Given the Fourier transformed geometry factor distribution along the chordwise direction, the linear disturbance amplitude evolution for a finite size, distributed nonuniformity can be computed by evaluating the integral effects of both disturbance generation and linear amplification. The synergistic approach via the linear adjoint PSE for receptivity and nonlinear PSE for disturbance evolution downstream of the leading edge forms the basis for an integrated transition prediction tool. Eventually, such physics-based, high fidelity prediction methods could simulate the transition process from the disturbance generation through the nonlinear breakdown in a holistic manner.

The Covariant Formulation of Maxwell's Equations Expressed in a Form Independent of Specific Units

ERIC Educational Resources Information Center

Heras, Jose A.; Baez, G.

2009-01-01

The covariant formulation of Maxwell's equations can be expressed in a form independent of the usual systems of units by introducing the constants alpha, beta and gamma into these equations. Maxwell's equations involving these constants are then specialized to the most commonly used systems of units: Gaussian, SI and Heaviside-Lorentz by giving…

Efficient Low Dissipative High Order Schemes for Multiscale MHD Flows, I: Basic Theory

NASA Technical Reports Server (NTRS)

Sjoegreen, Bjoern; Yee, H. C.

2003-01-01

The objective of this paper is to extend our recently developed highly parallelizable nonlinear stable high order schemes for complex multiscale hydrodynamic applications to the viscous MHD equations. These schemes employed multiresolution wavelets as adaptive numerical dissipation controls t o limit the amount of and to aid the selection and/or blending of the appropriate types of dissipation to be used. The new scheme is formulated for both the conservative and non-conservative form of the MHD equations in curvilinear grids. The four advantages of the present approach over existing MHD schemes reported in the open literature are as follows. First, the scheme is constructed for long-time integrations of shock/turbulence/combustion MHD flows. Available schemes are too diffusive for long-time integrations and/or turbulence/combustion problems. Second, unlike exist- ing schemes for the conservative MHD equations which suffer from ill-conditioned eigen- decompositions, the present scheme makes use of a well-conditioned eigen-decomposition obtained from a minor modification of the eigenvectors of the non-conservative MHD equations t o solve the conservative form of the MHD equations. Third, this approach of using the non-conservative eigensystem when solving the conservative equations also works well in the context of standard shock-capturing schemes for the MHD equations. Fourth, a new approach to minimize the numerical error of the divergence-free magnetic condition for high order schemes is introduced. Numerical experiments with typical MHD model problems revealed the applicability of the newly developed schemes for the MHD equations.

Time domain viscoelastic full waveform inversion

NASA Astrophysics Data System (ADS)

Fabien-Ouellet, Gabriel; Gloaguen, Erwan; Giroux, Bernard

2017-06-01

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity-stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity-stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity-stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO2 geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO2 plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

Numerical method for predicting flow characteristics and performance of nonaxisymmetric nozzles, theory

NASA Technical Reports Server (NTRS)

Thomas, P. D.

1979-01-01

The theoretical foundation and formulation of a numerical method for predicting the viscous flowfield in and about isolated three dimensional nozzles of geometrically complex configuration are presented. High Reynolds number turbulent flows are of primary interest for any combination of subsonic, transonic, and supersonic flow conditions inside or outside the nozzle. An alternating-direction implicit (ADI) numerical technique is employed to integrate the unsteady Navier-Stokes equations until an asymptotic steady-state solution is reached. Boundary conditions are computed with an implicit technique compatible with the ADI technique employed at interior points of the flow region. The equations are formulated and solved in a boundary-conforming curvilinear coordinate system. The curvilinear coordinate system and computational grid is generated numerically as the solution to an elliptic boundary value problem. A method is developed that automatically adjusts the elliptic system so that the interior grid spacing is controlled directly by the a priori selection of the grid spacing on the boundaries of the flow region.

Full-Scale Direct Numerical Simulation of Two- and Three-Dimensional Instabilities and Rivulet Formulation in Heated Falling Films

NASA Technical Reports Server (NTRS)

Krishnamoorthy, S.; Ramaswamy, B.; Joo, S. W.

1995-01-01

A thin film draining on an inclined plate has been studied numerically using finite element method. Three-dimensional governing equations of continuity, momentum and energy with a moving boundary are integrated in an arbitrary Lagrangian Eulerian frame of reference. Kinematic equation is solved to precisely update interface location. Rivulet formation based on instability mechanism has been simulated using full-scale computation. Comparisons with long-wave theory are made to validate the numerical scheme. Detailed analysis of two- and three-dimensional nonlinear wave formation and spontaneous rupture forming rivulets under the influence of combined thermocapillary and surface-wave instabilities is performed.

A nonperturbative light-front coupled-cluster method

NASA Astrophysics Data System (ADS)

Hiller, J. R.

2012-10-01

The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential-operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from standard coupled-cluster theory, to obtain form factors and other observables.

Propagation Path Effects for Rayleigh and Love Waves

DTIC Science & Technology

1981-05-01

The method finally selected is similar in some respects to the integral equation formulation. It is the Parker- Oldenburg -iuestis method (Parker, 1972...REGIONAL ANOMALIE S BASIN MODEL DEPTH RESIDUALS AND STRIPPING BOREHOLE DATA PROCESS DATA mots" 12 THE PARKER- OLDENBURG -HUESTIS POTENTIAL INVERSION Parker...series convergLnce and other properties are given by Parker (1972), Parker and Huestis (1974), and Oldenburg (1974). A discussion of this theory from

Regularization in Orbital Mechanics; Theory and Practice

NASA Astrophysics Data System (ADS)

Roa, Javier

2017-09-01

Regularized equations of motion can improve numerical integration for the propagation of orbits, and simplify the treatment of mission design problems. This monograph discusses standard techniques and recent research in the area. While each scheme is derived analytically, its accuracy is investigated numerically. Algebraic and topological aspects of the formulations are studied, as well as their application to practical scenarios such as spacecraft relative motion and new low-thrust trajectories.

Path integral Monte Carlo ground state approach: formalism, implementation, and applications

NASA Astrophysics Data System (ADS)

Yan, Yangqian; Blume, D.

2017-11-01

Monte Carlo techniques have played an important role in understanding strongly correlated systems across many areas of physics, covering a wide range of energy and length scales. Among the many Monte Carlo methods applicable to quantum mechanical systems, the path integral Monte Carlo approach with its variants has been employed widely. Since semi-classical or classical approaches will not be discussed in this review, path integral based approaches can for our purposes be divided into two categories: approaches applicable to quantum mechanical systems at zero temperature and approaches applicable to quantum mechanical systems at finite temperature. While these two approaches are related to each other, the underlying formulation and aspects of the algorithm differ. This paper reviews the path integral Monte Carlo ground state (PIGS) approach, which solves the time-independent Schrödinger equation. Specifically, the PIGS approach allows for the determination of expectation values with respect to eigen states of the few- or many-body Schrödinger equation provided the system Hamiltonian is known. The theoretical framework behind the PIGS algorithm, implementation details, and sample applications for fermionic systems are presented.

A Poisson equation formulation for pressure calculations in penalty finite element models for viscous incompressible flows

NASA Technical Reports Server (NTRS)

Sohn, J. L.; Heinrich, J. C.

1990-01-01

The calculation of pressures when the penalty-function approximation is used in finite-element solutions of laminar incompressible flows is addressed. A Poisson equation for the pressure is formulated that involves third derivatives of the velocity field. The second derivatives appearing in the weak formulation of the Poisson equation are calculated from the C0 velocity approximation using a least-squares method. The present scheme is shown to be efficient, free of spurious oscillations, and accurate. Examples of applications are given and compared with results obtained using mixed formulations.

Development of a defect stream function, law of the wall/wake method for compressible turbulent boundary layers. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Wahls, Richard A.

1990-01-01

The method presented is designed to improve the accuracy and computational efficiency of existing numerical methods for the solution of flows with compressible turbulent boundary layers. A compressible defect stream function formulation of the governing equations assuming an arbitrary turbulence model is derived. This formulation is advantageous because it has a constrained zero-order approximation with respect to the wall shear stress and the tangential momentum equation has a first integral. Previous problems with this type of formulation near the wall are eliminated by using empirically based analytic expressions to define the flow near the wall. The van Driest law of the wall for velocity and the modified Crocco temperature-velocity relationship are used. The associated compressible law of the wake is determined and it extends the valid range of the analytical expressions beyond the logarithmic region of the boundary layer. The need for an inner-region eddy viscosity model is completely avoided. The near-wall analytic expressions are patched to numerically computed outer region solutions at a point determined during the computation. A new boundary condition on the normal derivative of the tangential velocity at the surface is presented; this condition replaces the no-slip condition and enables numerical integration to the surface with a relatively coarse grid using only an outer region turbulence model. The method was evaluated for incompressible and compressible equilibrium flows and was implemented into an existing Navier-Stokes code using the assumption of local equilibrium flow with respect to the patching. The method has proven to be accurate and efficient.

Resonance line polarization and the Hanle effect in optically thick media. I - Formulation for the two-level atom

NASA Astrophysics Data System (ADS)

Landi Degl'Innocenti, E.; Bommier, V.; Sahal-Brechot, S.

1990-08-01

A general formalism is presented to describe resonance line polarization for a two-level atom in an optically thick, three-dimensional medium embedded in an arbitrary varying magnetic field and irradiated by an arbitrary radiation field. The magnetic field is supposed sufficiently small to induce a Zeeman splitting much smaller than the typical line width. By neglecting atomic polarization in the lower level and stimulated emission, an integral equation is derived for the multipole moments of the density matrix of the upper level. This equation shows how the multipole moments at any assigned point of the medium are coupled to the multipole moments relative at a different point as a consequence of the propagation of polarized radiation between the two points. The equation also accounts for the effect of the magnetic field, described by a kernel locally connecting multipole moments of the same rank, and for the role of inelastic and elastic (or depolarizing) collisions. After having given its formal derivation for the general case, the integral equation is particularized to the one-dimensional and two-dimensional cases. For the one-dimensional case of a plane parallel atmosphere, neglecting both the magnetic field and depolarizing collisions, the equation here derived reduces to a previous one given by Rees (1978).

{lambda} elements for singular problems in CFD: Viscoelastic fluids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wong, K.K.; Surana, K.S.

1996-10-01

This paper presents two dimensional {lambda} element formulation for viscoelastic fluid flow containing point singularities in the flow field. The flow of viscoelastic fluid even without singularities are a difficult class of problems for increasing Deborah number or Weissenburg number due to increased dominance of convective terms and thus increased hyperbolicity. In the present work the equations of fluid motion and the constitutive laws are recast in the form of a first order system of coupled equations with the use of auxiliary variables. The velocity, pressure and stresses are interpolated using equal order C{sup 0} {lambda} element approximations. The Leastmore » Squares Finite Element Method (LSFEM) is used to construct the integral form (error functional I) corresponding to these equations. The error functional is constructed by taking the integrated sum of the squares of the errors or residuals (over the whole discretization) resulting when the element approximation is substituted into these equations. The conditions resulting from the minimization of the error functional are satisfied by using Newton`s method with line search. LSFEM has much superior performance when dealing with non-linear and convection dominated problems.« less

An evaluation of three two-dimensional computational fluid dynamics codes including low Reynolds numbers and transonic Mach numbers

NASA Technical Reports Server (NTRS)

Hicks, Raymond M.; Cliff, Susan E.

1991-01-01

Full-potential, Euler, and Navier-Stokes computational fluid dynamics (CFD) codes were evaluated for use in analyzing the flow field about airfoils sections operating at Mach numbers from 0.20 to 0.60 and Reynolds numbers from 500,000 to 2,000,000. The potential code (LBAUER) includes weakly coupled integral boundary layer equations for laminar and turbulent flow with simple transition and separation models. The Navier-Stokes code (ARC2D) uses the thin-layer formulation of the Reynolds-averaged equations with an algebraic turbulence model. The Euler code (ISES) includes strongly coupled integral boundary layer equations and advanced transition and separation calculations with the capability to model laminar separation bubbles and limited zones of turbulent separation. The best experiment/CFD correlation was obtained with the Euler code because its boundary layer equations model the physics of the flow better than the other two codes. An unusual reversal of boundary layer separation with increasing angle of attack, following initial shock formation on the upper surface of the airfoil, was found in the experiment data. This phenomenon was not predicted by the CFD codes evaluated.

Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its solitary-wave solutions via mathematical methods

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-12-01

In this study, we presented the problem formulations of models for internal solitary waves in a stratified shear flow with a free surface. The nonlinear higher order of extended KdV equations for the free surface displacement is generated. We derived the coefficients of the nonlinear higher-order extended KdV equation in terms of integrals of the modal function for the linear long-wave theory. The wave amplitude potential and the fluid pressure of the extended KdV equation in the form of solitary-wave solutions are deduced. We discussed and analyzed the stability of the obtained solutions and the movement role of the waves by making graphs of the exact solutions.

Fem Formulation for Heat and Mass Transfer in Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Soudagar, Manzoor Elahi M.; Salman Ahmed, N. J.; Anjum Badruddin, Irfan

2017-08-01

Heat and mass transfer in porous medium can be modelled using three partial differential equations namely, momentum equation, energy equation and mass diffusion. These three equations are coupled to each other by some common terms that turn the whole phenomenon into a complex problem with inter-dependable variables. The current article describes the finite element formulation of heat and mass transfer in porous medium with respect to Cartesian coordinates. The problem under study is formulated into algebraic form of equations by using Galerkin's method with the help of two-node linear triangular element having three nodes. The domain is meshed with smaller sized elements near the wall region and bigger size away from walls.

Well-posedness, linear perturbations, and mass conservation for the axisymmetric Einstein equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dain, Sergio; Ortiz, Omar E.; Facultad de Matematica, Astronomia y Fisica, FaMAF, Universidad Nacional de Cordoba, Instituto de Fisica Enrique Gaviola, IFEG, CONICET, Ciudad Universitaria

2010-02-15

For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides a nonlinear control of the variables along the whole evolution. In this gauge, Einstein equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. As a first step in analyzing this system of equations we study linear perturbations on a flat background. We prove that the linear equations reduce to a very simple system of equations which provide, thoughmore » the mass formula, useful insight into the structure of the full system. However, the singular behavior of the coefficients at the axis makes the study of this linear system difficult from the analytical point of view. In order to understand the behavior of the solutions, we study the numerical evolution of them. We provide strong numerical evidence that the system is well-posed and that its solutions have the expected behavior. Finally, this linear system allows us to formulate a model problem which is physically interesting in itself, since it is connected with the linear stability of black hole solutions in axial symmetry. This model can contribute significantly to solve the nonlinear problem and at the same time it appears to be tractable.« less

Analysis of an Interface Crack for a Functionally Graded Strip Sandwiched between Two Homogeneous Layers of Finite Thickness

NASA Technical Reports Server (NTRS)

Shbeeh, N. I.; Binienda, W. K.

1999-01-01

The interface crack problem for a composite layer that consists of a homogeneous substrate, coating and a non-homogeneous interface was formulated for singular integral equations with Cauchy kernels and integrated using the Lobatto-Chebyshev collocation technique. Mixed-mode Stress Intensity Factors and Strain Energy Release Rates were calculated. The Stress Intensity Factors were compared for accuracy with relevant results previously published. The parametric studies were conducted for the various thickness of each layer and for various non-homogeneity ratios. Particular application to the Zirconia thermal barrier on steel substrate is demonstrated.

Scattering from finite bodies of translation - Plates, curved surfaces, and noncircular cylinders

NASA Astrophysics Data System (ADS)

Medgyesi-Mitschang, L. N.; Putnam, J. M.

1983-11-01

Electromagnetic scattering from finite, conducting bodies of translation (BOT) is examined using a formulation based on the electric field integral equation (EFIE) and solved by the method of moments (MM). The present approach provides a systematic, unified treatment for a wide class of finite, thin scatterers at all angles of illumination and polarization. Both concave and convex surfaces are considered. An entire-domain Galerkin expansion along one dimension of the body and a piecewise continuous one along the other are used to represent the unknown current variations. The scattering cross sections, obtained with this formulation, are compared with published results using more specialized methods and further confirmed by experimental measurements.

Implicit level set algorithms for modelling hydraulic fracture propagation.

PubMed

Peirce, A

2016-10-13

Hydraulic fractures are tensile cracks that propagate in pre-stressed solid media due to the injection of a viscous fluid. Developing numerical schemes to model the propagation of these fractures is particularly challenging due to the degenerate, hypersingular nature of the coupled integro-partial differential equations. These equations typically involve a singular free boundary whose velocity can only be determined by evaluating a distinguished limit. This review paper describes a class of numerical schemes that have been developed to use the multiscale asymptotic behaviour typically encountered near the fracture boundary as multiple physical processes compete to determine the evolution of the fracture. The fundamental concepts of locating the free boundary using the tip asymptotics and imposing the tip asymptotic behaviour in a weak form are illustrated in two quite different formulations of the governing equations. These formulations are the displacement discontinuity boundary integral method and the extended finite-element method. Practical issues are also discussed, including new models for proppant transport able to capture 'tip screen-out'; efficient numerical schemes to solve the coupled nonlinear equations; and fast methods to solve resulting linear systems. Numerical examples are provided to illustrate the performance of the numerical schemes. We conclude the paper with open questions for further research. This article is part of the themed issue 'Energy and the subsurface'. © 2016 The Author(s).

Implicit level set algorithms for modelling hydraulic fracture propagation

PubMed Central

2016-01-01

Hydraulic fractures are tensile cracks that propagate in pre-stressed solid media due to the injection of a viscous fluid. Developing numerical schemes to model the propagation of these fractures is particularly challenging due to the degenerate, hypersingular nature of the coupled integro-partial differential equations. These equations typically involve a singular free boundary whose velocity can only be determined by evaluating a distinguished limit. This review paper describes a class of numerical schemes that have been developed to use the multiscale asymptotic behaviour typically encountered near the fracture boundary as multiple physical processes compete to determine the evolution of the fracture. The fundamental concepts of locating the free boundary using the tip asymptotics and imposing the tip asymptotic behaviour in a weak form are illustrated in two quite different formulations of the governing equations. These formulations are the displacement discontinuity boundary integral method and the extended finite-element method. Practical issues are also discussed, including new models for proppant transport able to capture ‘tip screen-out’; efficient numerical schemes to solve the coupled nonlinear equations; and fast methods to solve resulting linear systems. Numerical examples are provided to illustrate the performance of the numerical schemes. We conclude the paper with open questions for further research.  This article is part of the themed issue ‘Energy and the subsurface’. PMID:27597787

New perspectives in the use of the Ffowcs Williams Hawkings equation for aeroacoustic analysis of rotating blades

NASA Astrophysics Data System (ADS)

Ianniello, S.

The Ffowcs Williams Hawkings equation represents a standard approach in the prediction of noise from rotating blades. It is widely used for linear aeroacoustic problems concerning helicopter rotors and aircraft propellers and over the last few years, through the use of the so called porous (or permeable) surface formulation, has replaced the Kirchhoff approach in the numerical solution of nonlinear problems. Nevertheless, because of numerical difficulties in evaluating the contribution from supersonic sources, most of the computing tools are still unable to treat the critical velocities at which the shock delocalization occurs. At those conditions, the attention is usually limited to the comparison between the noise prediction and the experimental data in the narrow time region where the pressure peak value is located, but there has been little attention paid to the singular behaviour of the governing equation at supersonic speeds. The aim of this paper is to couple the advantages of the porous formulation to an emission surface integration scheme in order to show if and how the singularities affect the noise prediction and to demonstrate a practical way to remove them. Such an analysis enables an investigation of some interesting and somewhat hidden features of the numerical solution of the governing equation and suggests a new solution approach to predicting the noise of a rotor at any rotational velocity.

Mixed finite-element formulations in piezoelectricity and flexoelectricity

PubMed Central

2016-01-01

Flexoelectricity, the linear coupling of strain gradient and electric polarization, is inherently a size-dependent phenomenon. The energy storage function for a flexoelectric material depends not only on polarization and strain, but also strain-gradient. Thus, conventional finite-element methods formulated solely on displacement are inadequate to treat flexoelectric solids since gradients raise the order of the governing differential equations. Here, we introduce a computational framework based on a mixed formulation developed previously by one of the present authors and a colleague. This formulation uses displacement and displacement-gradient as separate variables which are constrained in a ‘weighted integral sense’ to enforce their known relation. We derive a variational formulation for boundary-value problems for piezo- and/or flexoelectric solids. We validate this computational framework against available exact solutions. Our new computational method is applied to more complex problems, including a plate with an elliptical hole, stationary cracks, as well as tension and shear of solids with a repeating unit cell. Our results address several issues of theoretical interest, generate predictions of experimental merit and reveal interesting flexoelectric phenomena with potential for application. PMID:27436967

Mixed finite-element formulations in piezoelectricity and flexoelectricity.

PubMed

Mao, Sheng; Purohit, Prashant K; Aravas, Nikolaos

2016-06-01

Flexoelectricity, the linear coupling of strain gradient and electric polarization, is inherently a size-dependent phenomenon. The energy storage function for a flexoelectric material depends not only on polarization and strain, but also strain-gradient. Thus, conventional finite-element methods formulated solely on displacement are inadequate to treat flexoelectric solids since gradients raise the order of the governing differential equations. Here, we introduce a computational framework based on a mixed formulation developed previously by one of the present authors and a colleague. This formulation uses displacement and displacement-gradient as separate variables which are constrained in a 'weighted integral sense' to enforce their known relation. We derive a variational formulation for boundary-value problems for piezo- and/or flexoelectric solids. We validate this computational framework against available exact solutions. Our new computational method is applied to more complex problems, including a plate with an elliptical hole, stationary cracks, as well as tension and shear of solids with a repeating unit cell. Our results address several issues of theoretical interest, generate predictions of experimental merit and reveal interesting flexoelectric phenomena with potential for application.

Lattice properties of the Phase I BNL x-ray lithography source obtained from fits to magnetic measurement data

DOE Office of Scientific and Technical Information (OSTI.GOV)

Blumberg, L.N.; Murphy, J.B.; Reusch, M.F.

1991-01-01

The orbit, tune, chromaticity and {beta} values for the Phase 1 XLS ring were computed by numerical integration of equations of motion using fields obtained from the coefficients of the 3-dimensional solution of Laplace's Equation evaluated by fits to magnetic measurements. The results are in good agreement with available data. The method has been extended to higher order fits of TOSCA generated fields in planes normal to the reference axis using the coil configuration proposed for the Superconducting X-Ray Lithography Source. Agreement with results from numerical integration through fields given directly by TOSCA is excellent. The formulation of the normalmore » multipole expansion presented by Brown and Servranckx has been extended to include skew multipole terms. The method appears appropriate for analysis of magnetic measurements of the SXLS. 8 refs. , 2 figs., 2 tabs.« less

Further studies of propellant sloshing under low-gravity conditions

NASA Technical Reports Server (NTRS)

Dodge, F. T.

1971-01-01

A variational integral is formulated from Hamilton's Principle and is proved to be equivalent to the usual differential equations of low-gravity sloshing in ellipsoidal tanks. It is shown that for a zero-degree contact angle the contact line boundary condition corresponds to the stuck condition, a result that is due to the linearization of the equations and the ambiguity in the definition of the wave height at the wall. The variational integral is solved by a Rayleigh-Ritz technique. Results for slosh frequency when the free surface is not bent-over compare well with previous numerical solutions. When the free surface is bent over, however, the results for slosh frequency are considerably larger than those predicted by previous finite-difference, numerical approaches: the difference may be caused by the use of a zero degree contact angle in the present theory in contrast to the nonzero contact angle used in the numerical approaches.

On Babinet's principle and diffraction associated with an arbitrary particle.

PubMed

Sun, Bingqiang; Yang, Ping; Kattawar, George W; Mishchenko, Michael I

2017-12-01

Babinet's principle is widely used to compute the diffraction by a particle. However, the diffraction by a 3-D object is not totally the same as that simulated with Babinet's principle. This Letter uses a surface integral equation to exactly formulate the diffraction by an arbitrary particle and illustrate the condition for the applicability of Babinet's principle. The present results may serve to close the debate on the diffraction formalism.

Time Evolution of the Giant Molecular Cloud Mass Functions across Galactic Disks

NASA Astrophysics Data System (ADS)

Kobayashi, Masato I. N.; Inutsuka, Shu-Ichiro; Kobayashi, Hiroshi; Hasegawa, Kenji

2017-01-01

We formulate and conduct the time-integration of time evolution equation for the giant molecular cloud mass function (GMCMF) including the cloud-cloud collision (CCC) effect. Our results show that the CCC effect is only limited in the massive-end of the GMCMF and indicate that future high resolution and sensitivity radio observations may constrain giant molecular cloud (GMC) timescales by observing the GMCMF slope in the lower mass regime.

A spectral-finite difference solution of the Navier-Stokes equations in three dimensions

NASA Astrophysics Data System (ADS)

Alfonsi, Giancarlo; Passoni, Giuseppe; Pancaldo, Lea; Zampaglione, Domenico

1998-07-01

A new computational code for the numerical integration of the three-dimensional Navier-Stokes equations in their non-dimensional velocity-pressure formulation is presented. The system of non-linear partial differential equations governing the time-dependent flow of a viscous incompressible fluid in a channel is managed by means of a mixed spectral-finite difference method, in which different numerical techniques are applied: Fourier decomposition is used along the homogeneous directions, second-order Crank-Nicolson algorithms are employed for the spatial derivatives in the direction orthogonal to the solid walls and a fourth-order Runge-Kutta procedure is implemented for both the calculation of the convective term and the time advancement. The pressure problem, cast in the Helmholtz form, is solved with the use of a cyclic reduction procedure. No-slip boundary conditions are used at the walls of the channel and cyclic conditions are imposed at the other boundaries of the computing domain.Results are provided for different values of the Reynolds number at several time steps of integration and are compared with results obtained by other authors.

Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach

NASA Astrophysics Data System (ADS)

Norouzzadeh, A.; Ansari, R.; Rouhi, H.

2017-05-01

Differential form of Eringen's nonlocal elasticity theory is widely employed to capture the small-scale effects on the behavior of nanostructures. However, paradoxical results are obtained via the differential nonlocal constitutive relations in some cases such as in the vibration and bending analysis of cantilevers, and recourse must be made to the integral (original) form of Eringen's theory. Motivated by this consideration, a novel nonlocal formulation is developed herein based on the original formulation of Eringen's theory to study the buckling behavior of nanobeams. The governing equations are derived according to the Timoshenko beam theory, and are represented in a suitable vector-matrix form which is applicable to the finite-element analysis. In addition, an isogeometric analysis (IGA) is conducted for the solution of buckling problem. Construction of exact geometry using non-uniform rational B-splines and easy implementation of geometry refinement tools are the main advantages of IGA. A comparison study is performed between the predictions of integral and differential nonlocal models for nanobeams under different kinds of end conditions.

A highly accurate boundary integral equation method for surfactant-laden drops in 3D

NASA Astrophysics Data System (ADS)

Sorgentone, Chiara; Tornberg, Anna-Karin

2018-05-01

The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.

Stabilization of time domain acoustic boundary element method for the exterior problem avoiding the nonuniqueness.

PubMed

Jang, Hae-Won; Ih, Jeong-Guon

2013-03-01

The time domain boundary element method (TBEM) to calculate the exterior sound field using the Kirchhoff integral has difficulties in non-uniqueness and exponential divergence. In this work, a method to stabilize TBEM calculation for the exterior problem is suggested. The time domain CHIEF (Combined Helmholtz Integral Equation Formulation) method is newly formulated to suppress low order fictitious internal modes. This method constrains the surface Kirchhoff integral by forcing the pressures at the additional interior points to be zero when the shortest retarded time between boundary nodes and an interior point elapses. However, even after using the CHIEF method, the TBEM calculation suffers the exponential divergence due to the remaining unstable high order fictitious modes at frequencies higher than the frequency limit of the boundary element model. For complete stabilization, such troublesome modes are selectively adjusted by projecting the time response onto the eigenspace. In a test example for a transiently pulsating sphere, the final average error norm of the stabilized response compared to the analytic solution is 2.5%.

Robust global identifiability theory using potentials--Application to compartmental models.

PubMed

Wongvanich, N; Hann, C E; Sirisena, H R

2015-04-01

This paper presents a global practical identifiability theory for analyzing and identifying linear and nonlinear compartmental models. The compartmental system is prolonged onto the potential jet space to formulate a set of input-output equations that are integrals in terms of the measured data, which allows for robust identification of parameters without requiring any simulation of the model differential equations. Two classes of linear and non-linear compartmental models are considered. The theory is first applied to analyze the linear nitrous oxide (N2O) uptake model. The fitting accuracy of the identified models from differential jet space and potential jet space identifiability theories is compared with a realistic noise level of 3% which is derived from sensor noise data in the literature. The potential jet space approach gave a match that was well within the coefficient of variation. The differential jet space formulation was unstable and not suitable for parameter identification. The proposed theory is then applied to a nonlinear immunological model for mastitis in cows. In addition, the model formulation is extended to include an iterative method which allows initial conditions to be accurately identified. With up to 10% noise, the potential jet space theory predicts the normalized population concentration infected with pathogens, to within 9% of the true curve. Copyright © 2015 Elsevier Inc. All rights reserved.

Parallel Domain Decomposition Formulation and Software for Large-Scale Sparse Symmetrical/Unsymmetrical Aeroacoustic Applications

NASA Technical Reports Server (NTRS)

Nguyen, D. T.; Watson, Willie R. (Technical Monitor)

2005-01-01

The overall objectives of this research work are to formulate and validate efficient parallel algorithms, and to efficiently design/implement computer software for solving large-scale acoustic problems, arised from the unified frameworks of the finite element procedures. The adopted parallel Finite Element (FE) Domain Decomposition (DD) procedures should fully take advantages of multiple processing capabilities offered by most modern high performance computing platforms for efficient parallel computation. To achieve this objective. the formulation needs to integrate efficient sparse (and dense) assembly techniques, hybrid (or mixed) direct and iterative equation solvers, proper pre-conditioned strategies, unrolling strategies, and effective processors' communicating schemes. Finally, the numerical performance of the developed parallel finite element procedures will be evaluated by solving series of structural, and acoustic (symmetrical and un-symmetrical) problems (in different computing platforms). Comparisons with existing "commercialized" and/or "public domain" software are also included, whenever possible.

Fully unsteady subsonic and supersonic potential aerodynamics for complex aircraft configurations with applications to flutter

NASA Technical Reports Server (NTRS)

Tseng, K.; Morino, L.

1975-01-01

A general formulation is presented for the analysis of steady and unsteady, subsonic and supersonic aerodynamics for complex aircraft configurations. The theoretical formulation, the numerical procedure, the description of the program SOUSSA (steady, oscillatory and unsteady, subsonic and supersonic aerodynamics) and numerical results are included. In particular, generalized forces for fully unsteady (complex frequency) aerodynamics for a wing-body configuration, AGARD wing-tail interference in both subsonic and supersonic flows as well as flutter analysis results are included. The theoretical formulation is based upon an integral equation, which includes completely arbitrary motion. Steady and oscillatory aerodynamic flows are considered. Here small-amplitude, fully transient response in the time domain is considered. This yields the aerodynamic transfer function (Laplace transform of the fully unsteady operator) for frequency domain analysis. This is particularly convenient for the linear systems analysis of the whole aircraft.

Three-dimensional coupled thermoelastodynamic stress and flux induced wave propagation for isotropic half-space with scalar potential functions

NASA Astrophysics Data System (ADS)

Hayati, Yazdan; Eskandari-Ghadi, Morteza

2018-02-01

An asymmetric three-dimensional thermoelastodynamic wave propagation with scalar potential functions is presented for an isotropic half-space, in such a way that the wave may be originated from an arbitrary either traction or heat flux applied on a patch at the free surface of the half-space. The displacements, stresses and temperature are presented within the framework of Biot's coupled thermoelasticity formulations. By employing a complete representation for the displacement and temperature fields in terms of two scalar potential functions, the governing equations of coupled thermoelasticity are uncoupled into a sixth- and a second-order partial differential equation in cylindrical coordinate system. By virtue of Fourier expansion and Hankel integral transforms, the angular and radial variables are suppressed respectively, and a 6{th}- and a 2{nd}-order ordinary differential equation in terms of depth are received, which are solved readily, from which the displacement, stresses and temperature fields are derived in transformed space by satisfying both the regularity and boundary conditions. By applying the inverse Hankel integral transforms, the displacements and temperature are numerically evaluated to determine the solutions in the real space. The numerical evaluations are done for three specific cases of vertical and horizontal time-harmonic patch traction and a constant heat flux passing through a circular disc on the surface of the half-space. It has been previously proved that the potential functions used in this paper are applicable from elastostatics to thermoelastodynamics. Thus, the analytical solutions presented in this paper are verified by comparing the results of this study with two specific problems reported in the literature, which are an elastodynamic problem and an axisymmetric quasi-static thermoelastic problem. To show the accuracy of numerical results, the solution of this study is also compared with the solution for elastodynamics exists in the literature for surface excitation, where a very good agreement is achieved. The formulations presented in this study may be used as benchmark for other related researches and it may be implemented in the related boundary integral equations.

An efficient formulation of robot arm dynamics for control and computer simulation

NASA Astrophysics Data System (ADS)

Lee, C. S. G.; Nigam, R.

This paper describes an efficient formulation of the dynamic equations of motion of industrial robots based on the Lagrange formulation of d'Alembert's principle. This formulation, as applied to a PUMA robot arm, results in a set of closed form second order differential equations with cross product terms. They are not as efficient in computation as those formulated by the Newton-Euler method, but provide a better analytical model for control analysis and computer simulation. Computational complexities of this dynamic model together with other models are tabulated for discussion.

HYDRODYNAMIC SIMULATION OF THE UPPER POTOMAC ESTUARY.

USGS Publications Warehouse

Schaffranck, Raymond W.

1986-01-01

Hydrodynamics of the upper extent of the Potomac Estuary between Indian Head and Morgantown, Md. , are simulated using a two-dimensional model. The model computes water-surface elevations and depth-averaged velocities by numerically integrating finite-difference forms of the equations of mass and momentum conservation using the alternating direction implicit method. The fundamental, non-linear, unsteady-flow equations, upon which the model is formulated, include additional terms to account for Coriolis acceleration and meteorological influences. Preliminary model/prototype data comparisons show agreement to within 9% for tidal flow volumes and phase differences within the measured-data-recording interval. Use of the model to investigate the hydrodynamics and certain aspects of transport within this Potomac Estuary reach is demonstrated. Refs.

Tail shortening by discrete hydrodynamics

NASA Astrophysics Data System (ADS)

Kiefer, J.; Visscher, P. B.

1982-02-01

A discrete formulation of hydrodynamics was recently introduced, whose most important feature is that it is exactly renormalizable. Previous numerical work has found that it provides a more efficient and rapidly convergent method for calculating transport coefficients than the usual Green-Kubo method. The latter's convergence difficulties are due to the well-known "long-time tail" of the time correlation function which must be integrated over time. The purpose of the present paper is to present additional evidence that these difficulties are really absent in the discrete equation of motion approach. The "memory" terms in the equation of motion are calculated accurately, and shown to decay much more rapidly with time than the equilibrium time correlations do.

Nonlinear mode interaction in equal-leg angle struts susceptible to cellular buckling.

PubMed

Bai, L; Wang, F; Wadee, M A; Yang, J

2017-11-01

A variational model that describes the interactive buckling of a thin-walled equal-leg angle strut under pure axial compression is presented. A formulation combining the Rayleigh-Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Solving the equations using numerical continuation reveals progressive cellular buckling (or snaking) arising from the nonlinear interaction between the weak-axis flexural buckling mode and the strong-axis flexural-torsional buckling mode for the first time-the resulting behaviour being highly unstable. Physical experiments conducted on 10 cold-formed steel specimens are presented and the results show good agreement with the variational model.

A Generalized Fluid Formulation for Turbomachinery Computations

NASA Technical Reports Server (NTRS)

Merkle, Charles L.; Sankaran, Venkateswaran; Dorney, Daniel J.; Sondak, Douglas L.

2003-01-01

A generalized formulation of the equations of motion of an arbitrary fluid are developed for the purpose of defining a common iterative algorithm for computational procedures. The method makes use of the equations of motion in conservation form with separate pseudo-time derivatives used for defining the numerical flux for a Riemann solver and the convergence algorithm. The partial differential equations are complemented by an thermodynamic and caloric equations of state of a complexity necessary for describing the fluid. Representative solutions with a new code based on this general equation formulation are provided for three turbomachinery problems. The first uses air as a working fluid while the second uses gaseous oxygen in a regime in which real gas effects are of little importance. These nearly perfect gas computations provide a basis for comparing with existing perfect gas code computations. The third case is for the flow of liquid oxygen through a turbine where real gas effects are significant. Vortex shedding predictions with the LOX formulations reduce the discrepancy between perfect gas computations and experiment by approximately an order of magnitude, thereby verifying the real gas formulation as well as providing an effective case where its capabilities are necessary.

Water resources planning for rivers draining into mobile bay

NASA Technical Reports Server (NTRS)

Ng, S.; April, G. C.

1976-01-01

A hydrodynamic model describing water movement and tidal elevation is formulated, computed, and used to provide basic data about water quality in natural systems. The hydrodynamic model is based on two-dimensional, unsteady flow equations. The water mass is considered to be reasonably mixed such that integration (averaging) in the depth direction is a valid restriction. Convective acceleration, the Coriolis force, wind and bottom interactions are included as contributing terms in the momentum equations. The solution of the equations is applied to Mobile Bay, and used to investigate the influence that river discharge rate, wind direction and speed, and tidal condition have on water circulation and holdup within the bay. Storm surge conditions, oil spill transport, artificial island construction, dredging, and areas subject to flooding are other topics which could be investigated using the mathematical modeling approach.

PIV-based estimation of unsteady loads on a flat plate at high angle of attack using momentum equation approaches

NASA Astrophysics Data System (ADS)

Guissart, Amandine; Bernal, Luis; Dimitriadis, Gregorios; Terrapon, Vincent

2015-11-01

The direct measurement of loads with force balance can become challenging when the forces are small or when the body is moving. An alternative is the use of Particle Image Velocimetry (PIV) velocity fields to indirectly obtain the aerodynamic coefficients. This can be done by the use of control volume approaches which lead to the integration of velocities, and other fields deriving from them, on a contour surrounding the studied body and its supporting surface. This work exposes and discusses results obtained with two different methods: the direct use of the integral formulation of the Navier-Stokes equations and the so-called Noca's method. The latter is a reformulation of the integral Navier-Stokes equations in order to get rid of the pressure. Results obtained using the two methods are compared and the influence of different parameters is discussed. The methods are applied to PIV data obtained from water channel testing for the flow around a 16:1 plate. Two cases are considered: a static plate at high angle of attack and a large amplitude imposed pitching motion. Two-dimensional PIV velocity fields are used to compute the aerodynamic forces. Direct measurements of dynamic loads are also carried out in order to assess the quality of the indirectly calculated coefficients.

A finite-volume Eulerian-Lagrangian Localized Adjoint Method for solution of the advection-dispersion equation

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1993-01-01

A new mass-conservative method for solution of the one-dimensional advection-dispersion equation is derived and discussed. Test results demonstrate that the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) outperforms standard finite-difference methods, in terms of accuracy and efficiency, for solute transport problems that are dominated by advection. For dispersion-dominated problems, the performance of the method is similar to that of standard methods. Like previous ELLAM formulations, FVELLAM systematically conserves mass globally with all types of boundary conditions. FVELLAM differs from other ELLAM approaches in that integrated finite differences, instead of finite elements, are used to approximate the governing equation. This approach, in conjunction with a forward tracking scheme, greatly facilitates mass conservation. The mass storage integral is numerically evaluated at the current time level, and quadrature points are then tracked forward in time to the next level. Forward tracking permits straightforward treatment of inflow boundaries, thus avoiding the inherent problem in backtracking, as used by most characteristic methods, of characteristic lines intersecting inflow boundaries. FVELLAM extends previous ELLAM results by obtaining mass conservation locally on Lagrangian space-time elements. Details of the integration, tracking, and boundary algorithms are presented. Test results are given for problems in Cartesian and radial coordinates.

Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

NASA Astrophysics Data System (ADS)

Pipkins, Daniel Scott

Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially.

On Formulations of Discontinuous Galerkin and Related Methods for Conservation Laws

NASA Technical Reports Server (NTRS)

Huynh, H. T.

2014-01-01

A formulation for the discontinuous Galerkin (DG) method that leads to solutions using the differential form of the equation (as opposed to the standard integral form) is presented. The formulation includes (a) a derivative calculation that involves only data within each cell with no data interaction among cells, and (b) for each cell, corrections to this derivative that deal with the jumps in fluxes at the cell boundaries and allow data across cells to interact. The derivative with no interaction is obtained by a projection, but for nodal-type methods, evaluating this derivative by interpolation at the nodal points is more economical. The corrections are derived using the approximate (Dirac) delta functions. The formulation results in a family of schemes: different approximate delta functions give rise to different methods. It is shown that the current formulation is essentially equivalent to the flux reconstruction (FR) formulation. Due to the use of approximate delta functions, an energy stability proof simpler than that of Vincent, Castonguay, and Jameson (2011) for a family of schemes is derived. Accuracy and stability of resulting schemes are discussed via Fourier analyses. Similar to FR, the current formulation provides a unifying framework for high-order methods by recovering the DG, spectral difference (SD), and spectral volume (SV) schemes. It also yields stable, accurate, and economical methods.

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

NASA Astrophysics Data System (ADS)

Läuter, Matthias; Giraldo, Francis X.; Handorf, Dörthe; Dethloff, Klaus

2008-12-01

A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R3, are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step. The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211-224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of O(Δx) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step Δt is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k.

A numerical model for density-and-viscosity-dependent flows in two-dimensional variably saturated porous media

NASA Astrophysics Data System (ADS)

Boufadel, Michel C.; Suidan, Makram T.; Venosa, Albert D.

1999-04-01

We present a formulation for water flow and solute transport in two-dimensional variably saturated media that accounts for the effects of the solute on water density and viscosity. The governing equations are cast in a dimensionless form that depends on six dimensionless groups of parameters. These equations are discretized in space using the Galerkin finite element formulation and integrated in time using the backward Euler scheme with mass lumping. The modified Picard method is used to linearize the water flow equation. The resulting numerical model, the MARUN model, is verified by comparison to published numerical results. It is then used to investigate beach hydraulics at seawater concentration (about 30 g l -1) in the context of nutrients delivery for bioremediation of oil spills on beaches. Numerical simulations that we conducted in a rectangular section of a hypothetical beach revealed that buoyancy in the unsaturated zone is significant in soils that are fine textured, with low anisotropy ratio, and/or exhibiting low physical dispersion. In such situations, application of dissolved nutrients to a contaminated beach in a freshwater solution is superior to their application in a seawater solution. Concentration-engendered viscosity effects were negligible with respect to concentration-engendered density effects for the cases that we considered.

TESS: A RELATIVISTIC HYDRODYNAMICS CODE ON A MOVING VORONOI MESH

DOE Office of Scientific and Technical Information (OSTI.GOV)

Duffell, Paul C.; MacFadyen, Andrew I., E-mail: pcd233@nyu.edu, E-mail: macfadyen@nyu.edu

2011-12-01

We have generalized a method for the numerical solution of hyperbolic systems of equations using a dynamic Voronoi tessellation of the computational domain. The Voronoi tessellation is used to generate moving computational meshes for the solution of multidimensional systems of conservation laws in finite-volume form. The mesh-generating points are free to move with arbitrary velocity, with the choice of zero velocity resulting in an Eulerian formulation. Moving the points at the local fluid velocity makes the formulation effectively Lagrangian. We have written the TESS code to solve the equations of compressible hydrodynamics and magnetohydrodynamics for both relativistic and non-relativistic fluidsmore » on a dynamic Voronoi mesh. When run in Lagrangian mode, TESS is significantly less diffusive than fixed mesh codes and thus preserves contact discontinuities to high precision while also accurately capturing strong shock waves. TESS is written for Cartesian, spherical, and cylindrical coordinates and is modular so that auxiliary physics solvers are readily integrated into the TESS framework and so that this can be readily adapted to solve general systems of equations. We present results from a series of test problems to demonstrate the performance of TESS and to highlight some of the advantages of the dynamic tessellation method for solving challenging problems in astrophysical fluid dynamics.« less

Elastic-viscoplastic modeling of soft biological tissues using a mixed finite element formulation based on the relative deformation gradient.

PubMed

Weickenmeier, J; Jabareen, M

2014-11-01

The characteristic highly nonlinear, time-dependent, and often inelastic material response of soft biological tissues can be expressed in a set of elastic-viscoplastic constitutive equations. The specific elastic-viscoplastic model for soft tissues proposed by Rubin and Bodner (2002) is generalized with respect to the constitutive equations for the scalar quantity of the rate of inelasticity and the hardening parameter in order to represent a general framework for elastic-viscoplastic models. A strongly objective integration scheme and a new mixed finite element formulation were developed based on the introduction of the relative deformation gradient-the deformation mapping between the last converged and current configurations. The numerical implementation of both the generalized framework and the specific Rubin and Bodner model is presented. As an example of a challenging application of the new model equations, the mechanical response of facial skin tissue is characterized through an experimental campaign based on the suction method. The measurement data are used for the identification of a suitable set of model parameters that well represents the experimentally observed tissue behavior. Two different measurement protocols were defined to address specific tissue properties with respect to the instantaneous tissue response, inelasticity, and tissue recovery. Copyright © 2014 John Wiley & Sons, Ltd.

Turbulent kinetic energy and a possible hierarchy of length scales in a generalization of the Navier-Stokes alpha theory.

PubMed

Fried, Eliot; Gurtin, Morton E

2007-05-01

We present a continuum-mechanical formulation and generalization of the Navier-Stokes alpha theory based on a general framework for fluid-dynamical theories with gradient dependencies. Our flow equation involves two additional problem-dependent length scales alpha and beta. The first of these scales enters the theory through the internal kinetic energy, per unit mass, alpha2|D|2, where D is the symmetric part of the gradient of the filtered velocity. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity. When alpha and beta are equal, our flow equation reduces to the Navier-Stokes alpha equation. In contrast to the original derivation of the Navier-Stokes alpha equation, which relies on Lagrangian averaging, our formulation delivers boundary conditions. For a confined flow, our boundary conditions involve an additional length scale l characteristic of the eddies found near walls. Based on a comparison with direct numerical simulations for fully developed turbulent flow in a rectangular channel of height 2h, we find that alphabeta approximately Re(0.470) and lh approximately Re(-0.772), where Re is the Reynolds number. The first result, which arises as a consequence of identifying the internal kinetic energy with the turbulent kinetic energy, indicates that the choice alpha=beta required to reduce our flow equation to the Navier-Stokes alpha equation is likely to be problematic. The second result evinces the classical scaling relation eta/L approximately Re(-3/4) for the ratio of the Kolmogorov microscale eta to the integral length scale L . The numerical data also suggests that l < or = beta . We are therefore led to conjecture a tentative hierarchy, l < or = beta < alpha , involving the three length scales entering our theory.

Stochastic Mixing Model with Power Law Decay of Variance

NASA Technical Reports Server (NTRS)

Fedotov, S.; Ihme, M.; Pitsch, H.

2003-01-01

Here we present a simple stochastic mixing model based on the law of large numbers (LLN). The reason why the LLN is involved in our formulation of the mixing problem is that the random conserved scalar c = c(t,x(t)) appears to behave as a sample mean. It converges to the mean value mu, while the variance sigma(sup 2)(sub c) (t) decays approximately as t(exp -1). Since the variance of the scalar decays faster than a sample mean (typically is greater than unity), we will introduce some non-linear modifications into the corresponding pdf-equation. The main idea is to develop a robust model which is independent from restrictive assumptions about the shape of the pdf. The remainder of this paper is organized as follows. In Section 2 we derive the integral equation from a stochastic difference equation describing the evolution of the pdf of a passive scalar in time. The stochastic difference equation introduces an exchange rate gamma(sub n) which we model in a first step as a deterministic function. In a second step, we generalize gamma(sub n) as a stochastic variable taking fluctuations in the inhomogeneous environment into account. In Section 3 we solve the non-linear integral equation numerically and analyze the influence of the different parameters on the decay rate. The paper finishes with a conclusion.

A clamped rectangular plate containing a crack

NASA Technical Reports Server (NTRS)

Tang, R.; Erdogan, F.

1985-01-01

The general problem of a rectangular plate clamped along two parallel sides and containing a crack parallel to the clamps is considered. The problem is formulated in terms of a system of singular integral equations and the asymptotic behavior of the stress state near the corners is investigated. Numerical examples are considered for a clamped plate without a crack and with a centrally located crack, and the stress intensity factors and the stresses along the clamps are calculated.

An Advanced One-Dimensional Finite Element Model for Incompressible Thermally Expandable Flow

DOE PAGES

Hu, Rui

2017-03-27

Here, this paper provides an overview of a new one-dimensional finite element flow model for incompressible but thermally expandable flow. The flow model was developed for use in system analysis tools for whole-plant safety analysis of sodium fast reactors. Although the pressure-based formulation was implemented, the use of integral equations in the conservative form ensured the conservation laws of the fluid. A stabilization scheme based on streamline-upwind/Petrov-Galerkin and pressure-stabilizing/Petrov-Galerkin formulations is also introduced. The flow model and its implementation have been verified by many test problems, including density wave propagation, steep gradient problems, discharging between tanks, and the conjugate heatmore » transfer in a heat exchanger.« less

An Advanced One-Dimensional Finite Element Model for Incompressible Thermally Expandable Flow

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hu, Rui

Here, this paper provides an overview of a new one-dimensional finite element flow model for incompressible but thermally expandable flow. The flow model was developed for use in system analysis tools for whole-plant safety analysis of sodium fast reactors. Although the pressure-based formulation was implemented, the use of integral equations in the conservative form ensured the conservation laws of the fluid. A stabilization scheme based on streamline-upwind/Petrov-Galerkin and pressure-stabilizing/Petrov-Galerkin formulations is also introduced. The flow model and its implementation have been verified by many test problems, including density wave propagation, steep gradient problems, discharging between tanks, and the conjugate heatmore » transfer in a heat exchanger.« less

Numerical Simulation of Delamination Growth in Composite Materials

NASA Technical Reports Server (NTRS)

Camanho, P. P.; Davila, C. G.; Ambur, D. R.

2001-01-01

The use of decohesion elements for the simulation of delamination in composite materials is reviewed. The test methods available to measure the interfacial fracture toughness used in the formulation of decohesion elements are described initially. After a brief presentation of the virtual crack closure technique, the technique most widely used to simulate delamination growth, the formulation of interfacial decohesion elements is described. Problems related with decohesion element constitutive equations, mixed-mode crack growth, element numerical integration and solution procedures are discussed. Based on these investigations, it is concluded that the use of interfacial decohesion elements is a promising technique that avoids the need for a pre-existing crack and pre-defined crack paths, and that these elements can be used to simulate both delamination onset and growth.

Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes

NASA Astrophysics Data System (ADS)

Dumbser, Michael; Guercilena, Federico; Köppel, Sven; Rezzolla, Luciano; Zanotti, Olindo

2018-04-01

We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential equations system is written in nonconservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations.

An accurate boundary element method for the exterior elastic scattering problem in two dimensions

NASA Astrophysics Data System (ADS)

Bao, Gang; Xu, Liwei; Yin, Tao

2017-11-01

This paper is concerned with a Galerkin boundary element method solving the two dimensional exterior elastic wave scattering problem. The original problem is first reduced to the so-called Burton-Miller [1] boundary integral formulation, and essential mathematical features of its variational form are discussed. In numerical implementations, a newly-derived and analytically accurate regularization formula [2] is employed for the numerical evaluation of hyper-singular boundary integral operator. A new computational approach is employed based on the series expansions of Hankel functions for the computation of weakly-singular boundary integral operators during the reduction of corresponding Galerkin equations into a discrete linear system. The effectiveness of proposed numerical methods is demonstrated using several numerical examples.

Scalar/Vector potential formulation for compressible viscous unsteady flows

NASA Technical Reports Server (NTRS)

Morino, L.

1985-01-01

A scalar/vector potential formulation for unsteady viscous compressible flows is presented. The scalar/vector potential formulation is based on the classical Helmholtz decomposition of any vector field into the sum of an irrotational and a solenoidal field. The formulation is derived from fundamental principles of mechanics and thermodynamics. The governing equations for the scalar potential and vector potential are obtained, without restrictive assumptions on either the equation of state or the constitutive relations or the stress tensor and the heat flux vector.

The mechanical and chemical equations of motion of muscle contraction

NASA Astrophysics Data System (ADS)

Shiner, J. S.; Sieniutycz, Stanislaw

1997-11-01

Up to now no formulation of muscle contraction has provided both the chemical kinetic equations for the reactions responsible for the contraction and the mechanical equation of motion for the muscle. This has most likely been due to the lack of general formalisms for nonlinear systems with chemical-nonchemical coupling valid under the far from equilibrium conditions under which muscle operates physiologically. We have recently developed such formalisms and apply them here to the formulation of muscle contraction to obtain both the chemical and the mechanical equations. The standard formulation up to now has yielded only the dynamic equations for the chemical variables and has considered these to be functions of both time and an appropriate mechanical variable. The macroscopically observable quantities were then obtained by averaging over the mechanical variable. When attempting to derive the dynamics equations for both the chemistry and mechanics this choice of variables leads to conflicting results for the mechanical equation of motion when two different general formalisms are applied. The conflict can be resolved by choosing the variables such that both the chemical variables and the mechanical variables are considered to be functions of time alone. This adds one equation to the set of differential equations to be solved but is actually a simplification of the problem, since these equations are ordinary differential equations, not the partial differential equations of the now standard formulation, and since in this choice of variables the variables themselves are the macroscopic observables the procedure of averaging over the mechanical variable is eliminated. Furthermore, the parameters occurring in the equations at this level of description should be accessible to direct experimental determination.

Recent advances in computational-analytical integral transforms for convection-diffusion problems

NASA Astrophysics Data System (ADS)

Cotta, R. M.; Naveira-Cotta, C. P.; Knupp, D. C.; Zotin, J. L. Z.; Pontes, P. C.; Almeida, A. P.

2017-10-01

An unifying overview of the Generalized Integral Transform Technique (GITT) as a computational-analytical approach for solving convection-diffusion problems is presented. This work is aimed at bringing together some of the most recent developments on both accuracy and convergence improvements on this well-established hybrid numerical-analytical methodology for partial differential equations. Special emphasis is given to novel algorithm implementations, all directly connected to enhancing the eigenfunction expansion basis, such as a single domain reformulation strategy for handling complex geometries, an integral balance scheme in dealing with multiscale problems, the adoption of convective eigenvalue problems in formulations with significant convection effects, and the direct integral transformation of nonlinear convection-diffusion problems based on nonlinear eigenvalue problems. Then, selected examples are presented that illustrate the improvement achieved in each class of extension, in terms of convergence acceleration and accuracy gain, which are related to conjugated heat transfer in complex or multiscale microchannel-substrate geometries, multidimensional Burgers equation model, and diffusive metal extraction through polymeric hollow fiber membranes. Numerical results are reported for each application and, where appropriate, critically compared against the traditional GITT scheme without convergence enhancement schemes and commercial or dedicated purely numerical approaches.

Nonlocal Electrostatics in Spherical Geometries Using Eigenfunction Expansions of Boundary-Integral Operators.

PubMed

Bardhan, Jaydeep P; Knepley, Matthew G; Brune, Peter

2015-01-01

In this paper, we present an exact, infinite-series solution to Lorentz nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact that their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for calculations in separable geometries, we first re-derive Kirkwood's classic results for a protein surrounded concentrically by a pure-water ion-exclusion (Stern) layer and then a dilute electrolyte, which is modeled with the linearized Poisson-Boltzmann equation. The eigenfunction-expansion approach provides a computationally efficient way to test some implications of nonlocal models, including estimating the reasonable range of the nonlocal length-scale parameter λ. Our results suggest that nonlocal solvent response may help to reduce the need for very high dielectric constants in calculating pH-dependent protein behavior, though more sophisticated nonlocal models are needed to resolve this question in full. An open-source MATLAB implementation of our approach is freely available online.

Nonlocal Electrostatics in Spherical Geometries Using Eigenfunction Expansions of Boundary-Integral Operators

PubMed Central

Bardhan, Jaydeep P.; Knepley, Matthew G.; Brune, Peter

2015-01-01

In this paper, we present an exact, infinite-series solution to Lorentz nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact that their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for calculations in separable geometries, we first re-derive Kirkwood’s classic results for a protein surrounded concentrically by a pure-water ion-exclusion (Stern) layer and then a dilute electrolyte, which is modeled with the linearized Poisson–Boltzmann equation. The eigenfunction-expansion approach provides a computationally efficient way to test some implications of nonlocal models, including estimating the reasonable range of the nonlocal length-scale parameter λ. Our results suggest that nonlocal solvent response may help to reduce the need for very high dielectric constants in calculating pH-dependent protein behavior, though more sophisticated nonlocal models are needed to resolve this question in full. An open-source MATLAB implementation of our approach is freely available online. PMID:26273581

A new numerical approach to solve Thomas-Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming.

PubMed

Raja, Muhammad Asif Zahoor; Zameer, Aneela; Khan, Aziz Ullah; Wazwaz, Abdul Majid

2016-01-01

In this study, a novel bio-inspired computing approach is developed to analyze the dynamics of nonlinear singular Thomas-Fermi equation (TFE) arising in potential and charge density models of an atom by exploiting the strength of finite difference scheme (FDS) for discretization and optimization through genetic algorithms (GAs) hybrid with sequential quadratic programming. The FDS procedures are used to transform the TFE differential equations into a system of nonlinear equations. A fitness function is constructed based on the residual error of constituent equations in the mean square sense and is formulated as the minimization problem. Optimization of parameters for the system is carried out with GAs, used as a tool for viable global search integrated with SQP algorithm for rapid refinement of the results. The design scheme is applied to solve TFE for five different scenarios by taking various step sizes and different input intervals. Comparison of the proposed results with the state of the art numerical and analytical solutions reveals that the worth of our scheme in terms of accuracy and convergence. The reliability and effectiveness of the proposed scheme are validated through consistently getting optimal values of statistical performance indices calculated for a sufficiently large number of independent runs to establish its significance.

Cones of localized shear strain in incompressible elasticity with prestress: Green's function and integral representations

PubMed Central

Argani, L. P.; Bigoni, D.; Capuani, D.; Movchan, N. V.

2014-01-01

The infinite-body three-dimensional Green's function set (for incremental displacement and mean stress) is derived for the incremental deformation of a uniformly strained incompressible, nonlinear elastic body. Particular cases of the developed formulation are the Mooney–Rivlin elasticity and the J2-deformation theory of plasticity. These Green's functions are used to develop a boundary integral equation framework, by introducing an ad hoc potential, which paves the way for a boundary element formulation of three-dimensional problems of incremental elasticity. Results are used to investigate the behaviour of a material deformed near the limit of ellipticity and to reveal patterns of shear failure. In fact, within the investigated three-dimensional framework, localized deformations emanating from a perturbation are shown to be organized in conical geometries rather than in planar bands, so that failure is predicted to develop through curved and thin surfaces of intense shearing, as can for instance be observed in the cup–cone rupture of ductile metal bars. PMID:25197258

Modelling electro-active polymers with a dispersion-type anisotropy

NASA Astrophysics Data System (ADS)

Hossain, Mokarram; Steinmann, Paul

2018-02-01

We propose a novel constitutive framework for electro-active polymers (EAPs) that can take into account anisotropy with a chain dispersion. To enhance actuation behaviour, particle-filled EAPs become promising candidates nowadays. Recent studies suggest that particle-filled EAPs, which can be cured under an electric field during the manufacturing time, do not necessarily form perfect anisotropic composites, rather they create composites with dispersed chains. Hence in this contribution, an electro-mechanically coupled constitutive model is devised that considers the chain dispersion with a probability distribution function in an integral form. To obtain relevant quantities in discrete form, numerical integration over the unit sphere is utilized. Necessary constitutive equations are derived exploiting the basic laws of thermodynamics that result in a thermodynamically consistent formulation. To demonstrate the performance of the proposed electro-mechanically coupled framework, we analytically solve a non-homogeneous boundary value problem, the extension and inflation of an axisymmetric cylindrical tube under electro-mechanically coupled load. The results capture various electro-mechanical couplings with the formulation proposed for EAP composites.

Mathematical Metaphors: Problem Reformulation and Analysis Strategies

NASA Technical Reports Server (NTRS)

Thompson, David E.

2005-01-01

This paper addresses the critical need for the development of intelligent or assisting software tools for the scientist who is working in the initial problem formulation and mathematical model representation stage of research. In particular, examples of that representation in fluid dynamics and instability theory are discussed. The creation of a mathematical model that is ready for application of certain solution strategies requires extensive symbolic manipulation of the original mathematical model. These manipulations can be as simple as term reordering or as complicated as discovery of various symmetry groups embodied in the equations, whereby Backlund-type transformations create new determining equations and integrability conditions or create differential Grobner bases that are then solved in place of the original nonlinear PDEs. Several examples are presented of the kinds of problem formulations and transforms that can be frequently encountered in model representation for fluids problems. The capability of intelligently automating these types of transforms, available prior to actual mathematical solution, is advocated. Physical meaning and assumption-understanding can then be propagated through the mathematical transformations, allowing for explicit strategy development.

Fully coupled methods for multiphase morphodynamics

NASA Astrophysics Data System (ADS)

Michoski, C.; Dawson, C.; Mirabito, C.; Kubatko, E. J.; Wirasaet, D.; Westerink, J. J.

2013-09-01

We present numerical methods for a system of equations consisting of the two dimensional Saint-Venant shallow water equations (SWEs) fully coupled to a completely generalized Exner formulation of hydrodynamically driven sediment discharge. This formulation is implemented by way of a discontinuous Galerkin (DG) finite element method, using a Roe Flux for the advective components and the unified form for the dissipative components. We implement a number of Runge-Kutta time integrators, including a family of strong stability preserving (SSP) schemes, and Runge-Kutta Chebyshev (RKC) methods. A brief discussion is provided regarding implementational details for generalizable computer algebra tokenization using arbitrary algebraic fluxes. We then run numerical experiments to show standard convergence rates, and discuss important mathematical and numerical nuances that arise due to prominent features in the coupled system, such as the emergence of nondifferentiable and sharp zero crossing functions, radii of convergence in manufactured solutions, and nonconservative product (NCP) formalisms. Finally we present a challenging application model concerning hydrothermal venting across metalliferous muds in the presence of chemical reactions occurring in low pH environments.

Collocation and Galerkin Time-Stepping Methods

NASA Technical Reports Server (NTRS)

Huynh, H. T.

2011-01-01

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

Burton-Miller-type singular boundary method for acoustic radiation and scattering

NASA Astrophysics Data System (ADS)

Fu, Zhuo-Jia; Chen, Wen; Gu, Yan

2014-08-01

This paper proposes the singular boundary method (SBM) in conjunction with Burton and Miller's formulation for acoustic radiation and scattering. The SBM is a strong-form collocation boundary discretization technique using the singular fundamental solutions, which is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors (SIFs) to eliminate the singularities of the fundamental solutions. Therefore, it avoids singular numerical integrals in the boundary element method (BEM) and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions (MFS). In the present method, we derive the SIFs of exterior Helmholtz equation by means of the SIFs of exterior Laplace equation owing to the same order of singularities between the Laplace and Helmholtz fundamental solutions. In conjunction with the Burton-Miller formulation, the SBM enhances the quality of the solution, particularly in the vicinity of the corresponding interior eigenfrequencies. Numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2D and 3D unbounded domains in comparison with the analytical solutions, the boundary element solutions and Dirichlet-to-Neumann finite element solutions.

Hamiltonian formulation of the KdV equation

NASA Astrophysics Data System (ADS)

Nutku, Y.

1984-06-01

We consider the canonical formulation of Whitham's variational principle for the KdV equation. This Lagrangian is degenerate and we have found it necessary to use Dirac's theory of constrained systems in constructing the Hamiltonian. Earlier discussions of the Hamiltonian structure of the KdV equation were based on various different decompositions of the field which is avoided by this new approach.

A computer program to generate equations of motion matrices, L217 (EOM). Volume 1: Engineering and usage

NASA Technical Reports Server (NTRS)

Kroll, R. I.; Clemmons, R. E.

1979-01-01

The equations of motion program L217 formulates the matrix coefficients for a set of second order linear differential equations that describe the motion of an airplane relative to its level equilibrium flight condition. Aerodynamic data from FLEXSTAB or Doublet Lattice (L216) programs can be used to derive the equations for quasi-steady or full unsteady aerodynamics. The data manipulation and the matrix coefficient formulation are described.

Formulation of the relativistic moment implicit particle-in-cell method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Noguchi, Koichi; Tronci, Cesare; Zuccaro, Gianluca

2007-04-15

A new formulation is presented for the implicit moment method applied to the time-dependent relativistic Vlasov-Maxwell system. The new approach is based on a specific formulation of the implicit moment method that allows us to retain the same formalism that is valid in the classical case despite the formidable complication introduced by the nonlinear nature of the relativistic equations of motion. To demonstrate the validity of the new formulation, an implicit finite difference algorithm is developed to solve the Maxwell's equations and equations of motion. A number of benchmark problems are run: two stream instability, ion acoustic wave damping, Weibelmore » instability, and Poynting flux acceleration. The numerical results are all in agreement with analytical solutions.« less

Lagrangian formulation for penny-shaped and Perkins-Kern geometry models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lee, W.S.

1989-09-01

This paper discusses basic theories for vertical penny-shaped and Perkins-Kern (PK) geometry models developed with a Lagrangian formulation combined with a virtual-work analysis. The Lagrangian formulation yields a pair of nonlinear equations in R/sub f/ or L/sub f/ and b/sub f/, the fracture radius or length and half-width. By introduction of a virtual-work analysis, a simple equation is obtained that can be solved numerically. This equation is written in a form that can be used to determine fracture geometry when the fluid-loss coefficient of the fracturing fluid is known. Also, this equation, coupled with a material-balance equation after shut-in, canmore » be used to analyze pressure-decline data after shut-in to determine the effective fluid-loss coefficient and fracture geometry.« less

Three-Dimensional High-Order Spectral Volume Method for Solving Maxwell's Equations on Unstructured Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel; Wang, Z. J.

2004-01-01

A three-dimensional, high-order, conservative, and efficient discontinuous spectral volume (SV) method for the solutions of Maxwell's equations on unstructured grids is presented. The concept of discontinuous 2nd high-order loca1 representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) method, but instead of using a Galerkin finite-element formulation, the SV method is based on a finite-volume approach to attain a simpler formulation. Conventional unstructured finite-volume methods require data reconstruction based on the least-squares formulation using neighboring cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In the SV method, one starts with a relatively coarse grid of triangles or tetrahedra, called spectral volumes (SVs), and partition each SV into a number of structured subcells, called control volumes (CVs), that support a polynomial expansion of a desired degree of precision. The unknowns are cell averages over CVs. If all the SVs are partitioned in a geometrically similar manner, the reconstruction becomes universal as a weighted sum of unknowns, and only a few universal coefficients need to be stored for the surface integrals over CV faces. Since the solution is discontinuous across the SV boundaries, a Riemann solver is thus necessary to maintain conservation. In the paper, multi-parameter and symmetric SV partitions, up to quartic for triangle and cubic for tetrahedron, are first presented. The corresponding weight coefficients for CV face integrals in terms of CV cell averages for each partition are analytically determined. These discretization formulas are then applied to the integral form of the Maxwell equations. All numerical procedures for outer boundary, material interface, zonal interface, and interior SV face are unified with a single characteristic formulation. The load balancing in a massive parallel computing environment is therefore easier to achieve. A parameter is introduced in the Riemann solver to control the strength of the smoothing term. Important aspects of the data structure and its effects to communication and the optimum use of cache memory are discussed. Results will be presented for plane TE and TM waves incident on a perfectly conducting cylinder for up to fifth order of accuracy, and a plane wave incident on a perfectly conducting sphere for up to fourth order of accuracy. Comparisons are made with exact solutions for these cases.

An energy-stable method for solving the incompressible Navier-Stokes equations with non-slip boundary condition

NASA Astrophysics Data System (ADS)

Lee, Byungjoon; Min, Chohong

2018-05-01

We introduce a stable method for solving the incompressible Navier-Stokes equations with variable density and viscosity. Our method is stable in the sense that it does not increase the total energy of dynamics that is the sum of kinetic energy and potential energy. Instead of velocity, a new state variable is taken so that the kinetic energy is formulated by the L2 norm of the new variable. Navier-Stokes equations are rephrased with respect to the new variable, and a stable time discretization for the rephrased equations is presented. Taking into consideration the incompressibility in the Marker-And-Cell (MAC) grid, we present a modified Lax-Friedrich method that is L2 stable. Utilizing the discrete integration-by-parts in MAC grid and the modified Lax-Friedrich method, the time discretization is fully discretized. An explicit CFL condition for the stability of the full discretization is given and mathematically proved.

Efficient propagation-inside-layer expansion algorithm for solving the scattering from three-dimensional nested homogeneous dielectric bodies with arbitrary shape.

PubMed

Bellez, Sami; Bourlier, Christophe; Kubické, Gildas

2015-03-01

This paper deals with the evaluation of electromagnetic scattering from a three-dimensional structure consisting of two nested homogeneous dielectric bodies with arbitrary shape. The scattering problem is formulated in terms of a set of Poggio-Miller-Chang-Harrington-Wu integral equations that are afterwards converted into a system of linear equations (impedance matrix equation) by applying the Galerkin method of moments (MoM) with Rao-Wilton-Glisson basis functions. The MoM matrix equation is then solved by deploying the iterative propagation-inside-layer expansion (PILE) method in order to obtain the unknown surface current densities, which are thereafter used to handle the radar cross-section (RCS) patterns. Some numerical results for various structures including canonical geometries are presented and compared with those of the FEKO software in order to validate the PILE-based approach as well as to show its efficiency to analyze the full-polarized RCS patterns.

Two modified symplectic partitioned Runge-Kutta methods for solving the elastic wave equation

NASA Astrophysics Data System (ADS)

Su, Bo; Tuo, Xianguo; Xu, Ling

2017-08-01

Based on a modified strategy, two modified symplectic partitioned Runge-Kutta (PRK) methods are proposed for the temporal discretization of the elastic wave equation. The two symplectic schemes are similar in form but are different in nature. After the spatial discretization of the elastic wave equation, the ordinary Hamiltonian formulation for the elastic wave equation is presented. The PRK scheme is then applied for time integration. An additional term associated with spatial discretization is inserted into the different stages of the PRK scheme. Theoretical analyses are conducted to evaluate the numerical dispersion and stability of the two novel PRK methods. A finite difference method is used to approximate the spatial derivatives since the two schemes are independent of the spatial discretization technique used. The numerical solutions computed by the two new schemes are compared with those computed by a conventional symplectic PRK. The numerical results, which verify the new method, are superior to those generated by traditional conventional methods in seismic wave modeling.

An analytical method for the inverse Cauchy problem of Lame equation in a rectangle

NASA Astrophysics Data System (ADS)

Grigor’ev, Yu

2018-04-01

In this paper, we present an analytical computational method for the inverse Cauchy problem of Lame equation in the elasticity theory. A rectangular domain is frequently used in engineering structures and we only consider the analytical solution in a two-dimensional rectangle, wherein a missing boundary condition is recovered from the full measurement of stresses and displacements on an accessible boundary. The essence of the method consists in solving three independent Cauchy problems for the Laplace and Poisson equations. For each of them, the Fourier series is used to formulate a first-kind Fredholm integral equation for the unknown function of data. Then, we use a Lavrentiev regularization method, and the termwise separable property of kernel function allows us to obtain a closed-form regularized solution. As a result, for the displacement components, we obtain solutions in the form of a sum of series with three regularization parameters. The uniform convergence and error estimation of the regularized solutions are proved.

A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

NASA Technical Reports Server (NTRS)

Gerritsen, Margot; Olsson, Pelle

1996-01-01

We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.

Non-Linear Steady State Vibrations of Beams Excited by Vortex Shedding

NASA Astrophysics Data System (ADS)

LEWANDOWSKI, R.

2002-05-01

In this paper the non-linear vibrations of beams excited by vortex-shedding are considered. In particular, the steady state responses of beams near the synchronization region are taken into account. The main aerodynamic properties of wind are described by using the semi-empirical model proposed by Hartlen and Currie. The finite element method and the strip method are used to formulate the equation of motion of the system treated. The harmonic balance method is adopted to derive the amplitude equations. These equations are solved with the help of the continuation method which is very convenient to perform the parametric studies of the problem and to determine the response curve in the synchronization region. Moreover, the equations of motion are also integrated using the Newmark method. The results of calculations of several example problems are also shown to confirm the efficiency and accuracy of the presented method. The results obtained by the harmonic balance method and by the Newmark methods are in good agreement with each other.

Application of linear multifrequency-grey acceleration to preconditioned Krylov iterations for thermal radiation transport

DOE PAGES

Till, Andrew T.; Warsa, James S.; Morel, Jim E.

2018-06-15

The thermal radiative transfer (TRT) equations comprise a radiation equation coupled to the material internal energy equation. Linearization of these equations produces effective, thermally-redistributed scattering through absorption-reemission. In this paper, we investigate the effectiveness and efficiency of Linear-Multi-Frequency-Grey (LMFG) acceleration that has been reformulated for use as a preconditioner to Krylov iterative solution methods. We introduce two general frameworks, the scalar flux formulation (SFF) and the absorption rate formulation (ARF), and investigate their iterative properties in the absence and presence of true scattering. SFF has a group-dependent state size but may be formulated without inner iterations in the presence ofmore » scattering, while ARF has a group-independent state size but requires inner iterations when scattering is present. We compare and evaluate the computational cost and efficiency of LMFG applied to these two formulations using a direct solver for the preconditioners. Finally, this work is novel because the use of LMFG for the radiation transport equation, in conjunction with Krylov methods, involves special considerations not required for radiation diffusion.« less

Development and evaluation of Ketoprofen sustained release matrix tablet using Hibiscus rosa-sinensis leaves mucilage.

PubMed

Kaleemullah, M; Jiyauddin, K; Thiban, E; Rasha, S; Al-Dhalli, S; Budiasih, S; Gamal, O E; Fadli, A; Eddy, Y

2017-07-01

Currently, the use of natural gums and mucilage is of increasing importance in pharmaceutical formulations as valuable drug excipient. Natural plant-based materials are economic, free of side effects, biocompatible and biodegradable. Therefore, Ketoprofen matrix tablets were formulated by employing Hibiscus rosa-sinensis leaves mucilage as natural polymer and HPMC (K100M) as a synthetic polymer to sustain the drug release from matrix system. Direct compression method was used to develop sustained released matrix tablets. The formulated matrix tablets were evaluated in terms of physical appearance, weight variation, thickness, diameter, hardness, friability and in vitro drug release. The difference between the natural and synthetic polymers was investigated concurrently. Matrix tablets developed from each formulation passed all standard physical evaluation tests. The dissolution studies of formulated tablets revealed sustained drug release up to 24 h compared to the reference drug Apo Keto® SR tablets. The dissolution data later were fitted into kinetic models such as zero order equation, first order equation, Higuchi equation, Hixson Crowell equation and Korsmeyer-Peppas equation to study the release of drugs from each formulation. The best formulations were selected based on the similarity factor ( f 2 ) value of 50% and more. Through the research, it is found that by increasing the polymers concentration, the rate of drug release decreased for both natural and synthetic polymers. The best formulation was found to be F3 which contained 40% Hibiscus rosa-sinensis mucilage polymer and showed comparable dissolution profile to the reference drug with f 2 value of 78.03%. The release kinetics of this formulation has shown to follow non-Fickian type which involved both diffusion and erosion mechanism. Additionally, the statistical results indicated that there was no significant difference (p > 0.05) between the F3 and reference drug in terms of MDT and T50% with p-values of 1.00 and 0.995 respectively.

Application of variational and Galerkin equations to linear and nonlinear finite element analysis

NASA Technical Reports Server (NTRS)

Yu, Y.-Y.

1974-01-01

The paper discusses the application of the variational equation to nonlinear finite element analysis. The problem of beam vibration with large deflection is considered. The variational equation is shown to be flexible in both the solution of a general problem and in the finite element formulation. Difficulties are shown to arise when Galerkin's equations are used in the consideration of the finite element formulation of two-dimensional linear elasticity and of the linear classical beam.

Unified connected theory of few-body reaction mechanisms in N-body scattering theory

NASA Technical Reports Server (NTRS)

Polyzou, W. N.; Redish, E. F.

1978-01-01

A unified treatment of different reaction mechanisms in nonrelativistic N-body scattering is presented. The theory is based on connected kernel integral equations that are expected to become compact for reasonable constraints on the potentials. The operators T/sub +-//sup ab/(A) are approximate transition operators that describe the scattering proceeding through an arbitrary reaction mechanism A. These operators are uniquely determined by a connected kernel equation and satisfy an optical theorem consistent with the choice of reaction mechanism. Connected kernel equations relating T/sub +-//sup ab/(A) to the full T/sub +-//sup ab/ allow correction of the approximate solutions for any ignored process to any order. This theory gives a unified treatment of all few-body reaction mechanisms with the same dynamic simplicity of a model calculation, but can include complicated reaction mechanisms involving overlapping configurations where it is difficult to formulate models.

Numerical Simulation and Quantitative Uncertainty Assessment of Microchannel Flow

NASA Astrophysics Data System (ADS)

Debusschere, Bert; Najm, Habib; Knio, Omar; Matta, Alain; Ghanem, Roger; Le Maitre, Olivier

2002-11-01

This study investigates the effect of uncertainty in physical model parameters on computed electrokinetic flow of proteins in a microchannel with a potassium phosphate buffer. The coupled momentum, species transport, and electrostatic field equations give a detailed representation of electroosmotic and pressure-driven flow, including sample dispersion mechanisms. The chemistry model accounts for pH-dependent protein labeling reactions as well as detailed buffer electrochemistry in a mixed finite-rate/equilibrium formulation. To quantify uncertainty, the governing equations are reformulated using a pseudo-spectral stochastic methodology, which uses polynomial chaos expansions to describe uncertain/stochastic model parameters, boundary conditions, and flow quantities. Integration of the resulting equations for the spectral mode strengths gives the evolution of all stochastic modes for all variables. Results show the spatiotemporal evolution of uncertainties in predicted quantities and highlight the dominant parameters contributing to these uncertainties during various flow phases. This work is supported by DARPA.

Adaptive boundary concentration control using Zakai equation

NASA Astrophysics Data System (ADS)

Tenno, R.; Mendelson, A.

2010-06-01

A mean-variance control problem is formulated with respect to a partially observed nonlinear system that includes unknown constant parameters. A physical prototype of the system is the cathode surface reaction in an electrolysis cell, where the controller aim is to keep the boundary concentration of species in the near vicinity of the cathode surface low but not zero. The boundary concentration is a diffusion-controlled process observed through the measured current density and, in practice, controlled through the applied voltage. The former incomplete data control problem is converted to complete data-to the so-called separated control problem whose solution is given by the infinite-dimensional Zakai equation. In this article, the separated control problem is solved numerically using pathwise integration of the Zakai equation. This article demonstrates precise tracking of the target trajectory with a rapid convergence of estimates to unknown parameters, which take place simultaneously with control.

The terminal area simulation system. Volume 1: Theoretical formulation

NASA Technical Reports Server (NTRS)

Proctor, F. H.

1987-01-01

A three-dimensional numerical cloud model was developed for the general purpose of studying convective phenomena. The model utilizes a time splitting integration procedure in the numerical solution of the compressible nonhydrostatic primitive equations. Turbulence closure is achieved by a conventional first-order diagnostic approximation. Open lateral boundaries are incorporated which minimize wave reflection and which do not induce domain-wide mass trends. Microphysical processes are governed by prognostic equations for potential temperature water vapor, cloud droplets, ice crystals, rain, snow, and hail. Microphysical interactions are computed by numerous Orville-type parameterizations. A diagnostic surface boundary layer is parameterized assuming Monin-Obukhov similarity theory. The governing equation set is approximated on a staggered three-dimensional grid with quadratic-conservative central space differencing. Time differencing is approximated by the second-order Adams-Bashforth method. The vertical grid spacing may be either linear or stretched. The model domain may translate along with a convective cell, even at variable speeds.

The development of a three-dimensional partially elliptic flow computer program for combustor research

NASA Technical Reports Server (NTRS)

Pan, Y. S.

1978-01-01

A three dimensional, partially elliptic, computer program was developed. Without requiring three dimensional computer storage locations for all flow variables, the partially elliptic program is capable of predicting three dimensional combustor flow fields with large downstream effects. The program requires only slight increase of computer storage over the parabolic flow program from which it was developed. A finite difference formulation for a three dimensional, fully elliptic, turbulent, reacting, flow field was derived. Because of the negligible diffusion effects in the main flow direction in a supersonic combustor, the set of finite-difference equations can be reduced to a partially elliptic form. Only the pressure field was governed by an elliptic equation and requires three dimensional storage; all other dependent variables are governed by parabolic equations. A numerical procedure which combines a marching integration scheme with an iterative scheme for solving the elliptic pressure was adopted.

Lagrangian equations of motion of particles and photons in a Schwarzschild field

NASA Astrophysics Data System (ADS)

Ritus, V. I.

2015-11-01

The equations of motion of a particle in the gravitational field of a black hole are considered in a formulation that uses generalized coordinates, velocities, and accelerations and is convenient for finding the integrals of motion. The equations are rewritten in terms of the physical velocities and accelerations measured in the Schwarzschild frame by a stationary observer using proper local length and time standards. The attractive force due to the field and the centripetal acceleration of a particle is proportional to the particle kinetic energy m/\\sqrt{1 - v^2}, consistently with the fact that the particle kinetic energy and the photon energy \\hbarω in the field increase by the same factor compared with their values without a field. The attraction exerted on particles and photons by a gravitational field source is proportional to their kinetic energies. The particle trajectory in the ultrarelativistic limit v \\to 1 coincides with the photon trajectory.

Element Library for Three-Dimensional Stress Analysis by the Integrated Force Method

NASA Technical Reports Server (NTRS)

Kaljevic, Igor; Patnaik, Surya N.; Hopkins, Dale A.

1996-01-01

The Integrated Force Method, a recently developed method for analyzing structures, is extended in this paper to three-dimensional structural analysis. First, a general formulation is developed to generate the stress interpolation matrix in terms of complete polynomials of the required order. The formulation is based on definitions of the stress tensor components in term of stress functions. The stress functions are written as complete polynomials and substituted into expressions for stress components. Then elimination of the dependent coefficients leaves the stress components expressed as complete polynomials whose coefficients are defined as generalized independent forces. Such derived components of the stress tensor identically satisfy homogenous Navier equations of equilibrium. The resulting element matrices are invariant with respect to coordinate transformation and are free of spurious zero-energy modes. The formulation provides a rational way to calculate the exact number of independent forces necessary to arrive at an approximation of the required order for complete polynomials. The influence of reducing the number of independent forces on the accuracy of the response is also analyzed. The stress fields derived are used to develop a comprehensive finite element library for three-dimensional structural analysis by the Integrated Force Method. Both tetrahedral- and hexahedral-shaped elements capable of modeling arbitrary geometric configurations are developed. A number of examples with known analytical solutions are solved by using the developments presented herein. The results are in good agreement with the analytical solutions. The responses obtained with the Integrated Force Method are also compared with those generated by the standard displacement method. In most cases, the performance of the Integrated Force Method is better overall.

A geometrically exact formulation for three-dimensional numerical simulation of the umbilical cable in a deep-sea ROV system

NASA Astrophysics Data System (ADS)

Quan, Wei-cai; Zhang, Zhu-ying; Zhang, Ai-qun; Zhang, Qi-feng; Tian, Yu

2015-04-01

This paper proposes a geometrically exact formulation for three-dimensional static and dynamic analyses of the umbilical cable in a deep-sea remotely operated vehicle (ROV) system. The presented formulation takes account of the geometric nonlinearities of large displacement, effects of axial load and bending stiffness for modeling of slack cables. The resulting nonlinear second-order governing equations are discretized spatially by the finite element method and solved temporally by the generalized- α implicit time integration algorithm, which is adapted to the case of varying coefficient matrices. The ability to consider three-dimensional union action of ocean current and ship heave motion upon the umbilical cable is the key feature of this analysis. The presented formulation is firstly validated, and then three numerical examples for the umbilical cable in a deep-sea ROV system are demonstrated and discussed, including the steady configurations only under the action of depth-dependent ocean current, the dynamic responses in the case of the only ship heave motion, and in the case of the combined action of the ship heave motion and ocean current.

Modeling of fatigue crack induced nonlinear ultrasonics using a highly parallelized explicit local interaction simulation approach

NASA Astrophysics Data System (ADS)

Shen, Yanfeng; Cesnik, Carlos E. S.

2016-04-01

This paper presents a parallelized modeling technique for the efficient simulation of nonlinear ultrasonics introduced by the wave interaction with fatigue cracks. The elastodynamic wave equations with contact effects are formulated using an explicit Local Interaction Simulation Approach (LISA). The LISA formulation is extended to capture the contact-impact phenomena during the wave damage interaction based on the penalty method. A Coulomb friction model is integrated into the computation procedure to capture the stick-slip contact shear motion. The LISA procedure is coded using the Compute Unified Device Architecture (CUDA), which enables the highly parallelized supercomputing on powerful graphic cards. Both the explicit contact formulation and the parallel feature facilitates LISA's superb computational efficiency over the conventional finite element method (FEM). The theoretical formulations based on the penalty method is introduced and a guideline for the proper choice of the contact stiffness is given. The convergence behavior of the solution under various contact stiffness values is examined. A numerical benchmark problem is used to investigate the new LISA formulation and results are compared with a conventional contact finite element solution. Various nonlinear ultrasonic phenomena are successfully captured using this contact LISA formulation, including the generation of nonlinear higher harmonic responses. Nonlinear mode conversion of guided waves at fatigue cracks is also studied.

Advanced Concepts in Quantum Mechanics

NASA Astrophysics Data System (ADS)

Esposito, Giampiero; Marmo, Giuseppe; Miele, Gennaro; Sudarshan, George

2014-11-01

Preface; 1. Introduction: the need for a quantum theory; 2. Experimental foundations of quantum theory; 3. Waves and particles; 4. Schrödinger picture, Heisenberg picture and probabilistic aspects; 5. Integrating the equations of motion; 6. Elementary applications: 1-dimensional problems; 7. Elementary applications: multidimensional problems; 8. Coherent states and related formalism; 9. Introduction to spin; 10. Symmetries in quantum mechanics; 11. Approximation methods; 12. Modern pictures of quantum mechanics; 13. Formulations of quantum mechanics and their physical implications; 14. Exam problems; Glossary of geometric concepts; References; Index.

Time dependence of breakdown in a global fiber-bundle model with continuous damage.

PubMed

Moral, L; Moreno, Y; Gómez, J B; Pacheco, A F

2001-06-01

A time-dependent global fiber-bundle model of fracture with continuous damage is formulated in terms of a set of coupled nonlinear differential equations. A first integral of this set is analytically obtained. The time evolution of the system is studied by applying a discrete probabilistic method. Several results are discussed emphasizing their differences with the standard time-dependent model. The results obtained show that with this simple model a variety of experimental observations can be qualitatively reproduced.

Artificial immune system for effective properties optimization of magnetoelectric composites

NASA Astrophysics Data System (ADS)

Poteralski, Arkadiusz; Dziatkiewicz, Grzegorz

2018-01-01

The optimization problem of the effective properties for magnetoelectric composites is considered. The effective properties are determined by the semi-analytical Mori-Tanaka approach. The generalized Eshelby tensor components are calculated numerically by using the Gauss quadrature method for the integral representation of the inclusion problem. The linear magnetoelectric constitutive equation is used. The effect of orientation of the electromagnetic materials components is taken into account. The optimization problem of the design is formulated and the artificial immune system is applied to solve it.

Correlation of nonorthogonality of best hybrid bond orbitals with bond strength of orthogonal orbitals

PubMed Central

Pauling, Linus

1976-01-01

An expression is derived for the bond length of two spd orbitals with maximum values in two directions forming a given bond angle by consideration of the nonorthogonality integral of two best orbitals in these directions. This equation is equivalent to the expression derived by formulating the pair of orthogonal orbitals. Similar expressions are derived for spdf orbitals. Applications are made to icosahedral and cuboctahedral bonds and to the packing of nucleons in atomic nuclei. PMID:16578736

Correlation of nonorthogonality of best hybrid bond orbitals with bond strength of orthogonal orbitals.

PubMed

Pauling, L

1976-02-01

An expression is derived for the bond length of two spd orbitals with maximum values in two directions forming a given bond angle by consideration of the nonorthogonality integral of two best orbitals in these directions. This equation is equivalent to the expression derived by formulating the pair of orthogonal orbitals. Similar expressions are derived for spdf orbitals. Applications are made to icosahedral and cuboctahedral bonds and to the packing of nucleons in atomic nuclei.

The elasticity problem for a thick-walled cylinder containing a circumferential crack

NASA Technical Reports Server (NTRS)

Nied, H. F.; Erdogan, F.

1983-01-01

The elasticity problem for a long hollow circular cylinder containing an axisymmetric circumferential crack subjected to general nonaxisymmetric external loads is considered. The problem is formulated in terms of a system of singular integral equations with the Fourier coefficients of the derivative of the crack surface displacement as density functions. The stress intensity factors and the crack opening displacement are calculated for a cylinder under uniform tension, bending by end couples, and self-equilibrating residual stresses.

Quantum field theory in the presence of a medium: Green's function expansions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kheirandish, Fardin; Salimi, Shahriar

2011-12-15

Starting from a Lagrangian and using functional-integration techniques, series expansions of Green's function of a real scalar field and electromagnetic field, in the presence of a medium, are obtained. The parameter of expansion in these series is the susceptibility function of the medium. Relativistic and nonrelativistic Langevin-type equations are derived. Series expansions for Lifshitz energy in finite temperature and for an arbitrary matter distribution are derived. Covariant formulations for both scalar and electromagnetic fields are introduced. Two illustrative examples are given.

The elasticity problem for a thick-walled cylinder containing a circumferential crack

NASA Technical Reports Server (NTRS)

Nied, H. F.; Erdogan, F.

1982-01-01

The elasticity problem for a long hollow circular cylinder containing an axisymmetric circumferential crack subjected to general nonaxisymmetric external loads is considered. The problem is formulated in terms of a system of singular integral equations with the Fourier coefficients of the derivative of the crack surface displacement as density functions. The stress intensity factors and the crack opening displacement are calculated for a cylinder under uniform tension, bending by end couples, and self-equilibrating residual stresses.

The effects of anisotropy on the nonlinear behavior of bridged cracks in long strips

NASA Technical Reports Server (NTRS)

Ballarini, R.; Luo, H. A.

1994-01-01

A model which can be used to predict the two-dimensional nonlinear behavior of bridged cracks in orthotropic strips is presented. The results obtained using a singular integral equation formulation which incorporates the anisotropy rigorously show that, although the effects of anisotropy are significant, the nondimensional quantities employed by Cox and Marshall can generate nearly universal results (R-curves, for example) for different levels of relative anisotropy. The role of composite constituent properties in the behavior of bridged cracks is clarified.

DISCHARGE AND DEPTH BEHIND A PARTIALLY BREACHED DAM.

USGS Publications Warehouse

Chen, Cheng-lung

1987-01-01

The role that the velocity-distribution correction factor plays in the determination of the flood discharge and corresponding flow depth behind a partially breached dam is investigated. Assumption of a uniformly progressive flow for an established dam-break flood in a rectangular channel of infinite extent leads to the formulation of a theoretical relation between the depth and velocity of flow expressed in differential form. Integrating this ordinary differential equation, one can express the velocity in terms of the depth.

Spheroidal Integral Equations for Geodetic Inversion of Geopotential Gradients

NASA Astrophysics Data System (ADS)

Novák, Pavel; Šprlák, Michal

2018-03-01

The static Earth's gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth's surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet's boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green's kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth.

A Curved, Elastostatic Boundary Element for Plane Anisotropic Structures

NASA Technical Reports Server (NTRS)

Smeltzer, Stanley S.; Klang, Eric C.

2001-01-01

The plane-stress equations of linear elasticity are used in conjunction with those of the boundary element method to develop a novel curved, quadratic boundary element applicable to structures composed of anisotropic materials in a state of plane stress or plane strain. The curved boundary element is developed to solve two-dimensional, elastostatic problems of arbitrary shape, connectivity, and material type. As a result of the anisotropy, complex variables are employed in the fundamental solution derivations for a concentrated unit-magnitude force in an infinite elastic anisotropic medium. Once known, the fundamental solutions are evaluated numerically by using the known displacement and traction boundary values in an integral formulation with Gaussian quadrature. All the integral equations of the boundary element method are evaluated using one of two methods: either regular Gaussian quadrature or a combination of regular and logarithmic Gaussian quadrature. The regular Gaussian quadrature is used to evaluate most of the integrals along the boundary, and the combined scheme is employed for integrals that are singular. Individual element contributions are assembled into the global matrices of the standard boundary element method, manipulated to form a system of linear equations, and the resulting system is solved. The interior displacements and stresses are found through a separate set of auxiliary equations that are derived using an Airy-type stress function in terms of complex variables. The capabilities and accuracy of this method are demonstrated for a laminated-composite plate with a central, elliptical cutout that is subjected to uniform tension along one of the straight edges of the plate. Comparison of the boundary element results for this problem with corresponding results from an analytical model show a difference of less than 1%.

Entrainment of bed material by Earth-surface mass flows: review and reformulation of depth-integrated theory

USGS Publications Warehouse

Iverson, Richard M.; Chaojun Ouyang,

2015-01-01

Earth-surface mass flows such as debris flows, rock avalanches, and dam-break floods can grow greatly in size and destructive potential by entraining bed material they encounter. Increasing use of depth-integrated mass- and momentum-conservation equations to model these erosive flows motivates a review of the underlying theory. Our review indicates that many existing models apply depth-integrated conservation principles incorrectly, leading to spurious inferences about the role of mass and momentum exchanges at flow-bed boundaries. Model discrepancies can be rectified by analyzing conservation of mass and momentum in a two-layer system consisting of a moving upper layer and static lower layer. Our analysis shows that erosion or deposition rates at the interface between layers must in general satisfy three jump conditions. These conditions impose constraints on valid erosion formulas, and they help determine the correct forms of depth-integrated conservation equations. Two of the three jump conditions are closely analogous to Rankine-Hugoniot conditions that describe the behavior of shocks in compressible gasses, and the third jump condition describes shear traction discontinuities that necessarily exist across eroding boundaries. Grain-fluid mixtures commonly behave as compressible materials as they undergo entrainment, because changes in bulk density occur as the mixtures mobilize and merge with an overriding flow. If no bulk density change occurs, then only the shear-traction jump condition applies. Even for this special case, however, accurate formulation of depth-integrated momentum equations requires a clear distinction between boundary shear tractions that exist in the presence or absence of bed erosion.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Astrophysics Data System (ADS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30 degree inclination angle.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Technical Reports Server (NTRS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

1981-01-01

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30 degree inclination angle.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Astrophysics Data System (ADS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

1981-03-01

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center-line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30-deg inclination angle.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Technical Reports Server (NTRS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

1981-01-01

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center-line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30-deg inclination angle.

Galileon string measure and other modified measure extended objects

NASA Astrophysics Data System (ADS)

Vulfs, T. O.; Guendelman, E. I.

2017-12-01

We show that it is possible to formulate string theory as a “Galileon string theory”. The Galileon field χ enters in the definition of the integration measure in the action. Following the methods of the modified measure string theory, we find that the final equations are again those of the sigma-model. Moreover, the string tension appears again as an additional dynamical degree of freedom. At the same time, the theory satisfies all requirements of the Galileon higher derivative theory at the action level while the equations of motion are still of the second-order. A Galileon symmetry is displayed explicitly in the conformal string worldsheet frame. Also, we define the Galileon gauge transformations. Generalizations to branes with other modified measures are discussed.

Derivation of Formulations 1 and 1A of Farassat

NASA Technical Reports Server (NTRS)

Farassat, F.

2007-01-01

Formulations 1 and 1A are the solutions of the Ffowcs Williams-Hawkings (FW-H) equation with surface sources only when the surface moves at subsonic speed. Both formulations have been successfully used for helicopter rotor and propeller noise prediction for many years although we now recommend using Formulation 1A for this purpose. Formulation 1 has an observer time derivative that is taken numerically, and thus, increasing execution time on a computer and reducing the accuracy of the results. After some discussion of the Green's function of the wave equation, we derive Formulation 1 which is the basis of deriving Formulation 1A. We will then show how to take this observer time derivative analytically to get Formulation 1A. We give here the most detailed derivation of these formulations. Once you see the whole derivation, you will ask yourself why you did not do it yourself!

Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics

NASA Technical Reports Server (NTRS)

Farassat, F.

1994-01-01

Generalized functions have many applications in science and engineering. One useful aspect is that discontinuous functions can be handled as easily as continuous or differentiable functions and provide a powerful tool in formulating and solving many problems of aerodynamics and acoustics. Furthermore, generalized function theory elucidates and unifies many ad hoc mathematical approaches used by engineers and scientists. We define generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support, then introduce the concept of generalized differentiation. Generalized differentiation is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some results of classical analysis, are derived with the generalized function theory. Other applications of the generalized function theory in aerodynamics discussed here are the derivations of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of the finite part of divergent integrals, the derivation of the Oswatitsch integral equation of transonic flow, and the analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics include the derivation of the Kirchhoff formula for moving surfaces, the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Numerical Modeling of Saturated Boiling in a Heated Tube

NASA Technical Reports Server (NTRS)

Majumdar, Alok; LeClair, Andre; Hartwig, Jason

2017-01-01

This paper describes a mathematical formulation and numerical solution of boiling in a heated tube. The mathematical formulation involves a discretization of the tube into a flow network consisting of fluid nodes and branches and a thermal network consisting of solid nodes and conductors. In the fluid network, the mass, momentum and energy conservation equations are solved and in the thermal network, the energy conservation equation of solids is solved. A pressure-based, finite-volume formulation has been used to solve the equations in the fluid network. The system of equations is solved by a hybrid numerical scheme which solves the mass and momentum conservation equations by a simultaneous Newton-Raphson method and the energy conservation equation by a successive substitution method. The fluid network and thermal network are coupled through heat transfer between the solid and fluid nodes which is computed by Chen's correlation of saturated boiling heat transfer. The computer model is developed using the Generalized Fluid System Simulation Program and the numerical predictions are compared with test data.

Flow past a rotating cylinder

NASA Astrophysics Data System (ADS)

Mittal, Sanjay; Kumar, Bhaskar

2003-02-01

Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, [alpha] (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for [alpha] < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < [alpha] < 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of [alpha] a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

Fluctuating volume-current formulation of electromagnetic fluctuations in inhomogeneous media: Incandescence and luminescence in arbitrary geometries

NASA Astrophysics Data System (ADS)

Polimeridis, Athanasios G.; Reid, M. T. H.; Jin, Weiliang; Johnson, Steven G.; White, Jacob K.; Rodriguez, Alejandro W.

2015-10-01

We describe a fluctuating volume-current formulation of electromagnetic fluctuations that extends our recent work on heat exchange and Casimir interactions between arbitrarily shaped homogeneous bodies [A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, Phys. Rev. B 88, 054305 (2013), 10.1103/PhysRevB.88.054305] to situations involving incandescence and luminescence problems, including thermal radiation, heat transfer, Casimir forces, spontaneous emission, fluorescence, and Raman scattering, in inhomogeneous media. Unlike previous scattering formulations based on field and/or surface unknowns, our work exploits powerful techniques from the volume-integral equation (VIE) method, in which electromagnetic scattering is described in terms of volumetric, current unknowns throughout the bodies. The resulting trace formulas (boxed equations) involve products of well-studied VIE matrices and describe power and momentum transfer between objects with spatially varying material properties and fluctuation characteristics. We demonstrate that thanks to the low-rank properties of the associated matrices, these formulas are susceptible to fast-trace computations based on iterative methods, making practical calculations tractable. We apply our techniques to study thermal radiation, heat transfer, and fluorescence in complicated geometries, checking our method against established techniques best suited for homogeneous bodies as well as applying it to obtain predictions of radiation from complex bodies with spatially varying permittivities and/or temperature profiles.

Plastic zone size and crack tip opening displacement of a Dugdale crack interacting with a coated circular inclusion

NASA Astrophysics Data System (ADS)

Hoh, H. J.; Xiao, Z. M.; Luo, J.

2010-09-01

An analytical investigation on the plastic zone size of a crack near a coated circular inclusion under three different loading conditions of uniaxial tension, uniform tension and pure shear was carried out. Both the crack and coated circular inclusion are embedded in an infinite matrix, with the crack oriented along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. With the Dugdale model of small-scale yielding [J. Mech. Phys. Solids 8 (1960) p. 100], two thin strips of yielded plastic zones are introduced at both crack tips. Using the solution for a coated circular inclusion interacting with a single dislocation as the Green's function, the physical problem is formulated into a set of singular integral equations. Using the method of Erdogan and Gupta [Q. J. Appl. Math. 29 (1972) p. 525] and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement.

Construction of optimum controls and trajectories of motion of the center of masses of a spacecraft equipped with the solar sail and low-thrust engine, using quaternions and Kustaanheimo-Stiefel variables

NASA Astrophysics Data System (ADS)

Sapunkov, Ya. G.; Chelnokov, Yu. N.

2014-11-01

The problem of optimum rendezvous of a controllable spacecraft (SC) with an uncontrollable spacecraft, moving over a Keplerian elliptic orbit in the gravitational field of the Sun, is considered. Control of the SC is performed using a solar sail and low-thrust engine. For solving the problem, the regular quaternion equations of the two-body problem with the Kustaanheimo-Stiefel variables and the Pontryagin maximum principle are used. The combined integral quality functional, which characterizes energy consumption for controllable SC transition from an initial to final state and the time spent for this transition, is used as a minimized functional. The differential boundary-value optimization problems are formulated, and their first integrals are found. Examples of numerical solution of problems are presented. The paper develops the application [1-6] of quaternion regular equations with the Kustaanheimo-Stiefel variables in the space flight mechanics.

A statistical approach to the brittle fracture of a multi-phase solid

NASA Technical Reports Server (NTRS)

Liu, W. K.; Lua, Y. I.; Belytschko, T.

1991-01-01

A stochastic damage model is proposed to quantify the inherent statistical distribution of the fracture toughness of a brittle, multi-phase solid. The model, based on the macrocrack-microcrack interaction, incorporates uncertainties in locations and orientations of microcracks. Due to the high concentration of microcracks near the macro-tip, a higher order analysis based on traction boundary integral equations is formulated first for an arbitrary array of cracks. The effects of uncertainties in locations and orientations of microcracks at a macro-tip are analyzed quantitatively by using the boundary integral equations method in conjunction with the computer simulation of the random microcrack array. The short range interactions resulting from surrounding microcracks closet to the main crack tip are investigated. The effects of microcrack density parameter are also explored in the present study. The validity of the present model is demonstrated by comparing its statistical output with the Neville distribution function, which gives correct fits to sets of experimental data from multi-phase solids.

Multi-scale diffuse interface modeling of multi-component two-phase flow with partial miscibility

NASA Astrophysics Data System (ADS)

Kou, Jisheng; Sun, Shuyu

2016-08-01

In this paper, we introduce a diffuse interface model to simulate multi-component two-phase flow with partial miscibility based on a realistic equation of state (e.g. Peng-Robinson equation of state). Because of partial miscibility, thermodynamic relations are used to model not only interfacial properties but also bulk properties, including density, composition, pressure, and realistic viscosity. As far as we know, this effort is the first time to use diffuse interface modeling based on equation of state for modeling of multi-component two-phase flow with partial miscibility. In numerical simulation, the key issue is to resolve the high contrast of scales from the microscopic interface composition to macroscale bulk fluid motion since the interface has a nanoscale thickness only. To efficiently solve this challenging problem, we develop a multi-scale simulation method. At the microscopic scale, we deduce a reduced interfacial equation under reasonable assumptions, and then we propose a formulation of capillary pressure, which is consistent with macroscale flow equations. Moreover, we show that Young-Laplace equation is an approximation of this capillarity formulation, and this formulation is also consistent with the concept of Tolman length, which is a correction of Young-Laplace equation. At the macroscopical scale, the interfaces are treated as discontinuous surfaces separating two phases of fluids. Our approach differs from conventional sharp-interface two-phase flow model in that we use the capillary pressure directly instead of a combination of surface tension and Young-Laplace equation because capillarity can be calculated from our proposed capillarity formulation. A compatible condition is also derived for the pressure in flow equations. Furthermore, based on the proposed capillarity formulation, we design an efficient numerical method for directly computing the capillary pressure between two fluids composed of multiple components. Finally, numerical tests are carried out to verify the effectiveness of the proposed multi-scale method.

Multi-scale diffuse interface modeling of multi-component two-phase flow with partial miscibility

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kou, Jisheng; Sun, Shuyu, E-mail: shuyu.sun@kaust.edu.sa; School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049

2016-08-01

In this paper, we introduce a diffuse interface model to simulate multi-component two-phase flow with partial miscibility based on a realistic equation of state (e.g. Peng–Robinson equation of state). Because of partial miscibility, thermodynamic relations are used to model not only interfacial properties but also bulk properties, including density, composition, pressure, and realistic viscosity. As far as we know, this effort is the first time to use diffuse interface modeling based on equation of state for modeling of multi-component two-phase flow with partial miscibility. In numerical simulation, the key issue is to resolve the high contrast of scales from themore » microscopic interface composition to macroscale bulk fluid motion since the interface has a nanoscale thickness only. To efficiently solve this challenging problem, we develop a multi-scale simulation method. At the microscopic scale, we deduce a reduced interfacial equation under reasonable assumptions, and then we propose a formulation of capillary pressure, which is consistent with macroscale flow equations. Moreover, we show that Young–Laplace equation is an approximation of this capillarity formulation, and this formulation is also consistent with the concept of Tolman length, which is a correction of Young–Laplace equation. At the macroscopical scale, the interfaces are treated as discontinuous surfaces separating two phases of fluids. Our approach differs from conventional sharp-interface two-phase flow model in that we use the capillary pressure directly instead of a combination of surface tension and Young–Laplace equation because capillarity can be calculated from our proposed capillarity formulation. A compatible condition is also derived for the pressure in flow equations. Furthermore, based on the proposed capillarity formulation, we design an efficient numerical method for directly computing the capillary pressure between two fluids composed of multiple components. Finally, numerical tests are carried out to verify the effectiveness of the proposed multi-scale method.« less

Drekar v.2.0

DOE Office of Scientific and Technical Information (OSTI.GOV)

Seefeldt, Ben; Sondak, David; Hensinger, David M.

Drekar is an application code that solves partial differential equations for fluids that can be optionally coupled to electromagnetics. Drekar solves low-mach compressible and incompressible computational fluid dynamics (CFD), compressible and incompressible resistive magnetohydrodynamics (MHD), and multiple species plasmas interacting with electromagnetic fields. Drekar discretization technology includes continuous and discontinuous finite element formulations, stabilized finite element formulations, mixed integration finite element bases (nodal, edge, face, volume) and an initial arbitrary Lagrangian Eulerian (ALE) capability. Drekar contains the implementation of the discretized physics and leverages the open source Trilinos project for both parallel solver capabilities and general finite element discretization tools.more » The code will be released open source under a BSD license. The code is used for fundamental research for simulation of fluids and plasmas on high performance computing environments.« less

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

DOE PAGES

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul; ...

2017-12-20

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

The Swift-Hohenberg equation with a nonlocal nonlinearity

NASA Astrophysics Data System (ADS)

Morgan, David; Dawes, Jonathan H. P.

2014-03-01

It is well known that aspects of the formation of localised states in a one-dimensional Swift-Hohenberg equation can be described by Ginzburg-Landau-type envelope equations. This paper extends these multiple scales analyses to cases where an additional nonlinear integral term, in the form of a convolution, is present. The presence of a kernel function introduces a new lengthscale into the problem, and this results in additional complexity in both the derivation of envelope equations and in the bifurcation structure. When the kernel is short-range, weakly nonlinear analysis results in envelope equations of standard type but whose coefficients are modified in complicated ways by the nonlinear nonlocal term. Nevertheless, these computations can be formulated quite generally in terms of properties of the Fourier transform of the kernel function. When the lengthscale associated with the kernel is longer, our method leads naturally to the derivation of two different, novel, envelope equations that describe aspects of the dynamics in these new regimes. The first of these contains additional bifurcations, and unexpected loops in the bifurcation diagram. The second of these captures the stretched-out nature of the homoclinic snaking curves that arises due to the nonlocal term.

Formulation of boundary conditions for the multigrid acceleration of the Euler and Navier Stokes equations

NASA Technical Reports Server (NTRS)

Jentink, Thomas Neil; Usab, William J., Jr.

1990-01-01

An explicit, Multigrid algorithm was written to solve the Euler and Navier-Stokes equations with special consideration given to the coarse mesh boundary conditions. These are formulated in a manner consistent with the interior solution, utilizing forcing terms to prevent coarse-mesh truncation error from affecting the fine-mesh solution. A 4-Stage Hybrid Runge-Kutta Scheme is used to advance the solution in time, and Multigrid convergence is further enhanced by using local time-stepping and implicit residual smoothing. Details of the algorithm are presented along with a description of Jameson's standard Multigrid method and a new approach to formulating the Multigrid equations.

Wavelet-like bases for thin-wire integral equations in electromagnetics

NASA Astrophysics Data System (ADS)

Francomano, E.; Tortorici, A.; Toscano, E.; Ala, G.; Viola, F.

2005-03-01

In this paper, wavelets are used in solving, by the method of moments, a modified version of the thin-wire electric field integral equation, in frequency domain. The time domain electromagnetic quantities, are obtained by using the inverse discrete fast Fourier transform. The retarded scalar electric and vector magnetic potentials are employed in order to obtain the integral formulation. The discretized model generated by applying the direct method of moments via point-matching procedure, results in a linear system with a dense matrix which have to be solved for each frequency of the Fourier spectrum of the time domain impressed source. Therefore, orthogonal wavelet-like basis transform is used to sparsify the moment matrix. In particular, dyadic and M-band wavelet transforms have been adopted, so generating different sparse matrix structures. This leads to an efficient solution in solving the resulting sparse matrix equation. Moreover, a wavelet preconditioner is used to accelerate the convergence rate of the iterative solver employed. These numerical features are used in analyzing the transient behavior of a lightning protection system. In particular, the transient performance of the earth termination system of a lightning protection system or of the earth electrode of an electric power substation, during its operation is focused. The numerical results, obtained by running a complex structure, are discussed and the features of the used method are underlined.

The Reduction of Ducted Fan Engine Noise Via A Boundary Integral Equation Method

NASA Technical Reports Server (NTRS)

Tweed, J.; Dunn, M.

1997-01-01

The development of a Boundary Integral Equation Method (BIEM) for the prediction of ducted fan engine noise is discussed. The method is motivated by the need for an efficient and versatile computational tool to assist in parametric noise reduction studies. In this research, the work in reference 1 was extended to include passive noise control treatment on the duct interior. The BEM considers the scattering of incident sound generated by spinning point thrust dipoles in a uniform flow field by a thin cylindrical duct. The acoustic field is written as a superposition of spinning modes. Modal coefficients of acoustic pressure are calculated term by term. The BEM theoretical framework is based on Helmholtz potential theory. A boundary value problem is converted to a boundary integral equation formulation with unknown single and double layer densities on the duct wall. After solving for the unknown densities, the acoustic field is easily calculated. The main feature of the BIEM is the ability to compute any portion of the sound field without the need to compute the entire field. Other noise prediction methods such as CFD and Finite Element methods lack this property. Additional BIEM attributes include versatility, ease of use, rapid noise predictions, coupling of propagation and radiation both forward and aft, implementable on midrange personal computers, and valid over a wide range of frequencies.

Finite element-integral simulation of static and flight fan noise radiation from the JT15D turbofan engine

NASA Technical Reports Server (NTRS)

Baumeister, K. J.; Horowitz, S. J.

1982-01-01

An iterative finite element integral technique is used to predict the sound field radiated from the JT15D turbofan inlet. The sound field is divided into two regions: the sound field within and near the inlet which is computed using the finite element method and the radiation field beyond the inlet which is calculated using an integral solution technique. The velocity potential formulation of the acoustic wave equation was employed in the program. For some single mode JT15D data, the theory and experiment are in good agreement for the far field radiation pattern as well as suppressor attenuation. Also, the computer program is used to simulate flight effects that cannot be performed on a ground static test stand.

Shape and Stress Sensing of Multilayered Composite and Sandwich Structures Using an Inverse Finite Element Method

NASA Technical Reports Server (NTRS)

Cerracchio, Priscilla; Gherlone, Marco; Di Sciuva, Marco; Tessler, Alexander

2013-01-01

The marked increase in the use of composite and sandwich material systems in aerospace, civil, and marine structures leads to the need for integrated Structural Health Management systems. A key capability to enable such systems is the real-time reconstruction of structural deformations, stresses, and failure criteria that are inferred from in-situ, discrete-location strain measurements. This technology is commonly referred to as shape- and stress-sensing. Presented herein is a computationally efficient shape- and stress-sensing methodology that is ideally suited for applications to laminated composite and sandwich structures. The new approach employs the inverse Finite Element Method (iFEM) as a general framework and the Refined Zigzag Theory (RZT) as the underlying plate theory. A three-node inverse plate finite element is formulated. The element formulation enables robust and efficient modeling of plate structures instrumented with strain sensors that have arbitrary positions. The methodology leads to a set of linear algebraic equations that are solved efficiently for the unknown nodal displacements. These displacements are then used at the finite element level to compute full-field strains, stresses, and failure criteria that are in turn used to assess structural integrity. Numerical results for multilayered, highly heterogeneous laminates demonstrate the unique capability of this new formulation for shape- and stress-sensing.

Metriplectic Gyrokinetics and Discretization Methods for the Landau Collision Integral

NASA Astrophysics Data System (ADS)

Hirvijoki, Eero; Burby, Joshua W.; Kraus, Michael

2017-10-01

We present two important results for the kinetic theory and numerical simulation of warm plasmas: 1) We provide a metriplectic formulation of collisional electrostatic gyrokinetics that is fully consistent with the First and Second Laws of Thermodynamics. 2) We provide a metriplectic temporal and velocity-space discretization for the particle phase-space Landau collision integral that satisfies the conservation of energy, momentum, and particle densities to machine precision, as well as guarantees the existence of numerical H-theorem. The properties are demonstrated algebraically. These two result have important implications: 1) Numerical methods addressing the Vlasov-Maxwell-Landau system of equations, or its reduced gyrokinetic versions, should start from a metriplectic formulation to preserve the fundamental physical principles also at the discrete level. 2) The plasma physics community should search for a metriplectic reduction theory that would serve a similar purpose as the existing Lagrangian and Hamiltonian reduction theories do in gyrokinetics. The discovery of metriplectic formulation of collisional electrostatic gyrokinetics is strong evidence in favor of such theory and, if uncovered, the theory would be invaluable in constructing reduced plasma models. Supported by U.S. DOE Contract Nos. DE-AC02-09-CH11466 (EH) and DE-AC05-06OR23100 (JWB) and by European Union's Horizon 2020 research and innovation Grant No. 708124 (MK).

Eulerian Formulation of Spatially Constrained Elastic Rods

NASA Astrophysics Data System (ADS)

Huynen, Alexandre

Slender elastic rods are ubiquitous in nature and technology. For a vast majority of applications, the rod deflection is restricted by an external constraint and a significant part of the elastic body is in contact with a stiff constraining surface. The research work presented in this doctoral dissertation formulates a computational model for the solution of elastic rods constrained inside or around frictionless tube-like surfaces. The segmentation strategy adopted to cope with this complex class of problems consists in sequencing the global problem into, comparatively simpler, elementary problems either in continuous contact with the constraint or contact-free between their extremities. Within the conventional Lagrangian formulation of elastic rods, this approach is however associated with two major drawbacks. First, the boundary conditions specifying the locations of the rod centerline at both extremities of each elementary problem lead to the establishment of isoperimetric constraints, i.e., integral constraints on the unknown length of the rod. Second, the assessment of the unilateral contact condition requires, in principle, the comparison of two curves parametrized by distinct curvilinear coordinates, viz. the rod centerline and the constraint axis. Both conspire to burden the computations associated with the method. To streamline the solution along the elementary problems and rationalize the assessment of the unilateral contact condition, the rod governing equations are reformulated within the Eulerian framework of the constraint. The methodical exploration of both types of elementary problems leads to specific formulations of the rod governing equations that stress the profound connection between the mechanics of the rod and the geometry of the constraint surface. The proposed Eulerian reformulation, which restates the rod local equilibrium in terms of the curvilinear coordinate associated with the constraint axis, describes the rod deformed configuration by means of either its relative position with respect to the constraint axis (contact-free segments) or its angular position on the constraint surface (continuous contacts.) This formulation circumvents both drawbacks that afflict the conventional Lagrangian approach associated with the segmentation strategy. As the a priori unknown domain, viz. the rod length, is substituted for the known constraint axis, the free boundary problem and the associated isoperimetric constraints are converted into a classical two-point boundary value problem. Additionally, the description of the rod deflection by means of its eccentricity with respect to the constraint axis trivializes the assessment of the unilateral contact condition. Along continuous contacts, this formulation expresses the strain variables, measuring the rod change of shape, in terms of the geometric invariants of the constraint surface, and emphasizes the influence of the constraint local geometry on the reaction pressure. Formalizing the segmentation strategy, a computational model that exploits the Eulerian formulation of the rod governing equations is devised. To solve the quasi-static deflection of elastic rods constrained inside or around a tube-like surface, this computational model identifies the number of contacts, their nature (either discrete or continuous), and the rod configuration at the connections that satisfies the unilateral contact condition and preserves the rod integrity along the sequence of elementary problems.

An implicit boundary integral method for computing electric potential of macromolecules in solvent

NASA Astrophysics Data System (ADS)

Zhong, Yimin; Ren, Kui; Tsai, Richard

2018-04-01

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equation that arises in mathematical models for the electrostatics of molecules in solvent. The proposed method uses an implicit boundary integral formulation to derive a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separates the molecule and the solvent. The needed implicit surface is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.

Crossover from BCS to Bose superconductivity: A functional integral approach

DOE Office of Scientific and Technical Information (OSTI.GOV)

Randeria, M.; Sa de Melo, C.A.R.; Engelbrecht, J.R.

1993-04-01

We use a functional integral formulation to study the crossover from cooperative Cooper pairing to the formation and condensation of tightly bound pairs in a 3D continuum model of fermions with attractive interactions. The inadequacy of a saddle point approximation with increasing coupling is pointed out, and the importance of temporal (quantum) fluctuations for normal state properties at intermediate and strong coupling is emphasized. In addition to recovering the Nozieres-Schmitt-Pink interpolation scheme for T{sub c}, and the Leggett variational results for T = 0, we also present results for evolution of the time-dependent Ginzburg-Landau equation and collective mode spectrum asmore » a function of the coupling.« less

3D Higher Order Modeling in the BEM/FEM Hybrid Formulation

NASA Technical Reports Server (NTRS)

Fink, P. W.; Wilton, D. R.

2000-01-01

Higher order divergence- and curl-conforming bases have been shown to provide significant benefits, in both convergence rate and accuracy, in the 2D hybrid finite element/boundary element formulation (P. Fink and D. Wilton, National Radio Science Meeting, Boulder, CO, Jan. 2000). A critical issue in achieving the potential for accuracy of the approach is the accurate evaluation of all matrix elements. These involve products of high order polynomials and, in some instances, singular Green's functions. In the 2D formulation, the use of a generalized Gaussian quadrature method was found to greatly facilitate the computation and to improve the accuracy of the boundary integral equation self-terms. In this paper, a 3D, hybrid electric field formulation employing higher order bases and higher order elements is presented. The improvements in convergence rate and accuracy, compared to those resulting from lower order modeling, are established. Techniques developed to facilitate the computation of the boundary integral self-terms are also shown to improve the accuracy of these terms. Finally, simple preconditioning techniques are used in conjunction with iterative solution procedures to solve the resulting linear system efficiently. In order to handle the boundary integral singularities in the 3D formulation, the parent element- either a triangle or rectangle-is subdivided into a set of sub-triangles with a common vertex at the singularity. The contribution to the integral from each of the sub-triangles is computed using the Duffy transformation to remove the singularity. This method is shown to greatly facilitate t'pe self-term computation when the bases are of higher order. In addition, the sub-triangles can be further divided to achieve near arbitrary accuracy in the self-term computation. An efficient method for subdividing the parent element is presented. The accuracy obtained using higher order bases is compared to that obtained using lower order bases when the number of unknowns is approximately equal. Also, convergence rates obtained using higher order bases are compared to those obtained with lower order bases for selected sample

Finite Temperature Densities via the S-Function Method with Application to Electron Screening in Plasmas

NASA Astrophysics Data System (ADS)

Watrous, Mitchell James

1997-12-01

A new approach to the Green's-function method for the calculation of equilibrium densities within the finite temperature, Kohn-Sham formulation of density functional theory is presented, which extends the method to all temperatures. The contour of integration in the complex energy plane is chosen such that the density is given by a sum of Green's function differences evaluated at the Matsubara frequencies, rather than by the calculation and summation of Kohn-Sham single-particle wave functions. The Green's functions are written in terms of their spectral representation and are calculated as the solutions of their defining differential equations. These differential equations are boundary value problems as opposed to the standard eigenvalue problems. For large values of the complex energy, the differential equations are further simplified from second to first-order by writing the Green's functions in terms of logarithmic derivatives. An asymptotic expression for the Green's functions is derived, which allows the sum over Matsubara poles to be approximated. The method is applied to the screening of nuclei by electrons in finite temperature plasmas. To demonstrate the method's utility, and to illustrate its advantages, the results of previous wave function type calculations for protons and neon nuclei are reproduced. The method is also used to formulate a new screening model for fusion reactions in the solar core, and the predicted reaction rate enhancements factors are compared with existing models.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kokkoris, George; Boudouvis, Andreas G.; Gogolides, Evangelos

An integrated framework for the neutral flux calculation inside trenches and holes during plasma etching is described, and a comparison between the two types of structure in a number of applications is presented. First, a detailed and functional set of equations for the neutral and ion flux calculations inside long trenches and holes with cylindrical symmetry is explicitly formulated. This set is based on early works [T. S. Cale and G. B. Raupp, J. Vac. Sci. Technol. B 8, 1242 (1990); V. K. Singh et al., J. Vac. Sci. Technol. B 10, 1091 (1992)], and includes new equations for themore » case of holes with cylindrical symmetry. Second, a method for the solution of the respective numerical task, i.e., one or a set of linear or nonlinear integral equations, is described. This method includes a coupling algorithm with a surface chemistry model and resolves the singularity problem of the integral equations. Third, the fluxes inside trenches and holes are compared. The flux from reemission is the major portion of the local flux at the bottom of both types of structure. The framework is applied in SiO{sub 2} etching by fluorocarbon plasmas to predict the increased intensity of reactive ion etching lag in SiO{sub 2} holes compared to trenches. It is also applied in deep Si etching: By calculating the flux of F atoms at the bottom of very high aspect ratio (up to 150) Si trenches and holes during the gas chopping process, the aspect ratio at which the flux of F atoms is eliminated and etching practically stops is estimated.« less

A three-dimensional finite-volume Eulerian-Lagrangian Localized Adjoint Method (ELLAM) for solute-transport modeling

USGS Publications Warehouse

Heberton, C.I.; Russell, T.F.; Konikow, Leonard F.; Hornberger, G.Z.

2000-01-01

This report documents the U.S. Geological Survey Eulerian-Lagrangian Localized Adjoint Method (ELLAM) algorithm that solves an integral form of the solute-transport equation, incorporating an implicit-in-time difference approximation for the dispersive and sink terms. Like the algorithm in the original version of the U.S. Geological Survey MOC3D transport model, ELLAM uses a method of characteristics approach to solve the transport equation on the basis of the velocity field. The ELLAM algorithm, however, is based on an integral formulation of conservation of mass and uses appropriate numerical techniques to obtain global conservation of mass. The implicit procedure eliminates several stability criteria required for an explicit formulation. Consequently, ELLAM allows large transport time increments to be used. ELLAM can produce qualitatively good results using a small number of transport time steps. A description of the ELLAM numerical method, the data-input requirements and output options, and the results of simulator testing and evaluation are presented. The ELLAM algorithm was evaluated for the same set of problems used to test and evaluate Version 1 and Version 2 of MOC3D. These test results indicate that ELLAM offers a viable alternative to the explicit and implicit solvers in MOC3D. Its use is desirable when mass balance is imperative or a fast, qualitative model result is needed. Although accurate solutions can be generated using ELLAM, its efficiency relative to the two previously documented solution algorithms is problem dependent.

Differential renormalization-group generators for static and dynamic critical phenomena

NASA Astrophysics Data System (ADS)

Chang, T. S.; Vvedensky, D. D.; Nicoll, J. F.

1992-09-01

The derivation of differential renormalization-group (DRG) equations for applications to static and dynamic critical phenomena is reviewed. The DRG approach provides a self-contained closed-form representation of the Wilson renormalization group (RG) and should be viewed as complementary to the Callan-Symanzik equations used in field-theoretic approaches to the RG. The various forms of DRG equations are derived to illustrate the general mathematical structure of each approach and to point out the advantages and disadvantages for performing practical calculations. Otherwise, the review focuses upon the one-particle-irreducible DRG equations derived by Nicoll and Chang and by Chang, Nicoll, and Young; no attempt is made to provide a general treatise of critical phenomena. A few specific examples are included to illustrate the utility of the DRG approach: the large- n limit of the classical n-vector model (the spherical model), multi- or higher-order critical phenomena, and crit ical dynamics far from equilibrium. The large- n limit of the n-vector model is used to introduce the application of DRG equations to a well-known example, with exact solution obtained for the nonlinear trajectories, generating functions for nonlinear scaling fields, and the equation of state. Trajectory integrals and nonlinear scaling fields within the framework of ɛ-expansions are then discussed for tricritical crossover, and briefly for certain aspects of multi- or higher-order critical points, including the derivation of the Helmholtz free energy and the equation of state. The discussion then turns to critical dynamics with a development of the path integral formulation for general dynamic processes. This is followed by an application to a model far-from-equilibrium system that undergoes a phase transformation analogous to a second-order critical point, the Schlögl model for a chemical instability.

Soliton structure versus singularity analysis: Third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions

NASA Astrophysics Data System (ADS)

Fuchssteiner, Benno; Carillo, Sandra

1989-01-01

Bäcklund transformations between all known completely integrable third-order differential equations in (1 + 1)-dimensions are established and the corresponding transformations formulas for their hereditary operators and Hamiltonian formulations are exhibited. Some of these Bäcklund transformations are not injective; therefore additional non-commutative symmetry groups are found for some equations. These non-commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra A(1)1. New completely integrable third-order integro-differential equations, some depending explicitly on x, are given. These new equations give rise to nonin equation. Connections between the singularity equations (from the Painlevé analysis) and the nonlinear equations for interacting solitons are established. A common approach to singularity analysis and soliton structure is introduced. The Painlevé analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine-Gordon and nonlinear Schrödinger equation. A method to recover the Painlevé series from its constant level term is exhibit. The soliton-singularity transform is recognized to be connected to the Möbius group. This gives rise to a Darboux-like result for the spectral properties of the recursion operator. These connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons. Furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton-singularity transform. This then leads to the result that only for scaling degrees α = -1 and α = -2 the usual Painlevé analysis can be carried out. A new invariance principle, connected to kernels of differential operators is discovered. This new invariance, for example, connects the explicit solutions of the Liouville equation with the Miura transform. Simple methods are exhibited which allow to compute out of N-soliton solutions of the KdV (Bargman potentials) explicit solutions of equations like the Harry Dym equation. Certain solutions are plotted.

Free-form geometric modeling by integrating parametric and implicit PDEs.

PubMed

Du, Haixia; Qin, Hong

2007-01-01

Parametric PDE techniques, which use partial differential equations (PDEs) defined over a 2D or 3D parametric domain to model graphical objects and processes, can unify geometric attributes and functional constraints of the models. PDEs can also model implicit shapes defined by level sets of scalar intensity fields. In this paper, we present an approach that integrates parametric and implicit trivariate PDEs to define geometric solid models containing both geometric information and intensity distribution subject to flexible boundary conditions. The integrated formulation of second-order or fourth-order elliptic PDEs permits designers to manipulate PDE objects of complex geometry and/or arbitrary topology through direct sculpting and free-form modeling. We developed a PDE-based geometric modeling system for shape design and manipulation of PDE objects. The integration of implicit PDEs with parametric geometry offers more general and arbitrary shape blending and free-form modeling for objects with intensity attributes than pure geometric models.

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2006-01-01

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations

NASA Astrophysics Data System (ADS)

Husa, Sascha; Hinder, Ian; Lechner, Christiane

2006-06-01

We present a suite of Mathematica-based computer-algebra packages, termed "Kranc", which comprise a toolbox to convert certain (tensorial) systems of partial differential evolution equations to parallelized C or Fortran code for solving initial boundary value problems. Kranc can be used as a "rapid prototyping" system for physicists or mathematicians handling very complicated systems of partial differential equations, but through integration into the Cactus computational toolkit we can also produce efficient parallelized production codes. Our work is motivated by the field of numerical relativity, where Kranc is used as a research tool by the authors. In this paper we describe the design and implementation of both the Mathematica packages and the resulting code, we discuss some example applications, and provide results on the performance of an example numerical code for the Einstein equations. Program summaryTitle of program: Kranc Catalogue identifier: ADXS_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXS_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computer for which the program is designed and others on which it has been tested: General computers which run Mathematica (for code generation) and Cactus (for numerical simulations), tested under Linux Programming language used: Mathematica, C, Fortran 90 Memory required to execute with typical data: This depends on the number of variables and gridsize, the included ADM example requires 4308 KB Has the code been vectorized or parallelized: The code is parallelized based on the Cactus framework. Number of bytes in distributed program, including test data, etc.: 1 578 142 Number of lines in distributed program, including test data, etc.: 11 711 Nature of physical problem: Solution of partial differential equations in three space dimensions, which are formulated as an initial value problem. In particular, the program is geared towards handling very complex tensorial equations as they appear, e.g., in numerical relativity. The worked out examples comprise the Klein-Gordon equations, the Maxwell equations, and the ADM formulation of the Einstein equations. Method of solution: The method of numerical solution is finite differencing and method of lines time integration, the numerical code is generated through a high level Mathematica interface. Restrictions on the complexity of the program: Typical numerical relativity applications will contain up to several dozen evolution variables and thousands of source terms, Cactus applications have shown scaling up to several thousand processors and grid sizes exceeding 500 3. Typical running time: This depends on the number of variables and the grid size: the included ADM example takes approximately 100 seconds on a 1600 MHz Intel Pentium M processor. Unusual features of the program: based on Mathematica and Cactus

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution - Part I, second-order FVTD schemes

NASA Astrophysics Data System (ADS)

Balsara, Dinshaw S.; Taflove, Allen; Garain, Sudip; Montecinos, Gino

2017-11-01

While classic finite-difference time-domain (FDTD) solutions of Maxwell's equations have served the computational electrodynamics (CED) community very well, formulations based on Godunov methodology have begun to show advantages. We argue that the formulations presented so far are such that FDTD schemes and Godunov-based schemes each have their own unique advantages. However, there is currently not a single formulation that systematically integrates the strengths of both these major strains of development. While an early glimpse of such a formulation was offered in Balsara et al. [16], that paper focused on electrodynamics in plasma. Here, we present a synthesis that integrates the strengths of both FDTD and Godunov-based schemes into a robust single formulation for CED in material media. Three advances make this synthesis possible. First, from the FDTD method, we retain (but somewhat modify) a spatial staggering strategy for the primal variables. This provides a beneficial constraint preservation for the electric displacement and magnetic induction vector fields via reconstruction methods that were initially developed in some of the first author's papers for numerical magnetohydrodynamics (MHD). Second, from the Godunov method, we retain the idea of upwinding, except that this idea, too, has to be significantly modified to use the multi-dimensionally upwinded Riemann solvers developed by the first author. Third, we draw upon recent advances in arbitrary derivatives in space and time (ADER) time-stepping by the first author and his colleagues. We use the ADER predictor step to endow our method with sub-cell resolving capabilities so that the method can be stiffly stable and resolve significant sub-cell variation in the material properties within a zone. Overall, in this paper, we report a new scheme for numerically solving Maxwell's equations in material media, with special attention paid to a second-order-accurate formulation. Several numerical examples are presented to show that the proposed technique works. Because of its sub-cell resolving ability, the new method retains second-order accuracy even when material permeability and permittivity vary by an order-of-magnitude over just one or two zones. Furthermore, because the new method is also unconditionally stable in the presence of stiff source terms (i.e., in problems involving giant conductivity variations), it can handle several orders-of-magnitude variation in material conductivity over just one or two zones without any reduction of the time-step. Consequently, the CFL depends only on the propagation speed of light in the medium being studied.

Developing semi-analytical solution for multiple-zone transient storage model with spatially non-uniform storage

NASA Astrophysics Data System (ADS)

Deng, Baoqing; Si, Yinbing; Wang, Jia

2017-12-01

Transient storages may vary along the stream due to stream hydraulic conditions and the characteristics of storage. Analytical solutions of transient storage models in literature didn't cover the spatially non-uniform storage. A novel integral transform strategy is presented that simultaneously performs integral transforms to the concentrations in the stream and in storage zones by using the single set of eigenfunctions derived from the advection-diffusion equation of the stream. The semi-analytical solution of the multiple-zone transient storage model with the spatially non-uniform storage is obtained by applying the generalized integral transform technique to all partial differential equations in the multiple-zone transient storage model. The derived semi-analytical solution is validated against the field data in literature. Good agreement between the computed data and the field data is obtained. Some illustrative examples are formulated to demonstrate the applications of the present solution. It is shown that solute transport can be greatly affected by the variation of mass exchange coefficient and the ratio of cross-sectional areas. When the ratio of cross-sectional areas is big or the mass exchange coefficient is small, more reaches are recommended to calibrate the parameter.

Implementation of an integrated op-amp based chaotic neuron model and observation of its chaotic dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jung, Jinwoo; Lee, Jewon; Song, Hanjung

2011-03-15

This paper presents a fully integrated circuit implementation of an operational amplifier (op-amp) based chaotic neuron model with a bipolar output function, experimental measurements, and analyses of its chaotic behavior. The proposed chaotic neuron model integrated circuit consists of several op-amps, sample and hold circuits, a nonlinear function block for chaotic signal generation, a clock generator, a nonlinear output function, etc. Based on the HSPICE (circuit program) simulation results, approximated empirical equations for analyses were formulated. Then, the chaotic dynamical responses such as bifurcation diagrams, time series, and Lyapunov exponent were calculated using these empirical equations. In addition, we performedmore » simulations about two chaotic neuron systems with four synapses to confirm neural network connections and got normal behavior of the chaotic neuron such as internal state bifurcation diagram according to the synaptic weight variation. The proposed circuit was fabricated using a 0.8-{mu}m single poly complementary metal-oxide semiconductor technology. Measurements of the fabricated single chaotic neuron with {+-}2.5 V power supplies and a 10 kHz sampling clock frequency were carried out and compared with the simulated results.« less

Discontinuous Finite Element Quasidiffusion Methods

DOE PAGES

Anistratov, Dmitriy Yurievich; Warsa, James S.

2018-05-21

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

Discontinuous Finite Element Quasidiffusion Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Anistratov, Dmitriy Yurievich; Warsa, James S.

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

A well-posed and stable stochastic Galerkin formulation of the incompressible Navier–Stokes equations with random data

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pettersson, Per, E-mail: per.pettersson@uib.no; Nordström, Jan, E-mail: jan.nordstrom@liu.se; Doostan, Alireza, E-mail: alireza.doostan@colorado.edu

2016-02-01

We present a well-posed stochastic Galerkin formulation of the incompressible Navier–Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimatemore » for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined.« less

Criteria for Modeling in LES of Multicomponent Fuel Flow

NASA Technical Reports Server (NTRS)

Bellan, Josette; Selle, Laurent

2009-01-01

A report presents a study addressing the question of which large-eddy simulation (LES) equations are appropriate for modeling the flow of evaporating drops of a multicomponent liquid in a gas (e.g., a spray of kerosene or diesel fuel in air). The LES equations are obtained from the direct numerical simulation (DNS) equations in which the solution is computed at all flow length scales, by applying a spatial low-pass filter. Thus, in LES the small scales are removed and replaced by terms that cannot be computed from the LES solution and instead must be modeled to retain the effect of the small scales into the equations. The mathematical form of these models is a subject of contemporary research. For a single-component liquid, there is only one LES formulation, but this study revealed that for a multicomponent liquid, there are two non-equivalent LES formulations for the conservation equations describing the composition of the vapor. Criteria were proposed for selecting the multicomponent LES formulation that gives the best accuracy and increased computational efficiency. These criteria were applied in examination of filtered DNS databases to compute the terms in the LES equations. The DNS databases are from mixing layers of diesel and kerosene fuels. The comparisons resulted in the selection of one of the multicomponent LES formulations as the most promising with respect to all criteria.

A framework for simultaneous aerodynamic design optimization in the presence of chaos

DOE Office of Scientific and Technical Information (OSTI.GOV)

Günther, Stefanie, E-mail: stefanie.guenther@scicomp.uni-kl.de; Gauger, Nicolas R.; Wang, Qiqi

Integrating existing solvers for unsteady partial differential equations into a simultaneous optimization method is challenging due to the forward-in-time information propagation of classical time-stepping methods. This paper applies the simultaneous single-step one-shot optimization method to a reformulated unsteady constraint that allows for both forward- and backward-in-time information propagation. Especially in the presence of chaotic and turbulent flow, solving the initial value problem simultaneously with the optimization problem often scales poorly with the time domain length. The new formulation relaxes the initial condition and instead solves a least squares problem for the discrete partial differential equations. This enables efficient one-shot optimizationmore » that is independent of the time domain length, even in the presence of chaos.« less

The application of the principles of invariance to the radiative transfer equation in plant canopies

NASA Technical Reports Server (NTRS)

Ganapol, B. D.; Myneni, R. B.

1992-01-01

Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satellite acquired information. The general one-dimensional two-angle transport problem for a finite copy of arbitrary leaf angle distribution is considered. Analytical solutions are obtained in terms of generalized Chandrasekhar's X- and Y-functions by invoking the principles of invariance. A critical step in the formulation involves the decomposition of the integral of the scattering phase function into a product of known functions of the incident and scattered photon directions. Several simplified cases previously considered in the literature are derived from the generalized solution. Various symmetries obeyed by the scattering operator and reciprocity relations are formally proved.

Simulations of surface winds at the Viking Lander sites using a one-level model

NASA Technical Reports Server (NTRS)

Bridger, Alison F. C.; Haberle, Robert M.

1992-01-01

The one-level model developed by Mass and Dempsey for use in predicting surface flows in regions of complex terrain was adapted to simulate surface flows at the Viking lander sites on Mars. In the one-level model, prediction equations for surface winds and temperatures are formulated and solved. Surface temperatures change with time in response to diabatic heating, horizontal advection, adiabatic heating and cooling effects, and horizontal diffusion. Surface winds can change in response to horizontal advection, pressure gradient forces, Coriolis forces, surface drag, and horizontal diffusion. Surface pressures are determined by integration of the hydrostatic equation from the surface to some reference level. The model has successfully simulated surface flows under a variety of conditions in complex-terrain regions on Earth.

Integrated surface/subsurface permafrost thermal hydrology: Model formulation and proof-of-concept simulations

DOE PAGES

Painter, Scott L.; Coon, Ethan T.; Atchley, Adam L.; ...

2016-08-11

The need to understand potential climate impacts and feedbacks in Arctic regions has prompted recent interest in modeling of permafrost dynamics in a warming climate. A new fine-scale integrated surface/subsurface thermal hydrology modeling capability is described and demonstrated in proof-of-concept simulations. The new modeling capability combines a surface energy balance model with recently developed three-dimensional subsurface thermal hydrology models and new models for nonisothermal surface water flows and snow distribution in the microtopography. Surface water flows are modeled using the diffusion wave equation extended to include energy transport and phase change of ponded water. Variation of snow depth in themore » microtopography, physically the result of wind scour, is also modeled heuristically with a diffusion wave equation. The multiple surface and subsurface processes are implemented by leveraging highly parallel community software. Fully integrated thermal hydrology simulations on the tilted open book catchment, an important test case for integrated surface/subsurface flow modeling, are presented. Fine-scale 100-year projections of the integrated permafrost thermal hydrological system on an ice wedge polygon at Barrow Alaska in a warming climate are also presented. Finally, these simulations demonstrate the feasibility of microtopography-resolving, process-rich simulations as a tool to help understand possible future evolution of the carbon-rich Arctic tundra in a warming climate.« less

Higher-order time integration of Coulomb collisions in a plasma using Langevin equations

DOE PAGES

Dimits, A. M.; Cohen, B. I.; Caflisch, R. E.; ...

2013-02-08

The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt 1/2)] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering ifmore » and only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.« less

Formulation and closure of compressible turbulence equations in the light of kinetic theory

NASA Technical Reports Server (NTRS)

Tsuge, S.; Sagara, K.

1976-01-01

Fluid-dynamic moment equations, based on a kinetic hierarchy system, are derived governing the interaction between turbulent and thermal fluctuations. The kinetic theory is shown to reduce the inherent complexity of the conventional formalism of compressible turbulence theory and to minimize arbitrariness in formulating the closure condition.

A conservative numerical scheme for modeling nonlinear acoustic propagation in thermoviscous homogeneous media

NASA Astrophysics Data System (ADS)

Diaz, Manuel A.; Solovchuk, Maxim A.; Sheu, Tony W. H.

2018-06-01

A nonlinear system of partial differential equations capable of describing the nonlinear propagation and attenuation of finite amplitude perturbations in thermoviscous media is presented. This system constitutes a full nonlinear wave model that has been formulated in the conservation form. Initially, this model is investigated analytically in the inviscid limit where it has been found that the resulting flux function fulfills the Lax-Wendroff theorem, and the scheme can match the solutions of the Westervelt and Burgers equations numerically. Here, high-order numerical descriptions of strongly nonlinear wave propagations become of great interest. For that matter we consider finite difference formulations of the weighted essentially non-oscillatory (WENO) schemes associated with explicit strong stability preserving Runge-Kutta (SSP-RK) time integration methods. Although this strategy is known to be computationally demanding, it is found to be effective when implemented to be solved in graphical processing units (GPUs). As we consider wave propagations in unbounded domains, perfectly matching layers (PML) have been also considered in this work. The proposed system model is validated and illustrated by using one- and two-dimensional benchmark test cases proposed in the literature for nonlinear acoustic propagation in homogeneous thermoviscous media.

Stress intensity factors in two bonded elastic layers containing cracks perpendicular to and on the interface. Part 1: Analysis

NASA Technical Reports Server (NTRS)

Lu, M. C.; Erdogan, F.

1980-01-01

The basic crack problem which is essential for the study of subcritical crack propagation and fracture of layered structural materials is considered. Because of the apparent analytical difficulties, the problem is idealized as one of plane strain or plane stress. An additional simplifying assumption is made by restricting the formulation of the problem to crack geometries and loading conditions which have a plane of symmetry perpendicular to the interface. The general problem is formulated in terms of a coupled system of four integral equations. For each relevant crack configuration of practical interest, the singular behavior of the solution near and at the ends and points of intersection of the cracks is investigated and the related characteristic equations are obtained. The edge crack terminating at and crossing the interface, the T-shaped crack consisting of a broken layer and a delamination crack, the cross-shaped crack which consists of a delamination crack intersecting a crack which is perpendicular to the interface, and a delamination crack initiating from a stress-free boundary of the bonded layers are some of the practical crack geometries considered.

Dynamics of Proton Spin: Role of qqq Force

NASA Astrophysics Data System (ADS)

Mitra, A. N.

The analytic structure of the qqq wave function, obtained recently in the high momentum regime of QCD, is employed for the formulation of baryonic transition amplitudes via quark loops. A new aspect of this study is the role of a direct (Y -shaped, Mercedes-Benz type) qqq force in generating the qqq wave function The dynamics is that of a Salpeter-like equation (3D support for the kernel) formulated covariantly on the light front, a la Markov-Yukawa Transversality Principle (MYTP) which warrants a 2-way interconnection between the 3D and 4D Bethe-Salpeter (BSE) forms for 2 as well as 3 fermion quarks. The dynamics of this 3-body force shows up through a characteristic singularity in the hypergeometric differential equation for the 3D wave function ϕ, corresponding to a negative eigenvalue of the spin operator iσ1·σ2 × σ3 which is an integral part of the qqq force. As a first application of this wave function to the problem of the proton spin anomaly, the two-gluon contribution to the anomaly yields an estimate of the right sign, although somewhat smaller in magnitude.

Variational approach to the volume viscosity of fluids

NASA Astrophysics Data System (ADS)

Zuckerwar, Allan J.; Ash, Robert L.

2006-04-01

The variational principle of Hamilton is applied to develop an analytical formulation to describe the volume viscosity in fluids. The procedure described here differs from those used in the past in that a dissipative process is represented by the chemical affinity and progress variable (sometimes called "order parameter") of a reacting species. These state variables appear in the variational integral in two places: first, in the expression for the internal energy, and second, in a subsidiary condition accounting for the conservation of the reacting species. As a result of the variational procedure, two dissipative terms appear in the Navier-Stokes equation. The first is the traditional volume viscosity term, proportional to the dilatational component of velocity; the second term is proportional to the material time derivative of the pressure gradient. Values of the respective volume viscosity coefficients are determined by applying the resulting volume-viscous Navier-Stokes equation to the case of acoustical propagation and then comparing expressions for the dispersion and absorption of sound. The formulation includes the special case of equilibration of the translational degrees of freedom. As examples, values are tabulated for dry and humid air, argon, and sea water.

Morphodynamic Modeling of the Lower Yellow River, China: Flux (Equilibrium) Form or Entrainment (Nonequilibrium) Form of Sediment Mass Conservation?

NASA Astrophysics Data System (ADS)

An, C.; Parker, G.; Ma, H.; Naito, K.; Moodie, A. J.; Fu, X.

2017-12-01

Models of river morphodynamics consist of three elements: (1) a treatment of flow hydraulics, (2) a formulation relating some aspect of sediment transport to flow hydraulics, and (3) a description of sediment conservation. In the case of unidirectional river flow, the Exner equation of sediment conservation is commonly described in terms of a flux-based formulation, in which bed elevation variation is related to the streamwise gradient of sediment transport rate. An alternate formulation of the Exner equation, however, is the entrainment-based formulation in which bed elevation variation is related to the difference between the entrainment rate of bed sediment into suspension and the deposition rate of suspended sediment onto the bed. In the flux-based formulation, sediment transport is regarded to be in a local equilibrium state (i.e., sediment transport rate locally equals sediment transport capacity). However, the entrainment-based formulation does not require this constraint; the sediment transport rate may lag in space and time behind the changing flow conditions. In modeling the fine-grained Lower Yellow River, it is usual to treat sediment conservation in terms of an entrainment-based (nonequilibrium) rather than a flux-based (equilibrium) formulation with the consideration that fine-grained sediment may be entrained at one place but deposited only at some distant location downstream. However, the differences in prediction between the two formulations are still not well known, and the entrainment formulation may not always be necessary for the Lower Yellow River. Here we study this problem by comparing the results of flux-based and entrainment-based morphodynamics under conditions typical of the Yellow River, using sediment transport equations specifically designed for the Lower Yellow River. We find, somewhat unexpectedly, that in a treatment of a 200-km reach using uniform sediment, there is little difference between the two formulations unless the sediment fall velocity is arbitrarily greatly reduced. A consideration of sediment mixtures, however, shows that the two formulations give very different patterns of grain sorting. We explain this in terms of the structures of the two Exner equations for sediment mixtures, and define conditions for applicability of each formulation.

Singularities in Dromo formulation. Analysis of deep flybys

NASA Astrophysics Data System (ADS)

Roa, Javier; Sanjurjo-Rivo, Manuel; Peláez, Jesús

2015-08-01

The singularities in Dromo are characterized in this paper, both from an analytical and a numerical perspective. When the angular momentum vanishes, Dromo may encounter a singularity in the evolution equations. The cancellation of the angular momentum occurs in very specific situations and may be caused by the action of strong perturbations. The gravitational attraction of a perturbing planet may lead to rapid changes in the angular momentum of the particle. In practice, this situation may be encountered during deep planetocentric flybys. The performance of Dromo is evaluated in different scenarios. First, Dromo is validated for integrating the orbit of Near Earth Asteroids. Resulting errors are of the order of the diameter of the asteroid. Second, a set of theoretical flybys are designed for analyzing the performance of the formulation in the vicinity of the singularity. New sets of Dromo variables are proposed in order to minimize the dependency of Dromo on the angular momentum. A slower time scale is introduced, leading to a more stable description of the flyby phase. Improvements in the overall performance of the algorithm are observed when integrating orbits close to the singularity.

Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions

USGS Publications Warehouse

Rubin, Jacob

1983-01-01

Examples involving six broad reaction classes show that the nature of transport-affecting chemistry may have a profound effect on the mathematical character of solute transport problem formulation. Substantive mathematical diversity among such formulations is brought about principally by reaction properties that determine whether (1) the reaction can be regarded as being controlled by local chemical equilibria or whether it must be considered as being controlled by kinetics, (2) the reaction is homogeneous or heterogeneous, (3) the reaction is a surface reaction (adsorption, ion exchange) or one of the reactions of classical chemistry (e.g., precipitation, dissolution, oxidation, reduction, complex formation). These properties, as well as the choice of means to describe them, stipulate, for instance, (1) the type of chemical entities for which a formulation's basic, mass-balance equations should be written; (2) the nature of mathematical transformations needed to change the problem's basic equations into operational ones. These and other influences determine such mathematical features of problem formulations as the nature of the operational transport-equation system (e.g., whether it involves algebraic, partial-differential, or integro-partial-differential simultaneous equations), the type of nonlinearities of such a system, and the character of the boundaries (e.g., whether they are stationary or moving). Exploration of the reasons for the dependence of transport mathematics on transport chemistry suggests that many results of this dependence stem from the basic properties of the reactions' chemical-relation (i.e., equilibrium or rate) equations.

A three-dimensional meso-macroscopic model for Li-Ion intercalation batteries

DOE PAGES

Allu, S.; Kalnaus, S.; Simunovic, S.; ...

2016-06-09

Through this study, we present a three-dimensional computational formulation for electrode-electrolyte-electrode system of Li-Ion batteries. The physical consistency between electrical, thermal and chemical equations is enforced at each time increment by driving the residual of the resulting coupled system of nonlinear equations to zero. The formulation utilizes a rigorous volume averaging approach typical of multiphase formulations used in other fields and recently extended to modeling of supercapacitors [1]. Unlike existing battery modeling methods which use segregated solution of conservation equations and idealized geometries, our unified approach can model arbitrary battery and electrode configurations. The consistency of multi-physics solution also allowsmore » for consideration of a wide array of initial conditions and load cases. The formulation accounts for spatio-temporal variations of material and state properties such as electrode/void volume fractions and anisotropic conductivities. The governing differential equations are discretized using the finite element method and solved using a nonlinearly consistent approach that provides robust stability and convergence. The new formulation was validated for standard Li-ion cells and compared against experiments. Finally, its scope and ability to capture spatio-temporal variations of potential and lithium distribution is demonstrated on a prototypical three-dimensional electrode problem.« less

SYMBOD - A computer program for the automatic generation of symbolic equations of motion for systems of hinge-connected rigid bodies

NASA Technical Reports Server (NTRS)

Macala, G. A.

1983-01-01

A computer program is described that can automatically generate symbolic equations of motion for systems of hinge-connected rigid bodies with tree topologies. The dynamical formulation underlying the program is outlined, and examples are given to show how a symbolic language is used to code the formulation. The program is applied to generate the equations of motion for a four-body model of the Galileo spacecraft. The resulting equations are shown to be a factor of three faster in execution time than conventional numerical subroutines.

The method of projected characteristics for the evolution of magnetic arches

NASA Technical Reports Server (NTRS)

Nakagawa, Y.; Hu, Y. Q.; Wu, S. T.

1987-01-01

A numerical method of solving fully nonlinear MHD equation is described. In particular, the formulation based on the newly developed method of projected characteristics (Nakagawa, 1981) suitable to study the evolution of magnetic arches due to motions of their foot-points is presented. The final formulation is given in the form of difference equations; therefore, the analysis of numerical stability is also presented. Further, the most important derivation of physically self-consistent, time-dependent boundary conditions (i.e. the evolving boundary equations) is given in detail, and some results obtained with such boundary equations are reported.

Application of perturbation theory to lattice calculations based on method of cyclic characteristics

NASA Astrophysics Data System (ADS)

Assawaroongruengchot, Monchai

Perturbation theory is a technique used for the estimation of changes in performance functionals, such as linear reaction rate ratio and eigenvalue affected by small variations in reactor core compositions. Here the algorithm of perturbation theory is developed for the multigroup integral neutron transport problems in 2D fuel assemblies with isotropic scattering. The integral transport equation is used in the perturbative formulation because it represents the interconnecting neutronic systems of the lattice assemblies via the tracking lines. When the integral neutron transport equation is used in the formulation, one needs to solve the resulting integral transport equations for the flux importance and generalized flux importance functions. The relationship between the generalized flux importance and generalized source importance functions is defined in order to transform the generalized flux importance transport equations into the integro-differential equations for the generalized adjoints. Next we develop the adjoint and generalized adjoint transport solution algorithms based on the method of cyclic characteristics (MOCC) in DRAGON code. In the MOCC method, the adjoint characteristics equations associated with a cyclic tracking line are formulated in such a way that a closed form for the adjoint angular function can be obtained. The MOCC method then requires only one cycle of scanning over the cyclic tracking lines in each spatial iteration. We also show that the source importance function by CP method is mathematically equivalent to the adjoint function by MOCC method. In order to speed up the MOCC solution algorithm, a group-reduction and group-splitting techniques based on the structure of the adjoint scattering matrix are implemented. A combined forward flux/adjoint function iteration scheme, based on the group-splitting technique and the common use of a large number of variables storing tracking-line data and exponential values, is proposed to reduce the computing time when both direct and adjoint solutions are required. A problem that arises for the generalized adjoint problem is that the direct use of the negative external generalized adjoint sources in the adjoint solution algorithm results in negative generalized adjoint functions. A coupled flux biasing/decontamination scheme is applied to make the generalized adjoint functions positive using the adjoint functions in such a way that it can be used for the multigroup rebalance technique. Next we consider the application of the perturbation theory to the reactor problems. Since the coolant void reactivity (CVR) is a important factor in reactor safety analysis, we have decided to select this parameter for optimization studies. We consider the optimization and adjoint sensitivity techniques for the adjustments of CVR at beginning of burnup cycle (BOC) and k eff at end of burnup cycle (EOC) for a 2D Advanced CANDU Reactor (ACR) lattice. The sensitivity coefficients are evaluated using the perturbation theory based on the integral transport equations. Three sets of parameters for CVR-BOC and keff-EOC adjustments are studied: (1) Dysprosium density in the central pin with Uranium enrichment in the outer fuel rings, (2) Dysprosium density and Uranium enrichment both in the central pin, and (3) the same parameters as in the first case but the objective is to obtain a negative checkerboard CVR at beginning of cycle (CBCVR-BOC). To approximate the sensitivity coefficient at EOC, we perform constant-power burnup/depletion calculations for 600 full power days (FPD) using a slightly perturbed nuclear library and the unperturbed neutron fluxes to estimate the variation of nuclide densities at EOC. Sensitivity analyses of CVR and eigenvalue are included in the study. In addition the optimization and adjoint sensitivity techniques are applied to the CBCVR-BOC and keff-EOC adjustment of the ACR lattices with Gadolinium in the central pin. Finally we apply these techniques to the CVR-BOC, CVR-EOC and keff-EOC adjustment of a CANDU lattice of which the burnup period is extended from 300 to 450 FPDs. The cases with the central pin containing either Dysprosium or Gadolinium in the natural Uranium are considered in our study. (Abstract shortened by UMI.)

Assessments of a Turbulence Model Based on Menter's Modification to Rotta's Two-Equation Model

NASA Technical Reports Server (NTRS)

Abdol-Hamid, Khaled S.

2013-01-01

The main objective of this paper is to construct a turbulence model with a more reliable second equation simulating length scale. In the present paper, we assess the length scale equation based on Menter s modification to Rotta s two-equation model. Rotta shows that a reliable second equation can be formed in an exact transport equation from the turbulent length scale L and kinetic energy. Rotta s equation is well suited for a term-by-term modeling and shows some interesting features compared to other approaches. The most important difference is that the formulation leads to a natural inclusion of higher order velocity derivatives into the source terms of the scale equation, which has the potential to enhance the capability of Reynolds-averaged Navier-Stokes (RANS) to simulate unsteady flows. The model is implemented in the PAB3D solver with complete formulation, usage methodology, and validation examples to demonstrate its capabilities. The detailed studies include grid convergence. Near-wall and shear flows cases are documented and compared with experimental and Large Eddy Simulation (LES) data. The results from this formulation are as good or better than the well-known SST turbulence model and much better than k-epsilon results. Overall, the study provides useful insights into the model capability in predicting attached and separated flows.

Near-wall modelling of compressible turbulent flows

NASA Technical Reports Server (NTRS)

So, Ronald M. C.

1990-01-01

Work was carried out to formulate near-wall models for the equations governing the transport of the temperature-variance and its dissipation rate. With these equations properly modeled, a foundation is laid for their extension together with the heat-flux equations to compressible flows. This extension is carried out in a manner similar to that used to extend the incompressible near-wall Reynolds-stress models to compressible flows. The methodology used to accomplish the extension of the near-wall Reynolds-stress models is examined and the actual extension of the models for the Reynolds-stress equations and the near-wall dissipation-rate equation to compressible flows is given. Then the formulation of the near-wall models for the equations governing the transport of the temperature variance and its dissipation rate is discussed. Finally, a sample calculation of a flat plate compressible turbulent boundary-layer flow with adiabatic wall boundary condition and a free-stream Mach number of 2.5 using a two-equation near-wall closure is presented. The results show that the near-wall two-equation closure formulated for compressible flows is quite valid and the calculated properties are in good agreement with measurements. Furthermore, the near-wall behavior of the turbulence statistics and structure parameters is consistent with that found in incompressible flows.

Reference Ellipsoid and Geoid in Chronometric Geodesy

NASA Astrophysics Data System (ADS)

Kopeikin, Sergei M.

2016-02-01

Chronometric geodesy applies general relativity to study the problem of the shape of celestial bodies including the earth, and their gravitational field. The present paper discusses the relativistic problem of construction of a background geometric manifold that is used for describing a reference ellipsoid, geoid, the normal gravity field of the earth and for calculating geoid's undulation (height). We choose the perfect fluid with an ellipsoidal mass distribution uniformly rotating around a fixed axis as a source of matter generating the geometry of the background manifold through the Einstein equations. We formulate the post-Newtonian hydrodynamic equations of the rotating fluid to find out the set of algebraic equations defining the equipotential surface of the gravity field. In order to solve these equations we explicitly perform all integrals characterizing the interior gravitational potentials in terms of elementary functions depending on the parameters defining the shape of the body and the mass distribution. We employ the coordinate freedom of the equations to choose these parameters to make the shape of the rotating fluid configuration to be an ellipsoid of rotation. We derive expressions of the post-Newtonian mass and angular momentum of the rotating fluid as functions of the rotational velocity and the parameters of the ellipsoid including its bare density, eccentricity and semi-major axes. We formulate the post-Newtonian Pizzetti and Clairaut theorems that are used in geodesy to connect the parameters of the reference ellipsoid to the polar and equatorial values of force of gravity. We expand the post-Newtonian geodetic equations characterizing the reference ellipsoid into the Taylor series with respect to the eccentricity of the ellipsoid, and discuss the small-eccentricity approximation. Finally, we introduce the concept of relativistic geoid and its undulation with respect to the reference ellipsoid, and discuss how to calculate it in chronometric geodesy by making use of the anomalous gravity potential.

Minimal string theories and integrable hierarchies

NASA Astrophysics Data System (ADS)

Iyer, Ramakrishnan

Well-defined, non-perturbative formulations of the physics of string theories in specific minimal or superminimal model backgrounds can be obtained by solving matrix models in the double scaling limit. They provide us with the first examples of completely solvable string theories. Despite being relatively simple compared to higher dimensional critical string theories, they furnish non-perturbative descriptions of interesting physical phenomena such as geometrical transitions between D-branes and fluxes, tachyon condensation and holography. The physics of these theories in the minimal model backgrounds is succinctly encoded in a non-linear differential equation known as the string equation, along with an associated hierarchy of integrable partial differential equations (PDEs). The bosonic string in (2,2m-1) conformal minimal model backgrounds and the type 0A string in (2,4 m) superconformal minimal model backgrounds have the Korteweg-de Vries system, while type 0B in (2,4m) backgrounds has the Zakharov-Shabat system. The integrable PDE hierarchy governs flows between backgrounds with different m. In this thesis, we explore this interesting connection between minimal string theories and integrable hierarchies further. We uncover the remarkable role that an infinite hierarchy of non-linear differential equations plays in organizing and connecting certain minimal string theories non-perturbatively. We are able to embed the type 0A and 0B (A,A) minimal string theories into this single framework. The string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy. We find that there are several other string-like limits of the system, and conjecture that some of them are type IIA and IIB (A,D) minimal string backgrounds. We explain how these and several other string-like special points arise and are connected. In some cases, the framework endows the theories with a non-perturbative definition for the first time. Notably, we discover that the Painleve IV equation plays a key role in organizing the string theory physics, joining its siblings, Painleve I and II, whose roles have previously been identified in this minimal string context. We then present evidence that the conjectured type II theories have smooth non-perturbative solutions, connecting two perturbative asymptotic regimes, in a 't Hooft limit. Our technique also demonstrates evidence for new minimal string theories that are not apparent in a perturbative analysis.

Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

NASA Astrophysics Data System (ADS)

Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.

2012-10-01

A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.

Computer simulation of two-dimensional unsteady flows in estuaries and embayments by the method of characteristics : basic theory and the formulation of the numerical method

USGS Publications Warehouse

Lai, Chintu

1977-01-01

Two-dimensional unsteady flows of homogeneous density in estuaries and embayments can be described by hyperbolic, quasi-linear partial differential equations involving three dependent and three independent variables. A linear combination of these equations leads to a parametric equation of characteristic form, which consists of two parts: total differentiation along the bicharacteristics and partial differentiation in space. For its numerical solution, the specified-time-interval scheme has been used. The unknown, partial space-derivative terms can be eliminated first by suitable combinations of difference equations, converted from the corresponding differential forms and written along four selected bicharacteristics and a streamline. Other unknowns are thus made solvable from the known variables on the current time plane. The computation is carried to the second-order accuracy by using trapezoidal rule of integration. Means to handle complex boundary conditions are developed for practical application. Computer programs have been written and a mathematical model has been constructed for flow simulation. The favorable computer outputs suggest further exploration and development of model worthwhile. (Woodard-USGS)

Forced pitch motion of wind turbines

NASA Astrophysics Data System (ADS)

Leble, V.; Barakos, G.

2016-09-01

The possibility of a wind turbine entering vortex ring state during pitching oscillations is explored in this paper. The aerodynamic performance of the rotor was computed using the Helicopter Multi-Block flow solver. This code solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. A 10-MW wind turbine was put to perform yawing and pitching oscillations suggesting the partial vortex ring state during pitching motion. The results also show the strong effect of the frequency and amplitude of oscillations on the wind turbine performance.

A penny shaped crack in a filament-reinforced matrix. 2: The crack problem

NASA Technical Reports Server (NTRS)

Pacella, A. H.; Erdogan, F.

1973-01-01

The elastostatic interaction problem between a penny-shaped crack and a slender inclusion or filament in an elastic matrix was formulated. For a single filament as well as multiple identical filaments located symmetrically around the crack the problem is shown to reduce to a singular integral equation. The solution of the problem is obtained for various geometries and filament-to-matrix stiffness ratios, and the results relating to the angular variation of the stress intensity factor and the maximum filament stress are presented.

EChem++--an object-oriented problem solving environment for electrochemistry. 2. The kinetic facilities of Ecco--a compiler for (electro-)chemistry.

PubMed

Ludwig, Kai; Speiser, Bernd

2004-01-01

We describe a modeling software component Ecco, implemented in the C++ programming language. It assists in the formulation of physicochemical systems including, in particular, electrochemical processes within general geometries. Ecco's kinetic part then translates any user defined reaction mechanism into an object-oriented representation and generates the according mathematical model equations. The input language, its grammar, the object-oriented design of Ecco, based on design patterns, and its integration into the open source software project EChem++ are discussed. Application Strategies are given.

Effects of unsteady free stream velocity and free stream turbulence on stagnation point heat transfer

NASA Technical Reports Server (NTRS)

Gorla, R. S. R.

1984-01-01

The combined effects of transient free stream velocity and free stream turbulence on heat transfer at a stagnation point over a cylinder situated in a crossflow are studied. An eddy diffusivity model was formulated and the governing momentum and energy equations are integrated by means of the steepest descent method. The numerical results for the wall shear stress and heat transfer rate are correlated by a turbulence parameter. The wall friction and heat transfer rate increase with increasing free stream turbulence intensity.

A statistical formulation of one-dimensional electron fluid turbulence

NASA Technical Reports Server (NTRS)

Fyfe, D.; Montgomery, D.

1977-01-01

A one-dimensional electron fluid model is investigated using the mathematical methods of modern fluid turbulence theory. Non-dissipative equilibrium canonical distributions are determined in a phase space whose co-ordinates are the real and imaginary parts of the Fourier coefficients for the field variables. Spectral densities are calculated, yielding a wavenumber electric field energy spectrum proportional to k to the negative second power for large wavenumbers. The equations of motion are numerically integrated and the resulting spectra are found to compare well with the theoretical predictions.

Boundary-element modelling of dynamics in external poroviscoelastic problems

NASA Astrophysics Data System (ADS)

Igumnov, L. A.; Litvinchuk, S. Yu; Ipatov, A. A.; Petrov, A. N.

2018-04-01

A problem of a spherical cavity in porous media is considered. Porous media are assumed to be isotropic poroelastic or isotropic poroviscoelastic. The poroviscoelastic formulation is treated as a combination of Biot’s theory of poroelasticity and elastic-viscoelastic correspondence principle. Such viscoelastic models as Kelvin–Voigt, Standard linear solid, and a model with weakly singular kernel are considered. Boundary field study is employed with the help of the boundary element method. The direct approach is applied. The numerical scheme is based on the collocation method, regularized boundary integral equation, and Radau stepped scheme.

Frequency-domain method for discrete frequency noise prediction of rotors in arbitrary steady motion

NASA Astrophysics Data System (ADS)

Gennaretti, M.; Testa, C.; Bernardini, G.

2012-12-01

A novel frequency-domain formulation for the prediction of the tonal noise emitted by rotors in arbitrary steady motion is presented. It is derived from Farassat's 'Formulation 1A', that is a time-domain boundary integral representation for the solution of the Ffowcs-Williams and Hawkings equation, and represents noise as harmonic response to body kinematics and aerodynamic loads via frequency-response-function matrices. The proposed frequency-domain solver is applicable to rotor configurations for which sound pressure levels of discrete tones are much higher than those of broadband noise. The numerical investigation concerns the analysis of noise produced by an advancing helicopter rotor in blade-vortex interaction conditions, as well as the examination of pressure disturbances radiated by the interaction of a marine propeller with a non-uniform inflow.

Broadband Noise Prediction When Turbulence Simulation Is Available - Derivation of Formulation 2B and Its Statistical Analysis

NASA Technical Reports Server (NTRS)

Farassat, Fereidoun; Casper, Jay H.

2012-01-01

We show that a simple modification of Formulation 1 of Farassat results in a new analytic expression that is highly suitable for broadband noise prediction when extensive turbulence simulation is available. This result satisfies all the stringent requirements, such as permitting the use of the exact geometry and kinematics of the moving body, that we have set as our goal in the derivation of useful acoustic formulas for the prediction of rotating blade and airframe noise. We also derive a simple analytic expression for the autocorrelation of the acoustic pressure that is valid in the near and far fields. Our analysis is based on the time integral of the acoustic pressure that can easily be obtained at any resolution for any observer time interval and digitally analyzed for broadband noise prediction. We have named this result as Formulation 2B of Farassat. One significant consequence of Formulation 2B is the derivation of the acoustic velocity potential for the thickness and loading terms of the Ffowcs Williams-Hawkings (FW-H) equation. This will greatly enhance the usefulness of the Fast Scattering Code (FSC) by providing a high fidelity boundary condition input for scattering predictions.

How students process equations in solving quantitative synthesis problems? Role of mathematical complexity in students' mathematical performance

NASA Astrophysics Data System (ADS)

Ibrahim, Bashirah; Ding, Lin; Heckler, Andrew F.; White, Daniel R.; Badeau, Ryan

2017-12-01

We examine students' mathematical performance on quantitative "synthesis problems" with varying mathematical complexity. Synthesis problems are tasks comprising multiple concepts typically taught in different chapters. Mathematical performance refers to the formulation, combination, and simplification of equations. Generally speaking, formulation and combination of equations require conceptual reasoning; simplification of equations requires manipulation of equations as computational tools. Mathematical complexity is operationally defined by the number and the type of equations to be manipulated concurrently due to the number of unknowns in each equation. We use two types of synthesis problems, namely, sequential and simultaneous tasks. Sequential synthesis tasks require a chronological application of pertinent concepts, and simultaneous synthesis tasks require a concurrent application of the pertinent concepts. A total of 179 physics major students from a second year mechanics course participated in the study. Data were collected from written tasks and individual interviews. Results show that mathematical complexity negatively influences the students' mathematical performance on both types of synthesis problems. However, for the sequential synthesis tasks, it interferes only with the students' simplification of equations. For the simultaneous synthesis tasks, mathematical complexity additionally impedes the students' formulation and combination of equations. Several reasons may explain this difference, including the students' different approaches to the two types of synthesis problems, cognitive load, and the variation of mathematical complexity within each synthesis type.

Saturation Length of Erodible Sediment Beds Subject to Shear Flow

NASA Astrophysics Data System (ADS)

Casler, D. M.; Kahn, B. P.; Furbish, D. J.; Schmeeckle, M. W.

2016-12-01

We examine the initial growth and wavelength selection of sand ripples based on probabilistic formulations of the flux and the Exner equation. Current formulations of this problem as a linear stability analysis appeal to the idea of a saturation length-the lag between the bed stress and the flux-as a key stabilizing influence leading to selection of a finite wavelength. We present two contrasting formulations. The first is based on the Fokker-Planck approximation of the divergence form of the Exner equation, and thus involves particle diffusion associated with variations in particle activity, in addition to the conventionally assumed advective term. The role of a saturation length associated with the particle activity is similar to previous analyses. However, particle diffusion provides an attenuating influence on the growth of small wavelengths, independent of a saturation length, and is thus a sufficient, if not necessary, condition contributing to selection of a finite wavelength. The second formulation is based on the entrainment form of the Exner equation. As a precise, probabilistic formulation of conservation, this form of the Exner equation does not distinguish between advection and diffusion, and, because it directly accounts for all particle motions via a convolution of the distribution of particle hop distances, it pays no attention to the idea of a saturation length. The formulation and resulting description of initial ripple growth and wavelength selection thus inherently subsume the effects embodied in the ideas of advection, diffusion, and a saturation length as used in other formulations. Moreover, the formulation does not distinguish between bed load and suspended load, and is thus broader in application. The analysis reveals that the length scales defined by the distribution of hop distances are more fundamental than the saturation length in determining the initial growth or decay of bedforms. Formulations involving the saturation length coincide with the special case of an exponential distribution of hop distance, where the saturation length is equal to the mean hop distance.

Modeling of high pressure arc-discharge with a fully-implicit Navier-Stokes stabilized finite element flow solver

NASA Astrophysics Data System (ADS)

Sahai, A.; Mansour, N. N.; Lopez, B.; Panesi, M.

2017-05-01

This work addresses the modeling of high pressure electric discharge in an arc-heated wind tunnel. The combined numerical solution of Poisson’s equation, radiative transfer equations, and the set of Favre-averaged thermochemical nonequilibrium Navier-Stokes equations allows for the determination of the electric, radiation, and flow fields, accounting for their mutual interaction. Semi-classical statistical thermodynamics is used to determine the plasma thermodynamic properties, while transport properties are obtained from kinetic principles with the Chapman-Enskog method. A multi-temperature formulation is used to account for thermal non-equilibrium. Finally, the turbulence closure of the flow equations is obtained by means of the Spalart-Allmaras model, which requires the solution of an additional scalar transport equation. A Streamline upwind Petrov-Galerkin stabilized finite element formulation is employed to solve the Navier-Stokes equation. The electric field equation is solved using the standard Galerkin formulation. A stable formulation for the radiative transfer equations is obtained using the least-squares finite element method. The developed simulation framework has been applied to investigate turbulent plasma flows in the 20 MW Aerodynamic Heating Facility at NASA Ames Research Center. The current model is able to predict the process of energy addition and re-distribution due to Joule heating and thermal radiation, resulting in a hot central core surrounded by colder flow. The use of an unsteady three-dimensional treatment also allows the asymmetry due to a dynamic electric arc attachment point in the cathode chamber to be captured accurately. The current work paves the way for detailed estimation of operating characteristics for arc-heated wind tunnels which are critical in testing thermal protection systems.

Kinetic theory of pattern formation in mixtures of microtubules and molecular motors

NASA Astrophysics Data System (ADS)

Maryshev, Ivan; Marenduzzo, Davide; Goryachev, Andrew B.; Morozov, Alexander

2018-02-01

In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006), 10.1103/PhysRevE.74.031915] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behavior by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behavior than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.

Correlations of RMT characteristic polynomials and integrability: Hermitean matrices

DOE Office of Scientific and Technical Information (OSTI.GOV)

Osipov, Vladimir Al., E-mail: Vladimir.Osipov@uni-due.d; Kanzieper, Eugene, E-mail: Eugene.Kanzieper@hit.ac.i; Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100

Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of {tau} functions, we (i) identify a zoo of hierarchical relations satisfied by {tau} functions in an abstract infinite-dimensional space and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasismore » is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.« less

A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries.

PubMed

Ge, Liang; Sotiropoulos, Fotis

2007-08-01

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries

PubMed Central

Ge, Liang; Sotiropoulos, Fotis

2008-01-01

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus. PMID:19194533

On the Geometry of the Hamilton-Jacobi Equation and Generating Functions

NASA Astrophysics Data System (ADS)

Ferraro, Sebastián; de León, Manuel; Marrero, Juan Carlos; Martín de Diego, David; Vaquero, Miguel

2017-10-01

In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80-88, 1991), Ge (Indiana Univ. Math. J. 39(3):859-876, 1990), Ge and Marsden (Phys Lett A 133(3):134-139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.

Experimental triplet and quadruplet fluctuation densities and spatial distribution function integrals for pure liquids.

PubMed

Ploetz, Elizabeth A; Karunaweera, Sadish; Smith, Paul E

2015-01-28

Fluctuation solution theory has provided an alternative view of many liquid mixture properties in terms of particle number fluctuations. The particle number fluctuations can also be related to integrals of the corresponding two body distribution functions between molecular pairs in order to provide a more physical picture of solution behavior and molecule affinities. Here, we extend this type of approach to provide expressions for higher order triplet and quadruplet fluctuations, and thereby integrals over the corresponding distribution functions, all of which can be obtained from available experimental thermodynamic data. The fluctuations and integrals are then determined using the International Association for the Properties of Water and Steam Formulation 1995 (IAPWS-95) equation of state for the liquid phase of pure water. The results indicate small, but significant, deviations from a Gaussian distribution for the molecules in this system. The pressure and temperature dependence of the fluctuations and integrals, as well as the limiting behavior as one approaches both the triple point and the critical point, are also examined.

Experimental triplet and quadruplet fluctuation densities and spatial distribution function integrals for pure liquids

NASA Astrophysics Data System (ADS)

Ploetz, Elizabeth A.; Karunaweera, Sadish; Smith, Paul E.

2015-01-01

Fluctuation solution theory has provided an alternative view of many liquid mixture properties in terms of particle number fluctuations. The particle number fluctuations can also be related to integrals of the corresponding two body distribution functions between molecular pairs in order to provide a more physical picture of solution behavior and molecule affinities. Here, we extend this type of approach to provide expressions for higher order triplet and quadruplet fluctuations, and thereby integrals over the corresponding distribution functions, all of which can be obtained from available experimental thermodynamic data. The fluctuations and integrals are then determined using the International Association for the Properties of Water and Steam Formulation 1995 (IAPWS-95) equation of state for the liquid phase of pure water. The results indicate small, but significant, deviations from a Gaussian distribution for the molecules in this system. The pressure and temperature dependence of the fluctuations and integrals, as well as the limiting behavior as one approaches both the triple point and the critical point, are also examined.

Pseudospectral collocation methods for fourth order differential equations

NASA Technical Reports Server (NTRS)

Malek, Alaeddin; Phillips, Timothy N.

1994-01-01

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.

An overview of methods to identify and manage uncertainty for modelling problems in the water-environment-agriculture cross-sector

DOE PAGES

Jakeman, Anthony J.; Jakeman, John Davis

2018-03-14

Uncertainty pervades the representation of systems in the water–environment–agriculture cross-sector. Successful methods to address uncertainties have largely focused on standard mathematical formulations of biophysical processes in a single sector, such as partial or ordinary differential equations. More attention to integrated models of such systems is warranted. Model components representing the different sectors of an integrated model can have less standard, and different, formulations to one another, as well as different levels of epistemic knowledge and data informativeness. Thus, uncertainty is not only pervasive but also crosses boundaries and propagates between system components. Uncertainty assessment (UA) cries out for more eclecticmore » treatment in these circumstances, some of it being more qualitative and empirical. Here in this paper, we discuss the various sources of uncertainty in such a cross-sectoral setting and ways to assess and manage them. We have outlined a fast-growing set of methodologies, particularly in the computational mathematics literature on uncertainty quantification (UQ), that seem highly pertinent for uncertainty assessment. There appears to be considerable scope for advancing UA by integrating relevant UQ techniques into cross-sectoral problem applications. Of course this will entail considerable collaboration between domain specialists who often take first ownership of the problem and computational methods experts.« less

An overview of methods to identify and manage uncertainty for modelling problems in the water-environment-agriculture cross-sector

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jakeman, Anthony J.; Jakeman, John Davis

Uncertainty pervades the representation of systems in the water–environment–agriculture cross-sector. Successful methods to address uncertainties have largely focused on standard mathematical formulations of biophysical processes in a single sector, such as partial or ordinary differential equations. More attention to integrated models of such systems is warranted. Model components representing the different sectors of an integrated model can have less standard, and different, formulations to one another, as well as different levels of epistemic knowledge and data informativeness. Thus, uncertainty is not only pervasive but also crosses boundaries and propagates between system components. Uncertainty assessment (UA) cries out for more eclecticmore » treatment in these circumstances, some of it being more qualitative and empirical. Here in this paper, we discuss the various sources of uncertainty in such a cross-sectoral setting and ways to assess and manage them. We have outlined a fast-growing set of methodologies, particularly in the computational mathematics literature on uncertainty quantification (UQ), that seem highly pertinent for uncertainty assessment. There appears to be considerable scope for advancing UA by integrating relevant UQ techniques into cross-sectoral problem applications. Of course this will entail considerable collaboration between domain specialists who often take first ownership of the problem and computational methods experts.« less

Stochastic Models for Laser Propagation in Atmospheric Turbulence.

NASA Astrophysics Data System (ADS)

Leland, Robert Patton

In this dissertation, stochastic models for laser propagation in atmospheric turbulence are considered. A review of the existing literature on laser propagation in the atmosphere and white noise theory is presented, with a view toward relating the white noise integral and Ito integral approaches. The laser beam intensity is considered as the solution to a random Schroedinger equation, or forward scattering equation. This model is formulated in a Hilbert space context as an abstract bilinear system with a multiplicative white noise input, as in the literature. The model is also modeled in the Banach space of Fresnel class functions to allow the plane wave case and the application of path integrals. Approximate solutions to the Schroedinger equation of the Trotter-Kato product form are shown to converge for each white noise sample path. The product forms are shown to be physical random variables, allowing an Ito integral representation. The corresponding Ito integrals are shown to converge in mean square, providing a white noise basis for the Stratonovich correction term associated with this equation. Product form solutions for Ornstein -Uhlenbeck process inputs were shown to converge in mean square as the input bandwidth was expanded. A digital simulation of laser propagation in strong turbulence was used to study properties of the beam. Empirical distributions for the irradiance function were estimated from simulated data, and the log-normal and Rice-Nakagami distributions predicted by the classical perturbation methods were seen to be inadequate. A gamma distribution fit the simulated irradiance distribution well in the vicinity of the boresight. Statistics of the beam were seen to converge rapidly as the bandwidth of an Ornstein-Uhlenbeck process was expanded to its white noise limit. Individual trajectories of the beam were presented to illustrate the distortion and bending of the beam due to turbulence. Feynman path integrals were used to calculate an approximate expression for the mean of the beam intensity without using the Markov, or white noise, assumption, and to relate local variations in the turbulence field to the behavior of the beam by means of two approximations.

On the mechanics of stress analysis of fiber-reinforced composites

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lee, V.G.

A general mathematical formulation is developed for the three-dimensional inclusion and inhomogeneity problems, which are practically important in many engineering applications such as fiber pullout of reinforced composites, load transfer behavior in the stiffened structural components, and material defects and impurities existing in engineering materials. First, the displacement field (Green's function) for an elastic solid subjected to various distributions of ring loading is derived in closed form using the Papkovich-Neuber displacement potentials and the Hankel transforms. The Green's functions are used to derive the displacement and stress fields due to a finite cylindrical inclusion of prescribed dilatational eigenstrain such asmore » thermal expansion caused by an internal heat source. Unlike an elliptical inclusion, the interior stress field in the cylindrical inclusion is not uniform. Next, the three-dimensional inhomogeneity problem of a cylindrical fiber embedded in an infinite matrix of different material properties is considered to study load transfer of a finite fiber to an elastic medium. By using the equivalent inclusion method, the fiber is modeled as an inclusion with distributed eigenstrains of unknown strength, and the inhomogeneity problem can be treated as an equivalent inclusion problem. The eigenstrains are determined to simulate the disturbance due to the existing fiber. The equivalency of elastic field between inhomogeneity and inclusion problems leads to a set of integral equations. To solve the integral equations, the inclusion domain is discretized into a finite number of sub-inclusions with uniform eigenstrains, and the integral equations are reduced to a set of algebraic equations. The distributions of eigenstrains, interior stress field and axial force along the fiber are presented for various fiber lengths and the ratio of material properties of the fiber relative to the matrix.« less

Investigation of an artificial intelligence technology--Model trees. Novel applications for an immediate release tablet formulation database.

PubMed

Shao, Q; Rowe, R C; York, P

2007-06-01

This study has investigated an artificial intelligence technology - model trees - as a modelling tool applied to an immediate release tablet formulation database. The modelling performance was compared with artificial neural networks that have been well established and widely applied in the pharmaceutical product formulation fields. The predictability of generated models was validated on unseen data and judged by correlation coefficient R(2). Output from the model tree analyses produced multivariate linear equations which predicted tablet tensile strength, disintegration time, and drug dissolution profiles of similar quality to neural network models. However, additional and valuable knowledge hidden in the formulation database was extracted from these equations. It is concluded that, as a transparent technology, model trees are useful tools to formulators.

An investigation of the accuracy of the Merkel equation for evaporative cooling tower calculations. Waste heat management

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yadigaroglu, G.; Pastor, E.J.

1974-01-01

The exact differential equations governing heat and mass transfer and air flow in an evaporative, natural-draft cooling tower are presented. The Merkel equation is then derived starting from this exact formulation and showing all the approximations involved. The Merkel formulation lumps the sensible and the latent heat transfer together and considers a single enthalpy-difference driving force for the total heat transfer. The effect of the approximations inherent in the Merkel equation is investigated and analyzed by a series of parametric numerical calculations of cooling tower performance under various ambient conditions and load conditions.

Toroidal regularization of the guiding center Lagrangian

DOE PAGES

Burby, J. W.; Ellison, C. L.

2017-11-22

In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v ∥∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, themore » Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.« less

Role of Multiple Soliton Interactions in the Generation of Rogue Waves: The Modified Korteweg-de Vries Framework.

PubMed

Slunyaev, A V; Pelinovsky, E N

2016-11-18

The role of multiple soliton and breather interactions in the formation of very high waves is disclosed within the framework of the integrable modified Korteweg-de Vries (MKdV) equation. Optimal conditions for the focusing of many solitons are formulated explicitly. Namely, trains of ordered solitons with alternate polarities evolve to huge strongly localized transient waves. The focused wave amplitude is exactly the sum of the focusing soliton heights; the maximum wave inherits the polarity of the fastest soliton in the train. The focusing of several solitary waves or/and breathers may naturally occur in a soliton gas and will lead to rogue-wave-type dynamics; hence, it represents a new nonlinear mechanism of rogue wave generation. The discovered scenario depends crucially on the soliton polarities (phases), and is not taken into account by existing kinetic theories. The performance of the soliton mechanism of rogue wave generation is shown for the example of the focusing MKdV equation, when solitons possess "frozen" phases (certain polarities), though the approach is efficient in some other integrable systems which admit soliton and breather solutions.

ALESEP. Part 2: A computer program for the analysis of leading edge separation bubbles on infinite swept wings

NASA Technical Reports Server (NTRS)

Davis, R. L.

1986-01-01

A program called ALESEP is presented for the analysis of the inviscid-viscous interaction which occurs due to the presence of a closed laminar-transitional separation bubble on an airfoil or infinite swept wing. The ALESEP code provides an iterative solution of the boundary layer equations expressed in an inverse formulation coupled to a Cauchy integral representation of the inviscid flow. This interaction analysis is treated as a local perturbation to a known solution obtained from a global airfoil analysis; hence, part of the required input to the ALESEP code are the reference displacement thickness and tangential velocity distributions. Special windward differencing may be used in the reversed flow regions of the separation bubble to accurately account for the flow direction in the discretization of the streamwise convection of momentum. The ALESEP code contains a forced transition model based on a streamwise intermittency function, a natural transition model based on a solution of the integral form of the turbulent kinetic energy equation, and an empirical natural transition model.

Role of Multiple Soliton Interactions in the Generation of Rogue Waves: The Modified Korteweg-de Vries Framework

NASA Astrophysics Data System (ADS)

Slunyaev, A. V.; Pelinovsky, E. N.

2016-11-01

The role of multiple soliton and breather interactions in the formation of very high waves is disclosed within the framework of the integrable modified Korteweg-de Vries (MKdV) equation. Optimal conditions for the focusing of many solitons are formulated explicitly. Namely, trains of ordered solitons with alternate polarities evolve to huge strongly localized transient waves. The focused wave amplitude is exactly the sum of the focusing soliton heights; the maximum wave inherits the polarity of the fastest soliton in the train. The focusing of several solitary waves or/and breathers may naturally occur in a soliton gas and will lead to rogue-wave-type dynamics; hence, it represents a new nonlinear mechanism of rogue wave generation. The discovered scenario depends crucially on the soliton polarities (phases), and is not taken into account by existing kinetic theories. The performance of the soliton mechanism of rogue wave generation is shown for the example of the focusing MKdV equation, when solitons possess "frozen" phases (certain polarities), though the approach is efficient in some other integrable systems which admit soliton and breather solutions.

Toroidal regularization of the guiding center Lagrangian

DOE Office of Scientific and Technical Information (OSTI.GOV)

Burby, J. W.; Ellison, C. L.

In the Lagrangian theory of guiding center motion, an effective magnetic field B* = B+ (m/e)v ∥∇ x b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. Here, this letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, themore » Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.« less

A new stream function formulation for the Euler equations

NASA Technical Reports Server (NTRS)

Atkins, H. L.; Hassan, H. A.

1983-01-01

A new stream function formulation is developed for the solution of Euler's equations in the transonic flow region. The stream function and the density are the dependent variables in this method, while the governing equations for adiabatic flow are the momentum equations which are solved in the strong conservation law form. The application of this method does not require a knowledge of the vorticity. The algorithm is combined with the automatic grid solver (GRAPE) of Steger and Sorenson (1979) in order to study arbitrary geometries. Results of the application of this method are presented for the NACA 0012 airfoil at various Mach numbers and angles of attack, and cylinders. In addition, detailed comparisons are made with other solutions of the Euler equations.

Fem Formulation of Heat Transfer in Cylindrical Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Khaleed, H. M. T.; Soudagar, Manzoor Elahi M.

2017-08-01

Heat transfer in porous medium can be derived from the fundamental laws of flow in porous region ass given by Henry Darcy. The fluid flow and energy transport inside the porous medium can be described with the help of momentum and energy equations. The heat transfer in cylindrical porous medium differs from its counterpart in radial and axial coordinates. The present work is focused to discuss the finite element formulation of heat transfer in cylindrical porous medium. The basic partial differential equations are derived using Darcy law which is the converted into a set of algebraic equations with the help of finite element method. The resulting equations are solved by matrix method for two solution variables involved in the coupled equations.

Accurate orbit propagation in the presence of planetary close encounters

NASA Astrophysics Data System (ADS)

Amato, Davide; Baù, Giulio; Bombardelli, Claudio

2017-09-01

We present an efficient strategy for the numerical propagation of small Solar system objects undergoing close encounters with massive bodies. The trajectory is split into several phases, each of them being the solution of a perturbed two-body problem. Formulations regularized with respect to different primaries are employed in two subsequent phases. In particular, we consider the Kustaanheimo-Stiefel regularization and a novel set of non-singular orbital elements pertaining to the Dromo family. In order to test the proposed strategy, we perform ensemble propagations in the Earth-Sun Circular Restricted 3-Body Problem (CR3BP) using a variable step size and order multistep integrator and an improved version of Everhart's radau solver of 15th order. By combining the trajectory splitting with regularized equations of motion in short-term propagations (1 year), we gain up to six orders of magnitude in accuracy with respect to the classical Cowell's method for the same computational cost. Moreover, in the propagation of asteroid (99942) Apophis through its 2029 Earth encounter, the position error stays within 100 metres after 100 years. In general, as to improve the performance of regularized formulations, the trajectory must be split between 1.2 and 3 Hill radii from the Earth. We also devise a robust iterative algorithm to stop the integration of regularized equations of motion at a prescribed physical time. The results rigorously hold in the CR3BP, and similar considerations may apply when considering more complex models. The methods and algorithms are implemented in the naples fortran 2003 code, which is available online as a GitHub repository.

Mixed quantum/classical theory of rotationally and vibrationally inelastic scattering in space-fixed and body-fixed reference frames

NASA Astrophysics Data System (ADS)

Semenov, Alexander; Babikov, Dmitri

2013-11-01

We formulated the mixed quantum/classical theory for rotationally and vibrationally inelastic scattering process in the diatomic molecule + atom system. Two versions of theory are presented, first in the space-fixed and second in the body-fixed reference frame. First version is easy to derive and the resultant equations of motion are transparent, but the state-to-state transition matrix is complex-valued and dense. Such calculations may be computationally demanding for heavier molecules and/or higher temperatures, when the number of accessible channels becomes large. In contrast, the second version of theory requires some tedious derivations and the final equations of motion are rather complicated (not particularly intuitive). However, the state-to-state transitions are driven by real-valued sparse matrixes of much smaller size. Thus, this formulation is the method of choice from the computational point of view, while the space-fixed formulation can serve as a test of the body-fixed equations of motion, and the code. Rigorous numerical tests were carried out for a model system to ensure that all equations, matrixes, and computer codes in both formulations are correct.

An evaluation of solution algorithms and numerical approximation methods for modeling an ion exchange process

NASA Astrophysics Data System (ADS)

Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.

2010-07-01

The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

Modèle tridimensionnel pour coupler les équations magnétiques et électriques dans le cas de la magnétostatique

NASA Astrophysics Data System (ADS)

Piriou, F.; Razek, A.

1991-03-01

In this paper a 3D model for coupling of magnetic and electric equations is presented. The magnetic equations are solved with the help of finite element method using the magnetic vector potential formulation. To take into account the effects of magnetic saturation we use the Newton-Raphson algorithm. We develop the analysis permitting the coupling of magnetic and electric equations to obtain a difrerential system equations which can be solved with numerical integration. As example we model an iron core coil and the validity of our model is verified by a comparison of the obtained results with an analytical solution and a 2D code calculation. Dans cet article est présenté un modèle 3D qui permet de coupler les équations magnétiques et électriques. Les équations magnétiques sont résolues à l'aide de la méthode des éléments finis en utilisant une formulation en potentiel vecteur magnétique. Dans le modèle proposé les effets de la saturation du circuit magnétique sont pris en compte en utilisant l'algorithme de Newton-Raphson. On montre comment relier les équations magnétiques avec celles du circuit électrique pour aboutir à un système d'équations différentielles que l'on résout avec une intégration numérique. A titre d'exemple on modélise une bobine à noyau ferromagnétique et pour montrer la validité du modèle on compare les résultats obtenus avec une solution analytique et un code de calcul 2D.

A tensor formulation of the equation of transfer for spherically symmetric flows. [radiative transfer in seven dimensional Riemannian space

NASA Technical Reports Server (NTRS)

Haisch, B. M.

1976-01-01

A tensor formulation of the equation of radiative transfer is derived in a seven-dimensional Riemannian space such that the resulting equation constitutes a divergence in any coordinate system. After being transformed to a spherically symmetric comoving coordinate system, the transfer equation contains partial derivatives in angle and frequency, as well as optical depth due to the effects of aberration and the Doppler shift. However, by virtue of the divergence form of this equation, the divergence theorem may be applied to yield a numerical differencing scheme which is expected to be stable and to conserve luminosity. It is shown that the equation of transfer derived by this method in a Lagrangian coordinate system may be reduced to that given by Castor (1972), although it is, of course, desirable to leave the equation in divergence form.

Strong diffusion formulation of Markov chain ensembles and its optimal weaker reductions

NASA Astrophysics Data System (ADS)

Güler, Marifi

2017-10-01

Two self-contained diffusion formulations, in the form of coupled stochastic differential equations, are developed for the temporal evolution of state densities over an ensemble of Markov chains evolving independently under a common transition rate matrix. Our first formulation derives from Kurtz's strong approximation theorem of density-dependent Markov jump processes [Stoch. Process. Their Appl. 6, 223 (1978), 10.1016/0304-4149(78)90020-0] and, therefore, strongly converges with an error bound of the order of lnN /N for ensemble size N . The second formulation eliminates some fluctuation variables, and correspondingly some noise terms, within the governing equations of the strong formulation, with the objective of achieving a simpler analytic formulation and a faster computation algorithm when the transition rates are constant or slowly varying. There, the reduction of the structural complexity is optimal in the sense that the elimination of any given set of variables takes place with the lowest attainable increase in the error bound. The resultant formulations are supported by numerical simulations.