Numerical integration of asymptotic solutions of ordinary differential equations

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.

1989-01-01

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

Exponential Methods for the Time Integration of Schroedinger Equation

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

Cano, B.; Gonzalez-Pachon, A.

2010-09-30

We consider exponential methods of second order in time in order to integrate the cubic nonlinear Schroedinger equation. We are interested in taking profit of the special structure of this equation. Therefore, we look at symmetry, symplecticity and approximation of invariants of the proposed methods. That will allow to integrate till long times with reasonable accuracy. Computational efficiency is also our aim. Therefore, we make numerical computations in order to compare the methods considered and so as to conclude that explicit Lawson schemes projected on the norm of the solution are an efficient tool to integrate this equation.

On the solution of integral equations with strongly singular kernels

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1986-01-01

Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

On the solution of integral equations with strong ly singular kernels

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1985-01-01

In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

On the solution of integral equations with strongly singular kernels

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1987-01-01

Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

Extension of the KLI approximation toward the exact optimized effective potential.

PubMed

Iafrate, G J; Krieger, J B

2013-03-07

The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

Extension of the KLI approximation toward the exact optimized effective potential

NASA Astrophysics Data System (ADS)

Iafrate, G. J.; Krieger, J. B.

2013-03-01

The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

Modular Expression Language for Ordinary Differential Equation Editing

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

Blake, Robert C.

MELODEEis a system for describing systems of initial value problem ordinary differential equations, and a compiler for the language that produces optimized code to integrate the differential equations. Features include rational polynomial approximation for expensive functions and automatic differentiation for symbolic jacobians

A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation

NASA Astrophysics Data System (ADS)

Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohamed A.; Hafez, Ramy M.

2014-02-01

This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers' equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers' equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.

Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

NASA Technical Reports Server (NTRS)

Sidi, A.; Israeli, M.

1986-01-01

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.

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.

Orientation-dependent integral equation theory for a two-dimensional model of water

NASA Astrophysics Data System (ADS)

Urbič, T.; Vlachy, V.; Kalyuzhnyi, Yu. V.; Dill, K. A.

2003-03-01

We develop an integral equation theory that applies to strongly associating orientation-dependent liquids, such as water. In an earlier treatment, we developed a Wertheim integral equation theory (IET) that we tested against NPT Monte Carlo simulations of the two-dimensional Mercedes Benz model of water. The main approximation in the earlier calculation was an orientational averaging in the multidensity Ornstein-Zernike equation. Here we improve the theory by explicit introduction of an orientation dependence in the IET, based upon expanding the two-particle angular correlation function in orthogonal basis functions. We find that the new orientation-dependent IET (ODIET) yields a considerable improvement of the predicted structure of water, when compared to the Monte Carlo simulations. In particular, ODIET predicts more long-range order than the original IET, with hexagonal symmetry, as expected for the hydrogen bonded ice in this model. The new theoretical approximation still errs in some subtle properties; for example, it does not predict liquid water's density maximum with temperature or the negative thermal expansion coefficient.

An accurate method for evaluating the kernel of the integral equation relating lift to downwash in unsteady potential flow

NASA Technical Reports Server (NTRS)

Desmarais, R. N.

1982-01-01

The method is capable of generating approximations of arbitrary accuracy. It is based on approximating the algebraic part of the nonelementary integrals in the kernel by exponential functions and then integrating termwise. The exponent spacing in the approximation is a geometric sequence. The coefficients and exponent multiplier of the exponential approximation are computed by least squares so the method is completely automated. Exponential approximates generated in this manner are two orders of magnitude more accurate than the exponential approximation that is currently most often used for this purpose. The method can be used to generate approximations to attain any desired trade-off between accuracy and computing cost.

A spectral boundary integral equation method for the 2-D Helmholtz equation

NASA Technical Reports Server (NTRS)

Hu, Fang Q.

1994-01-01

In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the nonsmoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the nonsmoothness of the integral kernels in the spectral implementation. The present method is robust for a general boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. A numerical example of wave scattering is given in which the exponential accuracy of the present numerical method is demonstrated.

Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions.

PubMed

Ankiewicz, Adrian; Wang, Yan; Wabnitz, Stefan; Akhmediev, Nail

2014-01-01

We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specific constraints on the parameters of higher-order terms provide integrability of the resulting equation, providing a corresponding Lax pair. Particular cases of this equation are the Hirota and the Lakshmanan-Porsezian-Daniel equations. The resulting integrable equation admits exact rogue wave solutions. In particular cases, mentioned above, these solutions are reduced to the rogue wave solutions of the corresponding equations.

Analytical expressions for the correlation function of a hard sphere dimer fluid

NASA Astrophysics Data System (ADS)

Kim, Soonho; Chang, Jaeeon; Kim, Hwayong

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.

Analytical expression for the correlation function of a hard sphere chain fluid

NASA Astrophysics Data System (ADS)

Chang, Jaeeon; Kim, Hwayong

A closed form expression is given for the correlation function of flexible hard sphere chain fluid. A set of integral equations obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with the polymer Percus-Yevick ideal chain approximation is considered. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of individual correlation functions are obtained. By inverse Laplace transformation the inter- and intramolecular radial distribution functions (RDFs) are obtained in closed forms up to 3D(D is segment diameter). These analytical expressions for the RDFs would be useful in developing the perturbation theory of chain fluids.

Time domain convergence properties of Lyapunov stable penalty methods

NASA Technical Reports Server (NTRS)

Kurdila, A. J.; Sunkel, John

1991-01-01

Linear hyperbolic partial differential equations are analyzed using standard techniques to show that a sequence of solutions generated by the Liapunov stable penalty equations approaches the solution of the differential-algebraic equations governing the dynamics of multibody problems arising in linear vibrations. The analysis does not require that the system be conservative and does not impose any specific integration scheme. Variational statements are derived which bound the error in approximation by the norm of the constraint violation obtained in the approximate solutions.

Calculation of the Full Scattering Amplitude without Partial Wave Decomposition. 2; Inclusion of Exchange

NASA Technical Reports Server (NTRS)

Shertzer, Janine; Temkin, Aaron

2004-01-01

The development of a practical method of accurately calculating the full scattering amplitude, without making a partial wave decomposition is continued. The method is developed in the context of electron-hydrogen scattering, and here exchange is dealt with by considering e-H scattering in the static exchange approximation. The Schroedinger equation in this approximation can be simplified to a set of coupled integro-differential equations. The equations are solved numerically for the full scattering wave function. The scattering amplitude can most accurately be calculated from an integral expression for the amplitude; that integral can be formally simplified, and then evaluated using the numerically determined wave function. The results are essentially identical to converged partial wave results.

An alternative explicit model expression equivalent to the integrated michaelis-menten equation and its application to nonlinear saturation pharmacokinetics.

PubMed

Goličnik, Marko

2011-06-01

Many pharmacodynamic processes can be described by the nonlinear saturation kinetics that are most frequently based on the hyperbolic Michaelis-Menten equation. Thus, various time-dependent solutions for drugs obeying such kinetics can be expressed in terms of the Lambert W(x)-omega function. However, unfortunately, computer programs that can perform the calculations for W(x) are not widely available. To avoid this problem, the replacement of the integrated Michaelis-Menten equation with an empiric integrated 1--exp alternative model equation was proposed recently by Keller et al. (Ther Drug Monit. 2009;31:783-785), although, as shown here, it was not necessary. Simulated concentrations of model drugs obeying Michaelis-Menten elimination kinetics were generated by two approaches: 1) calculation of time-course data based on an approximation equation W2*(x) performed using Microsoft Excel; and 2) calculation of reference time-course data based on an exact W(x) function built in to the Wolfram Mathematica. I show here that the W2*(x) function approximates the actual W(x) accurately. W2*(x) is expressed in terms of elementary mathematical functions and, consequently, it can be easily implemented using any of the widely available software. Hence, with the example of a hypothetical drug, I demonstrate here that an equation based on this approximation is far better, because it is nearly equivalent to the original solution, whereas the same characteristics cannot be fully confirmed for the 1--exp model equation. The W2*(x) equation proposed here might have an important role as a useful shortcut in optional software to estimate kinetic parameters from experimental data for drugs, and it might represent an easy and universal analytical tool for simulating and designing dosing regimens.

High-frequency Born synthetic seismograms based on coupled normal modes

USGS Publications Warehouse

Pollitz, Fred F.

2011-01-01

High-frequency and full waveform synthetic seismograms on a 3-D laterally heterogeneous earth model are simulated using the theory of coupled normal modes. The set of coupled integral equations that describe the 3-D response are simplified into a set of uncoupled integral equations by using the Born approximation to calculate scattered wavefields and the pure-path approximation to modulate the phase of incident and scattered wavefields. This depends upon a decomposition of the aspherical structure into smooth and rough components. The uncoupled integral equations are discretized and solved in the frequency domain, and time domain results are obtained by inverse Fourier transform. Examples show the utility of the normal mode approach to synthesize the seismic wavefields resulting from interaction with a combination of rough and smooth structural heterogeneities. This approach is applied to an ∼4 Hz shallow crustal wave propagation around the site of the San Andreas Fault Observatory at Depth (SAFOD).

The thermal-vortex equations

NASA Technical Reports Server (NTRS)

Shebalin, John V.

1987-01-01

The Boussinesq approximation is extended so as to explicitly account for the transfer of fluid energy through viscous action into thermal energy. Ideal and dissipative integral invariants are discussed, in addition to the general equations for thermal-fluid motion.

Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics

NASA Technical Reports Server (NTRS)

Pratt, D. T.

1984-01-01

Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.

Topographical scattering of gravity waves

NASA Astrophysics Data System (ADS)

Miles, J. W.; Chamberlain, P. G.

1998-04-01

A systematic hierarchy of partial differential equations for linear gravity waves in water of variable depth is developed through the expansion of the average Lagrangian in powers of [mid R:][nabla del, Hamilton operator][mid R:] (h=depth, [nabla del, Hamilton operator]h=slope). The first and second members of this hierarchy, the Helmholtz and conventional mild-slope equations, are second order. The third member is fourth order but may be approximated by Chamberlain & Porter's (1995) ‘modified mild-slope’ equation, which is second order and comprises terms in [nabla del, Hamilton operator]2h and ([nabla del, Hamilton operator]h)2 that are absent from the mild-slope equation. Approximate solutions of the mild-slope and modified mild-slope equations for topographical scattering are determined through an iterative sequence, starting from a geometrical-optics approximation (which neglects reflection), then a quasi-geometrical-optics approximation, and on to higher-order results. The resulting reflection coefficient for a ramp of uniform slope is compared with the results of numerical integrations of each of the mild-slope equation (Booij 1983), the modified mild-slope equation (Porter & Staziker 1995), and the full linear equations (Booij 1983). Also considered is a sequence of sinusoidal sandbars, for which Bragg resonance may yield rather strong reflection and for which the modified mild-slope approximation is in close agreement with Mei's (1985) asymptotic approximation.

Accuracy of the Generalized Self-Consistent Method in Modelling the Elastic Behaviour of Periodic Composites

NASA Technical Reports Server (NTRS)

Walker, Kevin P.; Freed, Alan D.; Jordan, Eric H.

1993-01-01

Local stress and strain fields in the unit cell of an infinite, two-dimensional, periodic fibrous lattice have been determined by an integral equation approach. The effect of the fibres is assimilated to an infinite two-dimensional array of fictitious body forces in the matrix constituent phase of the unit cell. By subtracting a volume averaged strain polarization term from the integral equation we effectively embed a finite number of unit cells in a homogenized medium in which the overall stress and strain correspond to the volume averaged stress and strain of the constrained unit cell. This paper demonstrates that the zeroth term in the governing integral equation expansion, which embeds one unit cell in the homogenized medium, corresponds to the generalized self-consistent approximation. By comparing the zeroth term approximation with higher order approximations to the integral equation summation, both the accuracy of the generalized self-consistent composite model and the rate of convergence of the integral summation can be assessed. Two example composites are studied. For a tungsten/copper elastic fibrous composite the generalized self-consistent model is shown to provide accurate, effective, elastic moduli and local field representations. The local elastic transverse stress field within the representative volume element of the generalized self-consistent method is shown to be in error by much larger amounts for a composite with periodically distributed voids, but homogenization leads to a cancelling of errors, and the effective transverse Young's modulus of the voided composite is shown to be in error by only 23% at a void volume fraction of 75%.

On randomized algorithms for numerical solution of applied Fredholm integral equations of the second kind

NASA Astrophysics Data System (ADS)

Voytishek, Anton V.; Shipilov, Nikolay M.

2017-11-01

In this paper, the systematization of numerical (implemented on a computer) randomized functional algorithms for approximation of a solution of Fredholm integral equation of the second kind is carried out. Wherein, three types of such algorithms are distinguished: the projection, the mesh and the projection-mesh methods. The possibilities for usage of these algorithms for solution of practically important problems is investigated in detail. The disadvantages of the mesh algorithms, related to the necessity of calculation values of the kernels of integral equations in fixed points, are identified. On practice, these kernels have integrated singularities, and calculation of their values is impossible. Thus, for applied problems, related to solving Fredholm integral equation of the second kind, it is expedient to use not mesh, but the projection and the projection-mesh randomized algorithms.

General relaxation schemes in multigrid algorithms for higher order singularity methods

NASA Technical Reports Server (NTRS)

Oskam, B.; Fray, J. M. J.

1981-01-01

Relaxation schemes based on approximate and incomplete factorization technique (AF) are described. The AF schemes allow construction of a fast multigrid method for solving integral equations of the second and first kind. The smoothing factors for integral equations of the first kind, and comparison with similar results from the second kind of equations are a novel item. Application of the MD algorithm shows convergence to the level of truncation error of a second order accurate panel method.

The Investigation of Optimal Discrete Approximations for Real Time Flight Simulations

NASA Technical Reports Server (NTRS)

Parrish, E. A.; Mcvey, E. S.; Cook, G.; Henderson, K. C.

1976-01-01

The results are presented of an investigation of discrete approximations for real time flight simulation. Major topics discussed include: (1) consideration of the particular problem of approximation of continuous autopilots by digital autopilots; (2) use of Bode plots and synthesis of transfer functions by asymptotic fits in a warped frequency domain; (3) an investigation of the various substitution formulas, including the effects of nonlinearities; (4) use of pade approximation to the solution of the matrix exponential arising from the discrete state equations; and (5) an analytical integration of the state equation using interpolated input.

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.

Involution and Difference Schemes for the Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

Gerdt, Vladimir P.; Blinkov, Yuri A.

In the present paper we consider the Navier-Stokes equations for the two-dimensional viscous incompressible fluid flows and apply to these equations our earlier designed general algorithmic approach to generation of finite-difference schemes. In doing so, we complete first the Navier-Stokes equations to involution by computing their Janet basis and discretize this basis by its conversion into the integral conservation law form. Then we again complete the obtained difference system to involution with eliminating the partial derivatives and extracting the minimal Gröbner basis from the Janet basis. The elements in the obtained difference Gröbner basis that do not contain partial derivatives of the dependent variables compose a conservative difference scheme. By exploiting arbitrariness in the numerical integration approximation we derive two finite-difference schemes that are similar to the classical scheme by Harlow and Welch. Each of the two schemes is characterized by a 5×5 stencil on an orthogonal and uniform grid. We also demonstrate how an inconsistent difference scheme with a 3×3 stencil is generated by an inappropriate numerical approximation of the underlying integrals.

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

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

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

DOE PAGES

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris; ...

2018-04-23

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

Reply to "Comment on 'Defocusing complex short-pulse equation and its multi-dark-soliton solution' ".

PubMed

Feng, Bao-Feng; Ling, Liming; Zhu, Zuonong

2017-08-01

Our paper [Phys. Rev. E 93, 052227 (2016)PREHBM2470-004510.1103/PhysRevE.93.052227], proposing an integrable model for the propagation of ultrashort pulses, has recently received a Comment by Youssoufa et al. [Phys. Rev. E 96, 026201 (2017)10.1103/PhysRevE.96.026201] about a possible flaw in its derivation. We point out that their claim is incorrect since we have stated explicitly that a term is neglected to derive our model equation in our paper. Furthermore, the integrable model is validated by comparing with the normalized Maxwell equation and other known integrable models. Moreover, we show that a similar approximation has to be performed in deriving the same integrable equation as explained in the Comment.

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.

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.

Time-temperature effect in adhesively bonded joints

NASA Technical Reports Server (NTRS)

Delale, F.; Erdogan, F.

1981-01-01

The viscoelastic analysis of an adhesively bonded lap joint was reconsidered. The adherends are approximated by essentially Reissner plates and the adhesive is linearly viscoelastic. The hereditary integrals are used to model the adhesive. A linear integral differential equations system for the shear and the tensile stress in the adhesive is applied. The equations have constant coefficients and are solved by using Laplace transforms. It is shown that if the temperature variation in time can be approximated by a piecewise constant function, then the method of Laplace transforms can be used to solve the problem. A numerical example is given for a single lap joint under various loading conditions.

ALGORITHM TO REDUCE APPROXIMATION ERROR FROM THE COMPLEX-VARIABLE BOUNDARY-ELEMENT METHOD APPLIED TO SOIL FREEZING.

USGS Publications Warehouse

Hromadka, T.V.; Guymon, G.L.

1985-01-01

An algorithm is presented for the numerical solution of the Laplace equation boundary-value problem, which is assumed to apply to soil freezing or thawing. The Laplace equation is numerically approximated by the complex-variable boundary-element method. The algorithm aids in reducing integrated relative error by providing a true measure of modeling error along the solution domain boundary. This measure of error can be used to select locations for adding, removing, or relocating nodal points on the boundary or to provide bounds for the integrated relative error of unknown nodal variable values along the boundary.

Interpreting the Coulomb-field approximation for generalized-Born electrostatics using boundary-integral equation theory.

PubMed

Bardhan, Jaydeep P

2008-10-14

The importance of molecular electrostatic interactions in aqueous solution has motivated extensive research into physical models and numerical methods for their estimation. The computational costs associated with simulations that include many explicit water molecules have driven the development of implicit-solvent models, with generalized-Born (GB) models among the most popular of these. In this paper, we analyze a boundary-integral equation interpretation for the Coulomb-field approximation (CFA), which plays a central role in most GB models. This interpretation offers new insights into the nature of the CFA, which traditionally has been assessed using only a single point charge in the solute. The boundary-integral interpretation of the CFA allows the use of multiple point charges, or even continuous charge distributions, leading naturally to methods that eliminate the interpolation inaccuracies associated with the Still equation. This approach, which we call boundary-integral-based electrostatic estimation by the CFA (BIBEE/CFA), is most accurate when the molecular charge distribution generates a smooth normal displacement field at the solute-solvent boundary, and CFA-based GB methods perform similarly. Conversely, both methods are least accurate for charge distributions that give rise to rapidly varying or highly localized normal displacement fields. Supporting this analysis are comparisons of the reaction-potential matrices calculated using GB methods and boundary-element-method (BEM) simulations. An approximation similar to BIBEE/CFA exhibits complementary behavior, with superior accuracy for charge distributions that generate rapidly varying normal fields and poorer accuracy for distributions that produce smooth fields. This approximation, BIBEE by preconditioning (BIBEE/P), essentially generates initial guesses for preconditioned Krylov-subspace iterative BEMs. Thus, iterative refinement of the BIBEE/P results recovers the BEM solution; excellent agreement is obtained in only a few iterations. The boundary-integral-equation framework may also provide a means to derive rigorous results explaining how the empirical correction terms in many modern GB models significantly improve accuracy despite their simple analytical forms.

Fractional calculus in hydrologic modeling: A numerical perspective

PubMed Central

Benson, David A.; Meerschaert, Mark M.; Revielle, Jordan

2013-01-01

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus. PMID:23524449

High-frequency Born synthetic seismograms based on coupled normal modes

USGS Publications Warehouse

Pollitz, F.

2011-01-01

High-frequency and full waveform synthetic seismograms on a 3-D laterally heterogeneous earth model are simulated using the theory of coupled normal modes. The set of coupled integral equations that describe the 3-D response are simplified into a set of uncoupled integral equations by using the Born approximation to calculate scattered wavefields and the pure-path approximation to modulate the phase of incident and scattered wavefields. This depends upon a decomposition of the aspherical structure into smooth and rough components. The uncoupled integral equations are discretized and solved in the frequency domain, and time domain results are obtained by inverse Fourier transform. Examples show the utility of the normal mode approach to synthesize the seismic wavefields resulting from interaction with a combination of rough and smooth structural heterogeneities. This approach is applied to an ~4 Hz shallow crustal wave propagation around the site of the San Andreas Fault Observatory at Depth (SAFOD). ?? The Author Geophysical Journal International ?? 2011 RAS.

Approximation for limit cycles and their isochrons.

PubMed

Demongeot, Jacques; Françoise, Jean-Pierre

2006-12-01

Local analysis of trajectories of dynamical systems near an attractive periodic orbit displays the notion of asymptotic phase and isochrons. These notions are quite useful in applications to biosciences. In this note, we give an expression for the first approximation of equations of isochrons in the setting of perturbations of polynomial Hamiltonian systems. This method can be generalized to perturbations of systems that have a polynomial integral factor (like the Lotka-Volterra equation).

Pseudodynamic systems approach based on a quadratic approximation of update equations for diffuse optical tomography.

PubMed

Biswas, Samir Kumar; Kanhirodan, Rajan; Vasu, Ram Mohan; Roy, Debasish

2011-08-01

We explore a pseudodynamic form of the quadratic parameter update equation for diffuse optical tomographic reconstruction from noisy data. A few explicit and implicit strategies for obtaining the parameter updates via a semianalytical integration of the pseudodynamic equations are proposed. Despite the ill-posedness of the inverse problem associated with diffuse optical tomography, adoption of the quadratic update scheme combined with the pseudotime integration appears not only to yield higher convergence, but also a muted sensitivity to the regularization parameters, which include the pseudotime step size for integration. These observations are validated through reconstructions with both numerically generated and experimentally acquired data.

New integrable model of propagation of the few-cycle pulses in an anisotropic microdispersed medium

NASA Astrophysics Data System (ADS)

Sazonov, S. V.; Ustinov, N. V.

2018-03-01

We investigate the propagation of the few-cycle electromagnetic pulses in the anisotropic microdispersed medium. The effects of the anisotropy and spatial dispersion of the medium are created by the two sorts of the two-level atoms. The system of the material equations describing an evolution of the states of the atoms and the wave equations for the ordinary and extraordinary components of the pulses is derived. By applying the approximation of the sudden excitation to exclude the material variables, we reduce this system to the single nonlinear wave equation that generalizes the modified sine-Gordon equation and the Rabelo-Fokas equation. It is shown that this equation is integrable by means of the inverse scattering transformation method if an additional restriction on the parameters is imposed. The multisoliton solutions of this integrable generalization are constructed and investigated.

Modeling rainfall infiltration on hillslopes using Flux-concentration relation and time compression approximation

NASA Astrophysics Data System (ADS)

Wang, Jie; Chen, Li; Yu, Zhongbo

2018-02-01

Rainfall infiltration on hillslopes is an important issue in hydrology, which is related to many environmental problems, such as flood, soil erosion, and nutrient and contaminant transport. This study aimed to improve the quantification of infiltration on hillslopes under both steady and unsteady rainfalls. Starting from Darcy's law, an analytical integral infiltrability equation was derived for hillslope infiltration by use of the flux-concentration relation. Based on this equation, a simple scaling relation linking the infiltration times on hillslopes and horizontal planes was obtained which is applicable for both small and large times and can be used to simplify the solution procedure of hillslope infiltration. The infiltrability equation also improved the estimation of ponding time for infiltration under rainfall conditions. For infiltration after ponding, the time compression approximation (TCA) was applied together with the infiltrability equation. To improve the computational efficiency, the analytical integral infiltrability equation was approximated with a two-term power-like function by nonlinear regression. Procedures of applying this approach to both steady and unsteady rainfall conditions were proposed. To evaluate the performance of the new approach, it was compared with the Green-Ampt model for sloping surfaces by Chen and Young (2006) and Richards' equation. The proposed model outperformed the sloping Green-Ampt, and both ponding time and infiltration predictions agreed well with the solutions of Richards' equation for various soil textures, slope angles, initial water contents, and rainfall intensities for both steady and unsteady rainfalls.

The Green's matrix and the boundary integral equations for analysis of time-harmonic dynamics of elastic helical springs.

PubMed

Sorokin, Sergey V

2011-03-01

Helical springs serve as vibration isolators in virtually any suspension system. Various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions. The method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time-harmonic dynamics of elastic springs in the framework of Bernoulli-Euler beam theory. In this paper, the derivations of the Green's matrix, of the Somigliana's identities, and of the boundary integral equations are presented. The vibrational power transmission in an infinitely long spring is analyzed by means of the Green's matrix. The eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations. In the course of analysis, the essential features and advantages of the method of boundary integral equations are highlighted. The reported analytical results may be used to study the time-harmonic motion in any wave guide governed by a system of linear differential equations in a single spatial coordinate along its axis. © 2011 Acoustical Society of America

Nonlinear Solver Approaches for the Diffusive Wave Approximation to the Shallow Water Equations

NASA Astrophysics Data System (ADS)

Collier, N.; Knepley, M.

2015-12-01

The diffusive wave approximation to the shallow water equations (DSW) is a doubly-degenerate, nonlinear, parabolic partial differential equation used to model overland flows. Despite its challenges, the DSW equation has been extensively used to model the overland flow component of various integrated surface/subsurface models. The equation's complications become increasingly problematic when ponding occurs, a feature which becomes pervasive when solving on large domains with realistic terrain. In this talk I discuss the various forms and regularizations of the DSW equation and highlight their effect on the solvability of the nonlinear system. In addition to this analysis, I present results of a numerical study which tests the applicability of a class of composable nonlinear algebraic solvers recently added to the Portable, Extensible, Toolkit for Scientific Computation (PETSc).

Implementation of parallel moment equations in NIMROD

NASA Astrophysics Data System (ADS)

Lee, Hankyu Q.; Held, Eric D.; Ji, Jeong-Young

2017-10-01

As collisionality is low (the Knudsen number is large) in many plasma applications, kinetic effects become important, particularly in parallel dynamics for magnetized plasmas. Fluid models can capture some kinetic effects when integral parallel closures are adopted. The adiabatic and linear approximations are used in solving general moment equations to obtain the integral closures. In this work, we present an effort to incorporate non-adiabatic (time-dependent) and nonlinear effects into parallel closures. Instead of analytically solving the approximate moment system, we implement exact parallel moment equations in the NIMROD fluid code. The moment code is expected to provide a natural convergence scheme by increasing the number of moments. Work in collaboration with the PSI Center and supported by the U.S. DOE under Grant Nos. DE-SC0014033, DE-SC0016256, and DE-FG02-04ER54746.

Note on the eigensolution of a homogeneous equation with semi-infinite domain

NASA Technical Reports Server (NTRS)

Wadia, A. R.

1980-01-01

The 'variation-iteration' method using Green's functions to find the eigenvalues and the corresponding eigenfunctions of a homogeneous Fredholm integral equation is employed for the stability analysis of fluid hydromechanics problems with a semiinfinite (infinite) domain of application. The objective of the study is to develop a suitable numerical approach to the solution of such equations in order to better understand the full set of equations for 'real-world' flow models. The study involves a search for a suitable value of the length of the domain which is a fair finite approximation to infinity, which makes the eigensolution an approximation dependent on the length of the interval chosen. In the examples investigated y = 1 = a seems to be the best approximation of infinity; for y greater than unity this method fails due to the polynomial nature of Green's functions.

Brownian microhydrodynamics of active filaments.

PubMed

Laskar, Abhrajit; Adhikari, R

2015-12-21

Slender bodies capable of spontaneous motion in the absence of external actuation in an otherwise quiescent fluid are common in biological, physical and technological contexts. The interplay between the spontaneous fluid flow, Brownian motion, and the elasticity of the body presents a challenging fluid-structure interaction problem. Here, we model this problem by approximating the slender body as an elastic filament that can impose non-equilibrium velocities or stresses at the fluid-structure interface. We derive equations of motion for such an active filament by enforcing momentum conservation in the fluid-structure interaction and assuming slow viscous flow in the fluid. The fluid-structure interaction is obtained, to any desired degree of accuracy, through the solution of an integral equation. A simplified form of the equations of motion, which allows for efficient numerical solutions, is obtained by applying the Kirkwood-Riseman superposition approximation to the integral equation. We use this form of equation of motion to study dynamical steady states in free and hinged minimally active filaments. Our model provides the foundation to study collective phenomena in momentum-conserving, Brownian, active filament suspensions.

A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

NASA Astrophysics Data System (ADS)

Ishkhanyan, Tigran A.; Krainov, Vladimir P.; Ishkhanyan, Artur M.

2018-05-01

We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term x-1/2 with arbitrary strength and a repulsive centrifugal barrier core x-2 with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.

Use of source distributions for evaluating theoretical aerodynamics of thin finite wings at supersonic speeds

NASA Technical Reports Server (NTRS)

Evvard, John C

1950-01-01

A series of publications on the source-distribution methods for evaluating the aerodynamics of thin wings at supersonic speeds is summarized, extended, and unified. Included in the first part are the deviations of: (a) the linearized partial-differential equation for unsteady flow at a substantially constant Mach number. b) The source-distribution solution for the perturbation-velocity potential that satisfies the boundary conditions of tangential flow at the surface and in the plane of the wing; and (c) the integral equation for determining the strength and the location of sources to describe the interaction effects (as represented by upwash) of the bottom and top wing surfaces through the region between the finite wing boundary and the foremost Mach wave. The second part deals with steady-state thin-wing problems. The third part of the report approximates the integral equation for unsteady upwash and includes a solution of approximate equation. Expressions are then derived to evaluate the load distributions for time-dependent finite-wing motions.

A method for exponential propagation of large systems of stiff nonlinear differential equations

NASA Technical Reports Server (NTRS)

Friesner, Richard A.; Tuckerman, Laurette S.; Dornblaser, Bright C.; Russo, Thomas V.

1989-01-01

A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.

An Integrable Approximation for the Fermi Pasta Ulam Lattice

NASA Astrophysics Data System (ADS)

Rink, Bob

This contribution presents a review of results obtained from computations of approximate equations of motion for the Fermi-Pasta-Ulam lattice. These approximate equations are obtained as a finite-dimensional Birkhoff normal form. It turns out that in many cases, the Birkhoff normal form is suitable for application of the KAM theorem. In particular, this proves Nishida's 1971 conjecture stating that almost all low-energetic motions of the anharmonic Fermi-Pasta-Ulam lattice with fixed endpoints are quasi-periodic. The proof is based on the formal Birkhoff normal form computations of Nishida, the KAM theorem and discrete symmetry considerations.

Modified Chebyshev Picard Iteration for Efficient Numerical Integration of Ordinary Differential Equations

NASA Astrophysics Data System (ADS)

Macomber, B.; Woollands, R. M.; Probe, A.; Younes, A.; Bai, X.; Junkins, J.

2013-09-01

Modified Chebyshev Picard Iteration (MCPI) is an iterative numerical method for approximating solutions of linear or non-linear Ordinary Differential Equations (ODEs) to obtain time histories of system state trajectories. Unlike other step-by-step differential equation solvers, the Runge-Kutta family of numerical integrators for example, MCPI approximates long arcs of the state trajectory with an iterative path approximation approach, and is ideally suited to parallel computation. Orthogonal Chebyshev Polynomials are used as basis functions during each path iteration; the integrations of the Picard iteration are then done analytically. Due to the orthogonality of the Chebyshev basis functions, the least square approximations are computed without matrix inversion; the coefficients are computed robustly from discrete inner products. As a consequence of discrete sampling and weighting adopted for the inner product definition, Runge phenomena errors are minimized near the ends of the approximation intervals. The MCPI algorithm utilizes a vector-matrix framework for computational efficiency. Additionally, all Chebyshev coefficients and integrand function evaluations are independent, meaning they can be simultaneously computed in parallel for further decreased computational cost. Over an order of magnitude speedup from traditional methods is achieved in serial processing, and an additional order of magnitude is achievable in parallel architectures. This paper presents a new MCPI library, a modular toolset designed to allow MCPI to be easily applied to a wide variety of ODE systems. Library users will not have to concern themselves with the underlying mathematics behind the MCPI method. Inputs are the boundary conditions of the dynamical system, the integrand function governing system behavior, and the desired time interval of integration, and the output is a time history of the system states over the interval of interest. Examples from the field of astrodynamics are presented to compare the output from the MCPI library to current state-of-practice numerical integration methods. It is shown that MCPI is capable of out-performing the state-of-practice in terms of computational cost and accuracy.

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.

A method for computing the kernel of the downwash integral equation for arbitrary complex frequencies

NASA Technical Reports Server (NTRS)

Desmarais, R. N.; Rowe, W. S.

1984-01-01

For the design of active controls to stabilize flight vehicles, which requires the use of unsteady aerodynamics that are valid for arbitrary complex frequencies, algorithms are derived for evaluating the nonelementary part of the kernel of the integral equation that relates unsteady pressure to downwash. This part of the kernel is separated into an infinite limit integral that is evaluated using Bessel and Struve functions and into a finite limit integral that is expanded in series and integrated termwise in closed form. The developed series expansions gave reliable answers for all complex reduced frequencies and executed faster than exponential approximations for many pressure stations.

A multi-domain spectral method for time-fractional differential equations

NASA Astrophysics Data System (ADS)

Chen, Feng; Xu, Qinwu; Hesthaven, Jan S.

2015-07-01

This paper proposes an approach for high-order time integration within a multi-domain setting for time-fractional differential equations. Since the kernel is singular or nearly singular, two main difficulties arise after the domain decomposition: how to properly account for the history/memory part and how to perform the integration accurately. To address these issues, we propose a novel hybrid approach for the numerical integration based on the combination of three-term-recurrence relations of Jacobi polynomials and high-order Gauss quadrature. The different approximations used in the hybrid approach are justified theoretically and through numerical examples. Based on this, we propose a new multi-domain spectral method for high-order accurate time integrations and study its stability properties by identifying the method as a generalized linear method. Numerical experiments confirm hp-convergence for both time-fractional differential equations and time-fractional partial differential equations.

Generalization of Boundary-Layer Momentum-Integral Equations to Three-Dimensional Flows Including Those of Rotating System

NASA Technical Reports Server (NTRS)

Mager, Arthur

1952-01-01

The Navier-Stokes equations of motion and the equation of continuity are transformed so as to apply to an orthogonal curvilinear coordinate system rotating with a uniform angular velocity about an arbitrary axis in space. A usual simplification of these equations as consistent with the accepted boundary-layer theory and an integration of these equations through the boundary layer result in boundary-layer momentum-integral equations for three-dimensional flows that are applicable to either rotating or nonrotating fluid boundaries. These equations are simplified and an approximate solution in closed integral form is obtained for a generalized boundary-layer momentum-loss thickness and flow deflection at the wall in the turbulent case. A numerical evaluation of this solution carried out for data obtained in a curving nonrotating duct shows a fair quantitative agreement with the measures values. The form in which the equations are presented is readily adaptable to cases of steady, three-dimensional, incompressible boundary-layer flow like that over curved ducts or yawed wings; and it also may be used to describe the boundary-layer flow over various rotating surfaces, thus applying to turbomachinery, propellers, and helicopter blades.

Plane elasto-plastic analysis of v-notched plate under bending by boundary integral equation method. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Rzasnicki, W.

1973-01-01

A method of solution is presented, which, when applied to the elasto-plastic analysis of plates having a v-notch on one edge and subjected to pure bending, will produce stress and strain fields in much greater detail than presently available. Application of the boundary integral equation method results in two coupled Fredholm-type integral equations, subject to prescribed boundary conditions. These equations are replaced by a system of simultaneous algebraic equations and solved by a successive approximation method employing Prandtl-Reuss incremental plasticity relations. The method is first applied to number of elasto-static problems and the results compared with available solutions. Good agreement is obtained in all cases. The elasto-plastic analysis provides detailed stress and strain distributions for several cases of plates with various notch angles and notch depths. A strain hardening material is assumed and both plane strain and plane stress conditions are considered.

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.

Comparison of numerical techniques for integration of stiff ordinary differential equations arising in combustion chemistry

NASA Technical Reports Server (NTRS)

Radhakrishnan, K.

1984-01-01

The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.

Approximate solutions of acoustic 3D integral equation and their application to seismic modeling and full-waveform inversion

NASA Astrophysics Data System (ADS)

Malovichko, M.; Khokhlov, N.; Yavich, N.; Zhdanov, M.

2017-10-01

Over the recent decades, a number of fast approximate solutions of Lippmann-Schwinger equation, which are more accurate than classic Born and Rytov approximations, were proposed in the field of electromagnetic modeling. Those developments could be naturally extended to acoustic and elastic fields; however, until recently, they were almost unknown in seismology. This paper presents several solutions of this kind applied to acoustic modeling for both lossy and lossless media. We evaluated the numerical merits of those methods and provide an estimation of their numerical complexity. In our numerical realization we use the matrix-free implementation of the corresponding integral operator. We study the accuracy of those approximate solutions and demonstrate, that the quasi-analytical approximation is more accurate, than the Born approximation. Further, we apply the quasi-analytical approximation to the solution of the inverse problem. It is demonstrated that, this approach improves the estimation of the data gradient, comparing to the Born approximation. The developed inversion algorithm is based on the conjugate-gradient type optimization. Numerical model study demonstrates that the quasi-analytical solution significantly reduces computation time of the seismic full-waveform inversion. We also show how the quasi-analytical approximation can be extended to the case of elastic wavefield.

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.

Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations.

PubMed

Hasegawa, Chihiro; Duffull, Stephen B

2018-02-01

Pharmacokinetic-pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge-Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave. In circumstances when either a linear ODE is particularly desirable or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.

Exact and Approximate Solutions for the Decades-Old Michaelis-Menten Equation: Progress-Curve Analysis through Integrated Rate Equations

ERIC Educational Resources Information Center

Golicnik, Marko

2011-01-01

The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate "V", and the Michaelis constant "K"[subscript M]) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to…

Estimating the Gibbs energy of hydration from molecular dynamics trajectories obtained by integral equations of the theory of liquids in the RISM approximation

NASA Astrophysics Data System (ADS)

Tikhonov, D. A.; Sobolev, E. V.

2011-04-01

A method of integral equations of the theory of liquids in the reference interaction site model (RISM) approximation is used to estimate the Gibbs energy averaged over equilibrium trajectories computed by molecular mechanics. Peptide oxytocin is selected as the object of interest. The Gibbs energy is calculated using all chemical potential formulas introduced in the RISM approach for the excess chemical potential of solvation and is compared with estimates by the generalized Born model. Some formulas are shown to give the wrong sign of Gibbs energy changes when peptide passes from the gas phase into water environment; the other formulas give overestimated Gibbs energy changes with the right sign. Note that allowance for the repulsive correction in the approximate analytical expressions for the Gibbs energy derived by thermodynamic perturbation theory is not a remedy.

Multiparticle dynamics in an expanding universe

NASA Astrophysics Data System (ADS)

Anderson, James L.

1995-11-01

Approximate equations of motion for multiparticle systems in an expanding Einstein-deSitter universe are derived from the Einstein-Maxwell field equations using the Einstein-Infeld-Hoffmann surface integral method. At the Newtonian level of approximation one finds that, in comoving coordinates, both the Newtonian gravitational and Coulomb interactions in these equations are multiplied by the inverse third power of the scale factor R(t) appearing in the Einstein-deSitter field and they acquire a cosmic ``drag'' term. Nevertheless, both the period and luminosity size of bound two-body systems whose period is small compared to the Hubble time are found to be independent of t.

Optimized effective potential in real time: Problems and prospects in time-dependent density-functional theory

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

Mundt, Michael; Kuemmel, Stephan

2006-08-15

The integral equation for the time-dependent optimized effective potential (TDOEP) in time-dependent density-functional theory is transformed into a set of partial-differential equations. These equations only involve occupied Kohn-Sham orbitals and orbital shifts resulting from the difference between the exchange-correlation potential and the orbital-dependent potential. Due to the success of an analog scheme in the static case, a scheme that propagates orbitals and orbital shifts in real time is a natural candidate for an exact solution of the TDOEP equation. We investigate the numerical stability of such a scheme. An approximation beyond the Krieger-Li-Iafrate approximation for the time-dependent exchange-correlation potential ismore » analyzed.« less

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.

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.

The Equations of Oceanic Motions

NASA Astrophysics Data System (ADS)

Müller, Peter

2006-10-01

Modeling and prediction of oceanographic phenomena and climate is based on the integration of dynamic equations. The Equations of Oceanic Motions derives and systematically classifies the most common dynamic equations used in physical oceanography, from large scale thermohaline circulations to those governing small scale motions and turbulence. After establishing the basic dynamical equations that describe all oceanic motions, M|ller then derives approximate equations, emphasizing the assumptions made and physical processes eliminated. He distinguishes between geometric, thermodynamic and dynamic approximations and between the acoustic, gravity, vortical and temperature-salinity modes of motion. Basic concepts and formulae of equilibrium thermodynamics, vector and tensor calculus, curvilinear coordinate systems, and the kinematics of fluid motion and wave propagation are covered in appendices. Providing the basic theoretical background for graduate students and researchers of physical oceanography and climate science, this book will serve as both a comprehensive text and an essential reference.

Approximate analysis for repeated eigenvalue problems with applications to controls-structure integrated design

NASA Technical Reports Server (NTRS)

Kenny, Sean P.; Hou, Gene J. W.

1994-01-01

A method for eigenvalue and eigenvector approximate analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The approximate analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all orders). Examples are given which demonstrate the application of such equations for sensitivity and approximate analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.

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.

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.

Integral equation methods for computing likelihoods and their derivatives in the stochastic integrate-and-fire model.

PubMed

Paninski, Liam; Haith, Adrian; Szirtes, Gabor

2008-02-01

We recently introduced likelihood-based methods for fitting stochastic integrate-and-fire models to spike train data. The key component of this method involves the likelihood that the model will emit a spike at a given time t. Computing this likelihood is equivalent to computing a Markov first passage time density (the probability that the model voltage crosses threshold for the first time at time t). Here we detail an improved method for computing this likelihood, based on solving a certain integral equation. This integral equation method has several advantages over the techniques discussed in our previous work: in particular, the new method has fewer free parameters and is easily differentiable (for gradient computations). The new method is also easily adaptable for the case in which the model conductance, not just the input current, is time-varying. Finally, we describe how to incorporate large deviations approximations to very small likelihoods.

Nonlinear evolution equations for surface plasmons for nano-focusing at a Kerr/metallic interface and tapered waveguide

NASA Astrophysics Data System (ADS)

Crutcher, Sihon H.; Osei, Albert; Biswas, Anjan

2012-06-01

Maxwell's equations for a metallic and nonlinear Kerr interface waveguide at the nanoscale can be approximated to a (1+1) D Nonlinear Schrodinger type model equation (NLSE) with appropriate assumptions and approximations. Theoretically, without losses or perturbations spatial plasmon solitons profiles are easily produced. However, with losses, the amplitude or beam profile is no longer stationary and adiabatic parameters have to be considered to understand propagation. For this model, adiabatic parameters are calculated considering losses resulting in linear differential coupled integral equations with constant definite integral coefficients not dependent on the transverse and longitudinal coordinates. Furthermore, by considering another configuration, a waveguide that is an M-NL-M (metal-nonlinear Kerr-metal) that tapers, the tapering can balance the loss experienced at a non-tapered metal/nonlinear Kerr interface causing attenuation of the beam profile, so these spatial plasmon solitons can be produced. In this paper taking into consideration the (1+1)D NLSE model for a tapered waveguide, we derive a one soliton solution based on He's Semi-Inverse Variational Principle (HPV).

Coupling of Coastal Wave Transformation and Computational Fluid Dynamics Models for Seakeeping Analysis

DTIC Science & Technology

2017-04-03

setup in terms of temporal and spatial discretization . The second component was an extension of existing depth-integrated wave models to describe...equations (Abbott, 1976). Discretization schemes involve numerical dispersion and dissipation that distort the true character of the governing equations...represent a leading-order approximation of the Boussinesq-type equations. Tam and Webb (1993) proposed a wavenumber-based discretization scheme to preserve

The derivation and approximation of coarse-grained dynamics from Langevin dynamics

NASA Astrophysics Data System (ADS)

Ma, Lina; Li, Xiantao; Liu, Chun

2016-11-01

We present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. The focus is placed on the form of the memory kernel function, the colored noise, and the second fluctuation-dissipation theorem that connects them. Also presented is a hierarchy of approximations for the memory and random noise terms, using rational approximations in the Laplace domain. These approximations offer increasing accuracy. More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step. Direct sampling of the colored noise can also be avoided within this framework. Therefore, the numerical implementation of the generalized Langevin equation is much more efficient.

Volterra integral equation-factorisation method and nucleus-nucleus elastic scattering

NASA Astrophysics Data System (ADS)

Laha, U.; Majumder, M.; Bhoi, J.

2018-04-01

An approximate solution for the nuclear Hulthén plus atomic Hulthén potentials is constructed by solving the associated Volterra integral equation by series substitution method. Within the framework of supersymmetry-inspired factorisation method, this solution is exploited to construct higher partial wave interactions. The merit of our approach is examined by computing elastic scattering phases of the α {-}α system by the judicious use of phase function method. Reasonable agreements in phase shifts are obtained with standard data.

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.

A new approximation of Fermi-Dirac integrals of order 1/2 for degenerate semiconductor devices

NASA Astrophysics Data System (ADS)

AlQurashi, Ahmed; Selvakumar, C. R.

2018-06-01

There had been tremendous growth in the field of Integrated circuits (ICs) in the past fifty years. Scaling laws mandated both lateral and vertical dimensions to be reduced and a steady increase in doping densities. Most of the modern semiconductor devices have invariably heavily doped regions where Fermi-Dirac Integrals are required. Several attempts have been devoted to developing analytical approximations for Fermi-Dirac Integrals since numerical computations of Fermi-Dirac Integrals are difficult to use in semiconductor devices, although there are several highly accurate tabulated functions available. Most of these analytical expressions are not sufficiently suitable to be employed in semiconductor device applications due to their poor accuracy, the requirement of complicated calculations, and difficulties in differentiating and integrating. A new approximation has been developed for the Fermi-Dirac integrals of the order 1/2 by using Prony's method and discussed in this paper. The approximation is accurate enough (Mean Absolute Error (MAE) = 0.38%) and easy enough to be used in semiconductor device equations. The new approximation of Fermi-Dirac Integrals is applied to a more generalized Einstein Relation which is an important relation in semiconductor devices.

Correlations and the Ring-Kinetic Equation in Dense Sheared Granular Flows

NASA Astrophysics Data System (ADS)

Kumaran, V.

A formal way of deriving fluctuation-correlation relations in densesheared granular media, starting with the Enskog approximation for the collision integral in the Chapman-Enskog theory, is discussed. The correlation correction to the viscosity is obtained using the ring-kinetic equation, in terms of the correlations in the hydrodynamic modes of the linearised Enskog equation. It is shown that the Green-Kubo formula for the shear viscosity emerges from the two-body correlation function obtained from the ring-kinetic equation.

Trajectory characteristics and heating of hypervelocity projectiles having large ballistic coefficients

NASA Technical Reports Server (NTRS)

Tauber, Michael E.

1986-01-01

A simple, approximate equation describing the velocity-density relationship (or velocity-altitude) has been derived from the flight of large ballistic coefficient projectiles launched at high speeds. The calculations obtained by using the approximate equation compared well with results for numerical integrations of the exact equations of motion. The flightpath equation was used to parametrically calculate maximum body decelerations and stagnation pressures for initial velocities from 2 to 6 km/s. Expressions were derived for the stagnation-point convective heating rates and total heat loads. The stagnation-point heating was parametrically calculated for a nonablating wall and an ablating carbon surface. Although the heating rates were very high, the pulse decayed quickly. The total nose-region heat shield weight was conservatively estimated to be only about 1 percent of the body mass.

Numerical analysis of composite STEEL-CONCRETE SECTIONS using integral equation of Volterra

NASA Astrophysics Data System (ADS)

Partov, Doncho; Kantchev, Vesselin

2011-09-01

The paper presents analysis of the stress and deflections changes due to creep in statically determinate composite steel-concrete beam. The mathematical model involves the equation of equilibrium, compatibility and constitutive relationship, i.e. an elastic law for the steel part and an integral-type creep law of Boltzmann — Volterra for the concrete part. On the basis of the theory of the viscoelastic body of Arutyunian-Trost-Bažant for determining the redistribution of stresses in beam section between concrete plate and steel beam with respect to time "t", two independent Volterra integral equations of the second kind have been derived. Numerical method based on linear approximation of the singular kernal function in the integral equation is presented. Example with the model proposed is investigated. The creep functions is suggested by the model CEB MC90-99 and the "ACI 209R-92 model. The elastic modulus of concrete E c (t) is assumed to be constant in time `t'. The obtained results from the both models are compared.

High-precision numerical integration of equations in dynamics

NASA Astrophysics Data System (ADS)

Alesova, I. M.; Babadzanjanz, L. K.; Pototskaya, I. Yu.; Pupysheva, Yu. Yu.; Saakyan, A. T.

2018-05-01

An important requirement for the process of solving differential equations in Dynamics, such as the equations of the motion of celestial bodies and, in particular, the motion of cosmic robotic systems is high accuracy at large time intervals. One of effective tools for obtaining such solutions is the Taylor series method. In this connection, we note that it is very advantageous to reduce the given equations of Dynamics to systems with polynomial (in unknowns) right-hand sides. This allows us to obtain effective algorithms for finding the Taylor coefficients, a priori error estimates at each step of integration, and an optimal choice of the order of the approximation used. In the paper, these questions are discussed and appropriate algorithms are considered.

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.

Computational Modeling of Ultrafast Pulse Propagation in Nonlinear Optical Materials

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Agrawal, Govind P.; Kwak, Dochan (Technical Monitor)

1996-01-01

There is an emerging technology of photonic (or optoelectronic) integrated circuits (PICs or OEICs). In PICs, optical and electronic components are grown together on the same chip. rib build such devices and subsystems, one needs to model the entire chip. Accurate computer modeling of electromagnetic wave propagation in semiconductors is necessary for the successful development of PICs. More specifically, these computer codes would enable the modeling of such devices, including their subsystems, such as semiconductor lasers and semiconductor amplifiers in which there is femtosecond pulse propagation. Here, the computer simulations are made by solving the full vector, nonlinear, Maxwell's equations, coupled with the semiconductor Bloch equations, without any approximations. The carrier is retained in the description of the optical pulse, (i.e. the envelope approximation is not made in the Maxwell's equations), and the rotating wave approximation is not made in the Bloch equations. These coupled equations are solved to simulate the propagation of femtosecond optical pulses in semiconductor materials. The simulations describe the dynamics of the optical pulses, as well as the interband and intraband.

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

Thompson, S.

This report describes the use of several subroutines from the CORLIB core mathematical subroutine library for the solution of a model fluid flow problem. The model consists of the Euler partial differential equations. The equations are spatially discretized using the method of pseudo-characteristics. The resulting system of ordinary differential equations is then integrated using the method of lines. The stiff ordinary differential equation solver LSODE (2) from CORLIB is used to perform the time integration. The non-stiff solver ODE (4) is used to perform a related integration. The linear equation solver subroutines DECOMP and SOLVE are used to solve linearmore » systems whose solutions are required in the calculation of the time derivatives. The monotone cubic spline interpolation subroutines PCHIM and PCHFE are used to approximate water properties. The report describes the use of each of these subroutines in detail. It illustrates the manner in which modules from a standard mathematical software library such as CORLIB can be used as building blocks in the solution of complex problems of practical interest. 9 refs., 2 figs., 4 tabs.« less

Numerical Simulations of Light Bullets, Using The Full Vector, Time Dependent, Nonlinear Maxwell Equations

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1994-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.

Numerical Simulations of Light Bullets, Using The Full Vector, Time Dependent, Nonlinear Maxwell Equations

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1995-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that we currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Karr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.

Approximate and exact numerical integration of the gas dynamic equations

NASA Technical Reports Server (NTRS)

Lewis, T. S.; Sirovich, L.

1979-01-01

A highly accurate approximation and a rapidly convergent numerical procedure are developed for two dimensional steady supersonic flow over an airfoil. Examples are given for a symmetric airfoil over a range of Mach numbers. Several interesting features are found in the calculation of the tail shock and the flow behind the airfoil.

Formally exact integral equation theory of the exchange-only potential in density functional theory: Refined closure approximation

NASA Astrophysics Data System (ADS)

March, N. H.; Nagy, Á.

A fonnally exact integral equation theory for the exchange-only potential Vx(r) in density functional theory was recently set up by Howard and March [I.A. Howard, N.H. March, J. Chem. Phys. 119 (2003) 5789]. It involved a `closure' function P(r) satisfying the exact sum rule ∫ P(r) dr = 0. The simplest choice P(r) = 0 recovers then the approximation proposed by Della Sala and Görling [F. Della Sala, A. Görling, J. Chem. Phys. 115 (2001) 5718] and by Gritsenko and Baerends [O.V. Gritsenko, E.J. Baerends, Phys. Rev. A 64 (2001) 042506]. Here, refined choices of P(r) are proposed, the most direct being based on the KLI (Krieger-Li-Iafrate) approximation. A further choice given some attention is where P(r) involves frontier orbital properties. In particular, the introduction of the LUMO (lowest unoccupied molecular) orbital, along with the energy separation between HOMO (highest occupied molecular orbital) and LUMO levels, should prove a significant step beyond current approximations to the optimized potential method, all of which involve only single-particle occupied orbitals.

Piecewise linear approximation for hereditary control problems

NASA Technical Reports Server (NTRS)

Propst, Georg

1987-01-01

Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

Theoretical Prediction of Pressure Distributions on Nonlifting Airfoils at High Subsonic Speeds

NASA Technical Reports Server (NTRS)

Spreiter, John R; Alksne, Alberta

1955-01-01

Theoretical pressure distributions on nonlifting circular-arc airfoils in two-dimensional flows with high subsonic free-stream velocity are found by determining approximate solutions, through an iteration process, of an integral equation for transonic flow proposed by Oswatitsch. The integral equation stems directly from the small-disturbance theory for transonic flow. This method of analysis possesses the advantage of remaining in the physical, rather than the hodograph, variable and can be applied in airfoils having curved surfaces. After discussion of the derivation of the integral equation and qualitative aspects of the solution, results of calculations carried out for circular-arc airfoils in flows with free-stream Mach numbers up to unity are described. These results indicate most of the principal phenomena observed in experimental studies.

Quantum corrections of the truncated Wigner approximation applied to an exciton transport model.

PubMed

Ivanov, Anton; Breuer, Heinz-Peter

2017-04-01

We modify the path integral representation of exciton transport in open quantum systems such that an exact description of the quantum fluctuations around the classical evolution of the system is possible. As a consequence, the time evolution of the system observables is obtained by calculating the average of a stochastic difference equation which is weighted with a product of pseudoprobability density functions. From the exact equation of motion one can clearly identify the terms that are also present if we apply the truncated Wigner approximation. This description of the problem is used as a basis for the derivation of a new approximation, whose validity goes beyond the truncated Wigner approximation. To demonstrate this we apply the formalism to a donor-acceptor transport model.

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.

Characterizing traveling-wave collisions in granular chains starting from integrable limits: the case of the Korteweg-de Vries equation and the Toda lattice.

PubMed

Shen, Y; Kevrekidis, P G; Sen, S; Hoffman, A

2014-08-01

Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both one-soliton and two-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both copropagating and counterpropagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.

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.

Morphing Continuum Theory: A First Order Approximation to the Balance Laws

NASA Astrophysics Data System (ADS)

Wonnell, Louis; Cheikh, Mohamad Ibrahim; Chen, James

2017-11-01

Morphing Continuum Theory is constructed under the framework of Rational Continuum Mechanics (RCM) for fluid flows with inner structure. This multiscale theory has been successfully emplyed to model turbulent flows. The framework of RCM ensures the mathematical rigor of MCT, but contains new material constants related to the inner structure. The physical meanings of these material constants have yet to be determined. Here, a linear deviation from the zeroth-order Boltzmann-Curtiss distribution function is derived. When applied to the Boltzmann-Curtiss equation, a first-order approximation of the MCT governing equations is obtained. The integral equations are then related to the appropriate material constants found in the heat flux, Cauchy stress, and moment stress terms in the governing equations. These new material properties associated with the inner structure of the fluid are compared with the corresponding integrals, and a clearer physical interpretation of these coefficients emerges. The physical meanings of these material properties is determined by analyzing previous results obtained from numerical simulations of MCT for compressible and incompressible flows. The implications for the physics underlying the MCT governing equations will also be discussed. This material is based upon work supported by the Air Force Office of Scientific Research under Award Number FA9550-17-1-0154.

An exact sum-rule for the Hubbard model: an historical/pedagogical approach

NASA Astrophysics Data System (ADS)

Di Matteo, S.; Claveau, Y.

2017-07-01

The aim of the present article is to derive an exact integral equation for the Green function of the Hubbard model through an equation-of-motion procedure, like in the original Hubbard papers. Though our exact integral equation does not allow to solve the Hubbard model, it represents a strong constraint on its approximate solutions. An analogous sum rule has been already obtained in the literature, through the use of a spectral moment technique. We think however that our equation-of-motion procedure can be more easily related to the historical procedure of the original Hubbard papers. We also discuss examples of possible applications of the sum rule and propose and analyse a solution, fulfilling it, that can be used for a pedagogical introduction to the Mott-Hubbard metal-insulator transition.

Development of an Integrated Modeling Framework for Simulations of Coastal Processes in Deltaic Environments Using High-Performance Computing

DTIC Science & Technology

2008-01-01

exceeds the local water depth. The approximation eliminates the vertical dimension of the elliptic equation that is normally required for the fully non...used for vertical resolution. The shallow water equations (SWE) are a set of non-linear hyperbolic equations. As the equations are derived under...linear standing wave with a wavelength of 10 m in a square 10 m by 10 m basin. The still water depth is 0.5 m. In order to compare with the analytical

Approximation of the exponential integral (well function) using sampling methods

NASA Astrophysics Data System (ADS)

Baalousha, Husam Musa

2015-04-01

Exponential integral (also known as well function) is often used in hydrogeology to solve Theis and Hantush equations. Many methods have been developed to approximate the exponential integral. Most of these methods are based on numerical approximations and are valid for a certain range of the argument value. This paper presents a new approach to approximate the exponential integral. The new approach is based on sampling methods. Three different sampling methods; Latin Hypercube Sampling (LHS), Orthogonal Array (OA), and Orthogonal Array-based Latin Hypercube (OA-LH) have been used to approximate the function. Different argument values, covering a wide range, have been used. The results of sampling methods were compared with results obtained by Mathematica software, which was used as a benchmark. All three sampling methods converge to the result obtained by Mathematica, at different rates. It was found that the orthogonal array (OA) method has the fastest convergence rate compared with LHS and OA-LH. The root mean square error RMSE of OA was in the order of 1E-08. This method can be used with any argument value, and can be used to solve other integrals in hydrogeology such as the leaky aquifer integral.

Finite volume solution of the compressible boundary-layer equations

NASA Technical Reports Server (NTRS)

Loyd, B.; Murman, E. M.

1986-01-01

A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data.

Analytic approximations of Von Kármán plate under arbitrary uniform pressure—equations in integral form

NASA Astrophysics Data System (ADS)

Zhong, XiaoXu; Liao, ShiJun

2018-01-01

Analytic approximations of the Von Kármán's plate equations in integral form for a circular plate under external uniform pressure to arbitrary magnitude are successfully obtained by means of the homotopy analysis method (HAM), an analytic approximation technique for highly nonlinear problems. Two HAM-based approaches are proposed for either a given external uniform pressure Q or a given central deflection, respectively. Both of them are valid for uniform pressure to arbitrary magnitude by choosing proper values of the so-called convergence-control parameters c 1 and c 2 in the frame of the HAM. Besides, it is found that the HAM-based iteration approaches generally converge much faster than the interpolation iterative method. Furthermore, we prove that the interpolation iterative method is a special case of the first-order HAM iteration approach for a given external uniform pressure Q when c 1 = - θ and c 2 = -1, where θ denotes the interpolation iterative parameter. Therefore, according to the convergence theorem of Zheng and Zhou about the interpolation iterative method, the HAM-based approaches are valid for uniform pressure to arbitrary magnitude at least in the special case c 1 = - θ and c 2 = -1. In addition, we prove that the HAM approach for the Von Kármán's plate equations in differential form is just a special case of the HAM for the Von Kármán's plate equations in integral form mentioned in this paper. All of these illustrate the validity and great potential of the HAM for highly nonlinear problems, and its superiority over perturbation techniques.

Calculations of Laminar Heat Transfer Around Cylinders of Arbitrary Cross Section and Transpiration-Cooled Walls with Application to Turbine Blade Cooling

NASA Technical Reports Server (NTRS)

Eckert, E.R.G.; Livingood, John N.B.

1951-01-01

An approximate method for development of flow and thermal boundary layers in laminar regime on cylinders with arbitrary cross section and transpiration-cooled walls is obtained by use of Karman's integrated momentum equation and an analogous heat-flow equation. Incompressible flow with constant property values throughout boundary layer is assumed. Shape parameters for approximated velocity and temperature profiles and functions necessary for solution of boundary-layer equations are presented as charts, reducing calculations to a minimum. The method is applied to determine local heat-transfer coefficients and surface temperature-cooled turbine blades for a given flow rate. Coolant flow distributions necessary for maintaining uniform blade temperatures are also determined.

{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

Real-time approximate optimal guidance laws for the advanced launch system

NASA Technical Reports Server (NTRS)

Speyer, Jason L.; Feeley, Timothy; Hull, David G.

1989-01-01

An approach to optimal ascent guidance for a launch vehicle is developed using an expansion technique. The problem is to maximize the payload put into orbit subject to the equations of motion of a rocket over a rotating spherical earth. It is assumed that the thrust and gravitational forces dominate over the aerodynamic forces. It is shown that these forces can be separated by a small parameter epsilon, where epsilon is the ratio of the atmospheric scale height to the radius of the earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in a series where the zeroth-order term (epsilon = 0) can be obtained in closed form. The zeroth-order problem is that of putting maximum payload into orbit subject to the equations of motion of a rocket in a vacuum over a flat earth. The neglected inertial and aerodynamic terms are included in higher order terms of the expansion, which are determined from the solution of first-order linear partial differential equations requiring only quadrature integrations. These quadrature integrations can be performed rapidly, so that real-time approximate optimization can be used to construct the launch guidance law.

Bifurcations of large networks of two-dimensional integrate and fire neurons.

PubMed

Nicola, Wilten; Campbell, Sue Ann

2013-08-01

Recently, a class of two-dimensional integrate and fire models has been used to faithfully model spiking neurons. This class includes the Izhikevich model, the adaptive exponential integrate and fire model, and the quartic integrate and fire model. The bifurcation types for the individual neurons have been thoroughly analyzed by Touboul (SIAM J Appl Math 68(4):1045-1079, 2008). However, when the models are coupled together to form networks, the networks can display bifurcations that an uncoupled oscillator cannot. For example, the networks can transition from firing with a constant rate to burst firing. This paper introduces a technique to reduce a full network of this class of neurons to a mean field model, in the form of a system of switching ordinary differential equations. The reduction uses population density methods and a quasi-steady state approximation to arrive at the mean field system. Reduced models are derived for networks with different topologies and different model neurons with biologically derived parameters. The mean field equations are able to qualitatively and quantitatively describe the bifurcations that the full networks display. Extensions and higher order approximations are discussed.

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.

Eigen solutions and entropic system for Hellmann potential in the presence of the Schrödinger equation

NASA Astrophysics Data System (ADS)

Onate, C. A.; Onyeaju, M. C.; Ikot, A. N.; Ebomwonyi, O.

2017-11-01

By using the supersymmetric approach, we studied the approximate analytic solutions of the three-dimensional Schrödinger equation with the Hellmann potential by applying a suitable approximation scheme to the centrifugal term. The solutions of other useful potentials, such as Coulomb potential and Yukawa potential, are obtained by transformation of variables from the Hellmann potential. Finally, we calculated the Tsallis entropy and Rényi entropy both in position and momentum spaces under the Hellmann potential using integral method. The effects of these entropies on the angular momentum quantum number are investigated in detail.

Galerkin's Method and the Double P$sub 1$ approximation for Thermal Flux Calculation; IL METODO DI GALERKIN E LA DOPPIA P$sub 1$ PER IL CALCOLO DEL FLUSSO TERMICO

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

Daneri, A.; Daneri, A.

1964-01-01

The program DESTHEC DP, in FORTRAN MONITOR for the IBM 7090, solves the transport equation for thermal neutrons in slab geometry. For the energy, Galerkin's method with the double P/sub 1/ approximation is used, Comparison shows good agreement between DESTHEC DP results and results obtained by the THERMOS program, which solves the transport equation in integral form. The theory is presented, and input and output are discussed. Numerical results are included, as well as the program listing. (D.C.W.)

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

NASA Astrophysics Data System (ADS)

Tracy, Craig A.; Widom, Harold

The small t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, , have the representation where Kk$ are integral operators. This class includes the n-periodic cylindrical Toda equations. For n=2 our results reduce to the previously computed asymptotics of the 2D radial sinh-Gordon equation and for n=3 (and with an additional symmetry constraint) they reduce to earlier results for the radial Bullough-Dodd equation. Both of these special cases are examples of Painlevé III and have arisen in various applications. The asymptotics of are derived by computing the small t asymptotics where explicit formulas are given for the quantities ak and bk. The method consists of showing that the resolvent operator of Kk has an approximation in terms of resolvents of certain Wiener-Hopf operators, for which there are explicit integral formulas.

Analysis of Three-Dimensional, Nonlinear Development of Wave-Like Structure in a Compressible Round Jet

NASA Technical Reports Server (NTRS)

Dahl, Milo D.; Mankbadi, Reda R.

2002-01-01

An analysis of the nonlinear development of the large-scale structures or instability waves in compressible round jets was conducted using the integral energy method. The equations of motion were decomposed into two sets of equations; one set governing the mean flow motion and the other set governing the large-scale structure motion. The equations in each set were then combined to derive kinetic energy equations that were integrated in the radial direction across the jet after the boundary-layer approximations were applied. Following the application of further assumptions regarding the radial shape of the mean flow and the large structures, equations were derived that govern the nonlinear, streamwise development of the large structures. Using numerically generated mean flows, calculations show the energy exchanges and the effects of the initial amplitude on the coherent structure development in the jet.

A new treatment of nonlocality in scattering process

NASA Astrophysics Data System (ADS)

Upadhyay, N. J.; Bhagwat, A.; Jain, B. K.

2018-01-01

Nonlocality in the scattering potential leads to an integro-differential equation. In this equation nonlocality enters through an integral over the nonlocal potential kernel. The resulting Schrödinger equation is usually handled by approximating r,{r}{\\prime }-dependence of the nonlocal kernel. The present work proposes a novel method to solve the integro-differential equation. The method, using the mean value theorem of integral calculus, converts the nonhomogeneous term to a homogeneous term. The effective local potential in this equation turns out to be energy independent, but has relative angular momentum dependence. This method is accurate and valid for any form of nonlocality. As illustrative examples, the total and differential cross sections for neutron scattering off 12C, 56Fe and 100Mo nuclei are calculated with this method in the low energy region (up to 10 MeV) and are found to be in reasonable accord with the experiments.

Application of the Green's function method for 2- and 3-dimensional steady transonic flows

NASA Technical Reports Server (NTRS)

Tseng, K.

1984-01-01

A Time-Domain Green's function method for the nonlinear time-dependent three-dimensional aerodynamic potential equation is presented. The Green's theorem is being used to transform the partial differential equation into an integro-differential-delay equation. Finite-element and finite-difference methods are employed for the spatial and time discretizations to approximate the integral equation by a system of differential-delay equations. Solution may be obtained by solving for this nonlinear simultaneous system of equations in time. This paper discusses the application of the method to the Transonic Small Disturbance Equation and numerical results for lifting and nonlifting airfoils and wings in steady flows are presented.

Size-dependent resonance frequencies of cantilevered and bridged nanosensors

NASA Astrophysics Data System (ADS)

Shi, W.; Zou, J.; Lee, K. Y.; Li, X. F.

2018-03-01

This paper studies transverse vibration of nanoscale cantilevered and bridged sensors carrying a nanoparticle. The nanoscale sensors are modelled as Euler-Bernoulli beams with surface effect and nanoparticle as a concentrated mass. Frequency equations of cantilevered and bridged beam-mass system are derived and exact resonance frequencies are calculated. An alternative Fredholm integral equation method is used to obtain an approximate explicit expression for the fundamental frequency for both cases. A comparison between the approximate and analytical results is made and the approximation accuracy is satisfactory. The influences of the residual surface stress, surface elasticity, and attached mass on the resonance frequencies and mode shapes are discussed. These results are useful to illustrate the surface phenomena and are helpful to design micro-/nano-mechanical sensors.

Exact analytical formulae for linearly distributed vortex and source sheets in uence computation in 2D vortex methods

NASA Astrophysics Data System (ADS)

Kuzmina, K. S.; Marchevsky, I. K.; Ryatina, E. P.

2017-11-01

We consider the methodology of numerical schemes development for two-dimensional vortex method. We describe two different approaches to deriving integral equation for unknown vortex sheet intensity. We simulate the velocity of the surface line of an airfoil as the influence of attached vortex and source sheets. We consider a polygonal approximation of the airfoil and assume intensity distributions of free and attached vortex sheets and attached source sheet to be approximated with piecewise constant or piecewise linear (continuous or discontinuous) functions. We describe several specific numerical schemes that provide different accuracy and have a different computational cost. The study shows that a Galerkin-type approach to solving boundary integral equation requires computing several integrals and double integrals over the panels. We obtain exact analytical formulae for all the necessary integrals, which makes it possible to raise significantly the accuracy of vortex sheet intensity computation and improve the quality of velocity and vorticity field representation, especially in proximity to the surface line of the airfoil. All the formulae are written down in the invariant form and depend only on the geometric relationship between the positions of the beginnings and ends of the panels.

Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks.

PubMed

Bressloff, Paul C

2015-01-01

We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

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.

Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations

NASA Astrophysics Data System (ADS)

Gerdt, Vladimir P.; Blinkov, Yuri A.; Mozzhilkin, Vladimir V.

2006-05-01

In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.

A fast and well-conditioned spectral method for singular integral equations

NASA Astrophysics Data System (ADS)

Slevinsky, Richard Mikael; Olver, Sheehan

2017-03-01

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in O (m2 n) operations using an adaptive QR factorization, where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to O (mn) operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The JULIA software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface.

An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics

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

Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir; The Laboratory of Quantum Information Processing, Yazd University, Yazd; Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir

Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Errormore » analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.« less

Radiative transfer equation accounting for rotational Raman scattering and its solution by the discrete-ordinates method

NASA Astrophysics Data System (ADS)

Rozanov, Vladimir V.; Vountas, Marco

2014-01-01

Rotational Raman scattering of solar light in Earth's atmosphere leads to the filling-in of Fraunhofer and telluric lines observed in the reflected spectrum. The phenomenological derivation of the inelastic radiative transfer equation including rotational Raman scattering is presented. The different forms of the approximate radiative transfer equation with first-order rotational Raman scattering terms are obtained employing the Cabannes, Rayleigh, and Cabannes-Rayleigh scattering models. The solution of these equations is considered in the framework of the discrete-ordinates method using rigorous and approximate approaches to derive particular integrals. An alternative forward-adjoint technique is suggested as well. A detailed description of the model including the exact spectral matching and a binning scheme that significantly speeds up the calculations is given. The considered solution techniques are implemented in the radiative transfer software package SCIATRAN and a specified benchmark setup is presented to enable readers to compare with own results transparently.

Exact PDF equations and closure approximations for advective-reactive transport

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

Venturi, D.; Tartakovsky, Daniel M.; Tartakovsky, Alexandre M.

2013-06-01

Mathematical models of advection–reaction phenomena rely on advective flow velocity and (bio) chemical reaction rates that are notoriously random. By using functional integral methods, we derive exact evolution equations for the probability density function (PDF) of the state variables of the advection–reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions. These PDF equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly-correlated random field. Our analytical solution is then used to investigate the accuracy and robustness of the recentlymore » proposed large-eddy diffusivity (LED) closure approximation [1]. We find that the solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates, is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty.« less

Mean-force-field and mean-spherical approximations for the electric microfield distribution at a charged point in the charged-hard-particles fluid

NASA Astrophysics Data System (ADS)

Rosenfeld, Yaakov

1989-01-01

The linearized mean-force-field approximation, leading to a Gaussian distribution, provides an exact formal solution to the mean-spherical integral equation model for the electric microfield distribution at a charged point in the general charged-hard-particles fluid. Lado's explicit solution for plasmas immediately follows this general observation.

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.

Boundary Control of Linear Uncertain 1-D Parabolic PDE Using Approximate Dynamic Programming.

PubMed

Talaei, Behzad; Jagannathan, Sarangapani; Singler, John

2018-04-01

This paper develops a near optimal boundary control method for distributed parameter systems governed by uncertain linear 1-D parabolic partial differential equations (PDE) by using approximate dynamic programming. A quadratic surface integral is proposed to express the optimal cost functional for the infinite-dimensional state space. Accordingly, the Hamilton-Jacobi-Bellman (HJB) equation is formulated in the infinite-dimensional domain without using any model reduction. Subsequently, a neural network identifier is developed to estimate the unknown spatially varying coefficient in PDE dynamics. Novel tuning law is proposed to guarantee the boundedness of identifier approximation error in the PDE domain. A radial basis network (RBN) is subsequently proposed to generate an approximate solution for the optimal surface kernel function online. The tuning law for near optimal RBN weights is created, such that the HJB equation error is minimized while the dynamics are identified and closed-loop system remains stable. Ultimate boundedness (UB) of the closed-loop system is verified by using the Lyapunov theory. The performance of the proposed controller is successfully confirmed by simulation on an unstable diffusion-reaction process.

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.

Solution of a cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula

NASA Astrophysics Data System (ADS)

Remizov, I. D.

2012-07-01

The Cauchy problem for a class of diffusion equations in a Hilbert space is studied. It is proved that the Cauchy problem in well posed in the class of uniform limits of infinitely smooth bounded cylindrical functions on the Hilbert space, and the solution is presented in the form of the so-called Feynman formula, i.e., a limit of multiple integrals against a gaussian measure as the multiplicity tends to infinity. It is also proved that the solution of the Cauchy problem depends continuously on the diffusion coefficient. A process reducing an approximate solution of an infinite-dimensional diffusion equation to finding a multiple integral of a real function of finitely many real variables is indicated.

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.

Relativistic bound states in three space-time dimensions in Minkowski space

NASA Astrophysics Data System (ADS)

Gutierrez, C.; Gigante, V.; Frederico, T.; Tomio, Lauro

2016-01-01

With the aim to derive a workable framework for bound states in Minkowski space, we have investigated the Nakanishi perturbative integral representation of the Bethe-Salpeter (BS) amplitude in two-dimensions (2D) in space and time (2+1). The homogeneous BS amplitude, projected onto the light-front plane, is used to derive an equation for the Nakanishi weight function. The formal development is illustrated in detail and applied to the bound system composed by two scalar particles interacting through the exchange of a massive scalar. The explicit forms of the integral equations are obtained in ladder approximation.

Contact problem on indentation of an elastic half-plane with an inhomogeneous coating by a flat punch in the presence of tangential stresses on a surface

NASA Astrophysics Data System (ADS)

Volkov, Sergei S.; Vasiliev, Andrey S.; Aizikovich, Sergei M.; Sadyrin, Evgeniy V.

2018-05-01

Indentation of an elastic half-space with functionally graded coating by a rigid flat punch is studied. The half-plane is additionally subjected to distributed tangential stresses. Tangential stresses are represented in a form of Fourier series. The problem is reduced to the solution of two dual integral equations over even and odd functions describing distribution of unknown normal contact stresses. The solutions of these dual integral equations are constructed by the bilateral asymptotic method. Approximated analytical expressions for contact normal stresses are provided.

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.

2.5-D poroelastic wave modelling in double porosity media

NASA Astrophysics Data System (ADS)

Liu, Xu; Greenhalgh, Stewart; Wang, Yanghua

2011-09-01

To approximate seismic wave propagation in double porosity media, the 2.5-D governing equations of poroelastic waves are developed and numerically solved. The equations are obtained by taking a Fourier transform in the strike or medium-invariant direction over all of the field quantities in the 3-D governing equations. The new memory variables from the Zener model are suggested as a way to represent the sum of the convolution integrals for both the solid particle velocity and the macroscopic fluid flux in the governing equations. By application of the memory equations, the field quantities at every time step need not be stored. However, this approximation allows just two Zener relaxation times to represent the very complex double porosity and dual permeability attenuation mechanism, and thus reduce the difficulty. The 2.5-D governing equations are numerically solved by a time-splitting method for the non-stiff parts and an explicit fourth-order Runge-Kutta method for the time integration and a Fourier pseudospectral staggered-grid for handling the spatial derivative terms. The 2.5-D solution has the advantage of producing a 3-D wavefield (point source) for a 2-D model but is much more computationally efficient than the full 3-D solution. As an illustrative example, we firstly show the computed 2.5-D wavefields in a homogeneous single porosity model for which we reformulated an analytic solution. Results for a two-layer, water-saturated double porosity model and a laterally heterogeneous double porosity structure are also presented.

An efficient numerical method for the solution of the problem of elasticity for 3D-homogeneous elastic medium with cracks and inclusions

NASA Astrophysics Data System (ADS)

Kanaun, S.; Markov, A.

2017-06-01

An efficient numerical method for solution of static problems of elasticity for an infinite homogeneous medium containing inhomogeneities (cracks and inclusions) is developed. Finite number of heterogeneous inclusions and planar parallel cracks of arbitrary shapes is considered. The problem is reduced to a system of surface integral equations for crack opening vectors and volume integral equations for stress tensors inside the inclusions. For the numerical solution of these equations, a class of Gaussian approximating functions is used. The method based on these functions is mesh free. For such functions, the elements of the matrix of the discretized system are combinations of explicit analytical functions and five standard 1D-integrals that can be tabulated. Thus, the numerical integration is excluded from the construction of the matrix of the discretized problem. For regular node grids, the matrix of the discretized system has Toeplitz's properties, and Fast Fourier Transform technique can be used for calculation matrix-vector products of such matrices.

Thermodynamic aspect in using modified Boltzmann model as an acoustic probe for URu2Si2

NASA Astrophysics Data System (ADS)

Kwang-Hua, Chu Rainer

2018-05-01

The approximate system of equations describing ultrasonic attenuation propagating in many electrons of the heavy-fermion materials URu2Si2 under high magnetic fields were firstly derived and then calculated based on the modified Boltzmann model considering the microscopic contributions due to electronic fluids. A system of nonlinear partial differential coupled with integral equations were linearized firstly and approximately solved considering the perturbed thermodynamic equilibrium states. Our numerical data were compared with previous measurements using non-dimensional or normalized physical values. The rather good fit of our numerical calculations with experimental measurements confirms our present approach.

An approximate method for solution to variable moment of inertia problems

NASA Technical Reports Server (NTRS)

Beans, E. W.

1981-01-01

An approximation method is presented for reducing a nonlinear differential equation (for the 'weather vaning' motion of a wind turbine) to an equivalent constant moment of inertia problem. The integrated average of the moment of inertia is determined. Cycle time was found to be the equivalent cycle time if the rotating speed is 4 times greater than the system's minimum natural frequency.

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.

An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

PubMed Central

Almeida, Ricardo

2013-01-01

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type. PMID:24319382

An Extension of the Krieger-Li-Iafrate Approximation to the Optimized-Effective-Potential Method

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

Wilson, B.G.

1999-11-11

The Krieger-Li-Iafrate approximation can be expressed as the zeroth order result of an unstable iterative method for solving the integral equation form of the optimized-effective-potential method. By pre-conditioning the iterate a first order correction can be obtained which recovers the bulk of quantal oscillations missing in the zeroth order approximation. A comparison of calculated total energies are given with Krieger-Li-Iafrate, Local Density Functional, and Hyper-Hartree-Fock results for non-relativistic atoms and ions.

Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Cooke, C. H.; Blanchard, D. K.

1975-01-01

A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem.

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.

Discretization analysis of bifurcation based nonlinear amplifiers

NASA Astrophysics Data System (ADS)

Feldkord, Sven; Reit, Marco; Mathis, Wolfgang

2017-09-01

Recently, for modeling biological amplification processes, nonlinear amplifiers based on the supercritical Andronov-Hopf bifurcation have been widely analyzed analytically. For technical realizations, digital systems have become the most relevant systems in signal processing applications. The underlying continuous-time systems are transferred to the discrete-time domain using numerical integration methods. Within this contribution, effects on the qualitative behavior of the Andronov-Hopf bifurcation based systems concerning numerical integration methods are analyzed. It is shown exemplarily that explicit Runge-Kutta methods transform the truncated normalform equation of the Andronov-Hopf bifurcation into the normalform equation of the Neimark-Sacker bifurcation. Dependent on the order of the integration method, higher order terms are added during this transformation.A rescaled normalform equation of the Neimark-Sacker bifurcation is introduced that allows a parametric design of a discrete-time system which corresponds to the rescaled Andronov-Hopf system. This system approximates the characteristics of the rescaled Hopf-type amplifier for a large range of parameters. The natural frequency and the peak amplitude are preserved for every set of parameters. The Neimark-Sacker bifurcation based systems avoid large computational effort that would be caused by applying higher order integration methods to the continuous-time normalform equations.

Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications

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

Du, Qiang, E-mail: jyanghkbu@gmail.com; Yang, Jiang, E-mail: qd2125@columbia.edu

This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simplemore » ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models.« less

Studies of Coherent Synchrotron Radiation with the Discontinuous Galerkin Method

NASA Astrophysics Data System (ADS)

Bizzozero, David A.

In this thesis, we present methods for integrating Maxwell's equations in Frenet-Serret coordinates in several settings using discontinuous Galerkin (DG) finite element method codes in 1D, 2D, and 3D. We apply these routines to the study of coherent synchrotron radiation, an important topic in accelerator physics. We build upon the published computational work of T. Agoh and D. Zhou in solving Maxwell's equations in the frequency-domain using a paraxial approximation which reduces Maxwell's equations to a Schrodinger-like system. We also evolve Maxwell's equations in the time-domain using a Fourier series decomposition with 2D DG motivated by an experiment performed at the Canadian Light Source. A comparison between theory and experiment has been published (Phys. Rev. Lett. 114, 204801 (2015)). Lastly, we devise a novel approach to integrating Maxwell's equations with 3D DG using a Galilean transformation and demonstrate proof-of-concept. In the above studies, we examine the accuracy, efficiency, and convergence of DG.

On the derivation of the semiclassical approximation to the quantum propagator

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

Fischer, Stefan G., E-mail: stefan.fischer@physik.uni-freiburg.de; Buchleitner, Andreas

2015-07-15

In order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator, we extend an existing method originally devised to evaluate Gaussian path-integral expressions. Using a result which relates the determinant of symmetric block-tridiagonal matrices to the determinants of their blocks, two difference equations are obtained. The first one allows to establish the connection of the amplitude factor to Jacobi’s accessory equations in the continuous-time limit, while the second one leads to an additional factor which, however, contributes to the final result only in exceptional cases. In order to demonstrate the wide applicability of these differencemore » equations, we treat explicitly the case where the time-sliced Lagrangian is written in generalized coordinates, for which a general derivation has so far been unavailable.« less

Determination of the turbulence integral model parameters for a case of a coolant angular flow in regular rod-bundle

NASA Astrophysics Data System (ADS)

Bayaskhalanov, M. V.; Vlasov, M. N.; Korsun, A. S.; Merinov, I. G.; Philippov, M. Ph

2017-11-01

Research results of “k-ε” turbulence integral model (TIM) parameters dependence on the angle of a coolant flow in regular smooth cylindrical rod-bundle are presented. TIM is intended for the definition of efficient impulse and heat transport coefficients in the averaged equations of a heat and mass transfer in the regular rod structures in an anisotropic porous media approximation. The TIM equations are received by volume-averaging of the “k-ε” turbulence model equations on periodic cell of rod-bundle. The water flow across rod-bundle under angles from 15 to 75 degrees was simulated by means of an ANSYS CFX code. Dependence of the TIM parameters on flow angle was as a result received.

A Flexible and Efficient Method for Solving Ill-Posed Linear Integral Equations of the First Kind for Noisy Data

NASA Astrophysics Data System (ADS)

Antokhin, I. I.

2017-06-01

We propose an efficient and flexible method for solving Fredholm and Abel integral equations of the first kind, frequently appearing in astrophysics. These equations present an ill-posed problem. Our method is based on solving them on a so-called compact set of functions and/or using Tikhonov's regularization. Both approaches are non-parametric and do not require any theoretic model, apart from some very loose a priori constraints on the unknown function. The two approaches can be used independently or in a combination. The advantage of the method, apart from its flexibility, is that it gives uniform convergence of the approximate solution to the exact one, as the errors of input data tend to zero. Simulated and astrophysical examples are presented.

On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations

NASA Astrophysics Data System (ADS)

Doha, E. H.

2002-02-01

An analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by Phillips and Karageorghis (27 (1990) 823). A new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given. The tensor product of ultraspherical polynomials is used to approximate a function of more than one variable. Formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved. Some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described.

Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation

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

Leblond, Herve; Kremer, David; Mihalache, Dumitru

2010-03-15

By using a reductive perturbation method, we derive from Maxwell-Bloch equations a cubic generalized Kadomtsev-Petviashvili equation for ultrashort spatiotemporal optical pulse propagation in cubic (Kerr-like) media without the use of the slowly varying envelope approximation. We calculate the collapse threshold for the propagation of few-cycle spatiotemporal pulses described by the generic cubic generalized Kadomtsev-Petviashvili equation by a direct numerical method and compare it to analytic results based on a rigorous virial theorem. Besides, typical evolution of the spectrum (integrated over the transverse spatial coordinate) is given and a strongly asymmetric spectral broadening of ultrashort spatiotemporal pulses during collapse is evidenced.

Schwarzschild and linear potentials in Mannheim's model of conformal gravity

NASA Astrophysics Data System (ADS)

Phillips, Peter R.

2018-05-01

We study the equations of conformal gravity, as given by Mannheim, in the weak field limit, so that a linear approximation is adequate. Specialising to static fields with spherical symmetry, we obtain a second-order equation for one of the metric functions. We obtain the Green function for this equation, and represent the metric function in the form of integrals over the source. Near a compact source such as the Sun the solution no longer has a form that is compatible with observations. We conclude that a solution of Mannheim type (a Schwarzschild term plus a linear potential of galactic scale) cannot exist for these field equations.

On the extraction of pressure fields from PIV velocity measurements in turbines

NASA Astrophysics Data System (ADS)

Villegas, Arturo; Diez, Fancisco J.

2012-11-01

In this study, the pressure field for a water turbine is derived from particle image velocimetry (PIV) measurements. Measurements are performed in a recirculating water channel facility. The PIV measurements include calculating the tangential and axial forces applied to the turbine by solving the integral momentum equation around the airfoil. The results are compared with the forces obtained from the Blade Element Momentum theory (BEMT). Forces are calculated by using three different methods. In the first method, the pressure fields are obtained from PIV velocity fields by solving the Poisson equation. The boundary conditions are obtained from the Navier-Stokes momentum equations. In the second method, the pressure at the boundaries is determined by spatial integration of the pressure gradients along the boundaries. In the third method, applicable only to incompressible, inviscid, irrotational, and steady flow, the pressure is calculated using the Bernoulli equation. This approximated pressure is known to be accurate far from the airfoil and outside of the wake for steady flows. Additionally, the pressure is used to solve for the force from the integral momentum equation on the blade. From the three methods proposed to solve for pressure and forces from PIV measurements, the first one, which is solved by using the Poisson equation, provides the best match to the BEM theory calculations.

On exact solutions for disturbances to the asymptotic suction boundary layer: transformation of Barnes integrals to convolution integrals

NASA Astrophysics Data System (ADS)

Russell, John

2000-11-01

A modified Orr-Sommerfeld equation that applies to the asymptotic suction boundary layer was reported by Bussmann & Münz in a wartime report dated 1942 and by Hughes & Reid in J.F.M. ( 23, 1965, p715). Fundamental systems of exact solutions of the Orr-Sommerfeld equation for this mean velocity distribution were reported by D. Grohne in an unpublished typescript dated 1950. Exact solutions of the equation of Bussmann, Münz, Hughes, & Reid were reported by P. Baldwin in Mathematika ( 17, 1970, p206). Grohne and Baldwin noticed that these exact solutions may be expressed either as Barnes integrals or as convolution integrals. In a later paper (Phil. Trans. Roy. Soc. A, 399, 1985, p321), Baldwin applied the convolution integrals in the contruction of large-Reynolds number asymptotic approximations that hold uniformly. The present talk discusses the subtleties that arise in the construction of such convolution integrals, including several not reported by Grohne or Baldwin. The aim is to recover the full set of seven solutions (one well balanced, three balanced, and three dominant-recessive) postulated by W.H. Reid in various works on the uniformly valid solutions.

A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Lafitte, Pauline; Melis, Ward; Samaey, Giovanni

2017-07-01

We present a general, high-order, fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions. The scheme consists of two steps. In a first (relaxation) step, the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff BGK source term. Then, this kinetic equation is integrated in time using a projective integration method. After taking a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time step restriction on the outer time step is similar to the CFL condition for the hyperbolic conservation law. Moreover, the number of inner time steps is also independent of the stiffness of the BGK source term. We discuss stability and consistency, and illustrate with numerical results (linear advection, Burgers' equation and the shallow water and Euler equations) in one and two spatial dimensions.

Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind.

PubMed

Narayanamoorthy, S; Sathiyapriya, S P

2016-01-01

In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability.

Operational Solution to the Nonlinear Klein-Gordon Equation

NASA Astrophysics Data System (ADS)

Bengochea, G.; Verde-Star, L.; Ortigueira, M.

2018-05-01

We obtain solutions of the nonlinear Klein-Gordon equation using a novel operational method combined with the Adomian polynomial expansion of nonlinear functions. Our operational method does not use any integral transforms nor integration processes. We illustrate the application of our method by solving several examples and present numerical results that show the accuracy of the truncated series approximations to the solutions. Supported by Grant SEP-CONACYT 220603, the first author was supported by SEP-PRODEP through the project UAM-PTC-630, the third author was supported by Portuguese National Funds through the FCT Foundation for Science and Technology under the project PEst-UID/EEA/00066/2013

Divergence preserving discrete surface integral methods for Maxwell's curl equations using non-orthogonal unstructured grids

NASA Technical Reports Server (NTRS)

Madsen, Niel K.

1992-01-01

Several new discrete surface integral (DSI) methods for solving Maxwell's equations in the time-domain are presented. These methods, which allow the use of general nonorthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve divergence or charge. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) these methods allow more accurate modeling of non-rectangular structures and objects because the traditional stair-stepped boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented.

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.

Turbulence stress measurements in a nonadiabatic hypersonic boundary layer

NASA Technical Reports Server (NTRS)

Mikulla, V.; Horstman, C. C.

1975-01-01

Turbulent shear stress and direct turbulent total heat-flux measurements have been made across a nonadiabatic, zero pressure gradient, hypersonic boundary layer by using specially designed hot-wire probes free of strain-gauging and wire oscillation. Heat-flux measurements were in reasonably good agreement with values obtained by integrating the energy equation using measured profiles of velocity and temperature. The shear-stress values deduced from the measurements, by assuming zero correlation of velocity and pressure fluctuations, were lower than the values obtained by integrating the momentum equation. Statistical properties of the cross-correlations are similar to corresponding incompressible measurements at approximately the same momentum-thickness Reynolds number.

Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding

PubMed Central

Kreienkamp, Amelia B.; Liu, Lucy Y.; Minkara, Mona S.; Knepley, Matthew G.; Bardhan, Jaydeep P.; Radhakrishnan, Mala L.

2013-01-01

We analyze and suggest improvements to a recently developed approximate continuum-electrostatic model for proteins. The model, called BIBEE/I (boundary-integral based electrostatics estimation with interpolation), was able to estimate electrostatic solvation free energies to within a mean unsigned error of 4% on a test set of more than 600 proteins—a significant improvement over previous BIBEE models. In this work, we tested the BIBEE/I model for its capability to predict residue-by-residue interactions in protein–protein binding, using the widely studied model system of trypsin and bovine pancreatic trypsin inhibitor (BPTI). Finding that the BIBEE/I model performs surprisingly less well in this task than simpler BIBEE models, we seek to explain this behavior in terms of the models’ differing spectral approximations of the exact boundary-integral operator. Calculations of analytically solvable systems (spheres and tri-axial ellipsoids) suggest two possibilities for improvement. The first is a modified BIBEE/I approach that captures the asymptotic eigenvalue limit correctly, and the second involves the dipole and quadrupole modes for ellipsoidal approximations of protein geometries. Our analysis suggests that fast, rigorous approximate models derived from reduced-basis approximation of boundary-integral equations might reach unprecedented accuracy, if the dipole and quadrupole modes can be captured quickly for general shapes. PMID:24466561

Numerical Simulations of Self-Focused Pulses Using the Nonlinear Maxwell Equations

NASA Technical Reports Server (NTRS)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1994-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations. Abstract of a proposed paper for presentation at the meeting NONLINEAR OPTICS: Materials, Fundamentals, and Applications, Hyatt Regency Waikaloa, Waikaloa, Hawaii, July 24-29, 1994, Cosponsored by IEEE/Lasers and Electro-Optics Society and Optical Society of America

Evaluation of MOSTAS computer code for predicting dynamic loads in two-bladed wind turbines

NASA Technical Reports Server (NTRS)

Kaza, K. R. V.; Janetzke, D. C.; Sullivan, T. L.

1979-01-01

Calculated dynamic blade loads are compared with measured loads over a range of yaw stiffnesses of the DOE/NASA Mod-0 wind turbine to evaluate the performance of two versions of the MOSTAS computer code. The first version uses a time-averaged coefficient approximation in conjunction with a multiblade coordinate transformation for two-bladed rotors to solve the equations of motion by standard eigenanalysis. The results obtained with this approximate analysis do not agree with dynamic blade load amplifications at or close to resonance conditions. The results of the second version, which accounts for periodic coefficients while solving the equations by a time history integration, compare well with the measured data.

Fokker-Planck description of conductance-based integrate-and-fire neuronal networks

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

Kovacic, Gregor; Tao, Louis; Rangan, Aaditya V.

2009-08-15

Steady dynamics of coupled conductance-based integrate-and-fire neuronal networks in the limit of small fluctuations is studied via the equilibrium states of a Fokker-Planck equation. An asymptotic approximation for the membrane-potential probability density function is derived and the corresponding gain curves are found. Validity conditions are discussed for the Fokker-Planck description and verified via direct numerical simulations.

Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport.

PubMed

Garzó, Vicente; Dufty, James W; Hrenya, Christine M

2007-09-01

A hydrodynamic description for an s -component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

Sum rules and other properties involving resonance projection operators. [for optical potential description of electron scattering from atoms and ions

NASA Technical Reports Server (NTRS)

Berk, A.; Temkin, A.

1985-01-01

A sum rule is derived for the auxiliary eigenvalues of an equation whose eigenspectrum pertains to projection operators which describe electron scattering from multielectron atoms and ions. The sum rule's right-hand side depends on an integral involving the target system eigenfunctions. The sum rule is checked for several approximations of the two-electron target. It is shown that target functions which have a unit eigenvalue in their auxiliary eigenspectrum do not give rise to well-defined projection operators except through a limiting process. For Hylleraas target approximations, the auxiliary equations are shown to contain an infinite spectrum. However, using a Rayleigh-Ritz variational principle, it is shown that a comparatively simple aproximation can exhaust the sum rule to better than five significant figures. The auxiliary Hylleraas equation is greatly simplified by conversion to a square root equation containing the same eigenfunction spectrum and from which the required eigenvalues are trivially recovered by squaring.

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

Yanai, Takeshi; Fann, George I.; Beylkin, Gregory

Using the fully numerical method for time-dependent Hartree–Fock and density functional theory (TD-HF/DFT) with the Tamm–Dancoff (TD) approximation we use a multiresolution analysis (MRA) approach to present our findings. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. Moreover, the integral equation is efficiently and adaptively solved using amore » numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H 2, Be, N 2, H 2O, and C 2H 4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. Finally, we introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.« less

A higher order numerical method for time fractional partial differential equations with nonsmooth data

NASA Astrophysics Data System (ADS)

Xing, Yanyuan; Yan, Yubin

2018-03-01

Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [20] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 as in Gao et al. [11] (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O (k 3 - α), 0 < α < 1 for any fixed tn > 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

A differential equation for the Generalized Born radii.

PubMed

Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro

2013-06-28

The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear second-order partial differential equation for the Generalized Born radius, which may be solved using local iterative algorithms. The equation is derived under the assumption that the usual GB approximation to the reaction field obeys Laplace's equation. The equation admits as particular solutions the correct GB radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall agreement with other reference GB models and "perfect" Poisson-Boltzmann based values.

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.

Finite-surface method for the Maxwell equations with corner singularities

NASA Technical Reports Server (NTRS)

Vinokur, Marcel; Yarrow, Maurice

1994-01-01

The finite-surface method for the two-dimensional Maxwell equations in generalized coordinates is extended to treat perfect conductor boundaries with sharp corners. Known singular forms of the grid and the electromagnetic fields in the neighborhood of each corner are used to obtain accurate approximations to the surface and line integrals appearing in the method. Numerical results are presented for a harmonic plane wave incident on a finite flat plate. Comparisons with exact solutions show good agreement.

Glueball spectra from a matrix model of pure Yang-Mills theory

NASA Astrophysics Data System (ADS)

Acharyya, Nirmalendu; Balachandran, A. P.; Pandey, Mahul; Sanyal, Sambuddha; Vaidya, Sachindeo

2018-05-01

We present variational estimates for the low-lying energies of a simple matrix model that approximates SU(3) Yang-Mills theory on a three-sphere of radius R. By fixing the ground state energy, we obtain the (integrated) renormalization group (RG) equation for the Yang-Mills coupling g as a function of R. This RG equation allows to estimate the mass of other glueball states, which we find to be in excellent agreement with lattice simulations.

The most likely voltage path and large deviations approximations for integrate-and-fire neurons.

PubMed

Paninski, Liam

2006-08-01

We develop theory and numerical methods for computing the most likely subthreshold voltage path of a noisy integrate-and-fire (IF) neuron, given observations of the neuron's superthreshold spiking activity. This optimal voltage path satisfies a second-order ordinary differential (Euler-Lagrange) equation which may be solved analytically in a number of special cases, and which may be solved numerically in general via a simple "shooting" algorithm. Our results are applicable for both linear and nonlinear subthreshold dynamics, and in certain cases may be extended to correlated subthreshold noise sources. We also show how this optimal voltage may be used to obtain approximations to (1) the likelihood that an IF cell with a given set of parameters was responsible for the observed spike train; and (2) the instantaneous firing rate and interspike interval distribution of a given noisy IF cell. The latter probability approximations are based on the classical Freidlin-Wentzell theory of large deviations principles for stochastic differential equations. We close by comparing this most likely voltage path to the true observed subthreshold voltage trace in a case when intracellular voltage recordings are available in vitro.

Couple of the Variational Iteration Method and Fractional-Order Legendre Functions Method for Fractional Differential Equations

PubMed Central

Song, Junqiang; Leng, Hongze; Lu, Fengshun

2014-01-01

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique. PMID:24511303

Extension of the N-point Padé approximants solution of the Eliashberg equations to T ˜ T c

NASA Astrophysics Data System (ADS)

Leavens, C. R.; Ritchie, D. S.

1985-01-01

Vidberg and Serene introduced a very useful technique for calculating the low temperature (T « T c) gap function of a superconductor which bypasses the real-frequency singular integral equations of Eliashberg. Blashke and Blocksdorf recognized and resolved a difficulty with the technique thereby extending it to higher temperatures. We present a much simpler method of doing essentially the same thing and, for a strong-coupling superconductor at a temperature near T c, compare the gap functions calculated using these methods with the accurate one computed directly from the real-frequency equations.

Electromagnetic inverse scattering

NASA Technical Reports Server (NTRS)

Bojarski, N. N.

1972-01-01

A three-dimensional electromagnetic inverse scattering identity, based on the physical optics approximation, is developed for the monostatic scattered far field cross section of perfect conductors. Uniqueness of this inverse identity is proven. This identity requires complete scattering information for all frequencies and aspect angles. A nonsingular integral equation is developed for the arbitrary case of incomplete frequence and/or aspect angle scattering information. A general closed-form solution to this integral equation is developed, which yields the shape of the scatterer from such incomplete information. A specific practical radar solution is presented. The resolution of this solution is developed, yielding short-pulse target resolution radar system parameter equations. The special cases of two- and one-dimensional inverse scattering and the special case of a priori knowledge of scatterer symmetry are treated in some detail. The merits of this solution over the conventional radar imaging technique are discussed.

Calculation of the Full Scattering Amplitude without Partial Wave Decomposition II

NASA Technical Reports Server (NTRS)

Shertzer, J.; Temkin, A.

2003-01-01

As is well known, the full scattering amplitude can be expressed as an integral involving the complete scattering wave function. We have shown that the integral can be simplified and used in a practical way. Initial application to electron-hydrogen scattering without exchange was highly successful. The Schrodinger equation (SE) can be reduced to a 2d partial differential equation (pde), and was solved using the finite element method. We have now included exchange by solving the resultant SE, in the static exchange approximation. The resultant equation can be reduced to a pair of coupled pde's, to which the finite element method can still be applied. The resultant scattering amplitudes, both singlet and triplet, as a function of angle can be calculated for various energies. The results are in excellent agreement with converged partial wave results.

Effects of a descending lithospheric slab on yield estimates of underground nuclear tests. Final technical report, 8 Mar 88-31 Aug 90

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

Cormier, V.F.; Kim, W.; Mandal, B.

A method for computing seismic wavefields in a high-frequency approximation is proposed based on the integration of the kinematic ray tracing equations and a new set of differential equations for the dynamic properties of the wavefront, which the authors call the vicinity ray tracing (VRT) equations. These equations are directly obtained from the Hamiltonian in ray centered coordinates, using no paraxial approximations. This system is comparable to the standard dynamic ray tracing (DRT) system, but it is specified by fewer equations (four versus eight in 3-D) and only requires the specification of velocity and its first spacial derivative along amore » ray. The VRT equations describe the trajectory of a ray in ray centered coordinates of a reference ray. Quantities obtained from vicinity ray tracing can be used to determine wavefront curvature, geometric spreading, travel time to a receiver near the reference ray, and the KMAH index of the reference ray with greater numerical precision than is possible by differencing kinematically traced rays. Since second spatial derivatives of velocity are not required by the new technique, parameterization of the medium is simplified, and reflection and transmission of beams can be calculated by applying Snell's law to both vicinity and central rays. Conversation relations between VRT and DRT can be used to determine the paraxial vicinity of DRT, in which the errors of the paraxial approximations of DRT remain small. Because no paraxial approximations are made, the superposition of the Gaussian beams define from the vicinity rays should exhibit a much slower breakdown in accuracy as the scale length of the medium given by V/Delta v approaches the beamwidth.« less

Weak-noise limit of a piecewise-smooth stochastic differential equation.

PubMed

Chen, Yaming; Baule, Adrian; Touchette, Hugo; Just, Wolfram

2013-11-01

We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided the singularity of the path integral associated with the nonsmooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behavior of the nonsmooth SDE in the low-noise limit. Finally, we consider a smooth regularization of the piecewise-constant SDE and study to what extent this regularization can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.

An approximate analysis of the diffusing flow in a self-controlled heat pipe.

NASA Technical Reports Server (NTRS)

Somogyi, D.; Yen, H. H.

1973-01-01

Constant-density two-dimensional axisymmetric equations are presented for the diffusing flow of a class of self-controlled heat pipes. The analysis is restricted to the vapor space. Condensation of the vapor is related to its mass fraction at the wall by the gas kinetic formula. The Karman-Pohlhausen integral method is applied to obtain approximate solutions. Solutions are presented for a water heat pipe with neon control gas.

Shear and compression buckling analysis for anisotropic panels with centrally located elliptical cutouts

NASA Technical Reports Server (NTRS)

Britt, V. O.

1993-01-01

An approximate analysis for buckling of biaxial- and shear-loaded anisotropic panels with centrally located elliptical cutouts is presented in the present paper. The analysis is composed of two parts, a prebuckling analysis and a buckling analysis. The prebuckling solution is determined using Lekhnitskii's complex variable equations of plane elastostatics combined with a Laurent series approximation and a boundary collocation method. The buckling solution is obtained using the principle of minimum potential energy. A by-product of the minimum potential energy equation is an integral equation which is solved using Gaussian quadrature. Comparisons with documented experimental results and finite element analyses indicate that the approximate analysis accurately predicts the buckling loads of square biaxial- and shear-loaded panels having elliptical cutouts with major axes up to sixty percent of the panel width. Results of a parametric study are presented for shear- and compression-loaded rectangular anisotropic panels with elliptical cutouts. The effects of panel aspect ratio, cutout shape, cutout size, cutout orientation, laminate anisotropy, and combined loading on the buckling load are examined.

An integral turbulent kinetic energy analysis of free shear flows

NASA Technical Reports Server (NTRS)

Peters, C. E.; Phares, W. J.

1973-01-01

Mixing of coaxial streams is analyzed by application of integral techniques. An integrated turbulent kinetic energy (TKE) equation is solved simultaneously with the integral equations for the mean flow. Normalized TKE profile shapes are obtained from incompressible jet and shear layer experiments and are assumed to be applicable to all free turbulent flows. The shear stress at the midpoint of the mixing zone is assumed to be directly proportional to the local TKE, and dissipation is treated with a generalization of the model developed for isotropic turbulence. Although the analysis was developed for ducted flows, constant-pressure flows were approximated with the duct much larger than the jet. The axisymmetric flows under consideration were predicted with reasonable accuracy. Fairly good results were also obtained for the fully developed two-dimensional shear layers, which were computed as thin layers at the boundary of a large circular jet.

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

DOE PAGES

Steyer, Andrew J.; Van Vleck, Erik S.

2018-04-13

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly,more » exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.« less

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

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

Steyer, Andrew J.; Van Vleck, Erik S.

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly,more » exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.« less

The measurement of shear stress and total heat flux in a nonadiabatic turbulent hypersonic boundary layer

NASA Technical Reports Server (NTRS)

Mikulla, V.; Horstman, C. C.

1975-01-01

Turbulent shear stress and direct turbulent total heat-flux measurements have been made across a nonadiabatic, zero pressure gradient, hypersonic boundary layer by using specially designed hot-wire probes free of strain-gauging and wire oscillation. Heat-flux measurements were in reasonably good agreement with values obtained by integrating the energy equation using measured profiles of velocity and temperature. The shear-stress values deduced from the measurements, by assuming zero correlation of velocity and pressure fluctuations, were lower than the values obtained by integrating the momentum equation. Statistical properties of the cross-correlations are similar to corresponding incompressible measurements at approximately the same momentum-thickness Reynolds number.

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.

Exact and approximate solutions for the decades-old Michaelis-Menten equation: Progress-curve analysis through integrated rate equations.

PubMed

Goličnik, Marko

2011-01-01

The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate V, and the Michaelis constant K(M) ) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to understand fully, or can even be misunderstood, by students when based only on the differential form of the Michaelis-Menten equation, and the variety of methods available to calculate the kinetic constants from rate versus substrate concentration "textbook data." Consequently, enzyme kinetics can be confusing if an analytical solution of the Michaelis-Menten equation is not available. Therefore, the still rarely known exact solution to the Michaelis-Menten equation is presented here through the explicit closed-form equation in terms of the Lambert W(x) function. Unfortunately, as the W(x) is not available in standard curve-fitting computer programs, the practical use of this direct solution is limited for most life-science students. Thus, the purpose of this article is to provide analytical approximations to the equation for modeling Michaelis-Menten kinetics. The elementary and explicit nature of these approximations can provide students with direct and simple estimations of kinetic parameters from raw experimental time-course data. The Michaelis-Menten kinetics studied in the latter context can provide an ideal alternative to the 100-year-old problems of data transformation, graphical visualization, and data analysis of enzyme-catalyzed reactions. Hence, the content of the course presented here could gradually become an important component of the modern biochemistry curriculum in the 21st century. Copyright © 2011 Wiley Periodicals, Inc.

A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

USGS Publications Warehouse

Smith, Peter E.

2006-01-01

A semi-implicit, finite-difference method for the numerical solution of the three-dimensional equations for circulation in estuaries is presented and tested. The method uses a three-time-level, leapfrog-trapezoidal scheme that is essentially second-order accurate in the spatial and temporal numerical approximations. The three-time-level scheme is shown to be preferred over a two-time-level scheme, especially for problems with strong nonlinearities. The stability of the semi-implicit scheme is free from any time-step limitation related to the terms describing vertical diffusion and the propagation of the surface gravity waves. The scheme does not rely on any form of vertical/horizontal mode-splitting to treat the vertical diffusion implicitly. At each time step, the numerical method uses a double-sweep method to transform a large number of small tridiagonal equation systems and then uses the preconditioned conjugate-gradient method to solve a single, large, five-diagonal equation system for the water surface elevation. The governing equations for the multi-level scheme are prepared in a conservative form by integrating them over the height of each horizontal layer. The layer-integrated volumetric transports replace velocities as the dependent variables so that the depth-integrated continuity equation that is used in the solution for the water surface elevation is linear. Volumetric transports are computed explicitly from the momentum equations. The resulting method is mass conservative, efficient, and numerically accurate.

Direct numerical solution of the Ornstein-Zernike integral equation and spatial distribution of water around hydrophobic molecules

NASA Astrophysics Data System (ADS)

Ikeguchi, Mitsunori; Doi, Junta

1995-09-01

The Ornstein-Zernike integral equation (OZ equation) has been used to evaluate the distribution function of solvents around solutes, but its numerical solution is difficult for molecules with a complicated shape. This paper proposes a numerical method to directly solve the OZ equation by introducing the 3D lattice. The method employs no approximation the reference interaction site model (RISM) equation employed. The method enables one to obtain the spatial distribution of spherical solvents around solutes with an arbitrary shape. Numerical accuracy is sufficient when the grid-spacing is less than 0.5 Å for solvent water. The spatial water distribution around a propane molecule is demonstrated as an example of a nonspherical hydrophobic molecule using iso-value surfaces. The water model proposed by Pratt and Chandler is used. The distribution agrees with the molecular dynamics simulation. The distribution increases offshore molecular concavities. The spatial distribution of water around 5α-cholest-2-ene (C27H46) is visualized using computer graphics techniques and a similar trend is observed.

State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities.

PubMed

Korayem, M H; Nekoo, S R

2015-07-01

This work studies an optimal control problem using the state-dependent Riccati equation (SDRE) in differential form to track for time-varying systems with state and control nonlinearities. The trajectory tracking structure provides two nonlinear differential equations: the state-dependent differential Riccati equation (SDDRE) and the feed-forward differential equation. The independence of the governing equations and stability of the controller are proven along the trajectory using the Lyapunov approach. Backward integration (BI) is capable of solving the equations as a numerical solution; however, the forward solution methods require the closed-form solution to fulfill the task. A closed-form solution is introduced for SDDRE, but the feed-forward differential equation has not yet been obtained. Different ways of solving the problem are expressed and analyzed. These include BI, closed-form solution with corrective assumption, approximate solution, and forward integration. Application of the tracking problem is investigated to control robotic manipulators possessing rigid or flexible joints. The intention is to release a general program for automatic implementation of an SDDRE controller for any manipulator that obeys the Denavit-Hartenberg (D-H) principle when only D-H parameters are received as input data. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.

A cubic spline approximation for problems in fluid mechanics

NASA Technical Reports Server (NTRS)

Rubin, S. G.; Graves, R. A., Jr.

1975-01-01

A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

S-matrix method for the numerical determination of bound states.

NASA Technical Reports Server (NTRS)

Bhatia, A. K.; Madan, R. N.

1973-01-01

A rapid numerical technique for the determination of bound states of a partial-wave-projected Schroedinger equation is presented. First, one needs to integrate the equation only outwards as in the scattering case, and second, the number of trials necessary to determine the eigenenergy and the corresponding eigenfunction is considerably less than in the usual method. As a nontrivial example of the technique, bound states are calculated in the exchange approximation for the e-/He+ system and l equals 1 partial wave.

AFRRI (Armed Forces Radiobiology Research Institute) Reports, July, August, September 1988

DTIC Science & Technology

1988-11-01

samples. Asymmetry of phonon Using this approximation in equation (3) we can scattering rates in oriented DNA have been observed perform the space...integration and obtain the result. using Raman spectroscopy: ’" ’’ however, these low- frequency modes should not be very effective in Ifll(K) activating...Paretzke 1981) that is slowing down in a homogeneous material of unit density. This approximation, which is based on inelastic scattering of protons and

Approximations of Thermoelastic and Viscoelastic Control Systems

DTIC Science & Technology

1990-06-01

parabolic partial differential equations. The development of computational algorithms for designing controllers for such systems is an Immenselv complex...hereditary differential system on Rr , then approximate the "’historv" or -’memory- term (i.e.. the integral term in i.S)). In this paper we will use a... variation introduced by Fabiano and Ito ([FI]) of the averaging scheme considered by Banks and Burns ([BB]) for the second stage. The idea of the "’AVE

Two-dimensional Euler and Navier-Stokes Time accurate simulations of fan rotor flows

NASA Technical Reports Server (NTRS)

Boretti, A. A.

1990-01-01

Two numerical methods are presented which describe the unsteady flow field in the blade-to-blade plane of an axial fan rotor. These methods solve the compressible, time-dependent, Euler and the compressible, turbulent, time-dependent, Navier-Stokes conservation equations for mass, momentum, and energy. The Navier-Stokes equations are written in Favre-averaged form and are closed with an approximate two-equation turbulence model with low Reynolds number and compressibility effects included. The unsteady aerodynamic component is obtained by superposing inflow or outflow unsteadiness to the steady conditions through time-dependent boundary conditions. The integration in space is performed by using a finite volume scheme, and the integration in time is performed by using k-stage Runge-Kutta schemes, k = 2,5. The numerical integration algorithm allows the reduction of the computational cost of an unsteady simulation involving high frequency disturbances in both CPU time and memory requirements. Less than 200 sec of CPU time are required to advance the Euler equations in a computational grid made up of about 2000 grid during 10,000 time steps on a CRAY Y-MP computer, with a required memory of less than 0.3 megawords.

Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response

DOE PAGES

Yanai, Takeshi; Fann, George I.; Beylkin, Gregory; ...

2015-02-25

Using the fully numerical method for time-dependent Hartree–Fock and density functional theory (TD-HF/DFT) with the Tamm–Dancoff (TD) approximation we use a multiresolution analysis (MRA) approach to present our findings. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. Moreover, the integral equation is efficiently and adaptively solved using amore » numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H 2, Be, N 2, H 2O, and C 2H 4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. Finally, we introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.« less

Stochastic theory of polarized light in nonlinear birefringent media: An application to optical rotation

NASA Astrophysics Data System (ADS)

Tsuchida, Satoshi; Kuratsuji, Hiroshi

2018-05-01

A stochastic theory is developed for the light transmitting the optical media exhibiting linear and nonlinear birefringence. The starting point is the two-component nonlinear Schrödinger equation (NLSE). On the basis of the ansatz of “soliton” solution for the NLSE, the evolution equation for the Stokes parameters is derived, which turns out to be the Langevin equation by taking account of randomness and dissipation inherent in the birefringent media. The Langevin equation is converted to the Fokker-Planck (FP) equation for the probability distribution by employing the technique of functional integral on the assumption of the Gaussian white noise for the random fluctuation. The specific application is considered for the optical rotation, which is described by the ellipticity (third component of the Stokes parameters) alone: (i) The asymptotic analysis is given for the functional integral, which leads to the transition rate on the Poincaré sphere. (ii) The FP equation is analyzed in the strong coupling approximation, by which the diffusive behavior is obtained for the linear and nonlinear birefringence. These would provide with a basis of statistical analysis for the polarization phenomena in nonlinear birefringent media.

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations That Arise from Nonlinear Two-Point Boundary Value Problems

NASA Technical Reports Server (NTRS)

Sidi, Avram; Pennline, James A.

1999-01-01

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + definite integral of g(x, t)F(t,y(t))dt with limits between 0 and 1,0 less than or equal to x les than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integral equations arise, e.g., when one applied Green's function techniques to nonlinear two-point boundary value problems of the form y "(x) =f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and y(l) = y(sub l), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trepezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal rule, thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations that Arise from Nonlinear Two-Point Boundary Value Problems

NASA Technical Reports Server (NTRS)

Sidi, Avram; Pennline, James A.

1999-01-01

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + integral(0 to 1) g(x,t) F(t, y(t)) dt, 0 less than or equal to x less than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integrals equations arise, e.g., when one applies Green's function techniques to nonlinear two-point boundary value problems of the form U''(x) = f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and g(l) = y(sub 1), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trapezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

On the synchrotron radiation reaction in external magnetic field

NASA Astrophysics Data System (ADS)

Tursunov, Arman; Kološ, Martin

2017-12-01

We study the dynamics of point electric charges undergoing radiation reaction force due to synchrotron radiation in the presence of external uniform magnetic field. The radiation reaction force cannot be neglected in many physical situations and its presence modifies the equations of motion significantly. The exact form of the equation of motion known as the Lorentz-Dirac equation contains higher order Schott term which leads to the appearance of the runaway solutions. We demonstrate effective computational ways to avoid such unphysical solutions and perform numerical integration of the dynamical equations. We show that in the ultrarelativistic case the Schott term is small and does not have considerable effect to the trajectory of a particle. We compare results with the covariant Landau-Lifshitz equation which is the first iteration of the Lorentz-Dirac equation. Even though the Landau-Lifshitz equation is thought to be approximative solution, we show that in realistic scenarios both approaches lead to identical results.

Solution of the Fokker-Planck equation with mixing of angular harmonics by beam-beam charge exchange

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

Mikkelsen, D.R.

1989-09-01

A method for solving the linear Fokker-Planck equation with anisotropic beam-beam charge exchange loss is presented. The 2-D equation is transformed to a system of coupled 1-D equations which are solved iteratively as independent equations. Although isotropic approximations to the beam-beam losses lead to inaccurate fast ion distributions, typically only a few angular harmonics are needed to include accurately the effect of the beam-beam charge exchange loss on the usual integrals of the fast ion distribution. Consequently, the algorithm converges very rapidly and is much more efficient than a 2-D finite difference method. A convenient recursion formula for the couplingmore » coefficients is given and generalization of the method is discussed. 13 refs., 2 figs.« less

Electromagnetics. Volume 1, Number 4, October-December 1981.

DTIC Science & Technology

1981-01-01

terms. 1.6 Matrix and Operator Theory Integral equations have been cast in approximate numerical form by the moment method (MoM). In this numerical...introduced the eigenmode expansion method to find more properties of the SEM [3.4]. One defines eigenvalues and eigenmodes for the integral operator (kernel...exterior surface of the system. Mechanisms that play a role in the penetration are (1) diffusion through metal skins , (2) field leakage through

Steady and Oscillatory, Subsonic and Supersonic, Aerodynamic Pressure and Generalized Forces for Complex Aircraft Configurations and Applications to Flutter. M.S. Thesis

NASA Technical Reports Server (NTRS)

Chen, L. T.

1975-01-01

A general method for analyzing aerodynamic flows around complex configurations is presented. By applying the Green function method, a linear integral equation relating the unknown, small perturbation potential on the surface of the body, to the known downwash is obtained. The surfaces of the aircraft, wake and diaphragm (if necessary) are divided into small quadrilateral elements which are approximated with hyperboloidal surfaces. The potential and its normal derivative are assumed to be constant within each element. This yields a set of linear algebraic equations and the coefficients are evaluated analytically. By using Gaussian elimination method, equations are solved for the potentials at the centroids of elements. The pressure coefficient is evaluated by the finite different method; the lift and moment coefficients are evaluated by numerical integration. Numerical results are presented, and applications to flutter are also included.

Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1M Unknowns

NASA Technical Reports Server (NTRS)

Shaeffer, John

2008-01-01

Matrix methods for solving integral equations via direct solve LU factorization are presently limited to weeks to months of very expensive supercomputer time for problems sizes of several hundred thousand unknowns. This report presents matrix LU factor solutions for electromagnetic scattering problems for problem sizes to one million unknowns with thousands of right hand sides that run in mere days on PC level hardware. This EM solution is accomplished by utilizing the numerical low rank nature of spatially blocked unknowns using the Adaptive Cross Approximation for compressing the rank deficient blocks of the system Z matrix, the L and U factors, the right hand side forcing function and the final current solution. This compressed matrix solution is applied to a frequency domain EM solution of Maxwell's equations using standard Method of Moments approach. Compressed matrix storage and operations count leads to orders of magnitude reduction in memory and run time.

Connection between the regular approximation and the normalized elimination of the small component in relativistic quantum theory

NASA Astrophysics Data System (ADS)

Filatov, Michael; Cremer, Dieter

2005-02-01

The regular approximation to the normalized elimination of the small component (NESC) in the modified Dirac equation has been developed and presented in matrix form. The matrix form of the infinite-order regular approximation (IORA) expressions, obtained in [Filatov and Cremer, J. Chem. Phys. 118, 6741 (2003)] using the resolution of the identity, is the exact matrix representation and corresponds to the zeroth-order regular approximation to NESC (NESC-ZORA). Because IORA (=NESC-ZORA) is a variationally stable method, it was used as a suitable starting point for the development of the second-order regular approximation to NESC (NESC-SORA). As shown for hydrogenlike ions, NESC-SORA energies are closer to the exact Dirac energies than the energies from the fifth-order Douglas-Kroll approximation, which is much more computationally demanding than NESC-SORA. For the application of IORA (=NESC-ZORA) and NESC-SORA to many-electron systems, the number of the two-electron integrals that need to be evaluated (identical to the number of the two-electron integrals of a full Dirac-Hartree-Fock calculation) was drastically reduced by using the resolution of the identity technique. An approximation was derived, which requires only the two-electron integrals of a nonrelativistic calculation. The accuracy of this approach was demonstrated for heliumlike ions. The total energy based on the approximate integrals deviates from the energy calculated with the exact integrals by less than 5×10-9hartree units. NESC-ZORA and NESC-SORA can easily be implemented in any nonrelativistic quantum chemical program. Their application is comparable in cost with that of nonrelativistic methods. The methods can be run with density functional theory and any wave function method. NESC-SORA has the advantage that it does not imply a picture change.

Generalized recursive solutions to Ornstein-Zernike integral equations

NASA Astrophysics Data System (ADS)

Rossky, Peter J.; Dale, William D. T.

1980-09-01

Recursive procedures for the solution of a class of integral equations based on the Ornstein-Zernike equation are developed; the hypernetted chain and Percus-Yevick equations are two special cases of the class considered. It is shown that certain variants of the new procedures developed here are formally equivalent to those recently developed by Dale and Friedman, if the new recursive expressions are initialized in the same way as theirs. However, the computational solution of the new equations is significantly more efficient. Further, the present analysis leads to the identification of various graphical quantities arising in the earlier study with more familiar quantities related to pair correlation functions. The analysis is greatly facilitated by the use of several identities relating simple chain sums whose graphical elements can be written as a sum of two or more parts. In particular, the use of these identities permits renormalization of the equivalent series solution to the integral equation to be directly incorporated into the recursive solution in a straightforward manner. Formulas appropriate to renormalization with respect to long and short range parts of the pair potential, as well as more general components of the direct correlation function, are obtained. To further illustrate the utility of this approach, we show that a simple generalization of the hypernetted chain closure relation for the direct correlation function leads directly to the reference hypernetted chain (RHNC) equation due to Lado. The form of the correlation function used in the exponential approximation of Andersen and Chandler is then seen to be equivalent to the first estimate obtained from a renormalized RHNC equation.

Variable Step Integration Coupled with the Method of Characteristics Solution for Water-Hammer Analysis, A Case Study

NASA Technical Reports Server (NTRS)

Turpin, Jason B.

2004-01-01

One-dimensional water-hammer modeling involves the solution of two coupled non-linear hyperbolic partial differential equations (PDEs). These equations result from applying the principles of conservation of mass and momentum to flow through a pipe, and usually the assumption that the speed at which pressure waves propagate through the pipe is constant. In order to solve these equations for the interested quantities (i.e. pressures and flow rates), they must first be converted to a system of ordinary differential equations (ODEs) by either approximating the spatial derivative terms with numerical techniques or using the Method of Characteristics (MOC). The MOC approach is ideal in that no numerical approximation errors are introduced in converting the original system of PDEs into an equivalent system of ODEs. Unfortunately this resulting system of ODEs is bound by a time step constraint so that when integrating the equations the solution can only be obtained at fixed time intervals. If the fluid system to be modeled also contains dynamic components (i.e. components that are best modeled by a system of ODEs), it may be necessary to take extremely small time steps during certain points of the model simulation in order to achieve stability and/or accuracy in the solution. Coupled together, the fixed time step constraint invoked by the MOC, and the occasional need for extremely small time steps in order to obtain stability and/or accuracy, can greatly increase simulation run times. As one solution to this problem, a method for combining variable step integration (VSI) algorithms with the MOC was developed for modeling water-hammer in systems with highly dynamic components. A case study is presented in which reverse flow through a dual-flapper check valve introduces a water-hammer event. The predicted pressure responses upstream of the check-valve are compared with test data.

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.

An analytical theory of a scattering of radio waves on meteoric ionization - II. Solution of the integro-differential equation in case of backscatter

NASA Astrophysics Data System (ADS)

Pecina, P.

2016-12-01

The integro-differential equation for the polarization vector P inside the meteor trail, representing the analytical solution of the set of Maxwell equations, is solved for the case of backscattering of radio waves on meteoric ionization. The transversal and longitudinal dimensions of a typical meteor trail are small in comparison to the distances to both transmitter and receiver and so the phase factor appearing in the kernel of the integral equation is large and rapidly changing. This allows us to use the method of stationary phase to obtain an approximate solution of the integral equation for the scattered field and for the corresponding generalized radar equation. The final solution is obtained by expanding it into the complete set of Bessel functions, which results in solving a system of linear algebraic equations for the coefficients of the expansion. The time behaviour of the meteor echoes is then obtained using the generalized radar equation. Examples are given for values of the electron density spanning a range from underdense meteor echoes to overdense meteor echoes. We show that the time behaviour of overdense meteor echoes using this method is very different from the one obtained using purely numerical solutions of the Maxwell equations. Our results are in much better agreement with the observations performed e.g. by the Ondřejov radar.

A new operational approach for solving fractional variational problems depending on indefinite integrals

NASA Astrophysics Data System (ADS)

Ezz-Eldien, S. S.; Doha, E. H.; Bhrawy, A. H.; El-Kalaawy, A. A.; Machado, J. A. T.

2018-04-01

In this paper, we propose a new accurate and robust numerical technique to approximate the solutions of fractional variational problems (FVPs) depending on indefinite integrals with a type of fixed Riemann-Liouville fractional integral. The proposed technique is based on the shifted Chebyshev polynomials as basis functions for the fractional integral operational matrix (FIOM). Together with the Lagrange multiplier method, these problems are then reduced to a system of algebraic equations, which greatly simplifies the solution process. Numerical examples are carried out to confirm the accuracy, efficiency and applicability of the proposed algorithm

Adaptive focusing of laser radiation onto a rough reflecting surface through the turbulent and nonlinear atmosphere

NASA Astrophysics Data System (ADS)

Vorontsov, Mikhail A.; Kolosov, Valeriy V.

2004-12-01

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related with maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing outgoing wave propagation, and the equation describing evolution of the mutual coherence function (MCF) for the backscattered (returned) wave. The resulting evolution equation for the MCF is further simplified by the use of the smooth refractive index approximation. This approximation enables derivation of the transport equation for the returned wave brightness function, analyzed here using method characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wavefront sensors that perform sensing of speckle-averaged characteristics of the wavefront phase (TIL sensors). Analysis of the wavefront phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric turbulence-related phase aberrations. We also show that wavefront sensing results depend on the extended target shape, surface roughness, and the outgoing beam intensity distribution on the target surface.

Analysis of wave propagation and wavefront sensing in target-in-the-loop beam control systems

NASA Astrophysics Data System (ADS)

Vorontsov, Mikhail A.; Kolosov, Valeri V.

2004-10-01

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related with maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing outgoing wave propagation, and the equation describing evolution of the mutual intensity function (MIF) for the backscattered (returned) wave. The resulting evolution equation for the MIF is further simplified by the use of the smooth refractive index approximation. This approximation enables derivation of the transport equation for the returned wave brightness function, analyzed here using method characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wavefront sensors that perform sensing of speckle-averaged characteristics of the wavefront phase (TIL sensors). Analysis of the wavefront phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric turbulence-related phase aberrations. We also show that wavefront sensing results depend on the extended target shape, surface roughness, and the outgoing beam intensity distribution on the target surface.

An explicit asymptotic model for the surface wave in a viscoelastic half-space based on applying Rabotnov's fractional exponential integral operators

NASA Astrophysics Data System (ADS)

Wilde, M. V.; Sergeeva, N. V.

2018-05-01

An explicit asymptotic model extracting the contribution of a surface wave to the dynamic response of a viscoelastic half-space is derived. Fractional exponential Rabotnov's integral operators are used for describing of material properties. The model is derived by extracting the principal part of the poles corresponding to the surface waves after applying Laplace and Fourier transforms. The simplified equations for the originals are written by using power series expansions. Padè approximation is constructed to unite short-time and long-time models. The form of this approximation allows to formulate the explicit model using a fractional exponential Rabotnov's integral operator with parameters depending on the properties of surface wave. The applicability of derived models is studied by comparing with the exact solutions of a model problem. It is revealed that the model based on Padè approximation is highly effective for all the possible time domains.

A Formalism for Covariant Polarized Radiative Transport by Ray Tracing

NASA Astrophysics Data System (ADS)

Gammie, Charles F.; Leung, Po Kin

2012-06-01

We write down a covariant formalism for polarized radiative transfer appropriate for ray tracing through a turbulent plasma. The polarized radiation field is represented by the polarization tensor (coherency matrix) N αβ ≡ langa α k a*β k rang, where ak is a Fourier coefficient for the vector potential. Using Maxwell's equations, the Liouville-Vlasov equation, and the WKB approximation, we show that the transport equation in vacuo is k μ∇μ N αβ = 0. We show that this is equivalent to Broderick & Blandford's formalism based on invariant Stokes parameters and a rotation coefficient, and suggest a modification that may reduce truncation error in some situations. Finally, we write down several alternative approaches to integrating the transfer equation.

Evaluation of MOSTAS computer code for predicting dynamic loads in two bladed wind turbines

NASA Technical Reports Server (NTRS)

Kaza, K. R. V.; Janetzke, D. C.; Sullivan, T. L.

1979-01-01

Calculated dynamic blade loads were compared with measured loads over a range of yaw stiffnesses of the DOE/NASA Mod-O wind turbine to evaluate the performance of two versions of the MOSTAS computer code. The first version uses a time-averaged coefficient approximation in conjunction with a multi-blade coordinate transformation for two bladed rotors to solve the equations of motion by standard eigenanalysis. The second version accounts for periodic coefficients while solving the equations by a time history integration. A hypothetical three-degree of freedom dynamic model was investigated. The exact equations of motion of this model were solved using the Floquet-Lipunov method. The equations with time-averaged coefficients were solved by standard eigenanalysis.

A strictly Markovian expansion for plasma turbulence theory

NASA Technical Reports Server (NTRS)

Jones, F. C.

1976-01-01

The collision operator that appears in the equation of motion for a particle distribution function that was averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. An expansion is derived for the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest nontrivial order. The validity of this expansion is seen to be the same as that of the standard quasilinear expansion.

Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions.

PubMed

Salis, Howard; Kaznessis, Yiannis

2005-02-01

The dynamical solution of a well-mixed, nonlinear stochastic chemical kinetic system, described by the Master equation, may be exactly computed using the stochastic simulation algorithm. However, because the computational cost scales with the number of reaction occurrences, systems with one or more "fast" reactions become costly to simulate. This paper describes a hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the integral form of the "Next Reaction" variant of the stochastic simulation algorithm. The key innovation of this method is its mechanism of efficiently monitoring the occurrences of slow, discrete events while simultaneously simulating the dynamics of a continuous, stochastic or deterministic process. In addition, by introducing an approximation in which multiple slow reactions may occur within a time step of the numerical integration of the chemical Langevin equation, the hybrid stochastic method performs much faster with only a marginal decrease in accuracy. Multiple examples, including a biological pulse generator and a large-scale system benchmark, are simulated using the exact and proposed hybrid methods as well as, for comparison, a previous hybrid stochastic method. Probability distributions of the solutions are compared and the weak errors of the first two moments are computed. In general, these hybrid methods may be applied to the simulation of the dynamics of a system described by stochastic differential, ordinary differential, and Master equations.

Dimension reduction method for SPH equations

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

Tartakovsky, Alexandre M.; Scheibe, Timothy D.

2011-08-26

Smoothed Particle Hydrodynamics model of a complex multiscale processe often results in a system of ODEs with an enormous number of unknowns. Furthermore, a time integration of the SPH equations usually requires time steps that are smaller than the observation time by many orders of magnitude. A direct solution of these ODEs can be extremely expensive. Here we propose a novel dimension reduction method that gives an approximate solution of the SPH ODEs and provides an accurate prediction of the average behavior of the modeled system. The method consists of two main elements. First, effective equationss for evolution of averagemore » variables (e.g. average velocity, concentration and mass of a mineral precipitate) are obtained by averaging the SPH ODEs over the entire computational domain. These effective ODEs contain non-local terms in the form of volume integrals of functions of the SPH variables. Second, a computational closure is used to close the system of the effective equations. The computational closure is achieved via short bursts of the SPH model. The dimension reduction model is used to simulate flow and transport with mixing controlled reactions and mineral precipitation. An SPH model is used model transport at the porescale. Good agreement between direct solutions of the SPH equations and solutions obtained with the dimension reduction method for different boundary conditions confirms the accuracy and computational efficiency of the dimension reduction model. The method significantly accelerates SPH simulations, while providing accurate approximation of the solution and accurate prediction of the average behavior of the system.« less

Approximate Analytical Solutions for Hypersonic Flow Over Slender Power Law Bodies

NASA Technical Reports Server (NTRS)

Mirels, Harold

1959-01-01

Approximate analytical solutions are presented for two-dimensional and axisymmetric hypersonic flow over slender power law bodies. Both zero order (M approaches infinity) and first order (small but nonvanishing values of 1/(M(Delta)(sup 2) solutions are presented, where M is free-stream Mach number and Delta is a characteristic slope. These solutions are compared with exact numerical integration of the equations of motion and appear to be accurate particularly when the shock is relatively close to the body.

The Spiral of Life

NASA Astrophysics Data System (ADS)

Cajiao Vélez, F.; Kamiński, J. Z.; Krajewska, K.

2018-04-01

High-energy photoionization driven by short and circularly-polarized laser pulses is studied in the framework of the relativistic strong-field approximation. The saddle-point analysis of the integrals defining the probability amplitude is used to determine the general properties of the probability distributions. Additionally, an approximate solution to the saddle-point equation is derived. This leads to the concept of the three-dimensional spiral of life in momentum space, around which the ionization probability distribution is maximum. We demonstrate that such spiral is also obtained from a classical treatment.

Numerical realization of the variational method for generating self-trapped beams

NASA Astrophysics Data System (ADS)

Duque, Erick I.; Lopez-Aguayo, Servando; Malomed, Boris A.

2018-03-01

We introduce a numerical variational method based on the Rayleigh-Ritz optimization principle for predicting two-dimensional self-trapped beams in nonlinear media. This technique overcomes the limitation of the traditional variational approximation in performing analytical Lagrangian integration and differentiation. Approximate soliton solutions of a generalized nonlinear Schr\\"odinger equation are obtained, demonstrating robustness of the beams of various types (fundamental, vortices, multipoles, azimuthons) in the course of their propagation. The algorithm offers possibilities to produce more sophisticated soliton profiles in general nonlinear models.

Accuracy & Computational Considerations for Wide--Angle One--way Seismic Propagators and Multiple Scattering by Invariant Embedding

NASA Astrophysics Data System (ADS)

Thomson, C. J.

2004-12-01

Pseudodifferential operators (PSDOs) yield in principle exact one--way seismic wave equations, which are attractive both conceptually and for their promise of computational efficiency. The one--way operators can be extended to include multiple--scattering effects, again in principle exactly. In practice approximations must be made and, as an example, the variable--wavespeed Helmholtz equation for scalar waves in two space dimensions is here factorized to give the one--way wave equation. This simple case permits clear identification of a sequence of physically reasonable approximations to be used when the mathematically exact PSDO one--way equation is implemented on a computer. As intuition suggests, these approximations hinge on the medium gradients in the direction transverse to the main propagation direction. A key point is that narrow--angle approximations are to be avoided in the interests of accuracy. Another key consideration stems from the fact that the so--called ``standard--ordering'' PSDO indicates how lateral interpolation of the velocity structure can significantly reduce computational costs associated with the Fourier or plane--wave synthesis lying at the heart of the calculations. The decision on whether a slow or a fast Fourier transform code should be used rests upon how many lateral model parameters are truly distinct. A third important point is that the PSDO theory shows what approximations are necessary in order to generate an exponential one--way propagator for the laterally varying case, representing the intuitive extension of classical integral--transform solutions for a laterally homogeneous medium. This exponential propagator suggests the use of larger discrete step sizes, and it can also be used to approach phase--screen like approximations (though the latter are not the main interest here). Numerical comparisons with finite--difference solutions will be presented in order to assess the approximations being made and to gain an understanding of computation time differences. The ideas described extend to the three--dimensional, generally anisotropic case and to multiple scattering by invariant embedding.

Calculation of the Full Scattering Amplitude without Partial Wave Decomposition II: Inclusion of Exchange

NASA Technical Reports Server (NTRS)

Shertzer, Janine; Temkin, A.

2003-01-01

As is well known, the full scattering amplitude can be expressed as an integral involving the complete scattering wave function. We have shown that the integral can be simplified and used in a practical way. Initial application to electron-hydrogen scattering without exchange was highly successful. The Schrodinger equation (SE), which can be reduced to a 2d partial differential equation (pde), was solved using the finite element method. We have now included exchange by solving the resultant SE, in the static exchange approximation, which is reducible to a pair of coupled pde's. The resultant scattering amplitudes, both singlet and triplet, calculated as a function of energy are in excellent agreement with converged partial wave results.

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.

Achieving algorithmic resilience for temporal integration through spectral deferred corrections

DOE PAGES

Grout, Ray; Kolla, Hemanth; Minion, Michael; ...

2017-05-08

Spectral deferred corrections (SDC) is an iterative approach for constructing higher-order-accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited to recovering frommore » soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual of the first correction iteration and changes slowly between successive iterations. Here, we demonstrate the effectiveness of this strategy for both canonical test problems and a comprehensive situation involving a mature scientific application code that solves the reacting Navier-Stokes equations for combustion research.« less

Achieving algorithmic resilience for temporal integration through spectral deferred corrections

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

Grout, Ray; Kolla, Hemanth; Minion, Michael

2017-05-08

Spectral deferred corrections (SDC) is an iterative approach for constructing higher- order accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited tomore » recovering from soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual on the first correction iteration and changes slowly between successive iterations. We demonstrate the effectiveness of this strategy for both canonical test problems and a comprehen- sive situation involving a mature scientific application code that solves the reacting Navier-Stokes equations for combustion research.« less

Achieving algorithmic resilience for temporal integration through spectral deferred corrections

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

Grout, Ray; Kolla, Hemanth; Minion, Michael

2017-05-08

Spectral deferred corrections (SDC) is an iterative approach for constructing higher-order-accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited to recovering frommore » soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual of the first correction iteration and changes slowly between successive iterations. We demonstrate the effectiveness of this strategy for both canonical test problems and a comprehensive situation involving a mature scientific application code that solves the reacting Navier-Stokes equations for combustion research.« less

A path integral approach to the Hodgkin-Huxley model

NASA Astrophysics Data System (ADS)

Baravalle, Roman; Rosso, Osvaldo A.; Montani, Fernando

2017-11-01

To understand how single neurons process sensory information, it is necessary to develop suitable stochastic models to describe the response variability of the recorded spike trains. Spikes in a given neuron are produced by the synergistic action of sodium and potassium of the voltage-dependent channels that open or close the gates. Hodgkin and Huxley (HH) equations describe the ionic mechanisms underlying the initiation and propagation of action potentials, through a set of nonlinear ordinary differential equations that approximate the electrical characteristics of the excitable cell. Path integral provides an adequate approach to compute quantities such as transition probabilities, and any stochastic system can be expressed in terms of this methodology. We use the technique of path integrals to determine the analytical solution driven by a non-Gaussian colored noise when considering the HH equations as a stochastic system. The different neuronal dynamics are investigated by estimating the path integral solutions driven by a non-Gaussian colored noise q. More specifically we take into account the correlational structures of the complex neuronal signals not just by estimating the transition probability associated to the Gaussian approach of the stochastic HH equations, but instead considering much more subtle processes accounting for the non-Gaussian noise that could be induced by the surrounding neural network and by feedforward correlations. This allows us to investigate the underlying dynamics of the neural system when different scenarios of noise correlations are considered.

Three-dimensional simulation of vortex breakdown

NASA Technical Reports Server (NTRS)

Kuruvila, G.; Salas, M. D.

1990-01-01

The integral form of the complete, unsteady, compressible, three-dimensional Navier-Stokes equations in the conservation form, cast in generalized coordinate system, are solved, numerically, to simulate the vortex breakdown phenomenon. The inviscid fluxes are discretized using Roe's upwind-biased flux-difference splitting scheme and the viscous fluxes are discretized using central differencing. Time integration is performed using a backward Euler ADI (alternating direction implicit) scheme. A full approximation multigrid is used to accelerate the convergence to steady state.

Nonlinear storage models of unconfined flow through a shallow aquifer on an inclined base and their quasi-steady flow application

NASA Astrophysics Data System (ADS)

Varvaris, Ioannis; Gravanis, Elias; Koussis, Antonis; Akylas, Evangelos

2013-04-01

Hillslope processes involving flow through an inclined shallow aquifer range from subsurface stormflow to stream base flow (drought flow, or groundwater recession flow). In the case of recharge, the infiltrating water moves vertically as unsaturated flow until it reaches the saturated groundwater, where the flow is approximately parallel to the base of the aquifer. Boussinesq used the Dupuit-Forchheimer (D-F) hydraulic theory to formulate unconfined groundwater flow through a soil layer resting on an impervious inclined bed, deriving a nonlinear equation for the flow rate that consists of a linear gravity-driven component and a quadratic pressure-gradient component. Inserting that flow rate equation into the differential storage balance equation (volume conservation) Boussinesq obtained a nonlinear second-order partial differential equation for the depth. So far however, only few special solutions have been advanced for that governing equation. The nonlinearity of the equation of Boussinesq is the major obstacle to deriving a general analytical solution for the depth profile of unconfined flow on a sloping base with recharge (from which the discharges could be then determined). Henderson and Wooding (1964) were able to obtain an exact analytical solution for steady unconfined flow on a sloping base, with recharge, and their work deserves special note in the realm of solutions of the nonlinear equation of Boussinesq. However, the absence of a general solution for the transient case, which is of practical interest to hydrologists, has been the motivation for developing approximate solutions of the non-linear equation of Boussinesq. In this work, we derive the aquifer storage function by integrating analytically over the aquifer base the depth profiles resulting from the complete nonlinear Boussinesq equation for steady flow. This storage function consists of a linear and a nonlinear outflow-dependent term. Then, we use this physics-based storage function in the transient storage balance over the hillslope, obtaining analytical solutions of the outflow and the storage, for recharge and drainage, via a quasi-steady flow calculation. The hydraulically derived storage model is thus embedded in a quasi-steady approximation of transient unconfined flow in sloping aquifers. We generalise this hydrologic model of groundwater flow by modifying the storage function to be the weighted sum of the linear and the nonlinear storage terms, determining the weighting factor objectively from a known integral quantity of the flow (either an initial volume of water stored in the aquifer or a drained water volume). We demonstrate the validity of this model through comparisons with experimental data and simulation results.

The multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test: I

NASA Astrophysics Data System (ADS)

Santini, Paolo Maria

2010-01-01

We propose an algorithmic procedure (i) to study the 'distance' between an integrable PDE and any discretization of it, in the small lattice spacing epsilon regime, and, at the same time, (ii) to test the (asymptotic) integrability properties of such discretization. This method should provide, in particular, useful and concrete information on how good is any numerical scheme used to integrate a given integrable PDE. The procedure, illustrated on a fairly general ten-parameter family of discretizations of the nonlinear Schrödinger equation, consists of the following three steps: (i) the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in epsilon, following Degasperis et al (1997 Physica D 100 187-211) (ii) the application, to such an expansion, of the Degasperis-Procesi (DP) integrability test (Degasperis A and Procesi M 1999 Asymptotic integrability Symmetry and Perturbation Theory, SPT98, ed A Degasperis and G Gaeta (Singapore: World Scientific) pp 23-37 Degasperis A 2001 Multiscale expansion and integrability of dispersive wave equations Lectures given at the Euro Summer School: 'What is integrability?' (Isaac Newton Institute, Cambridge, UK, 13-24 August); Integrability (Lecture Notes in Physics vol 767) ed A Mikhailov (Berlin: Springer)), to test the asymptotic integrability properties of the discrete system and its 'distance' from its continuous limit; (iii) the use of the main output of the DP test to construct infinitely many approximate symmetries and constants of motion of the discrete system, through novel and simple formulas.

Computing aerodynamic sound using advanced statistical turbulence theories

NASA Technical Reports Server (NTRS)

Hecht, A. M.; Teske, M. E.; Bilanin, A. J.

1981-01-01

It is noted that the calculation of turbulence-generated aerodynamic sound requires knowledge of the spatial and temporal variation of Q sub ij (xi sub k, tau), the two-point, two-time turbulent velocity correlations. A technique is presented to obtain an approximate form of these correlations based on closure of the Reynolds stress equations by modeling of higher order terms. The governing equations for Q sub ij are first developed for a general flow. The case of homogeneous, stationary turbulence in a unidirectional constant shear mean flow is then assumed. The required closure form for Q sub ij is selected which is capable of qualitatively reproducing experimentally observed behavior. This form contains separation time dependent scale factors as parameters and depends explicitly on spatial separation. The approximate forms of Q sub ij are used in the differential equations and integral moments are taken over the spatial domain. The velocity correlations are used in the Lighthill theory of aerodynamic sound by assuming normal joint probability.

Using Equation-Free Computation to Accelerate Network-Free Stochastic Simulation of Chemical Kinetics.

PubMed

Lin, Yen Ting; Chylek, Lily A; Lemons, Nathan W; Hlavacek, William S

2018-06-21

The chemical kinetics of many complex systems can be concisely represented by reaction rules, which can be used to generate reaction events via a kinetic Monte Carlo method that has been termed network-free simulation. Here, we demonstrate accelerated network-free simulation through a novel approach to equation-free computation. In this process, variables are introduced that approximately capture system state. Derivatives of these variables are estimated using short bursts of exact stochastic simulation and finite differencing. The variables are then projected forward in time via a numerical integration scheme, after which a new exact stochastic simulation is initialized and the whole process repeats. The projection step increases efficiency by bypassing the firing of numerous individual reaction events. As we show, the projected variables may be defined as populations of building blocks of chemical species. The maximal number of connected molecules included in these building blocks determines the degree of approximation. Equation-free acceleration of network-free simulation is found to be both accurate and efficient.

Time-independent hybrid enrichment for finite element solution of transient conduction–radiation in diffusive grey media

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

Mohamed, M. Shadi, E-mail: m.s.mohamed@durham.ac.uk; Seaid, Mohammed; Trevelyan, Jon

2013-10-15

We investigate the effectiveness of the partition-of-unity finite element method for transient conduction–radiation problems in diffusive grey media. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media. The coupled equations are integrated in time using a semi-implicit method in the finite element framework. We show that for the considered problems, a combination of hyperbolic and exponential enrichment functions based on an approximation of the boundary layer leads to improved accuracy compared to the conventional finite element method. It is illustrated that this approach canmore » be more efficient than using h adaptivity to increase the accuracy of the finite element method near the boundary walls. The performance of the proposed partition-of-unity method is analyzed on several test examples for transient conduction–radiation problems in two space dimensions.« less

Role of nonlinear refraction in the generation of terahertz field pulses by light fields

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

Zabolotskii, A. A., E-mail: zabolotskii@iae.nsk.su

2013-07-15

The generation of microwave (terahertz) pulses without any envelope in a four-level quasi-resonant medium is considered. Two intense quasi-monochromatic laser fields lead to a partial upper-level population. Microwave field pulses cause the transition between these levels. For appropriately chosen scales, the evolution of the fields is shown to be described by the pseudo-spin evolution equations in a microwave field with the inclusion of nonlinear refraction caused by an adiabatic upper-level population. The evolution of terahertz field pulses is described outside the scope of the slow-envelope approximation. When a number of standard approximations are taken into account, this system of equationsmore » is shown to be equivalent to an integrable version of the generalized reduced Maxwell-Bloch equations or to the generalized three-wave mixing equations. The soliton solution found by the inverse scattering transform method is used as an example to show that nonlinear refraction leads to a strong compression of the microwave (terahertz) field soliton.« less

Post-Newtonian Circular Restricted 3-Body Problem: Schwarzschild primaries

NASA Astrophysics Data System (ADS)

Dubeibe, F. L.; Lora-Clavijo, F. D.; González, G. A.

2017-07-01

The restricted three-body problem (RTBP) has been extensively studied to investigate the stability of the solar system, extra-solar subsystems, asteroid capture, and the dynamics of two massive black holes orbited by a sun. In the present work, we study the stability of the planar circular restricted three-body problem in the context of post-Newtonian approximations. First of all, we review the results obtained from the post-Newtonian equations of motion calculated in the framework of the Einstein-Infeld-Hoffmann formalism (EIH). Therefore, using the Fodor-Hoenselers-Perjes formalism (FHP), we have performed an expansion of the gravitational potential for two primaries, deriving a new system of equations of motion, which unlike the EIH-approach, preserves the Jacobian integral of motion. Additionally, we have obtained approximate expressions for the Lagrange points in terms of a mass parameter μ, where it is found that the deviations from the classical regime are larger for the FHP than for the EIH equations.

Integration within the Felsenstein equation for improved Markov chain Monte Carlo methods in population genetics

PubMed Central

Hey, Jody; Nielsen, Rasmus

2007-01-01

In 1988, Felsenstein described a framework for assessing the likelihood of a genetic data set in which all of the possible genealogical histories of the data are considered, each in proportion to their probability. Although not analytically solvable, several approaches, including Markov chain Monte Carlo methods, have been developed to find approximate solutions. Here, we describe an approach in which Markov chain Monte Carlo simulations are used to integrate over the space of genealogies, whereas other parameters are integrated out analytically. The result is an approximation to the full joint posterior density of the model parameters. For many purposes, this function can be treated as a likelihood, thereby permitting likelihood-based analyses, including likelihood ratio tests of nested models. Several examples, including an application to the divergence of chimpanzee subspecies, are provided. PMID:17301231

Monotonic Derivative Correction for Calculation of Supersonic Flows

ERIC Educational Resources Information Center

Bulat, Pavel V.; Volkov, Konstantin N.

2016-01-01

Aim of the study: This study examines numerical methods for solving the problems in gas dynamics, which are based on an exact or approximate solution to the problem of breakdown of an arbitrary discontinuity (the Riemann problem). Results: Comparative analysis of finite difference schemes for the Euler equations integration is conducted on the…

SIVA/DIVA- INITIAL VALUE ORDINARY DIFFERENTIAL EQUATION SOLUTION VIA A VARIABLE ORDER ADAMS METHOD

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1994-01-01

The SIVA/DIVA package is a collection of subroutines for the solution of ordinary differential equations. There are versions for single precision and double precision arithmetic. These solutions are applicable to stiff or nonstiff differential equations of first or second order. SIVA/DIVA requires fewer evaluations of derivatives than other variable order Adams predictor-corrector methods. There is an option for the direct integration of second order equations which can make integration of trajectory problems significantly more efficient. Other capabilities of SIVA/DIVA include: monitoring a user supplied function which can be separate from the derivative; dynamically controlling the step size; displaying or not displaying output at initial, final, and step size change points; saving the estimated local error; and reverse communication where subroutines return to the user for output or computation of derivatives instead of automatically performing calculations. The user must supply SIVA/DIVA with: 1) the number of equations; 2) initial values for the dependent and independent variables, integration stepsize, error tolerance, etc.; and 3) the driver program and operational parameters necessary for subroutine execution. SIVA/DIVA contains an extensive diagnostic message library should errors occur during execution. SIVA/DIVA is written in FORTRAN 77 for batch execution and is machine independent. It has a central memory requirement of approximately 120K of 8 bit bytes. This program was developed in 1983 and last updated in 1987.

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.

Transforming parts of a differential equations system to difference equations as a method for run-time savings in NONMEM.

PubMed

Petersson, K J F; Friberg, L E; Karlsson, M O

2010-10-01

Computer models of biological systems grow more complex as computing power increase. Often these models are defined as differential equations and no analytical solutions exist. Numerical integration is used to approximate the solution; this can be computationally intensive, time consuming and be a large proportion of the total computer runtime. The performance of different integration methods depend on the mathematical properties of the differential equations system at hand. In this paper we investigate the possibility of runtime gains by calculating parts of or the whole differential equations system at given time intervals, outside of the differential equations solver. This approach was tested on nine models defined as differential equations with the goal to reduce runtime while maintaining model fit, based on the objective function value. The software used was NONMEM. In four models the computational runtime was successfully reduced (by 59-96%). The differences in parameter estimates, compared to using only the differential equations solver were less than 12% for all fixed effects parameters. For the variance parameters, estimates were within 10% for the majority of the parameters. Population and individual predictions were similar and the differences in OFV were between 1 and -14 units. When computational runtime seriously affects the usefulness of a model we suggest evaluating this approach for repetitive elements of model building and evaluation such as covariate inclusions or bootstraps.

Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law

NASA Astrophysics Data System (ADS)

Želi, Velibor; Zorica, Dušan

2018-02-01

Generalization of the heat conduction equation is obtained by considering the system of equations consisting of the energy balance equation and fractional-order constitutive heat conduction law, assumed in the form of the distributed-order Cattaneo type. The Cauchy problem for system of energy balance equation and constitutive heat conduction law is treated analytically through Fourier and Laplace integral transform methods, as well as numerically by the method of finite differences through Adams-Bashforth and Grünwald-Letnikov schemes for approximation derivatives in temporal domain and leap frog scheme for spatial derivatives. Numerical examples, showing time evolution of temperature and heat flux spatial profiles, demonstrate applicability and good agreement of both methods in cases of multi-term and power-type distributed-order heat conduction laws.

MEANS: python package for Moment Expansion Approximation, iNference and Simulation

PubMed Central

Fan, Sisi; Geissmann, Quentin; Lakatos, Eszter; Lukauskas, Saulius; Ale, Angelique; Babtie, Ann C.; Kirk, Paul D. W.; Stumpf, Michael P. H.

2016-01-01

Motivation: Many biochemical systems require stochastic descriptions. Unfortunately these can only be solved for the simplest cases and their direct simulation can become prohibitively expensive, precluding thorough analysis. As an alternative, moment closure approximation methods generate equations for the time-evolution of the system’s moments and apply a closure ansatz to obtain a closed set of differential equations; that can become the basis for the deterministic analysis of the moments of the outputs of stochastic systems. Results: We present a free, user-friendly tool implementing an efficient moment expansion approximation with parametric closures that integrates well with the IPython interactive environment. Our package enables the analysis of complex stochastic systems without any constraints on the number of species and moments studied and the type of rate laws in the system. In addition to the approximation method our package provides numerous tools to help non-expert users in stochastic analysis. Availability and implementation: https://github.com/theosysbio/means Contacts: m.stumpf@imperial.ac.uk or e.lakatos13@imperial.ac.uk Supplementary information: Supplementary data are available at Bioinformatics online. PMID:27153663

MEANS: python package for Moment Expansion Approximation, iNference and Simulation.

PubMed

Fan, Sisi; Geissmann, Quentin; Lakatos, Eszter; Lukauskas, Saulius; Ale, Angelique; Babtie, Ann C; Kirk, Paul D W; Stumpf, Michael P H

2016-09-15

Many biochemical systems require stochastic descriptions. Unfortunately these can only be solved for the simplest cases and their direct simulation can become prohibitively expensive, precluding thorough analysis. As an alternative, moment closure approximation methods generate equations for the time-evolution of the system's moments and apply a closure ansatz to obtain a closed set of differential equations; that can become the basis for the deterministic analysis of the moments of the outputs of stochastic systems. We present a free, user-friendly tool implementing an efficient moment expansion approximation with parametric closures that integrates well with the IPython interactive environment. Our package enables the analysis of complex stochastic systems without any constraints on the number of species and moments studied and the type of rate laws in the system. In addition to the approximation method our package provides numerous tools to help non-expert users in stochastic analysis. https://github.com/theosysbio/means m.stumpf@imperial.ac.uk or e.lakatos13@imperial.ac.uk Supplementary data are available at Bioinformatics online. © The Author 2016. Published by Oxford University Press.

Nuclear magnetic resonance signal dynamics of liquids in the presence of distant dipolar fields, revisited

PubMed Central

Barros, Wilson; Gochberg, Daniel F.; Gore, John C.

2009-01-01

The description of the nuclear magnetic resonance magnetization dynamics in the presence of long-range dipolar interactions, which is based upon approximate solutions of Bloch–Torrey equations including the effect of a distant dipolar field, has been revisited. New experiments show that approximate analytic solutions have a broader regime of validity as well as dependencies on pulse-sequence parameters that seem to have been overlooked. In order to explain these experimental results, we developed a new method consisting of calculating the magnetization via an iterative formalism where both diffusion and distant dipolar field contributions are treated as integral operators incorporated into the Bloch–Torrey equations. The solution can be organized as a perturbative series, whereby access to higher order terms allows one to set better boundaries on validity regimes for analytic first-order approximations. Finally, the method legitimizes the use of simple analytic first-order approximations under less demanding experimental conditions, it predicts new pulse-sequence parameter dependencies for the range of validity, and clarifies weak points in previous calculations. PMID:19425789

Electron-pair-production cross section in the tip region of the positron spectrum

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

Sud, K.K.; Sharma, D.K.

1984-11-01

The radial integrals for electron-pair production in a point Coulomb potential have been expressed by Sud, Sharma, and Sud in terms of the matrix generalization of the GAMMA function. Two new partial differential equations in photon energy satisfied by the matrix GAMMA function are obtained. We have obtained, on integrating the partial differential equations, accurate radial integrals as a function of photon energy for the pair production by intermediate-energy photons. The cross section in the tip region of the spectrum are calculated for photons of energy 5.0 to 10.0 MeV for /sup 92/U. The new technique results in extensive savingmore » in computer time as the basic radial integrals in terms of the hypergeometric function F/sub 2/ are computed at one photon energy for each pair of partial waves. The results of our calculations are compared with plane-wave Born-approximation results and with the calculations of Dugne and of Deck, Moroi, and Alling.« less

High Energy Laser Beam Propagation in the Atmosphere: The Integral Invariants of the Nonlinear Parabolic Equation and the Method of Moments

NASA Technical Reports Server (NTRS)

Manning, Robert M.

2012-01-01

The method of moments is used to define and derive expressions for laser beam deflection and beam radius broadening for high-energy propagation through the Earth s atmosphere. These expressions are augmented with the integral invariants of the corresponding nonlinear parabolic equation that describes the electric field of high-energy laser beam to propagation to yield universal equations for the aforementioned quantities; the beam deflection is a linear function of the propagation distance whereas the beam broadening is a quadratic function of distance. The coefficients of these expressions are then derived from a thin screen approximation solution of the nonlinear parabolic equation to give corresponding analytical expressions for a target located outside the Earth s atmospheric layer. These equations, which are graphically presented for a host of propagation scenarios, as well as the thin screen model, are easily amenable to the phase expansions of the wave front for the specification and design of adaptive optics algorithms to correct for the inherent phase aberrations. This work finds application in, for example, the analysis of beamed energy propulsion for space-based vehicles.

Assessment of time-dependent density functional theory with the restricted excitation space approximation for excited state calculations of large systems

NASA Astrophysics Data System (ADS)

Hanson-Heine, Magnus W. D.; George, Michael W.; Besley, Nicholas A.

2018-06-01

The restricted excitation subspace approximation is explored as a basis to reduce the memory storage required in linear response time-dependent density functional theory (TDDFT) calculations within the Tamm-Dancoff approximation. It is shown that excluding the core orbitals and up to 70% of the virtual orbitals in the construction of the excitation subspace does not result in significant changes in computed UV/vis spectra for large molecules. The reduced size of the excitation subspace greatly reduces the size of the subspace vectors that need to be stored when using the Davidson procedure to determine the eigenvalues of the TDDFT equations. Furthermore, additional screening of the two-electron integrals in combination with a reduction in the size of the numerical integration grid used in the TDDFT calculation leads to significant computational savings. The use of these approximations represents a simple approach to extend TDDFT to the study of large systems and make the calculations increasingly tractable using modest computing resources.

A Discrete Probability Function Method for the Equation of Radiative Transfer

NASA Technical Reports Server (NTRS)

Sivathanu, Y. R.; Gore, J. P.

1993-01-01

A discrete probability function (DPF) method for the equation of radiative transfer is derived. The DPF is defined as the integral of the probability density function (PDF) over a discrete interval. The derivation allows the evaluation of the PDF of intensities leaving desired radiation paths including turbulence-radiation interactions without the use of computer intensive stochastic methods. The DPF method has a distinct advantage over conventional PDF methods since the creation of a partial differential equation from the equation of transfer is avoided. Further, convergence of all moments of intensity is guaranteed at the basic level of simulation unlike the stochastic method where the number of realizations for convergence of higher order moments increases rapidly. The DPF method is described for a representative path with approximately integral-length scale-sized spatial discretization. The results show good agreement with measurements in a propylene/air flame except for the effects of intermittency resulting from highly correlated realizations. The method can be extended to the treatment of spatial correlations as described in the Appendix. However, information regarding spatial correlations in turbulent flames is needed prior to the execution of this extension.

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.

Discrete sudden perturbation theory for inelastic scattering. I. Quantum and semiclassical treatment

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

Cross, R.J.

1985-12-01

A double perturbation theory is constructed to treat rotationally and vibrationally inelastic scattering. It uses both the elastic scattering from the spherically averaged potential and the infinite-order sudden (IOS) approximation as the unperturbed solutions. First, a standard perturbation expansion is done to express the radial wave functions in terms of the elastic wave functions. The resulting coupled equations are transformed to the discrete-variable representation where the IOS equations are diagonal. Then, the IOS solutions are removed from the equations which are solved by an exponential perturbation approximation. The results for Ar+N/sub 2/ are very much more accurate than the IOSmore » and somewhat more accurate than a straight first-order exponential perturbation theory. The theory is then converted into a semiclassical, time-dependent form by using the WKB approximation. The result is an integral of the potential times a slowly oscillating factor over the classical trajectory. A method of interpolating the result is given so that the calculation is done at the average velocity for a given transition. With this procedure, the semiclassical version of the theory is more accurate than the quantum version and very much faster. Calculations on Ar+N/sub 2/ show the theory to be much more accurate than the infinite-order sudden (IOS) approximation and the exponential time-dependent perturbation theory.« less

Inclusion of exact exchange in the noniterative partial-differential-equation method of electron-molecule scattering - Application to e-N2

NASA Technical Reports Server (NTRS)

Weatherford, C. A.; Onda, K.; Temkin, A.

1985-01-01

The noniterative partial-differential-equation (PDE) approach to electron-molecule scattering of Onda and Temkin (1983) is modified to account for the effects of exchange explicitly. The exchange equation is reduced to a set of inhomogeneous equations containing no integral terms and solved noniteratively in a difference form; a method for propagating the solution to large values of r is described; the changes in the polarization potential of the original PDE method required by the inclusion of exact static exchange are indicated; and the results of computations for e-N2 scattering in the fixed-nuclei approximation are presented in tables and graphs and compared with previous calculations and experimental data. Better agreement is obtained using the modified PDE method.

A strictly Markovian expansion for plasma turbulence theory

NASA Technical Reports Server (NTRS)

Jones, F. C.

1978-01-01

The collision operator that appears in the equation of motion for a particle distribution function that has been averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. In this note we derive an expansion of the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest non-trivial order. The validity of this expansion is seen to be the same as that of the standard quasi-linear expansion.

Comparison of the Effect of Horizontal Vibrations on Interfacial Waves in a Two-Layer System of Inviscid Liquids to Effective Gravity Inversion

NASA Astrophysics Data System (ADS)

Pimenova, Anastasiya V.; Goldobin, Denis S.; Lyubimova, Tatyana P.

2018-02-01

We study the waves at the interface between two thin horizontal layers of immiscible liquids subject to high-frequency tangential vibrations. Nonlinear governing equations are derived for the cases of two- and three-dimensional flows and arbitrary ratio of layer thicknesses. The derivation is performed within the framework of the long-wavelength approximation, which is relevant as the linear instability of a thin-layers system is long-wavelength. The dynamics of equations is integrable and the equations themselves can be compared to the Boussinesq equation for the gravity waves in shallow water, which allows one to compare the action of the vibrational field to the action of the gravity and its possible effective inversion.

A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

NASA Astrophysics Data System (ADS)

Bhrawy, A. H.; Doha, E. H.; Baleanu, D.; Ezz-Eldien, S. S.

2015-07-01

In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.

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.

Iterative discrete ordinates solution of the equation for surface-reflected radiance

NASA Astrophysics Data System (ADS)

Radkevich, Alexander

2017-11-01

This paper presents a new method of numerical solution of the integral equation for the radiance reflected from an anisotropic surface. The equation relates the radiance at the surface level with BRDF and solutions of the standard radiative transfer problems for a slab with no reflection on its surfaces. It is also shown that the kernel of the equation satisfies the condition of the existence of a unique solution and the convergence of the successive approximations to that solution. The developed method features two basic steps: discretization on a 2D quadrature, and solving the resulting system of algebraic equations with successive over-relaxation method based on the Gauss-Seidel iterative process. Presented numerical examples show good coincidence between the surface-reflected radiance obtained with DISORT and the proposed method. Analysis of contributions of the direct and diffuse (but not yet reflected) parts of the downward radiance to the total solution is performed. Together, they represent a very good initial guess for the iterative process. This fact ensures fast convergence. The numerical evidence is given that the fastest convergence occurs with the relaxation parameter of 1 (no relaxation). An integral equation for BRDF is derived as inversion of the original equation. The potential of this new equation for BRDF retrievals is analyzed. The approach is found not viable as the BRDF equation appears to be an ill-posed problem, and it requires knowledge the surface-reflected radiance on the entire domain of both Sun and viewing zenith angles.

YORP on tumbling asteroids

NASA Astrophysics Data System (ADS)

Rozek, A.; Breiter, S.; Vokrouhlicky, D.

2011-10-01

A semi-analytical model of the Yarkovsky-O'Keefe- Radzievskii-Paddack (YORP) effect on an asteroid spin in non principal axis rotation state is presented. Assuming zero conductivity, the YORP torque is represented by spherical harmonics series with vector coefficients, allowing to use any degree and order of approximation. Within the quadrupole approximation of the illumination function we find the same first integrals involving rotational momentum, obliquity and dynamical inertia that were obtained by Cicaló and Scheeres [1]. The integrals do not exist when higher degree terms of illumination function are included and then the asymptotic states known from Vokrouhlický et al. [2] appear. This resolves an apparent contradiction between earlier results. Averaged equations of motion admit stable and unstable limit cycle solutions that were not detected previously.

Dissipative Particle Dynamics at Isothermal Conditions Using Shardlow-Like Splitting Algorithms

DTIC Science & Technology

2013-09-01

purely-repulsive models that contain attractive character further supports the need for effective integration schemes. However, the integration is a...defined in equations 5a and b, and further denote Ct∆φ and diss jit ,;∆φ ( )Nji ≤<≤1 as the solution operators or flow maps associated with...methods for SDEs.) An approximation of t∆φ for first-order splitting (18) can then be given as Ct diss NNt diss NNt diss jit diss t diss tt

Mercedes-Benz water molecules near hydrophobic wall: Integral equation theories vs Monte Carlo simulations

NASA Astrophysics Data System (ADS)

Urbic, T.; Holovko, M. F.

2011-10-01

Associative version of Henderson-Abraham-Barker theory is applied for the study of Mercedes-Benz model of water near hydrophobic surface. We calculated density profiles and adsorption coefficients using Percus-Yevick and soft mean spherical associative approximations. The results are compared with Monte Carlo simulation data. It is shown that at higher temperatures both approximations satisfactory reproduce the simulation data. For lower temperatures, soft mean spherical approximation gives good agreement at low and at high densities while in at mid range densities, the prediction is only qualitative. The formation of a depletion layer between water and hydrophobic surface was also demonstrated and studied.

Mercedes–Benz water molecules near hydrophobic wall: Integral equation theories vs Monte Carlo simulations

PubMed Central

Urbic, T.; Holovko, M. F.

2011-01-01

Associative version of Henderson-Abraham-Barker theory is applied for the study of Mercedes–Benz model of water near hydrophobic surface. We calculated density profiles and adsorption coefficients using Percus-Yevick and soft mean spherical associative approximations. The results are compared with Monte Carlo simulation data. It is shown that at higher temperatures both approximations satisfactory reproduce the simulation data. For lower temperatures, soft mean spherical approximation gives good agreement at low and at high densities while in at mid range densities, the prediction is only qualitative. The formation of a depletion layer between water and hydrophobic surface was also demonstrated and studied. PMID:21992334

Retrieval of the aerosol size distribution in the complex anomalous diffraction approximation

NASA Astrophysics Data System (ADS)

Franssens, Ghislain R.

This contribution reports some recently achieved results in aerosol size distribution retrieval in the complex anomalous diffraction approximation (ADA) to MIE scattering theory. This approximation is valid for spherical particles that are large compared to the wavelength and have a refractive index close to 1. The ADA kernel is compared with the exact MIE kernel. Despite being a simple approximation, the ADA seems to have some practical value for the retrieval of the larger modes of tropospheric and lower stratospheric aerosols. The ADA has the advantage over MIE theory that an analytic inversion of the associated Fredholm integral equation becomes possible. In addition, spectral inversion in the ADA can be formulated as a well-posed problem. In this way, a new inverse formula was obtained, which allows the direct computation of the size distribution as an integral over the spectral extinction function. This formula is valid for particles that both scatter and absorb light and it also takes the spectral dispersion of the refractive index into account. Some details of the numerical implementation of the inverse formula are illustrated using a modified gamma test distribution. Special attention is given to the integration of spectrally truncated discrete extinction data with errors.

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.

Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

NASA Astrophysics Data System (ADS)

Zhao, Hai-Sheng; Zhang, Yao; Lie, Seng-Tjhen

2018-02-01

Considerations of nonlocal elasticity and surface effects in micro- and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged-hinged, clamped-clamped and clamped-hinged ends. For a hinged-hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped-clamped and clamped-hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short, explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.

Towards efficient backward-in-time adjoint computations using data compression techniques

DOE PAGES

Cyr, E. C.; Shadid, J. N.; Wildey, T.

2014-12-16

In the context of a posteriori error estimation for nonlinear time-dependent partial differential equations, the state-of-the-practice is to use adjoint approaches which require the solution of a backward-in-time problem defined by a linearization of the forward problem. One of the major obstacles in the practical application of these approaches, we found, is the need to store, or recompute, the forward solution to define the adjoint problem and to evaluate the error representation. Our study considers the use of data compression techniques to approximate forward solutions employed in the backward-in-time integration. The development derives an error representation that accounts for themore » difference between the standard-approach and the compressed approximation of the forward solution. This representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data-compressed reconstructed solution (i.e. scalar quantities that can be evaluated as the forward problem is integrated). This approach is then compared with existing techniques, such as checkpointing and time-averaged adjoints. Lastly, we provide numerical results indicating the potential efficiency of our approach on a transient diffusion–reaction equation and on the Navier–Stokes equations. These results demonstrate memory compression ratios up to 450×450× while maintaining reasonable accuracy in the error-estimates.« less

Integral equation calculations for the photodisintegration process {sup 4}He({gamma},{ital n}){sup 3}He

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

Ellerkmann, G.; Sandhas, W.; Sofianos, S.A.

1996-06-01

Results obtained by solving Alt-Grassberger-Sandhas (AGS)-type integral equations for the photodisintegration of {sup 4}He, employing the Malfliet-Tjon potential, are compared with the latest experimental data. Good agreement between theory and experiment is found in electric dipole approximation for the total cross section, but the differential cross sections differ at higher energies. This discrepancy is reduced, but not fully removed by taking into account the electric quadrupole contributions. In order to get some feeling for the sensitivity to the underlying potential, we also show calculations based on the Yamaguchi potential. They differ from the Malfliet-Tjon results in a way which resemblesmore » the trends known from triton photodisintegration. {copyright} {ital 1996 The American Physical Society.}« less

A numerical scheme to solve unstable boundary value problems

NASA Technical Reports Server (NTRS)

Kalnay-Rivas, E.

1977-01-01

The considered scheme makes it possible to determine an unstable steady state solution in cases in which, because of lack of symmetry, such a solution cannot be obtained analytically, and other time integration or relaxation schemes, because of instability, fail to converge. The iterative solution of a single complex equation is discussed and a nonlinear system of equations is considered. Described applications of the scheme are related to a steady state solution with shear instability, an unstable nonlinear Ekman boundary layer, and the steady state solution of a baroclinic atmosphere with asymmetric forcing. The scheme makes use of forward and backward time integrations of the original spatial differential operators and of an approximation of the adjoint operators. Only two computations of the time derivative per iteration are required.

An interactive adaptive remeshing algorithm for the two-dimensional Euler equations

NASA Technical Reports Server (NTRS)

Slack, David C.; Walters, Robert W.; Lohner, R.

1990-01-01

An interactive adaptive remeshing algorithm utilizing a frontal grid generator and a variety of time integration schemes for the two-dimensional Euler equations on unstructured meshes is presented. Several device dependent interactive graphics interfaces have been developed along with a device independent DI-3000 interface which can be employed on any computer that has the supporting software including the Cray-2 supercomputers Voyager and Navier. The time integration methods available include: an explicit four stage Runge-Kutta and a fully implicit LU decomposition. A cell-centered finite volume upwind scheme utilizing Roe's approximate Riemann solver is developed. To obtain higher order accurate results a monotone linear reconstruction procedure proposed by Barth is utilized. Results for flow over a transonic circular arc and flow through a supersonic nozzle are examined.

Convergence of Galerkin approximations for operator Riccati equations: A nonlinear evolution equation approach

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1988-01-01

An approximation and convergence theory was developed for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. The Riccati equation was treated as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. A generic approximation result was proven for quasi-autonomous nonlinear evolution system involving accretive operators which was then used to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. The application of the results was illustrated in the context of a linear quadratic optimal control problem for a one dimensional heat equation.

A spectral geometric model for Compton single scatter in PET based on the single scatter simulation approximation

NASA Astrophysics Data System (ADS)

Kazantsev, I. G.; Olsen, U. L.; Poulsen, H. F.; Hansen, P. C.

2018-02-01

We investigate the idealized mathematical model of single scatter in PET for a detector system possessing excellent energy resolution. The model has the form of integral transforms estimating the distribution of photons undergoing a single Compton scattering with a certain angle. The total single scatter is interpreted as the volume integral over scatter points that constitute a rotation body with a football shape, while single scattering with a certain angle is evaluated as the surface integral over the boundary of the rotation body. The equations for total and sample single scatter calculations are derived using a single scatter simulation approximation. We show that the three-dimensional slice-by-slice filtered backprojection algorithm is applicable for scatter data inversion provided that the attenuation map is assumed to be constant. The results of the numerical experiments are presented.

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

Numerical realization of the variational method for generating self-trapped beams.

PubMed

Duque, Erick I; Lopez-Aguayo, Servando; Malomed, Boris A

2018-03-19

We introduce a numerical variational method based on the Rayleigh-Ritz optimization principle for predicting two-dimensional self-trapped beams in nonlinear media. This technique overcomes the limitation of the traditional variational approximation in performing analytical Lagrangian integration and differentiation. Approximate soliton solutions of a generalized nonlinear Schrödinger equation are obtained, demonstrating robustness of the beams of various types (fundamental, vortices, multipoles, azimuthons) in the course of their propagation. The algorithm offers possibilities to produce more sophisticated soliton profiles in general nonlinear models.

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.

Legendre-tau approximations for functional differential equations

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1986-01-01

The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

Legendre-Tau approximations for functional differential equations

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1983-01-01

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

A Unified Treatment of the Acoustic and Elastic Scattered Waves from Fluid-Elastic Media

NASA Astrophysics Data System (ADS)

Denis, Max Fernand

In this thesis, contributions are made to the numerical modeling of the scattering fields from fluid-filled poroelastic materials. Of particular interest are highly porous materials that demonstrate strong contrast to the saturating fluid. A Biot's analysis of porous medium serves as the starting point of the elastic-solid and pore-fluid governing equations of motion. The longitudinal scattering waves of the elastic-solid mode and the pore-fluid mode are modeled by the Kirchhoff-Helmholtz integral equation. The integral equation is evaluated using a series approximation, describing the successive perturbation of the material contrasts. To extended the series' validity into larger domains, rational fraction extrapolation methods are employed. The local Pade□ approximant procedure is a technique that allows one to extrapolate from a scattered field of small contrast into larger values, using Pade□ approximants. To ensure the accuracy of the numerical model, comparisons are made with the exact solution of scattering from a fluid sphere. Mean absolute error analyses, yield convergent and accurate results. In addition, the numerical model correctly predicts the Bragg peaks for a periodic lattice of fluid spheres. In the case of trabecular bones, the far-field scattering pressure attenuation is a superposition of the elastic-solid mode and the pore-fluid mode generated waves from the surrounding fluid and poroelastic boundaries. The attenuation is linearly dependent with frequency between 0.2 and 0.6MHz. The slope of the attenuation is nonlinear with porosity, and does not reflect the mechanical properties of the trabecular bone. The attenuation shows the anisotropic effects of the trabeculae structure. Thus, ultrasound can possibly be employed to non-invasively predict the principal structural orientation of trabecular bones.

Application of variational principles and adjoint integrating factors for constructing numerical GFD models

NASA Astrophysics Data System (ADS)

Penenko, Vladimir; Tsvetova, Elena; Penenko, Alexey

2015-04-01

The proposed method is considered on an example of hydrothermodynamics and atmospheric chemistry models [1,2]. In the development of the existing methods for constructing numerical schemes possessing the properties of total approximation for operators of multiscale process models, we have developed a new variational technique, which uses the concept of adjoint integrating factors. The technique is as follows. First, a basic functional of the variational principle (the integral identity that unites the model equations, initial and boundary conditions) is transformed using Lagrange's identity and the second Green's formula. As a result, the action of the operators of main problem in the space of state functions is transferred to the adjoint operators defined in the space of sufficiently smooth adjoint functions. By the choice of adjoint functions the order of the derivatives becomes lower by one than those in the original equations. We obtain a set of new balance relationships that take into account the sources and boundary conditions. Next, we introduce the decomposition of the model domain into a set of finite volumes. For multi-dimensional non-stationary problems, this technique is applied in the framework of the variational principle and schemes of decomposition and splitting on the set of physical processes for each coordinate directions successively at each time step. For each direction within the finite volume, the analytical solutions of one-dimensional homogeneous adjoint equations are constructed. In this case, the solutions of adjoint equations serve as integrating factors. The results are the hybrid discrete-analytical schemes. They have the properties of stability, approximation and unconditional monotony for convection-diffusion operators. These schemes are discrete in time and analytic in the spatial variables. They are exact in case of piecewise-constant coefficients within the finite volume and along the coordinate lines of the grid area in each direction on a time step. In each direction, they have tridiagonal structure. They are solved by the sweep method. An important advantage of the discrete-analytical schemes is that the values of derivatives at the boundaries of finite volume are calculated together with the values of the unknown functions. This technique is particularly attractive for problems with dominant convection, as it does not require artificial monotonization and limiters. The same idea of integrating factors is applied in temporal dimension to the stiff systems of equations describing chemical transformation models [2]. The proposed method is applicable for the problems involving convection-diffusion-reaction operators. The work has been partially supported by the Presidium of RAS under Program 43, and by the RFBR grants 14-01-00125 and 14-01-31482. References: 1. V.V. Penenko, E.A. Tsvetova, A.V. Penenko. Variational approach and Euler's integrating factors for environmental studies// Computers and Mathematics with Applications, (2014) V.67, Issue 12, P. 2240-2256. 2. V.V.Penenko, E.A.Tsvetova. Variational methods of constructing monotone approximations for atmospheric chemistry models // Numerical analysis and applications, 2013, V. 6, Issue 3, pp 210-220.

Tunneling effects in electromagnetic wave scattering by nonspherical particles: A comparison of the Debye series and physical-geometric optics approximations

NASA Astrophysics Data System (ADS)

Bi, Lei; Yang, Ping

2016-07-01

The accuracy of the physical-geometric optics (PG-O) approximation is examined for the simulation of electromagnetic scattering by nonspherical dielectric particles. This study seeks a better understanding of the tunneling effect on the phase matrix by employing the invariant imbedding method to rigorously compute the zeroth-order Debye series, from which the tunneling efficiency and the phase matrix corresponding to the diffraction and external reflection are obtained. The tunneling efficiency is shown to be a factor quantifying the relative importance of the tunneling effect over the Fraunhofer diffraction near the forward scattering direction. Due to the tunneling effect, different geometries with the same projected cross section might have different diffraction patterns, which are traditionally assumed to be identical according to the Babinet principle. For particles with a fixed orientation, the PG-O approximation yields the external reflection pattern with reasonable accuracy, but ordinarily fails to predict the locations of peaks and minima in the diffraction pattern. The larger the tunneling efficiency, the worse the PG-O accuracy is at scattering angles less than 90°. If the particles are assumed to be randomly oriented, the PG-O approximation yields the phase matrix close to the rigorous counterpart, primarily due to error cancellations in the orientation-average process. Furthermore, the PG-O approximation based on an electric field volume-integral equation is shown to usually be much more accurate than the Kirchhoff surface integral equation at side-scattering angles, particularly when the modulus of the complex refractive index is close to unity. Finally, tunneling efficiencies are tabulated for representative faceted particles.

Gaussian theory for spatially distributed self-propelled particles

NASA Astrophysics Data System (ADS)

Seyed-Allaei, Hamid; Schimansky-Geier, Lutz; Ejtehadi, Mohammad Reza

2016-12-01

Obtaining a reduced description with particle and momentum flux densities outgoing from the microscopic equations of motion of the particles requires approximations. The usual method, we refer to as truncation method, is to zero Fourier modes of the orientation distribution starting from a given number. Here we propose another method to derive continuum equations for interacting self-propelled particles. The derivation is based on a Gaussian approximation (GA) of the distribution of the direction of particles. First, by means of simulation of the microscopic model, we justify that the distribution of individual directions fits well to a wrapped Gaussian distribution. Second, we numerically integrate the continuum equations derived in the GA in order to compare with results of simulations. We obtain that the global polarization in the GA exhibits a hysteresis in dependence on the noise intensity. It shows qualitatively the same behavior as we find in particles simulations. Moreover, both global polarizations agree perfectly for low noise intensities. The spatiotemporal structures of the GA are also in agreement with simulations. We conclude that the GA shows qualitative agreement for a wide range of noise intensities. In particular, for low noise intensities the agreement with simulations is better as other approximations, making the GA to an acceptable candidates of describing spatially distributed self-propelled particles.

Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: A tutorial review

NASA Astrophysics Data System (ADS)

Kahnert, Michael

2016-07-01

Numerical solution methods for electromagnetic scattering by non-spherical particles comprise a variety of different techniques, which can be traced back to different assumptions and solution strategies applied to the macroscopic Maxwell equations. One can distinguish between time- and frequency-domain methods; further, one can divide numerical techniques into finite-difference methods (which are based on approximating the differential operators), separation-of-variables methods (which are based on expanding the solution in a complete set of functions, thus approximating the fields), and volume integral-equation methods (which are usually solved by discretisation of the target volume and invoking the long-wave approximation in each volume cell). While existing reviews of the topic often tend to have a target audience of program developers and expert users, this tutorial review is intended to accommodate the needs of practitioners as well as novices to the field. The required conciseness is achieved by limiting the presentation to a selection of illustrative methods, and by omitting many technical details that are not essential at a first exposure to the subject. On the other hand, the theoretical basis of numerical methods is explained with little compromises in mathematical rigour; the rationale is that a good grasp of numerical light scattering methods is best achieved by understanding their foundation in Maxwell's theory.

Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation.

PubMed

Augustin, Moritz; Ladenbauer, Josef; Baumann, Fabian; Obermayer, Klaus

2017-06-01

The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. This approach, however, leads to a model with an infinite-dimensional state space and non-standard boundary conditions. Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. Particularly the cascade-based models are overall most accurate and robust, especially in the sensitive region of rapidly changing input. For the mean-driven regime, when input fluctuations are not too strong and fast, however, the best performing model is based on the spectral decomposition. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback. The computational demands of the reduced models are very low but the implementation complexity differs between the different model variants. Therefore we have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models.

Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation

PubMed Central

Baumann, Fabian; Obermayer, Klaus

2017-01-01

The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. This approach, however, leads to a model with an infinite-dimensional state space and non-standard boundary conditions. Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. Particularly the cascade-based models are overall most accurate and robust, especially in the sensitive region of rapidly changing input. For the mean-driven regime, when input fluctuations are not too strong and fast, however, the best performing model is based on the spectral decomposition. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback. The computational demands of the reduced models are very low but the implementation complexity differs between the different model variants. Therefore we have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models. PMID:28644841

Equation of state of the one- and three-dimensional Bose-Bose gases

NASA Astrophysics Data System (ADS)

Chiquillo, Emerson

2018-06-01

We calculate the equation of state of Bose-Bose gases in one and three dimensions in the framework of an effective quantum field theory. The beyond-mean-field approximation at zero temperature and the one-loop finite-temperature results are obtained performing functional integration on a local effective action. The ultraviolet divergent zero-point quantum fluctuations are removed by means of dimensional regularization. We derive the nonlinear Schrödinger equation to describe one- and three-dimensional Bose-Bose mixtures and solve it analytically in the one-dimensional scenario. This equation supports self-trapped brightlike solitonic droplets and self-trapped darklike solitons. At low temperature, we also find that the pressure and the number of particles of symmetric quantum droplets have a nontrivial dependence on the chemical potential and the difference between the intra- and the interspecies coupling constants.

Quasi-optical simulation of the electron cyclotron plasma heating in a mirror magnetic trap

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

Shalashov, A. G., E-mail: ags@appl.sci-nnov.ru; Balakin, A. A.; Khusainov, T. A.

The resonance microwave plasma heating in a large-scale open magnetic trap is simulated taking into account all the basic wave effects during the propagation of short-wavelength wave beams (diffraction, dispersion, and aberration) within the framework of the consistent quasi-optical approximation of Maxwell’s equations. The quasi-optical method is generalized to the case of inhomogeneous media with absorption dispersion, a new form of the quasi-optical equation is obtained, the efficient method for numerical integration is found, and simulation results are verified on the GDT facility (Novosibirsk).

A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

DOE PAGES

Lehoucq, Richard B.; Rowe, Stephen T.

2016-02-01

We introduce a discretization for a nonlocal diffusion problem using a localized basis of radial basis functions. The stiffness matrix entries are assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, sparse, symmetric positive definite stiffness matrix. We demonstrate that both the continuum and discrete problems are well-posed and present numerical results for the convergence behavior of the radial basis function method. As a result, we explore approximating the solution to anisotropic differential equations by solving anisotropic nonlocal integral equations using the radial basis function method.

Semiempirical methods for computing turbulent flows

NASA Technical Reports Server (NTRS)

Belov, I. A.; Ginzburg, I. P.

1986-01-01

Two semiempirical theories which provide a basis for determining the turbulent friction and heat exchange near a wall are presented: (1) the Prandtl-Karman theory, and (2) the theory utilizing an equation for the energy of turbulent pulsations. A comparison is made between exact numerical methods and approximate integral methods for computing the turbulent boundary layers in the presence of pressure, blowing, or suction gradients. Using the turbulent flow around a plate as an example, it is shown that, when computing turbulent flows with external turbulence, it is preferable to construct a turbulence model based on the equation for energy of turbulent pulsations.

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.

A Closely Coupled Experimental and Numerical Approach for Hypersonic and High Enthalpy Flow Investigations Utilising the HEG Shock Tunnel and the DLR TAU Code

DTIC Science & Technology

2010-04-01

factorization scheme (Lower-Upper Symmetric Gauss- Seidel ) can be used for time integration. Additional convergence acceleration is achieved by the...of the full Stefan -Maxwell equations. The diffusive mass flux of species S is computed according to: for 1 for jS S S Sm j jm S j eS jd S S S j j j...approximate factorization scheme (Lower-Upper Symmetric Gauss- Seidel ). For steady state problems, equation (69) reduces to R=0 because ddU t

Scale matters

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

Margolin, L. G.

The applicability of Navier–Stokes equations is limited to near-equilibrium flows in which the gradients of density, velocity and energy are small. Here I propose an extension of the Chapman–Enskog approximation in which the velocity probability distribution function (PDF) is averaged in the coordinate phase space as well as the velocity phase space. I derive a PDF that depends on the gradients and represents a first-order generalization of local thermodynamic equilibrium. I then integrate this PDF to derive a hydrodynamic model. Finally, I discuss the properties of that model and its relation to the discrete equations of computational fluid dynamics.

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.

Scale matters

DOE PAGES

Margolin, L. G.

2018-03-19

The applicability of Navier–Stokes equations is limited to near-equilibrium flows in which the gradients of density, velocity and energy are small. Here I propose an extension of the Chapman–Enskog approximation in which the velocity probability distribution function (PDF) is averaged in the coordinate phase space as well as the velocity phase space. I derive a PDF that depends on the gradients and represents a first-order generalization of local thermodynamic equilibrium. I then integrate this PDF to derive a hydrodynamic model. Finally, I discuss the properties of that model and its relation to the discrete equations of computational fluid dynamics.

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

Sergyeyev, Artur; Krtous, Pavel; Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, V Holesovickach 2, Prague

We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in [J. High Energy Phys. 02 (2007) 004] and show that these operators, along with the first-order operators originating from the Killing vectors, form a complete set of commuting symmetry operators (i.e., integrals of motion) for the Klein-Gordon equation. Moreover, we demonstrate that the separated solutions of the Klein-Gordon equation obtained in [J. High Energy Phys. 02 (2007) 005] are joint eigenfunctions for all of thesemore » operators. We also present an explicit form of the zero mode for the Klein-Gordon equation with zero mass. In the semiclassical approximation we find that the separated solutions of the Hamilton-Jacobi equation for geodesic motion are also solutions for a set of Hamilton-Jacobi-type equations which correspond to the quadratic conserved quantities arising from the above Killing tensors.« less

ΛCDM Cosmology for Astronomers

NASA Astrophysics Data System (ADS)

Condon, J. J.; Matthews, A. M.

2018-07-01

The homogeneous, isotropic, and flat ΛCDM universe favored by observations of the cosmic microwave background can be described using only Euclidean geometry, locally correct Newtonian mechanics, and the basic postulates of special and general relativity. We present simple derivations of the most useful equations connecting astronomical observables (redshift, flux density, angular diameter, brightness, local space density, ...) with the corresponding intrinsic properties of distant sources (lookback time, distance, spectral luminosity, linear size, specific intensity, source counts, ...). We also present an analytic equation for lookback time that is accurate within 0.1% for all redshifts z. The exact equation for comoving distance is an elliptic integral that must be evaluated numerically, but we found a simple approximation with errors <0.2% for all redshifts up to z ≈ 50.

Computation of Large-Scale Structure Jet Noise Sources With Weak Nonlinear Effects Using Linear Euler

NASA Technical Reports Server (NTRS)

Dahl, Milo D.; Hixon, Ray; Mankbadi, Reda R.

2003-01-01

An approximate technique is presented for the prediction of the large-scale turbulent structure sound source in a supersonic jet. A linearized Euler equations code is used to solve for the flow disturbances within and near a jet with a given mean flow. Assuming a normal mode composition for the wave-like disturbances, the linear radial profiles are used in an integration of the Navier-Stokes equations. This results in a set of ordinary differential equations representing the weakly nonlinear self-interactions of the modes along with their interaction with the mean flow. Solutions are then used to correct the amplitude of the disturbances that represent the source of large-scale turbulent structure sound in the jet.

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.

Variational methods for direct/inverse problems of atmospheric dynamics and chemistry

NASA Astrophysics Data System (ADS)

Penenko, Vladimir; Penenko, Alexey; Tsvetova, Elena

2013-04-01

We present a variational approach for solving direct and inverse problems of atmospheric hydrodynamics and chemistry. It is important that the accurate matching of numerical schemes has to be provided in the chain of objects: direct/adjoint problems - sensitivity relations - inverse problems, including assimilation of all available measurement data. To solve the problems we have developed a new enhanced set of cost-effective algorithms. The matched description of the multi-scale processes is provided by a specific choice of the variational principle functionals for the whole set of integrated models. Then all functionals of variational principle are approximated in space and time by splitting and decomposition methods. Such approach allows us to separately consider, for example, the space-time problems of atmospheric chemistry in the frames of decomposition schemes for the integral identity sum analogs of the variational principle at each time step and in each of 3D finite-volumes. To enhance the realization efficiency, the set of chemical reactions is divided on the subsets related to the operators of production and destruction. Then the idea of the Euler's integrating factors is applied in the frames of the local adjoint problem technique [1]-[3]. The analytical solutions of such adjoint problems play the role of integrating factors for differential equations describing atmospheric chemistry. With their help, the system of differential equations is transformed to the equivalent system of integral equations. As a result we avoid the construction and inversion of preconditioning operators containing the Jacobi matrixes which arise in traditional implicit schemes for ODE solution. This is the main advantage of our schemes. At the same time step but on the different stages of the "global" splitting scheme, the system of atmospheric dynamic equations is solved. For convection - diffusion equations for all state functions in the integrated models we have developed the monotone and stable discrete-analytical numerical schemes [1]-[3] conserving the positivity of the chemical substance concentrations and possessing the properties of energy and mass balance that are postulated in the general variational principle for integrated models. All algorithms for solution of transport, diffusion and transformation problems are direct (without iterations). The work is partially supported by the Programs No 4 of Presidium RAS and No 3 of Mathematical Department of RAS, by RFBR project 11-01-00187 and Integrating projects of SD RAS No 8 and 35. Our studies are in the line with the goals of COST Action ES1004. References Penenko V., Tsvetova E. Discrete-analytical methods for the implementation of variational principles in environmental applications// Journal of computational and applied mathematics, 2009, v. 226, 319-330. Penenko A.V. Discrete-analytic schemes for solving an inverse coefficient heat conduction problem in a layered medium with gradient methods// Numerical Analysis and Applications, 2012, V. 5, pp 326-341. V. Penenko, E. Tsvetova. Variational methods for constructing the monotone approximations for atmospheric chemistry models //Numerical Analysis and Applications, 2013 (in press).

Integrated Conceptual Design of Joined-Wing SensorCraft Using Response Surface Models

DTIC Science & Technology

2006-11-01

vi Acknowledgements I would like to express my sincere appreciation to my thesis advisor, Dr. Robert Canfield for his guidance and...55 Raymer Approximate and Group Weights Sizing Methods....................................... 57 Finite Element Model Structural Weight...Empty Weight Fraction Equation ............................... 54 Figure 29 Response of Refined Weight to T/W and W/S Inputs for Model (2) Raymer ASW

Counterflow diffusion flames: effects of thermal expansion and non-unity Lewis numbers

NASA Astrophysics Data System (ADS)

Koundinyan, Sushilkumar P.; Matalon, Moshe; Stewart, D. Scott

2018-05-01

In this work we re-examine the counterflow diffusion flame problem focusing in particular on the flame-flow interactions due to thermal expansion and its influence on various flame properties such as flame location, flame temperature, reactant leakage and extinction conditions. The analysis follows two different procedures: an asymptotic approximation for large activation energy chemical reactions, and a direct numerical approach. The asymptotic treatment follows the general theory of Cheatham and Matalon, which consists of a free-boundary problem with jump conditions across the surface representing the reaction sheet, and is well suited for variable-density flows and for mixtures with non-unity and distinct Lewis numbers for the fuel and oxidiser. Due to density variations, the species and energy transport equations are coupled to the Navier-Stokes equations and the problem does not possess an analytical solution. We thus propose and implement a methodology for solving the free-boundary problem numerically. Results based on the asymptotic approximation are then verified against those obtained from the 'exact' numerical integration of the governing equations, comparing predictions of the various flame properties.

On the r-mode spectrum of relativistic stars in the low-frequency approximation

NASA Astrophysics Data System (ADS)

Ruoff, Johannes; Kokkotas, Kostas D.

2001-12-01

The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time-independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for l=2, it is nevertheless also possible to find discrete mode solutions (the r modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time-dependent equations. For stellar models admitting a physical r-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of l it seems that in certain cases there are no mode solutions at all.

A real-time approximate optimal guidance law for flight in a plane

NASA Technical Reports Server (NTRS)

Feeley, Timothy S.; Speyer, Jason L.

1990-01-01

A real-time guidance scheme is presented for the problem of maximizing the payload into orbit subject to the equations of motion of a rocket over a nonrotating spherical earth. The flight is constrained to a path in the equatorial plane while reaching an orbital altitude at orbital injection speeds. The dynamics of the problem can be separated into primary and perturbation effects by a small parameter, epsilon, which is the ratio of the atmospheric scale height to the radius of the earth. The Hamilton-Jacobi-Bellman or dynamic programming equation is expanded in an asymptotic series where the zeroth-order term (epsilon = 0) can be obtained in closed form. The neglected perturbation terms are included in the higher-order terms of the expansion, which are determined from the solution of first-order linear partial differential equations requiring only integrations which are quadratures. The quadratures can be performed rapidly with emerging computer capability, so that real-time approximate optimization can be used to construct the launch guidance law. The application of this technique to flight in three-dimensions is made apparent from the solution presented.

Towards information-optimal simulation of partial differential equations.

PubMed

Leike, Reimar H; Enßlin, Torsten A

2018-03-01

Most simulation schemes for partial differential equations (PDEs) focus on minimizing a simple error norm of a discretized version of a field. This paper takes a fundamentally different approach; the discretized field is interpreted as data providing information about a real physical field that is unknown. This information is sought to be conserved by the scheme as the field evolves in time. Such an information theoretic approach to simulation was pursued before by information field dynamics (IFD). In this paper we work out the theory of IFD for nonlinear PDEs in a noiseless Gaussian approximation. The result is an action that can be minimized to obtain an information-optimal simulation scheme. It can be brought into a closed form using field operators to calculate the appearing Gaussian integrals. The resulting simulation schemes are tested numerically in two instances for the Burgers equation. Their accuracy surpasses finite-difference schemes on the same resolution. The IFD scheme, however, has to be correctly informed on the subgrid correlation structure. In certain limiting cases we recover well-known simulation schemes like spectral Fourier-Galerkin methods. We discuss implications of the approximations made.

Mercedes-Benz water molecules near hydrophobic wall: integral equation theories vs Monte Carlo simulations.

PubMed

Urbic, T; Holovko, M F

2011-10-07

Associative version of Henderson-Abraham-Barker theory is applied for the study of Mercedes-Benz model of water near hydrophobic surface. We calculated density profiles and adsorption coefficients using Percus-Yevick and soft mean spherical associative approximations. The results are compared with Monte Carlo simulation data. It is shown that at higher temperatures both approximations satisfactory reproduce the simulation data. For lower temperatures, soft mean spherical approximation gives good agreement at low and at high densities while in at mid range densities, the prediction is only qualitative. The formation of a depletion layer between water and hydrophobic surface was also demonstrated and studied. © 2011 American Institute of Physics

A comparative study of Rosenbrock-type and implicit Runge-Kutta time integration for discontinuous Galerkin method for unsteady 3D compressible Navier-Stokes equations

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

Liu, Xiaodong; Xia, Yidong; Luo, Hong

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flowsmore » to DNS of turbulent flows, are presented to assess the performance of these schemes. Here, numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.« less

A comparative study of Rosenbrock-type and implicit Runge-Kutta time integration for discontinuous Galerkin method for unsteady 3D compressible Navier-Stokes equations

DOE PAGES

Liu, Xiaodong; Xia, Yidong; Luo, Hong; ...

2016-10-05

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flowsmore » to DNS of turbulent flows, are presented to assess the performance of these schemes. Here, numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.« less

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.

The Pendulum Equation

ERIC Educational Resources Information Center

Fay, Temple H.

2002-01-01

We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…

The Adams formulas for numerical integration of differential equations from 1st to 20th order

NASA Technical Reports Server (NTRS)

Kirkpatrick, J. C.

1976-01-01

The Adams Bashforth predictor coefficients and the Adams Moulton corrector coefficients for the integration of differential equations are presented for methods of 1st to 20th order. The order of the method as presented refers to the highest order difference formula used in Newton's backward difference interpolation formula, on which the Adams method is based. The Adams method is a polynomial approximation method derived from Newton's backward difference interpolation formula. The Newton formula is derived and expanded to 20th order. The Adams predictor and corrector formulas are derived and expressed in terms of differences of the derivatives, as well as in terms of the derivatives themselves. All coefficients are given to 18 significant digits. For the difference formula only, the ratio coefficients are given to 10th order.

Comparative Analysis of Models of the Earth's Gravity: 3. Accuracy of Predicting EAS Motion

NASA Astrophysics Data System (ADS)

Kuznetsov, E. D.; Berland, V. E.; Wiebe, Yu. S.; Glamazda, D. V.; Kajzer, G. T.; Kolesnikov, V. I.; Khremli, G. P.

2002-05-01

This paper continues a comparative analysis of modern satellite models of the Earth's gravity which we started in [6, 7]. In the cited works, the uniform norms of spherical functions were compared with their gradients for individual harmonics of the geopotential expansion [6] and the potential differences were compared with the gravitational accelerations obtained in various models of the Earth's gravity [7]. In practice, it is important to know how consistently the EAS motion is represented by various geopotential models. Unless otherwise stated, a model version in which the equations of motion are written using the classical Encke scheme and integrated together with the variation equations by the implicit one-step Everhart's algorithm [1] was used. When calculating coordinates and velocities on the integration step (at given instants of time), the approximate Everhart formula was employed.

Adiabatic particle motion in a nearly drift-free magnetic field: Application to the geomagnetic tail

NASA Technical Reports Server (NTRS)

Stern, D. P.

1977-01-01

The guiding center motion of particles in a nearly drift free magnetic field is analyzed in order to investigate the dependence of mean drift velocity on equatorial pitch angle, the variation of local drift velocity along the trajectory, and other properties. The mean drift for adiabatic particles is expressed by means of elliptic integrals. Approximations to the twice-averaged Hamiltonian W near z = O are derived, permitting simple representation of drift paths if an electric potential also exists. In addition, the use of W or of expressions for the longitudinal invariant allows the derivation of the twice averaged Liouville equation and of the corresponding Vlasov equation. Bounce times are calculated (using the drift-free approximation), as are instantaneous guiding center drift velocities, which are then used to provide a numerical check on the formulas for the mean drift.

New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

PubMed Central

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. PMID:25996369

An investigation of dynamic-analysis methods for variable-geometry structures

NASA Technical Reports Server (NTRS)

Austin, F.

1980-01-01

Selected space structure configurations were reviewed in order to define dynamic analysis problems associated with variable geometry. The dynamics of a beam being constructed from a flexible base and the relocation of the completed beam by rotating the remote manipulator system about the shoulder joint were selected. Equations of motion were formulated in physical coordinates for both of these problems, and FORTRAN programs were developed to generate solutions by numerically integrating the equations. These solutions served as a standard of comparison to gauge the accuracy of approximate solution techniques that were developed and studied. Good control was achieved in both problems. Unstable control system coupling with the system flexibility did not occur. An approximate method was developed for each problem to enable the analyst to investigate variable geometry effects during a short time span using standard fixed geometry programs such as NASTRAN. The average angle and average length techniques are discussed.

Microwave inversion of leaf area and inclination angle distributions from backscattered data

NASA Technical Reports Server (NTRS)

Lang, R. H.; Saleh, H. A.

1985-01-01

The backscattering coefficient from a slab of thin randomly oriented dielectric disks over a flat lossy ground is used to reconstruct the inclination angle and area distributions of the disks. The disks are employed to model a leafy agricultural crop, such as soybeans, in the L-band microwave region of the spectrum. The distorted Born approximation, along with a thin disk approximation, is used to obtain a relationship between the horizontal-like polarized backscattering coefficient and the joint probability density of disk inclination angle and disk radius. Assuming large skin depth reduces the relationship to a linear Fredholm integral equation of the first kind. Due to the ill-posed nature of this equation, a Phillips-Twomey regularization method with a second difference smoothing condition is used to find the inversion. Results are obtained in the presence of 1 and 10 percent noise for both leaf inclination angle and leaf radius densities.

New operational matrices for solving fractional differential equations on the half-line.

PubMed

Bhrawy, Ali H; Taha, Taha M; Alzahrani, Ebraheem O; Alzahrani, Ebrahim O; Baleanu, Dumitru; Alzahrani, Abdulrahim A

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

NASA Astrophysics Data System (ADS)

Bhrawy, A. H.; Zaky, M. A.

2015-01-01

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time-space fractional differential equation with Dirichlet boundary conditions. The shifted Jacobi operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives are presented. By using these operational matrices, we propose a shifted Jacobi tau method for both temporal and spatial discretizations, which allows us to present an efficient spectral method for solving such problem. Furthermore, the error is estimated and the proposed method has reasonable convergence rates in spatial and temporal discretizations. In addition, some known spectral tau approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters θ and ϑ. Finally, in order to demonstrate its accuracy, we compare our method with those reported in the literature.

An Analytic Approximation to Very High Specific Impulse and Specific Power Interplanetary Space Mission Analysis

NASA Technical Reports Server (NTRS)

Williams, Craig Hamilton

1995-01-01

A simple, analytic approximation is derived to calculate trip time and performance for propulsion systems of very high specific impulse (50,000 to 200,000 seconds) and very high specific power (10 to 1000 kW/kg) for human interplanetary space missions. The approach assumed field-free space, constant thrust/constant specific power, and near straight line (radial) trajectories between the planets. Closed form, one dimensional equations of motion for two-burn rendezvous and four-burn round trip missions are derived as a function of specific impulse, specific power, and propellant mass ratio. The equations are coupled to an optimizing parameter that maximizes performance and minimizes trip time. Data generated for hypothetical one-way and round trip human missions to Jupiter were found to be within 1% and 6% accuracy of integrated solutions respectively, verifying that for these systems, credible analysis does not require computationally intensive numerical techniques.

Dual-shaped offset reflector antenna designs from solutions of the geometrical optics first-order partial differential equations

NASA Technical Reports Server (NTRS)

Galindo-Israel, V.; Imbriale, W.; Shogen, K.; Mittra, R.

1990-01-01

In obtaining solutions to the first-order nonlinear partial differential equations (PDEs) for synthesizing offset dual-shaped reflectors, it is found that previously observed computational problems can be avoided if the integration of the PDEs is started from an inner projected perimeter and integrated outward rather than starting from an outer projected perimeter and integrating inward. This procedure, however, introduces a new parameter, the main reflector inner perimeter radius p(o), when given a subreflector inner angle 0(o). Furthermore, a desired outer projected perimeter (e.g., a circle) is no longer guaranteed. Stability of the integration is maintained if some of the initial parameters are determined first from an approximate solution to the PDEs. A one-, two-, or three-parameter optimization algorithm can then be used to obtain a best set of parameters yielding a close fit to the desired projected outer rim. Good low cross-polarization mapping functions are also obtained. These methods are illustrated by synthesis of a high-gain offset-shaped Cassegrainian antenna and a low-noise offset-shaped Gregorian antenna.

Path integral solution for a Klein-Gordon particle in vector and scalar deformed radial Rosen-Morse-type potentials

NASA Astrophysics Data System (ADS)

Khodja, A.; Kadja, A.; Benamira, F.; Guechi, L.

2017-12-01

The problem of a Klein-Gordon particle moving in equal vector and scalar Rosen-Morse-type potentials is solved in the framework of Feynman's path integral approach. Explicit path integration leads to a closed form for the radial Green's function associated with different shapes of the potentials. For q≤-1, and 1/2α ln | q|0, it is shown that the quantization conditions for the bound state energy levels E_{nr} are transcendental equations which can be solved numerically. Three special cases such as the standard radial Manning-Rosen potential (| q| =1), the standard radial Rosen-Morse potential (V2→ -V2,q=1) and the radial Eckart potential (V1→ -V1,q=1) are also briefly discussed.

An Explicit Upwind Algorithm for Solving the Parabolized Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Korte, John J.

1991-01-01

An explicit, upwind algorithm was developed for the direct (noniterative) integration of the 3-D Parabolized Navier-Stokes (PNS) equations in a generalized coordinate system. The new algorithm uses upwind approximations of the numerical fluxes for the pressure and convection terms obtained by combining flux difference splittings (FDS) formed from the solution of an approximate Riemann (RP). The approximate RP is solved using an extension of the method developed by Roe for steady supersonic flow of an ideal gas. Roe's method is extended for use with the 3-D PNS equations expressed in generalized coordinates and to include Vigneron's technique of splitting the streamwise pressure gradient. The difficulty associated with applying Roe's scheme in the subsonic region is overcome. The second-order upwind differencing of the flux derivatives are obtained by adding FDS to either an original forward or backward differencing of the flux derivative. This approach is used to modify an explicit MacCormack differencing scheme into an upwind differencing scheme. The second order upwind flux approximations, applied with flux limiters, provide a method for numerically capturing shocks without the need for additional artificial damping terms which require adjustment by the user. In addition, a cubic equation is derived for determining Vegneron's pressure splitting coefficient using the updated streamwise flux vector. Decoding the streamwise flux vector with the updated value of Vigneron's pressure splitting improves the stability of the scheme. The new algorithm is applied to 2-D and 3-D supersonic and hypersonic laminar flow test cases. Results are presented for the experimental studies of Holden and of Tracy. In addition, a flow field solution is presented for a generic hypersonic aircraft at a Mach number of 24.5 and angle of attack of 1 degree. The computed results compare well to both experimental data and numerical results from other algorithms. Computational times required for the upwind PNS code are approximately equal to an explicit PNS MacCormack's code and existing implicit PNS solvers.

The Laguerre finite difference one-way equation solver

NASA Astrophysics Data System (ADS)

Terekhov, Andrew V.

2017-05-01

This paper presents a new finite difference algorithm for solving the 2D one-way wave equation with a preliminary approximation of a pseudo-differential operator by a system of partial differential equations. As opposed to the existing approaches, the integral Laguerre transform instead of Fourier transform is used. After carrying out the approximation of spatial variables it is possible to obtain systems of linear algebraic equations with better computing properties and to reduce computer costs for their solution. High accuracy of calculations is attained at the expense of employing finite difference approximations of higher accuracy order that are based on the dispersion-relationship-preserving method and the Richardson extrapolation in the downward continuation direction. The numerical experiments have verified that as compared to the spectral difference method based on Fourier transform, the new algorithm allows one to calculate wave fields with a higher degree of accuracy and a lower level of numerical noise and artifacts including those for non-smooth velocity models. In the context of solving the geophysical problem the post-stack migration for velocity models of the types Syncline and Sigsbee2A has been carried out. It is shown that the images obtained contain lesser noise and are considerably better focused as compared to those obtained by the known Fourier Finite Difference and Phase-Shift Plus Interpolation methods. There is an opinion that purely finite difference approaches do not allow carrying out the seismic migration procedure with sufficient accuracy, however the results obtained disprove this statement. For the supercomputer implementation it is proposed to use the parallel dichotomy algorithm when solving systems of linear algebraic equations with block-tridiagonal matrices.

A refinement of the combination equations for evaporation

USGS Publications Warehouse

Milly, P.C.D.

1991-01-01

Most combination equations for evaporation rely on a linear expansion of the saturation vapor-pressure curve around the air temperature. Because the temperature at the surface may differ from this temperature by several degrees, and because the saturation vapor-pressure curve is nonlinear, this approximation leads to a certain degree of error in those evaporation equations. It is possible, however, to introduce higher-order polynomial approximations for the saturation vapor-pressure curve and to derive a family of explicit equations for evaporation, having any desired degree of accuracy. Under the linear approximation, the new family of equations for evaporation reduces, in particular cases, to the combination equations of H. L. Penman (Natural evaporation from open water, bare soil and grass, Proc. R. Soc. London, Ser. A193, 120-145, 1948) and of subsequent workers. Comparison of the linear and quadratic approximations leads to a simple approximate expression for the error associated with the linear case. Equations based on the conventional linear approximation consistently underestimate evaporation, sometimes by a substantial amount. ?? 1991 Kluwer Academic Publishers.

On the origins of approximations for stochastic chemical kinetics.

PubMed

Haseltine, Eric L; Rawlings, James B

2005-10-22

This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.

Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition

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

Jang, Seogjoo, E-mail: sjang@qc.cuny.edu

2016-06-07

This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functionalmore » but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.« less

Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition

NASA Astrophysics Data System (ADS)

Jang, Seogjoo

2016-06-01

This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.

Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability

NASA Technical Reports Server (NTRS)

Ito, K.

1984-01-01

The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

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.

Single-scatter vector-wave scattering from surfaces with infinite slopes using the Kirchhoff approximation.

PubMed

Bruce, Neil C

2008-08-01

This paper presents a new formulation of the 3D Kirchhoff approximation that allows calculation of the scattering of vector waves from 2D rough surfaces containing structures with infinite slopes. This type of surface has applications, for example, in remote sensing and in testing or imaging of printed circuits. Some preliminary calculations for rectangular-shaped grooves in a plane are presented for the 2D surface method and are compared with the equivalent 1D surface calculations for the Kirchhoff and integral equation methods. Good agreement is found between the methods.

An efficient computational method for the approximate solution of nonlinear Lane-Emden type equations arising in astrophysics

NASA Astrophysics Data System (ADS)

Singh, Harendra

2018-04-01

The key purpose of this article is to introduce an efficient computational method for the approximate solution of the homogeneous as well as non-homogeneous nonlinear Lane-Emden type equations. Using proposed computational method given nonlinear equation is converted into a set of nonlinear algebraic equations whose solution gives the approximate solution to the Lane-Emden type equation. Various nonlinear cases of Lane-Emden type equations like standard Lane-Emden equation, the isothermal gas spheres equation and white-dwarf equation are discussed. Results are compared with some well-known numerical methods and it is observed that our results are more accurate.

Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium

NASA Astrophysics Data System (ADS)

Horowitz, Jordan M.

2015-07-01

The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.

Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium.

PubMed

Horowitz, Jordan M

2015-07-28

The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.

Solvent dynamics and electron transfer reactions

NASA Astrophysics Data System (ADS)

Rasaiah, Jayendran C.; Zhu, Jianjun

1994-02-01

Recent experimental and theoretical studies of the influence of solvent dynamics on electron transfer (ET) reactions are discussed. It is seen that the survival probabilities of the reactants and products can be obtained as the solution to an integral equation using experimental or simulation data on the solvation dynamics. The theory developed for ET between thermally equilibrated reactants in solution, in which the ligand vibrations were treated classically, is extended to include quantum effects on the inner-shell ligand vibration and electron transfer from a nonequilibrium initial state prepared, for example, by laser excitation. This leads to a slight modification of the integral equation which is easily solved on a personal computer to provide results that can be directly compared with experiment. Analytic approximations to the solutions of the integral equation, ranging from a single exponential to multiexponential time dependence of the survival probabilities are discussed. The rate constant for the single exponential decay of the reactants interpolates between the thermal equilibrium rate constant kie (that is independent of solvent dynamics) and a diffusion controlled rate constant kid (determined by solvent dynamics) and also between the wide (A=0) and narrow (A=1) window limits dominated by inner-sphere ligand vibration and outer-sphere solvent reorganization respectively. The explicit dependence of the integral equation solutions on solvation dynamics S(t), the free energy of reaction ΔG0, the total reorganization energy λ and its partitioning between ligand vibration λq and solvent polarization fluctuations λ0, and the nature of the initial state should be useful in the analysis and design of ET experiments in different solvents.

Construction and accuracy of partial differential equation approximations to the chemical master equation.

PubMed

Grima, Ramon

2011-11-01

The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.

Test particle propagation in magnetostatic turbulence. 2: The local approximation method

NASA Technical Reports Server (NTRS)

Klimas, A. J.; Sandri, G.; Scudder, J. D.; Howell, D. R.

1976-01-01

An approximation method for statistical mechanics is presented and applied to a class of problems which contains a test particle propagation problem. All of the available basic equations used in statistical mechanics are cast in the form of a single equation which is integrodifferential in time and which is then used as the starting point for the construction of the local approximation method. Simplification of the integrodifferential equation is achieved through approximation to the Laplace transform of its kernel. The approximation is valid near the origin in the Laplace space and is based on the assumption of small Laplace variable. No other small parameter is necessary for the construction of this approximation method. The n'th level of approximation is constructed formally, and the first five levels of approximation are calculated explicitly. It is shown that each level of approximation is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The order in time of these partial differential equations is found to increase as n does. At n = 0 the most local first order partial differential equation which governs the Markovian limit is regained.

A recurrence matrix method for the analysis of longitudinal and torsional vibrations in non-uniform multibranch beams with variable boundary conditions

NASA Technical Reports Server (NTRS)

Davis, R. B.; Stephens, M. V.

1974-01-01

An approximate method for calculating the longitudinal and torsional natural frequencies and associated modal data of a beamlike, variable cross section multibranch structure is presented. The procedure described is the numerical integration of the first order differential equations that characterize the beam element in longitudinal motion and that satisfy the appropriate boundary conditions.

The Magnetic Field of a Finite Solenoid

NASA Technical Reports Server (NTRS)

Callaghan, Edmund E.; Maslen, Stephen H.

1960-01-01

The axial and radial fields at any point inside or outside a finite solenoid with infinitely thin walls are derived. Solution of the equations has been obtained in terms of tabulated complete elliptic integrals. For the axial field an accurate approximation is given in terms of elementary functions. Fields internal and external to the solenoid are presented in graphical form for a wide variety of solenoid lengths.

Viscous wing theory development. Volume 2: GRUMWING computer program user's manual

NASA Technical Reports Server (NTRS)

Chow, R. R.; Ogilvie, P. L.

1986-01-01

This report is a user's manual which describes the operation of the computer program, GRUMWING. The program computes the viscous transonic flow over three-dimensional wings using a boundary layer type viscid-inviscid interaction approach. The inviscid solution is obtained by an approximate factorization (AFZ)method for the full potential equation. The boundary layer solution is based on integral entrainment methods.

Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation.

PubMed

Liu, Jian; Miller, William H

2007-06-21

It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.

Accuracy and efficiency considerations for wide-angle wavefield extrapolators and scattering operators

NASA Astrophysics Data System (ADS)

Thomson, C. J.

2005-10-01

Several observations are made concerning the numerical implementation of wide-angle one-way wave equations, using for illustration scalar waves obeying the Helmholtz equation in two space dimensions. This simple case permits clear identification of a sequence of physically motivated approximations of use when the mathematically exact pseudo-differential operator (PSDO) one-way method is applied. As intuition suggests, these approximations largely depend on the medium gradients in the direction transverse to the main propagation direction. A key point is that narrow-angle approximations are to be avoided in the interests of accuracy. Another key consideration stems from the fact that the so-called `standard-ordering' PSDO indicates how lateral interpolation of the velocity structure can significantly reduce computational costs associated with the Fourier or plane-wave synthesis lying at the heart of the calculations. A third important point is that the PSDO theory shows what approximations are necessary in order to generate an exponential one-way propagator for the laterally varying case, representing the intuitive extension of classical integral-transform solutions for a laterally homogeneous medium. This exponential propagator permits larger forward stepsizes. Numerical comparisons with Helmholtz (i.e. full) wave-equation finite-difference solutions are presented for various canonical problems. These include propagation along an interfacial gradient, the effects of a compact inclusion and the formation of extended transmitted and backscattered wave trains by model roughness. The ideas extend to the 3-D, generally anisotropic case and to multiple scattering by invariant embedding. It is concluded that the method is very competitive, striking a new balance between simplifying approximations and computational labour. Complicated wave-scattering effects are retained without the need for expensive global solutions, providing a robust and flexible modelling tool.

Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel

1997-01-01

This paper is concerned with two important elements in the high-order accurate spatial discretization of finite volume equations over arbitrary grids. One element is the integration of basis functions over arbitrary domains, which is used in expressing various spatial integrals in terms of discrete unknowns. The other consists of quadrature approximations to those integrals. Only polynomial basis functions applied to polyhedral and polygonal grids are treated here. Non-triangular polygonal faces are subdivided into a union of planar triangular facets, and the resulting triangulated polyhedron is subdivided into a union of tetrahedra. The straight line segment, triangle, and tetrahedron are thus the fundamental shapes that are the building blocks for all integrations and quadrature approximations. Integrals of products up to the fifth order are derived in a unified manner for the three fundamental shapes in terms of the position vectors of vertices. Results are given both in terms of tensor products and products of Cartesian coordinates. The exact polynomial integrals are used to obtain symmetric quadrature approximations of any degree of precision up to five for arbitrary integrals over the three fundamental domains. Using a coordinate-free formulation, simple and rational procedures are developed to derive virtually all quadrature formulas, including some previously unpublished. Four symmetry groups of quadrature points are introduced to derive Gauss formulas, while their limiting forms are used to derive Lobatto formulas. Representative Gauss and Lobatto formulas are tabulated. The relative efficiency of their application to polyhedral and polygonal grids is detailed. The extension to higher degrees of precision is discussed.

Development of Finite Elements for Two-Dimensional Structural Analysis Using the Integrated Force Method

NASA Technical Reports Server (NTRS)

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

1996-01-01

The Integrated Force Method has been developed in recent years for the analysis of structural mechanics problems. This method treats all independent internal forces as unknown variables that can be calculated by simultaneously imposing equations of equilibrium and compatibility conditions. In this paper a finite element library for analyzing two-dimensional problems by the Integrated Force Method is presented. Triangular- and quadrilateral-shaped elements capable of modeling arbitrary domain configurations are presented. The element equilibrium and flexibility matrices are derived by discretizing the expressions for potential and complementary energies, respectively. The displacement and stress fields within the finite elements are independently approximated. The displacement field is interpolated as it is in the standard displacement method, and the stress field is approximated by using complete polynomials of the correct order. A procedure that uses the definitions of stress components in terms of an Airy stress function is developed to derive the stress interpolation polynomials. Such derived stress fields identically satisfy the equations of equilibrium. Moreover, the resulting element matrices are insensitive to the orientation of local coordinate systems. A method is devised to calculate the number of rigid body modes, and the present elements are shown to be free of spurious zero-energy modes. A number of example problems are solved by using the present library, and the results are compared with corresponding analytical solutions and with results from the standard displacement finite element method. The Integrated Force Method not only gives results that agree well with analytical and displacement method results but also outperforms the displacement method in stress calculations.

Theoretical study of interactions of BSA protein in a NaCl aqueous solution

NASA Astrophysics Data System (ADS)

Pellicane, Giuseppe; Cavero, Miguel

2013-03-01

Bovine Serum Albumine (BSA) aqueous solutions in the presence of NaCl are investigated for different protein concentrations and low to intermediate ionic strengths. Protein interactions are modeled via a charge-screened colloidal model, in which the range of the potential is determined by the Debye-Hückel constant. We use Monte Carlo computer simulations to calculate the structure factor, and assume an oblate ellipsoidal form factor for BSA. The theoretical scattered intensities are found in good agreement with the experimental small angle X-ray scattering intensities available in the literature. The performance of well-known integral equation closures to the Ornstein-Zernike equation, namely the mean spherical approximation, the Percus-Yevick, and the hypernetted chain equations, is also assessed with respect to computer simulation.

Scattering from arbitrarily shaped microstrip patch antennas

NASA Technical Reports Server (NTRS)

Shively, David G.; Deshpande, Manohar D.; Cockrell, Capers R.

1992-01-01

The scattering properties of arbitrarily shaped microstrip patch antennas are examined. The electric field integral equation for a current element on a grounded dielectric slab is developed for a rectangular geometry based on Galerkin's technique with subdomain rooftop basis functions. A shape function is introduced that allows a rectangular grid approximation to the arbitrarily shaped patch. The incident field on the patch is expressed as a function of incidence angle theta(i), phi(i). The resulting system of equations is then solved for the unknown current modes on the patch, and the electromagnetic scattering is calculated for a given angle. Comparisons are made with other calculated results as well as with measurements.

Simple iterative construction of the optimized effective potential for orbital functionals, including exact exchange.

PubMed

Kümmel, Stephan; Perdew, John P

2003-01-31

For exchange-correlation functionals that depend explicitly on the Kohn-Sham orbitals, the potential V(xcsigma)(r) must be obtained as the solution of the optimized effective potential (OEP) integral equation. This is very demanding and has limited the use of orbital functionals. We demonstrate that instead the OEP can be obtained iteratively by solving the partial differential equations for the orbital shifts that exactify the Krieger-Li-Iafrate approximation. Unoccupied orbitals do not need to be calculated. Accuracy and efficiency of the method are shown for atoms and clusters using the exact-exchange energy. Counterintuitive asymptotic limits of the exact OEP are presented.

On an integro-differential equation model for the study of the response of an acoustically coupled panel

NASA Technical Reports Server (NTRS)

Yen, D. H. Y.; Maestrello, L.; Padula, S.

1975-01-01

The response of a clamped panel to supersonically convected turbulence is considered. A theoretical model in the form of an integro-differential equation is employed that takes into account the coupling between the panel motion and the surrounding acoustic medium. The kernels of the integrals, which represent induced pressures due to the panel motion, are Green's functions for sound radiations under various moving and stationary sources. An approximate analysis is made by following a finite-element Ritz-Galerkin procedure. Preliminary numerical results, in agreement with experimental findings, indicate that the acoustic damping is the controlling mechanism of the response.

A molecular theory of liquid interfaces.

PubMed

Kovalenko, Andriy; Hirata, Fumio

2005-04-21

We propose a site site generalization of the Lovett-Mow-Buff-Wertheim integro-differential equation for the one-particle density distributions to polyatomic fluids. The method provides microscopic description of liquid interfaces of molecular fluids and solutions. It uses the inhomogeneous site-site direct correlation function of molecular fluid consistently constructed by nonlinear interpolation between the homogeneous ones. The site site correlations of the coexisting bulk phases are obtained from the reference interaction site model (RISM) integral equation with our closure approximation. For illustration, we calculated the structure of the planar liquid-vapor as well as liquid-liquid interfaces of n-hexane and methanol at ambient conditions.

Scale matters

NASA Astrophysics Data System (ADS)

Margolin, L. G.

2018-04-01

The applicability of Navier-Stokes equations is limited to near-equilibrium flows in which the gradients of density, velocity and energy are small. Here I propose an extension of the Chapman-Enskog approximation in which the velocity probability distribution function (PDF) is averaged in the coordinate phase space as well as the velocity phase space. I derive a PDF that depends on the gradients and represents a first-order generalization of local thermodynamic equilibrium. I then integrate this PDF to derive a hydrodynamic model. I discuss the properties of that model and its relation to the discrete equations of computational fluid dynamics. This article is part of the theme issue `Hilbert's sixth problem'.

Looping probabilities of elastic chains: a path integral approach.

PubMed

Cotta-Ramusino, Ludovica; Maddocks, John H

2010-11-01

We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e.g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end points. Our approximation formula for the looping pdf is of quite general applicability as, in contrast to most prior approaches, no assumption is made of either uniformity of the elastic chain, nor of a straight intrinsic shape. If the chain is uniform the Jacobi system evaluated at certain minimum energy configurations has constant coefficients. In such cases our approximate pdf can be evaluated in an entirely explicit, closed form. We illustrate our analysis with a planar example of this type and compute an approximate probability of cyclization, i.e., of forming a closed loop, from a uniform elastic chain whose intrinsic shape is an open circular arc.

Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium

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

Horowitz, Jordan M., E-mail: jordan.horowitz@umb.edu

The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochasticmore » thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.« less

Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry

NASA Astrophysics Data System (ADS)

Besse, Nicolas; Coulette, David

2016-08-01

Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov-Poisson and Vlasov-Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, "Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry" (submitted)] and were found to be surprisingly close to those for the original gyrokinetic-Vlasov equations. The purpose of the present paper is to make these new ideas accessible to two readerships: applied mathematicians and plasma physicists.

Perturbed invariant subspaces and approximate generalized functional variable separation solution for nonlinear diffusion-convection equations with weak source

NASA Astrophysics Data System (ADS)

Xia, Ya-Rong; Zhang, Shun-Li; Xin, Xiang-Peng

2018-03-01

In this paper, we propose the concept of the perturbed invariant subspaces (PISs), and study the approximate generalized functional variable separation solution for the nonlinear diffusion-convection equation with weak source by the approximate generalized conditional symmetries (AGCSs) related to the PISs. Complete classification of the perturbed equations which admit the approximate generalized functional separable solutions (AGFSSs) is obtained. As a consequence, some AGFSSs to the resulting equations are explicitly constructed by way of examples.

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.

Properties of the Boltzmann equation in the classical approximation

DOE PAGES

Epelbaum, Thomas; Gelis, François; Tanji, Naoto; ...

2014-12-30

We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since onemore » has also access to the non-approximated result for comparison.« less

Spin-diffusions and diffusive molecular dynamics

NASA Astrophysics Data System (ADS)

Farmer, Brittan; Luskin, Mitchell; Plecháč, Petr; Simpson, Gideon

2017-12-01

Metastable configurations in condensed matter typically fluctuate about local energy minima at the femtosecond time scale before transitioning between local minima after nanoseconds or microseconds. This vast scale separation limits the applicability of classical molecular dynamics (MD) methods and has spurned the development of a host of approximate algorithms. One recently proposed method is diffusive MD which aims at integrating a system of ordinary differential equations describing the likelihood of occupancy by one of two species, in the case of a binary alloy, while quasistatically evolving the locations of the atoms. While diffusive MD has shown itself to be efficient and provide agreement with observations, it is fundamentally a model, with unclear connections to classical MD. In this work, we formulate a spin-diffusion stochastic process and show how it can be connected to diffusive MD. The spin-diffusion model couples a classical overdamped Langevin equation to a kinetic Monte Carlo model for exchange amongst the species of a binary alloy. Under suitable assumptions and approximations, spin-diffusion can be shown to lead to diffusive MD type models. The key assumptions and approximations include a well-defined time scale separation, a choice of spin-exchange rates, a low temperature approximation, and a mean field type approximation. We derive several models from different assumptions and show their relationship to diffusive MD. Differences and similarities amongst the models are explored in a simple test problem.

Comment on "Symplectic integration of magnetic systems" by Stephen D. Webb [J. Comput. Phys. 270 (2014) 570-576

NASA Astrophysics Data System (ADS)

Zhang, Shuangxi; Jia, Yuesong; Sun, Qizhi

2015-02-01

Webb [1] proposed a method to get symplectic integrators of magnetic systems by Taylor expanding the discrete Euler-Lagrangian equations (DEL) which resulted from variational symplectic method by making the variation of the discrete action [2], and approximating the results to the order of O (h2), where h is the time step. And in that paper, Webb thought that the integrators obtained by that method are symplectic ones, especially, he treated Boris integrator (BI) as the symplectic one. However, we have questions about Webb's results. Theoretically the transformation of phase-space coordinates between two adjacent points induced by symplectic algorithm should conserve a symplectic 2-form [2-5]. As proved in Refs. [2,3], the transformations induced by the standard symplectic integrator derived from Hamilton and the variational symplectic integrator (VSI) [2,6] from Lagrangian should conserve a symplectic 2-forms. But the approximation of VSI to O (h2) obtained by that paper is hard to conserve a symplectic 2-form, contrary to the claim of [1]. In the next section, we will use BI as an example to support our point and will prove BI not to be a symplectic one but an integrator conserving discrete phase-space volume.

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.

Approximation of Quantum Stochastic Differential Equations for Input-Output Model Reduction

DTIC Science & Technology

2016-02-25

Approximation of Quantum Stochastic Differential Equations for Input-Output Model Reduction We have completed a short program of theoretical research...on dimensional reduction and approximation of models based on quantum stochastic differential equations. Our primary results lie in the area of...2211 quantum probability, quantum stochastic differential equations REPORT DOCUMENTATION PAGE 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 10. SPONSOR

Ring current-energy balance during intense magnetic storms

NASA Astrophysics Data System (ADS)

Clua de Gonzalez, A. L.; Gonzalez, W. D.

2013-12-01

The energy-rate balance that governs the storm-time ring current is analyzed in terms of the Burton-McPherron-Russell equation (Burton et al., 1975). This is a first order differential equation relating the time variation of the pressure corrected Dst index, with the energy input to the magnetosphere. Based on the Burton et al. equation, we have analyzed in detail the geomagnetic storm of February 11, 2004. The energy input is taken proportional to the interplanetary electric field, Q(t) = αBsV, where Bs is the southward component of the interplanetary magnetic field in GSM coordinates, V is the flow speed of the solar wind and α a constant. The equation is integrated using the OMNI-combined interplanetary data and, the value of the decay time is estimated from a best fit of the response to the observed curve. For this storm we also use a rectangular approximation for the energy input function, thus allowing an analytical solution of the Burton et al. equation. The results from this approximation are then compared to the numerical solution. The study is also extended to the geomagnetic storm of April 22, 2001. This analysis seems to indicate that the Burton et al. equation should contain also a corrective term proportional to the second time derivative of the Dst index. This corrective term might become important for intense storms, with an effect of counteracting the growth of |Dst| before the energy input from the interplanetary medium declines, such that the value of |Dst| starts to decrease instead of continuing to grow.

Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing

NASA Astrophysics Data System (ADS)

Vorontsov, Mikhail A.; Kolosov, Valeriy

2005-01-01

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related to maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive-index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing coherent outgoing-wave propagation, and the equation describing evolution of the mutual correlation function (MCF) for the backscattered wave (return wave). The resulting evolution equation for the MCF is further simplified by use of the smooth-refractive-index approximation. This approximation permits derivation of the transport equation for the return-wave brightness function, analyzed here by the method of characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wave-front sensors that perform sensing of speckle-averaged characteristics of the wave-front phase (TIL sensors). Analysis of the wave-front phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric-turbulence-related phase aberrations. We also show that wave-front sensing results depend on the extended target shape, surface roughness, and outgoing-beam intensity distribution on the target surface. For targets with smooth surfaces and nonflat shapes, the target-induced phase can contain aberrations. The presence of target-induced aberrations in the conjugated phase may result in a deterioration of adaptive system performance.

Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing.

PubMed

Vorontsov, Mikhail A; Kolosov, Valeriy

2005-01-01

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related to maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive-index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing coherent outgoing-wave propagation, and the equation describing evolution of the mutual correlation function (MCF) for the backscattered wave (return wave). The resulting evolution equation for the MCF is further simplified by use of the smooth-refractive-index approximation. This approximation permits derivation of the transport equation for the return-wave brightness function, analyzed here by the method of characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wave-front sensors that perform sensing of speckle-averaged characteristics of the wave-front phase (TIL sensors). Analysis of the wave-front phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric-turbulence-related phase aberrations. We also show that wave-front sensing results depend on the extended target shape, surface roughness, and outgoing-beam intensity distribution on the target surface. For targets with smooth surfaces and nonflat shapes, the target-induced phase can contain aberrations. The presence of target-induced aberrations in the conjugated phase may result in a deterioration of adaptive system performance.

Method of mechanical quadratures for solving singular integral equations of various types

NASA Astrophysics Data System (ADS)

Sahakyan, A. V.; Amirjanyan, H. A.

2018-04-01

The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.

CALL FOR PAPERS: Special issue on Symmetries and Integrability of Difference Equations

NASA Astrophysics Data System (ADS)

Doliwa, Adam; Korhonen, Risto; Lafortune, Stephane

2006-10-01

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/%7Eschief/side/side.html). Participants at that meeting, as well as other researchers working in the field of difference equations and discrete systems, are invited to submit a research paper to this issue. This meeting was the seventh of a series of biennial meetings devoted to the study of integrable difference equations and related topics. The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations, just as differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as: mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, quantum field theory, etc. It is thus crucial to develop tools to study and solve difference equations. While the theory of symmetry and integrability for differential equations is now well-established, this is not yet the case for discrete equations. The situation has undergone impressive development in recent years and has affected a broad range of fields, including the theory of special functions, quantum integrable systems, numerical analysis, cellular automata, representations of quantum groups, symmetries of difference equations, discrete (difference) geometry, etc. Consequently, the aim of the special issue is to benefit from the occasion offered by the SIDE VII meeting to provide a collection of papers which represent the state-of-the-art knowledge for studying integrability and symmetry properties of difference equations. Scope of the special issue The special issue will feature papers which deal with themes that were covered by the SIDE VII Conference. These are •Integrability testing •Discrete geometry and visualization •Laurent phenomena and cluster algebras •Ultra-discrete systems •Random matrix theory •Algebraic-geometric approaches to integrability •Yang-Baxter equations •Quantum and classical integrable systems •Difference Galois theory Editorial policy •The subject of the paper should relate to the subject of the meeting. The Guest Editors will reserve the right to judge whether a contribution fits the scope of the topic of the special issue. •Contributions will be refereed and processed according to the usual procedure of the journal. •Conference papers may be based on already published work but should either •contain significant additional new results and/or insights or •give a survey of the present state of the art, a critical assessment of the present understanding of a topic, and a discussion of open problems. •Papers submitted by non-participants should be original and contain substantial new results. Guidelines for preparation of contributions • The deadline for contributed papers will be 15 January 2007. •There is a page limit of 16 printed pages (approximately 9600 words) per contribution. For submitted papers exceeding this length the Guest Editors reserve the right to request a reduction in length. Further advice on document preparation can be found at www.iop.org/Journals/jphysa •Contributions to the special issue should if possible be submitted electronically by web upload at www.iop.org/Journals/jphysa, or by email to jphysa@iop.org, quoting 'J. Phys. A Special Issue: SIDE VII'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the website for further information on electronic submissions. •Authors unable to submit electronically may send hard-copy contributions to: Publishing Administrators, Journal of Physics A, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK, enclosing electronic code on floppy disk if available and quoting 'J. Phys. A Special Issue: SIDE VII'. • All contributions should be accompanied by a read-me file or covering letter giving the postal and email address for correspondence. The Publishing Office should be notified of any subsequent change of address. •The special issue will be published in the paper and online version of the journal. The corresponding author of each contribution will receive a complimentary copy of the issue.

Application of Krylov exponential propagation to fluid dynamics equations

NASA Technical Reports Server (NTRS)

Saad, Youcef; Semeraro, David

1991-01-01

An application of matrix exponentiation via Krylov subspace projection to the solution of fluid dynamics problems is presented. The main idea is to approximate the operation exp(A)v by means of a projection-like process onto a krylov subspace. This results in a computation of an exponential matrix vector product similar to the one above but of a much smaller size. Time integration schemes can then be devised to exploit this basic computational kernel. The motivation of this approach is to provide time-integration schemes that are essentially of an explicit nature but which have good stability properties.

S-193 scatterometer transfer function analysis for data processing

NASA Technical Reports Server (NTRS)

Johnson, L.

1974-01-01

A mathematical model for converting raw data measurements of the S-193 scatterometer into processed values of radar scattering coefficient is presented. The argument is based on an approximation derived from the Radar Equation and actual operating principles of the S-193 Scatterometer hardware. Possible error sources are inaccuracies in transmitted wavelength, range, antenna illumination integrals, and the instrument itself. The dominant source of error in the calculation of scattering coefficent is accuracy of the range. All other ractors with the possible exception of illumination integral are not considered to cause significant error in the calculation of scattering coefficient.

Internal hypersonic flow. [in thin shock layer

NASA Technical Reports Server (NTRS)

Lin, T. C.; Rubin, S. G.

1974-01-01

An approach for studying hypersonic internal flow with the aid of a thin-shock-layer approximation is discussed, giving attention to a comparison of thin-shock-layer results with the data obtained on the basis of the imposition theory or a finite-difference integration of the Euler equations. Relations in the case of strong interaction are considered together with questions of pressure distribution and aspects of the boundary-layer solution.

Unsteady Flow Simulation: A Numerical Challenge

DTIC Science & Technology

2003-03-01

drive to convergence the numerical unsteady term. The time marching procedure is based on the approximate implicit Newton method for systems of non...computed through analytical derivatives of S. The linear system stemming from equation (3) is solved at each integration step by the same iterative method...significant reduction of memory usage, thanks to the reduced dimensions of the linear system matrix during the implicit marching of the solution. The

A Conformal, Fully-Conservative Approach for Predicting Blast Effects on Ground Vehicles

DTIC Science & Technology

2014-04-01

time integration  Approximate Riemann Fluxes (HLLE, HLLC) ◦ Robust mixture model for multi-material flows  Multiple Equations of State ◦ Perfect Gas...Loci/CHEM: Chemically reacting compressible flow solver . ◦ Currently in production use by NASA for the simulation of rocket motors, plumes, and...vehicles  Loci/DROPLET: Eulerian and Lagrangian multiphase solvers  Loci/STREAM: pressure-based solver ◦ Developed by Streamline Numerics and

Excess Entropy Production in Quantum System: Quantum Master Equation Approach

NASA Astrophysics Data System (ADS)

Nakajima, Satoshi; Tokura, Yasuhiro

2017-12-01

For open systems described by the quantum master equation (QME), we investigate the excess entropy production under quasistatic operations between nonequilibrium steady states. The average entropy production is composed of the time integral of the instantaneous steady entropy production rate and the excess entropy production. We propose to define average entropy production rate using the average energy and particle currents, which are calculated by using the full counting statistics with QME. The excess entropy production is given by a line integral in the control parameter space and its integrand is called the Berry-Sinitsyn-Nemenman (BSN) vector. In the weakly nonequilibrium regime, we show that BSN vector is described by ln \\breve{ρ }_0 and ρ _0 where ρ _0 is the instantaneous steady state of the QME and \\breve{ρ }_0 is that of the QME which is given by reversing the sign of the Lamb shift term. If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, the excess entropy production approximately reduces to the difference between the von Neumann entropies of the system. Additionally, we point out that the expression of the entropy production obtained in the classical Markov jump process is different from our result and show that these are approximately equivalent only in the weakly nonequilibrium regime.

Bell-Plesset effects in Rayleigh-Taylor instability of finite-thickness spherical and cylindrical shells

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

Velikovich, A. L.; Schmit, P. F.

Bell-Plesset (BP) effects account for the influence of global convergence or divergence of the fluid flow on the evolution of the interfacial perturbations embedded in the flow. The development of the Rayleigh-Taylor instability in radiation-driven spherical capsules and magnetically-driven cylindrical liners necessarily includes a significant contribution from BP effects due to the time dependence of the radius, velocity, and acceleration of the unstable surfaces or interfaces. An analytical model is presented that, for an ideal incompressible fluid and small perturbation amplitudes, exactly evaluates the BP effects in finite-thickness shells through acceleration and deceleration phases. The time-dependent dispersion equations determining themore » “instantaneous growth rate” are derived. It is demonstrated that by integrating this approximate growth rate over time, one can accurately evaluate the number of perturbation e-foldings during the inward acceleration phase of the implosion. In the limit of small shell thickness, exact thin-shell perturbation equations and approximate thin-shell dispersion equations are obtained, generalizing the earlier results [E. G. Harris, Phys. Fluids 5, 1057 (1962); E. Ott, Phys. Rev. Lett. 29, 1429 (1972); A. B. Bud'ko et al., Phys. Fluids B 2, 1159 (1990)].« less

Virial Coefficients and Equations of State for Hard Polyhedron Fluids.

PubMed

Irrgang, M Eric; Engel, Michael; Schultz, Andrew J; Kofke, David A; Glotzer, Sharon C

2017-10-24

Hard polyhedra are a natural extension of the hard sphere model for simple fluids, but there is no general scheme for predicting the effect of shape on thermodynamic properties, even in moderate-density fluids. Only the second virial coefficient is known analytically for general convex shapes, so higher-order equations of state have been elusive. Here we investigate high-precision state functions in the fluid phase of 14 representative polyhedra with different assembly behaviors. We discuss historic efforts in analytically approximating virial coefficients up to B 4 and numerically evaluating them to B 8 . Using virial coefficients as inputs, we show the convergence properties for four equations of state for hard convex bodies. In particular, the exponential approximant of Barlow et al. (J. Chem. Phys. 2012, 137, 204102) is found to be useful up to the first ordering transition for most polyhedra. The convergence behavior we explore can guide choices in expending additional resources for improved estimates. Fluids of arbitrary hard convex bodies are too complicated to be described in a general way at high densities, so the high-precision state data we provide can serve as a reference for future work in calculating state data or as a basis for thermodynamic integration.

Quantum weak turbulence with applications to semiconductor lasers

NASA Astrophysics Data System (ADS)

Lvov, Yuri Victorovich

Based on a model Hamiltonian appropriate for the description of fermionic systems such as semiconductor lasers, we describe a natural asymptotic closure of the BBGKY hierarchy in complete analogy with that derived for classical weak turbulence. The main features of the interaction Hamiltonian are the inclusion of full Fermi statistics containing Pauli blocking and a simple, phenomenological, uniformly weak two particle interaction potential equivalent to the static screening approximation. The resulting asymytotic closure and quantum kinetic Boltzmann equation are derived in a self consistent manner without resorting to a priori statistical hypotheses or cumulant discard assumptions. We find a new class of solutions to the quantum kinetic equation which are analogous to the Kolmogorov spectra of hydrodynamics and classical weak turbulence. They involve finite fluxes of particles and energy across momentum space and are particularly relevant for describing the behavior of systems containing sources and sinks. We explore these solutions by using differential approximation to collision integral. We make a prima facie case that these finite flux solutions can be important in the context of semiconductor lasers. We show that semiconductor laser output efficiency can be improved by exciting these finite flux solutions. Numerical simulations of the semiconductor Maxwell Bloch equations support the claim.

Bell-Plesset effects in Rayleigh-Taylor instability of finite-thickness spherical and cylindrical shells

NASA Astrophysics Data System (ADS)

Velikovich, A. L.; Schmit, P. F.

2015-12-01

Bell-Plesset (BP) effects account for the influence of global convergence or divergence of the fluid flow on the evolution of the interfacial perturbations embedded in the flow. The development of the Rayleigh-Taylor instability in radiation-driven spherical capsules and magnetically-driven cylindrical liners necessarily includes a significant contribution from BP effects due to the time dependence of the radius, velocity, and acceleration of the unstable surfaces or interfaces. An analytical model is presented that, for an ideal incompressible fluid and small perturbation amplitudes, exactly evaluates the BP effects in finite-thickness shells through acceleration and deceleration phases. The time-dependent dispersion equations determining the "instantaneous growth rate" are derived. It is demonstrated that by integrating this approximate growth rate over time, one can accurately evaluate the number of perturbation e-foldings during the inward acceleration phase of the implosion. In the limit of small shell thickness, exact thin-shell perturbation equations and approximate thin-shell dispersion equations are obtained, generalizing the earlier results [E. G. Harris, Phys. Fluids 5, 1057 (1962); E. Ott, Phys. Rev. Lett. 29, 1429 (1972); A. B. Bud'ko et al., Phys. Fluids B 2, 1159 (1990)].

Bounding solutions of geometrically nonlinear viscoelastic problems

NASA Technical Reports Server (NTRS)

Stubstad, J. M.; Simitses, G. J.

1985-01-01

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Bounding solutions of geometrically nonlinear viscoelastic problems

NASA Technical Reports Server (NTRS)

Stubstad, J. M.; Simitses, G. J.

1986-01-01

Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

Application of the Parabolic Approximation to Predict Acoustical Propagation in the Ocean.

ERIC Educational Resources Information Center

McDaniel, Suzanne T.

1979-01-01

A simplified derivation of the parabolic approximation to the acoustical wave equation is presented. Exact solutions to this approximate equation are compared with solutions to the wave equation to demonstrate the applicability of this method to the study of underwater sound propagation. (Author/BB)

Effect of design selection on response surface performance

NASA Technical Reports Server (NTRS)

Carpenter, William C.

1993-01-01

The mathematical formulation of the engineering optimization problem is given. Evaluation of the objective function and constraint equations can be very expensive in a computational sense. Thus, it is desirable to use as few evaluations as possible in obtaining its solution. In solving the equation, one approach is to develop approximations to the objective function and/or restraint equations and then to solve the equation using the approximations in place of the original functions. These approximations are referred to as response surfaces. The desirability of using response surfaces depends upon the number of functional evaluations required to build the response surfaces compared to the number required in the direct solution of the equation without approximations. The present study is concerned with evaluating the performance of response surfaces so that a decision can be made as to their effectiveness in optimization applications. In particular, this study focuses on how the quality of approximations is effected by design selection. Polynomial approximations and neural net approximations are considered.

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

NASA Technical Reports Server (NTRS)

Mostrel, M. M.

1988-01-01

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Differential equation based method for accurate approximations in optimization

NASA Technical Reports Server (NTRS)

Pritchard, Jocelyn I.; Adelman, Howard M.

1990-01-01

This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.

Differential equation based method for accurate approximations in optimization

NASA Technical Reports Server (NTRS)

Pritchard, Jocelyn I.; Adelman, Howard M.

1990-01-01

A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.

Difference equation state approximations for nonlinear hereditary control problems

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1982-01-01

Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.

Distorted-wave methods for electron capture in ion-atom collisions

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

Burgdoerfer, J.; Taulbjerg, K.

1986-05-01

Distorted-wave methods for electron capture are discussed with emphasis on the surface term in the T matrix and on the properties of the associated integral equations. The surface term is generally nonvanishing if the distorted waves are sufficiently accurate to include parts of the considered physical process. Two examples are considered in detail. If distorted waves of the strong-potential Born-approximation (SPB) type are employed the surface term supplies the first-Born-approximation part of the T matrix. The surface term is shown to vanish in the continuum-distorted-wave (CDW) method. The integral kernel is in either case free of the dangerous disconnected termsmore » discussed by Greider and Dodd but the CDW theory is peculiar in the sense that its first-order approximation (CDW1) excludes a specific on-shell portion of the double-scattering term that is closely connected with the classical Thomas process. The latter is described by the second-order term in the CDW series. The distorted-wave Born approximation with SPB waves is shown to be free of divergences. In the limit of asymmetric collisions the DWB suggests a modification of the SPB approximation to avoid the divergence problem recently identified by Dewangan and Eichler.« less

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.

Legendre spectral-collocation method for solving some types of fractional optimal control problems

PubMed Central

Sweilam, Nasser H.; Al-Ajami, Tamer M.

2014-01-01

In this paper, the Legendre spectral-collocation method was applied to obtain approximate solutions for some types of fractional optimal control problems (FOCPs). The fractional derivative was described in the Caputo sense. Two different approaches were presented, in the first approach, necessary optimality conditions in terms of the associated Hamiltonian were approximated. In the second approach, the state equation was discretized first using the trapezoidal rule for the numerical integration followed by the Rayleigh–Ritz method to evaluate both the state and control variables. Illustrative examples were included to demonstrate the validity and applicability of the proposed techniques. PMID:26257937

Diffusion of strongly magnetized cosmic ray particles in a turbulent medium

NASA Technical Reports Server (NTRS)

Ptuskin, V. S.

1985-01-01

Cosmic ray (CR) propagation in a turbulent medium is usually considered in the diffusion approximation. Here, the diffusion equation is obtained for strongly magnetized particles in the general form. The influence of a large-scale random magnetic field on CR propagation in interstellar medium is discussed. Cosmic rays are assumed to propagate in a medium with a regular field H and an ensemble of random MHD waves. The energy density of waves on scales smaller than the free path 1 of CR particles is small. The collision integral of the general form which describes interaction between relativistic particles and waves in the quasilinear approximation is used.

Analysis of the theory of high energy ion transport

NASA Technical Reports Server (NTRS)

Wilson, J. W.

1977-01-01

Procedures for the approximation of the transport of high-energy ions are discussed on the basis of available data on ion nuclear reactions. A straightahead approximation appears appropriate for space applications. The assumption that the secondary-ion-fragment velocity is equal to that of the fragmenting nucleus is inferior to straightahead theory but is of sufficient accuracy if the primary ions display a broad energy spectrum. An iterative scheme for the solution of the inhomogenous integral transport equations holds promise for practical calculation. A model calculation shows that multiple charged ion fragments penetrate to greater depths in comparison with the free path of a primary heavy ion.

Multi-Component Diffusion with Application To Computational Aerothermodynamics

NASA Technical Reports Server (NTRS)

Sutton, Kenneth; Gnoffo, Peter A.

1998-01-01

The accuracy and complexity of solving multicomponent gaseous diffusion using the detailed multicomponent equations, the Stefan-Maxwell equations, and two commonly used approximate equations have been examined in a two part study. Part I examined the equations in a basic study with specified inputs in which the results are applicable for many applications. Part II addressed the application of the equations in the Langley Aerothermodynamic Upwind Relaxation Algorithm (LAURA) computational code for high-speed entries in Earth's atmosphere. The results showed that the presented iterative scheme for solving the Stefan-Maxwell equations is an accurate and effective method as compared with solutions of the detailed equations. In general, good accuracy with the approximate equations cannot be guaranteed for a species or all species in a multi-component mixture. 'Corrected' forms of the approximate equations that ensured the diffusion mass fluxes sum to zero, as required, were more accurate than the uncorrected forms. Good accuracy, as compared with the Stefan- Maxwell results, were obtained with the 'corrected' approximate equations in defining the heating rates for the three Earth entries considered in Part II.

Phase dynamics of single long Josephson junction in MgB2 superconductor

NASA Astrophysics Data System (ADS)

Chimouriya, Shanker Pd.; Ghimire, Bal Ram; Kim, Ju H.

2018-05-01

A system of perturbed sine Gordon equations is derived to a superconductor-insulator-superconductor (SIS) long Joseph-son junction as an extension of the Ambegaokar-Baratoff relation, following the long route of path integral formalism. A computer simulation is performed by discretizing the equations using finite difference approximation and applied to the MgB2 superconductor with SiO2 as the junction material. The solution of unperturbed sG equation is taken as the initial profile for the simulation and observed how the perturbation terms play the role to modify it. It is found initial profile deformed as time goes on. The variation of total Josephson current has also been observed. It is found that, the perturbation terms play the role for phase frustration. The phase frustration achieves quicker for high tunneling current.

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion

NASA Astrophysics Data System (ADS)

Congy, T.; Ivanov, S. K.; Kamchatnov, A. M.; Pavloff, N.

2017-08-01

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks, which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion.

PubMed

Congy, T; Ivanov, S K; Kamchatnov, A M; Pavloff, N

2017-08-01

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks, which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations.

Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment

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

Meng, Da; Zheng, Bin; Lin, Guang

2014-08-29

We have developed efficient numerical algorithms for the solution of 3D steady-state Poisson-Nernst-Planck equations (PNP) with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by finite difference scheme and solved iteratively by Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Algebraic multigrid method is then applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed which reduces computational complexity from O(N2) to O(NlogN) where N is themore » number of grid points. Integrals involving Dirac delta function are evaluated directly by coordinate transformation which yields more accurate result compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for Li ion batteries are shown to be in good agreement with the experimental data and the results from previous studies.« less

Falling films on flexible inclines

NASA Astrophysics Data System (ADS)

Matar, O. K.; Craster, R. V.; Kumar, S.

2007-11-01

The nonlinear stability and dynamic behavior of falling fluid films is studied for flow over a flexible substrate. We use asymptotic methods to deduce governing equations valid in various limits. Long-wave theory is used to derive Benney-like coupled equations for the film thickness and substrate deflection. Weakly nonlinear equations are then derived from these equations that, in the limit of large wall damping and/or large wall tension, reduce to the Kuramoto-Sivashinsky equation. These models break down when inertia becomes more significant, so we also use a long-wave approximation in conjunction with integral theory to derive three strongly coupled nonlinear evolution equations for the film thickness, substrate deflection, and film volumetric flow rate valid at higher Reynolds numbers. These equations, accounting for inertia, capillary, viscous, wall tension, and damping effects, are solved over a wide range of parameters. Our results suggest that decreasing wall damping and/or wall tension can promote the development of chaos in the weakly nonlinear regime and lead to severe substrate deformations in the strongly nonlinear regime; these can give rise to situations in which the free surface and underlying substrate come into contact in finite time.

Comparison of two integration methods for dynamic causal modeling of electrophysiological data.

PubMed

Lemaréchal, Jean-Didier; George, Nathalie; David, Olivier

2018-06-01

Dynamic causal modeling (DCM) is a methodological approach to study effective connectivity among brain regions. Based on a set of observations and a biophysical model of brain interactions, DCM uses a Bayesian framework to estimate the posterior distribution of the free parameters of the model (e.g. modulation of connectivity) and infer architectural properties of the most plausible model (i.e. model selection). When modeling electrophysiological event-related responses, the estimation of the model relies on the integration of the system of delay differential equations (DDEs) that describe the dynamics of the system. In this technical note, we compared two numerical schemes for the integration of DDEs. The first, and standard, scheme approximates the DDEs (more precisely, the state of the system, with respect to conduction delays among brain regions) using ordinary differential equations (ODEs) and solves it with a fixed step size. The second scheme uses a dedicated DDEs solver with adaptive step sizes to control error, making it theoretically more accurate. To highlight the effects of the approximation used by the first integration scheme in regard to parameter estimation and Bayesian model selection, we performed simulations of local field potentials using first, a simple model comprising 2 regions and second, a more complex model comprising 6 regions. In these simulations, the second integration scheme served as the standard to which the first one was compared. Then, the performances of the two integration schemes were directly compared by fitting a public mismatch negativity EEG dataset with different models. The simulations revealed that the use of the standard DCM integration scheme was acceptable for Bayesian model selection but underestimated the connectivity parameters and did not allow an accurate estimation of conduction delays. Fitting to empirical data showed that the models systematically obtained an increased accuracy when using the second integration scheme. We conclude that inference on connectivity strength and delay based on DCM for EEG/MEG requires an accurate integration scheme. Copyright © 2018 The Authors. Published by Elsevier Inc. All rights reserved.

Analysis of fluid-structure interaction in a frame pipe undergoing plastic deformations

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

Khamlichi, A.; Jezequel, L.; Jacques, Y.

1995-11-01

Water hammer pressure waves of sufficiently large magnitude can cause plastic flexural deformations in a frame pipe. In this study, the authors propose a modelization of this problem based on plane wave approximation for the fluid equations and approximation of the structure motion by a single-degree-of-freedom elastic-plastic oscillator. Direct analytical integration of elastic-plastic equations through pipe sections, then over the pipe length is performed in order to identify the oscillator parameters. Comparison of the global load-displacement relationship obtained with the finite element solution was considered and has shown good agreement. Fluid-structure coupling is achieved by assuming elbows to act likemore » plane monopole sources, where localized jumps of fluid velocity occur and where net pressure forces are exerted on the structure. The authors have applied this method to analyze the fluid-structure interaction in this range of deformations. Energy exchange between the fluid and the structure and energy dissipation are quantified.« less

Frozen Gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

NASA Astrophysics Data System (ADS)

Lorin, E.; Yang, X.; Antoine, X.

2016-06-01

The paper is devoted to develop efficient domain decomposition methods for the linear Schrödinger equation beyond the semiclassical regime, which does not carry a small enough rescaled Planck constant for asymptotic methods (e.g. geometric optics) to produce a good accuracy, but which is too computationally expensive if direct methods (e.g. finite difference) are applied. This belongs to the category of computing middle-frequency wave propagation, where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy. Motivated by recent works of the authors on absorbing boundary conditions (Antoine et al. (2014) [13] and Yang and Zhang (2014) [43]), we introduce Semiclassical Schwarz Waveform Relaxation methods (SSWR), which are seamless integrations of semiclassical approximation to Schwarz Waveform Relaxation methods. Two versions are proposed respectively based on Herman-Kluk propagation and geometric optics, and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods.

On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1988-01-01

An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included.

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.

Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

NASA Astrophysics Data System (ADS)

Kokurin, M. Yu.

2010-11-01

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.

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

Griffin, Brian M.; Larson, Vincent E.

Microphysical processes, such as the formation, growth, and evaporation of precipitation, interact with variability and covariances (e.g., fluxes) in moisture and heat content. For instance, evaporation of rain may produce cold pools, which in turn may trigger fresh convection and precipitation. These effects are usually omitted or else crudely parameterized at subgrid scales in weather and climate models.A more formal approach is pursued here, based on predictive, horizontally averaged equations for the variances, covariances, and fluxes of moisture and heat content. These higher-order moment equations contain microphysical source terms. The microphysics terms can be integrated analytically, given a suitably simplemore » warm-rain microphysics scheme and an approximate assumption about the multivariate distribution of cloud-related and precipitation-related variables. Performing the integrations provides exact expressions within an idealized context.A large-eddy simulation (LES) of a shallow precipitating cumulus case is performed here, and it indicates that the microphysical effects on (co)variances and fluxes can be large. In some budgets and altitude ranges, they are dominant terms. The analytic expressions for the integrals are implemented in a single-column, higher-order closure model. Interactive single-column simulations agree qualitatively with the LES. The analytic integrations form a parameterization of microphysical effects in their own right, and they also serve as benchmark solutions that can be compared to non-analytic integration methods.« less

Time integration algorithms for the two-dimensional Euler equations on unstructured meshes

NASA Technical Reports Server (NTRS)

Slack, David C.; Whitaker, D. L.; Walters, Robert W.

1994-01-01

Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.

Diffusion in random networks: Asymptotic properties, and numerical and engineering approximations

NASA Astrophysics Data System (ADS)

Padrino, Juan C.; Zhang, Duan Z.

2016-11-01

The ensemble phase averaging technique is applied to model mass transport by diffusion in random networks. The system consists of an ensemble of random networks, where each network is made of a set of pockets connected by tortuous channels. Inside a channel, we assume that fluid transport is governed by the one-dimensional diffusion equation. Mass balance leads to an integro-differential equation for the pores mass density. The so-called dual porosity model is found to be equivalent to the leading order approximation of the integration kernel when the diffusion time scale inside the channels is small compared to the macroscopic time scale. As a test problem, we consider the one-dimensional mass diffusion in a semi-infinite domain, whose solution is sought numerically. Because of the required time to establish the linear concentration profile inside a channel, for early times the similarity variable is xt- 1 / 4 rather than xt- 1 / 2 as in the traditional theory. This early time sub-diffusive similarity can be explained by random walk theory through the network. In addition, by applying concepts of fractional calculus, we show that, for small time, the governing equation reduces to a fractional diffusion equation with known solution. We recast this solution in terms of special functions easier to compute. Comparison of the numerical and exact solutions shows excellent agreement.

Integrated System-Level Optimization for Concurrent Engineering With Parametric Subsystem Modeling

NASA Technical Reports Server (NTRS)

Schuman, Todd; DeWeck, Oliver L.; Sobieski, Jaroslaw

2005-01-01

The introduction of concurrent design practices to the aerospace industry has greatly increased the productivity of engineers and teams during design sessions as demonstrated by JPL's Team X. Simultaneously, advances in computing power have given rise to a host of potent numerical optimization methods capable of solving complex multidisciplinary optimization problems containing hundreds of variables, constraints, and governing equations. Unfortunately, such methods are tedious to set up and require significant amounts of time and processor power to execute, thus making them unsuitable for rapid concurrent engineering use. This paper proposes a framework for Integration of System-Level Optimization with Concurrent Engineering (ISLOCE). It uses parametric neural-network approximations of the subsystem models. These approximations are then linked to a system-level optimizer that is capable of reaching a solution quickly due to the reduced complexity of the approximations. The integration structure is described in detail and applied to the multiobjective design of a simplified Space Shuttle external fuel tank model. Further, a comparison is made between the new framework and traditional concurrent engineering (without system optimization) through an experimental trial with two groups of engineers. Each method is evaluated in terms of optimizer accuracy, time to solution, and ease of use. The results suggest that system-level optimization, running as a background process during integrated concurrent engineering sessions, is potentially advantageous as long as it is judiciously implemented.

A Variational Nodal Approach to 2D/1D Pin Resolved Neutron Transport for Pressurized Water Reactors

DOE PAGES

Zhang, Tengfei; Lewis, E. E.; Smith, M. A.; ...

2017-04-18

A two-dimensional/one-dimensional (2D/1D) variational nodal approach is presented for pressurized water reactor core calculations without fuel-moderator homogenization. A 2D/1D approximation to the within-group neutron transport equation is derived and converted to an even-parity form. The corresponding nodal functional is presented and discretized to obtain response matrix equations. Within the nodes, finite elements in the x-y plane and orthogonal functions in z are used to approximate the spatial flux distribution. On the radial interfaces, orthogonal polynomials are employed; on the axial interfaces, piecewise constants corresponding to the finite elements eliminate the interface homogenization that has been a challenge for method ofmore » characteristics (MOC)-based 2D/1D approximations. The angular discretization utilizes an even-parity integral method within the nodes, and low-order spherical harmonics (P N) on the axial interfaces. The x-y surfaces are treated with high-order P N combined with quasi-reflected interface conditions. Furthermore, the method is applied to the C5G7 benchmark problems and compared to Monte Carlo reference calculations.« less

A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

NASA Astrophysics Data System (ADS)

Stelling, G. S.; Duinmeijer, S. P. A.

2003-12-01

This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large-scale inundations.

A Variational Nodal Approach to 2D/1D Pin Resolved Neutron Transport for Pressurized Water Reactors

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

Zhang, Tengfei; Lewis, E. E.; Smith, M. A.

A two-dimensional/one-dimensional (2D/1D) variational nodal approach is presented for pressurized water reactor core calculations without fuel-moderator homogenization. A 2D/1D approximation to the within-group neutron transport equation is derived and converted to an even-parity form. The corresponding nodal functional is presented and discretized to obtain response matrix equations. Within the nodes, finite elements in the x-y plane and orthogonal functions in z are used to approximate the spatial flux distribution. On the radial interfaces, orthogonal polynomials are employed; on the axial interfaces, piecewise constants corresponding to the finite elements eliminate the interface homogenization that has been a challenge for method ofmore » characteristics (MOC)-based 2D/1D approximations. The angular discretization utilizes an even-parity integral method within the nodes, and low-order spherical harmonics (P N) on the axial interfaces. The x-y surfaces are treated with high-order P N combined with quasi-reflected interface conditions. Furthermore, the method is applied to the C5G7 benchmark problems and compared to Monte Carlo reference calculations.« less

Real time estimation of ship motions using Kalman filtering techniques

NASA Technical Reports Server (NTRS)

Triantafyllou, M. S.; Bodson, M.; Athans, M.

1983-01-01

The estimation of the heave, pitch, roll, sway, and yaw motions of a DD-963 destroyer is studied, using Kalman filtering techniques, for application in VTOL aircraft landing. The governing equations are obtained from hydrodynamic considerations in the form of linear differential equations with frequency dependent coefficients. In addition, nonminimum phase characteristics are obtained due to the spatial integration of the water wave forces. The resulting transfer matrix function is irrational and nonminimum phase. The conditions for a finite-dimensional approximation are considered and the impact of the various parameters is assessed. A detailed numerical application for a DD-963 destroyer is presented and simulations of the estimations obtained from Kalman filters are discussed.

Difference equation state approximations for nonlinear hereditary control problems

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1984-01-01

Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589

Analytical approximate solutions for a general class of nonlinear delay differential equations.

PubMed

Căruntu, Bogdan; Bota, Constantin

2014-01-01

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

Correlation Energies from the Two-Component Random Phase Approximation.

PubMed

Kühn, Michael

2014-02-11

The correlation energy within the two-component random phase approximation accounting for spin-orbit effects is derived. The resulting plasmon equation is rewritten-analogously to the scalar relativistic case-in terms of the trace of two Hermitian matrices for (Kramers-restricted) closed-shell systems and then represented as an integral over imaginary frequency using the resolution of the identity approximation. The final expression is implemented in the TURBOMOLE program suite. The code is applied to the computation of equilibrium distances and vibrational frequencies of heavy diatomic molecules. The efficiency is demonstrated by calculation of the relative energies of the Oh-, D4h-, and C5v-symmetric isomers of Pb6. Results within the random phase approximation are obtained based on two-component Kohn-Sham reference-state calculations, using effective-core potentials. These values are finally compared to other two-component and scalar relativistic methods, as well as experimental data.

Radiative cooling in shock-heated hydrogen-helium plasmas. [for planetary entry probe heat shields

NASA Technical Reports Server (NTRS)

Poon, P. T. Y.; Stickford, G. H., Jr.

1978-01-01

Axial and off-axis radiative cooling of cylindrical shock-heated hydrogen-helium plasmas is investigated theoretically and experimentally. The coupled fluid dynamic-radiative transfer equations are solved by a combination of approximation techniques aimed at simplifying the computation of the flux divergence term, namely, the quasi-isothermal approximation and the exponential approximation developed for the solid angle integration. The accuracy of the approximation schemes has been assessed and found acceptable for applying the methods to the rapid computation of the radiatively coupled flow problem. Radiative cooling experiments were conducted in a 6-inch annular arc accelerator shock tube (ANAA) for an initial pressure of 1 torr and shock speeds from 35 to 45 Km/sec. The results indicate that the lateral cooling is small compared with the axial cooling, and that better agreement is achieved between the data and the theoretical results by inclusion of the lateral temperature gradient.

Constitutive Modelling of Resins in the Stiffness Domain

NASA Astrophysics Data System (ADS)

Klasztorny, M.

2004-09-01

An analytic method for inverting the constitutive compliance equations of viscoelasticity for resins is developed. These equations describe the HWKK/H rheological model, which makes it possible to simulate, with a good accuracy, short-, medium- and long-term viscoelastic processes in epoxy and polyester resins. These processes are of first-rank reversible isothermal type. The time histories of deviatoric stresses are simulated with three independent strain history functions of fractional and normal exponential types. The stiffness equations are described by two elastic and six viscoelastic constants having a clear physic meaning (three long-term relaxation coefficients and three relaxation times). The time histories of axiatoric stresses are simulated as perfectly elastic. The inversion method utilizes approximate constitutive stiffness equations of viscoelasticity for the HWKK/H model. The constitutive compliance equations for the model are a basis for determining the exact complex shear stiffness, whereas the approximate constitutive stiffness equations are used for determining the approximate complex shear stiffness. The viscoelastic constants in the stiffness domain are derived by equating the exact and approximate complex shear stiffnesses. The viscoelastic constants are obtained for Epidian 53 epoxy and Polimal 109 polyester resins. The accuracy of the approximate constitutive stiffness equations are assessed by comparing the approximate and exact complex shear stiffnesses. The constitutive stiffness equations for the HWKK/H model are presented in uncoupled (shear/bulk) and coupled forms. Formulae for converting the constants of shear viscoelasticity into the constants of coupled viscoelasticity are given as well.

Extending generalized Kubelka-Munk to three-dimensional radiative transfer.

PubMed

Sandoval, Christopher; Kim, Arnold D

2015-08-10

The generalized Kubelka-Munk (gKM) approximation is a linear transformation of the double spherical harmonics of order one (DP₁) approximation of the radiative transfer equation. Here, we extend the gKM approximation to study problems in three-dimensional radiative transfer. In particular, we derive the gKM approximation for the problem of collimated beam propagation and scattering in a plane-parallel slab composed of a uniform absorbing and scattering medium. The result is an 8×8 system of partial differential equations that is much easier to solve than the radiative transfer equation. We compare the solutions of the gKM approximation with Monte Carlo simulations of the radiative transfer equation to identify the range of validity for this approximation. We find that the gKM approximation is accurate for isotropic scattering media that are sufficiently thick and much less accurate for anisotropic, forward-peaked scattering media.

A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

NASA Technical Reports Server (NTRS)

Estes, R. H.

1977-01-01

A computer software system is described which computes global numerical solutions of the integro-differential Laplace tidal equations, including dissipation terms and ocean loading and self-gravitation effects, for arbitrary diurnal and semidiurnal tidal constituents. The integration algorithm features a successive approximation scheme for the integro-differential system, with time stepping forward differences in the time variable and central differences in spatial variables.

Solution of the Average-Passage Equations for the Incompressible Flow through Multiple-Blade-Row Turbomachinery

DTIC Science & Technology

1994-02-01

numerical treatment. An explicit numerical procedure based on Runqe-Kutta time stepping for cell-centered, hexahedral finite volumes is...An explicit numerical procedure based on Runge-Kutta time stepping for cell-centered, hexahedral finite volumes is outlined for the approximate...Discretization 16 3.1 Cell-Centered Finite -Volume Discretization in Space 16 3.2 Artificial Dissipation 17 3.3 Time Integration 21 3.4 Convergence

Computer-Aided Engineering of Semiconductor Integrated Circuits

DTIC Science & Technology

1979-07-01

equation using a five point finite difference approximation. Section 4.3.6 describes the numerical techniques and iterative algorithms which are used...neighbor points. This is generally referred to as a five point finite difference scheme on a rectangular grid, as described below. The finite difference ...problems in steady state have been analyzed by the finite difference method [4. 16 ] [4.17 3 or finite element method [4. 18 3, [4. 19 3 as reported last

Isogeometric Analysis of Boundary Integral Equations

DTIC Science & Technology

2015-04-21

methods, IgA relies on Non-Uniform Rational B- splines (NURBS) [43, 46], T- splines [55, 53] or subdivision surfaces [21, 48, 51] rather than piece- wise...structural dynamics [25, 26], plates and shells [15, 16, 27, 28, 37, 22, 23], phase-field models [17, 32, 33], and shape optimization [40, 41, 45, 59...polynomials for approximating the geometry and field variables. Thus, by replacing piecewise polynomials with NURBS or T- splines , one can develop

One-step methods for the prediction of orbital motion, taking its periodic components into account

NASA Astrophysics Data System (ADS)

Lavrov, K. N.

1988-03-01

The paper examines the design and analysis of the properties of implicit one-step integration methods which use the trigonometric approximation of ordinary differential equations containing periodic components. With reference to an orbital-motion prediction example, it is shown that the proposed schemes are more efficient in terms of computer memory than Everhart's (1974) approach. The results obtained make it possible to improve Everhart's method.

Computational Sciences.

DTIC Science & Technology

1987-11-01

III. - 7 1 11 1*25 4 11 - IN, I 61I’. UNCLASSIFIED MASTER COPY - FOR REPRODUCTION PURPOSES ) C . AD-A 190 ’PORT DOCUMENTATION PAGE ~~ 190 826 lb...E uations, University of Alabama, Birmingham, *AL.-7 N. Medhin, M. Sambandham, and C . K. Zoltani, Numerical Solution to a System of Random Volterra...Sambandham, and C . K. Zoltani, "Numerical Solution to a System of Random Volterra Integral Equations I: Successive Approximation Method’,"-submitted to

Asymptotic tracking and disturbance rejection of the blood glucose regulation system.

PubMed

Ashley, Brandon; Liu, Weijiu

2017-07-01

Type 1 diabetes patients need external insulin to maintain blood glucose within a narrow range from 65 to 108 mg/dl (3.6 to 6.0 mmol/l). A mathematical model for the blood glucose regulation is required for integrating a glucose monitoring system into insulin pump technology to form a closed-loop insulin delivery system on the feedback of the blood glucose, the so-called "artificial pancreas". The objective of this paper is to treat the exogenous glucose from food as a glucose disturbance and then develop a closed-loop feedback and feedforward control system for the blood glucose regulation system subject to the exogenous glucose disturbance. For this, a mathematical model for the glucose disturbance is proposed on the basis of experimental data, and then incorporated into an existing blood glucose regulation model. Because all the eigenvalues of the disturbance model have zero real parts, the center manifold theory is used to establish blood glucose regulator equations. We then use their solutions to synthesize a required feedback and feedforward controller to reject the disturbance and asymptotically track a constant glucose reference of 90  mg/dl. Since the regulator equations are nonlinear partial differential equations and usually impossible to solve analytically, a linear approximation solution is obtained. Our numerical simulations show that, under the linear approximate feedback and feedforward controller, the blood glucose asymptotically tracks its desired level of 90 mg/dl approximately. Copyright © 2017 Elsevier Inc. All rights reserved.

Eigenvalue and eigenvector sensitivity and approximate analysis for repeated eigenvalue problems

NASA Technical Reports Server (NTRS)

Hou, Gene J. W.; Kenny, Sean P.

1991-01-01

A set of computationally efficient equations for eigenvalue and eigenvector sensitivity analysis are derived, and a method for eigenvalue and eigenvector approximate analysis in the presence of repeated eigenvalues is presented. The method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem. Examples are given to demonstrate the application of such equations for sensitivity and approximate analysis.

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

Fleury, Pierre; Uzan, Jean-Philippe; Larena, Julien, E-mail: fleury@iap.fr, E-mail: j.larena@ru.ac.za, E-mail: uzan@iap.fr

On the scale of the light beams subtended by small sources, e.g. supernovae, matter cannot be accurately described as a fluid, which questions the applicability of standard cosmic lensing to those cases. In this article, we propose a new formalism to deal with small-scale lensing as a diffusion process: the Sachs and Jacobi equations governing the propagation of narrow light beams are treated as Langevin equations. We derive the associated Fokker-Planck-Kolmogorov equations, and use them to deduce general analytical results on the mean and dispersion of the angular distance. This formalism is applied to random Einstein-Straus Swiss-cheese models, allowing usmore » to: (1) show an explicit example of the involved calculations; (2) check the validity of the method against both ray-tracing simulations and direct numerical integration of the Langevin equation. As a byproduct, we obtain a post-Kantowski-Dyer-Roeder approximation, accounting for the effect of tidal distortions on the angular distance, in excellent agreement with numerical results. Besides, the dispersion of the angular distance is correctly reproduced in some regimes.« less

The theory of stochastic cosmological lensing

NASA Astrophysics Data System (ADS)

Fleury, Pierre; Larena, Julien; Uzan, Jean-Philippe

2015-11-01

On the scale of the light beams subtended by small sources, e.g. supernovae, matter cannot be accurately described as a fluid, which questions the applicability of standard cosmic lensing to those cases. In this article, we propose a new formalism to deal with small-scale lensing as a diffusion process: the Sachs and Jacobi equations governing the propagation of narrow light beams are treated as Langevin equations. We derive the associated Fokker-Planck-Kolmogorov equations, and use them to deduce general analytical results on the mean and dispersion of the angular distance. This formalism is applied to random Einstein-Straus Swiss-cheese models, allowing us to: (1) show an explicit example of the involved calculations; (2) check the validity of the method against both ray-tracing simulations and direct numerical integration of the Langevin equation. As a byproduct, we obtain a post-Kantowski-Dyer-Roeder approximation, accounting for the effect of tidal distortions on the angular distance, in excellent agreement with numerical results. Besides, the dispersion of the angular distance is correctly reproduced in some regimes.

Planar dynamics of a uniform beam with rigid bodies affixed to the ends

NASA Technical Reports Server (NTRS)

Storch, J.; Gates, S.

1983-01-01

The planar dynamics of a uniform elastic beam subject to a variety of geometric and natural boundary conditions and external excitations were analyzed. The beams are inextensible and capable of small transverse bending deformations only. Classical beam vibration eigenvalue problems for a cantilever with tip mass, a cantilever with tip body and an unconstrained beam with rigid bodies at each are examined. The characteristic equations, eigenfunctions and orthogonality relations for each are derived. The forced vibration of a cantilever with tip body subject to base acceleration is analyzed. The exact solution of the governing nonhomogeneous partial differential equation with time dependent boundary conditions is presented and compared with a Rayleigh-Ritz approximate solution. The arbitrary planar motion of an elastic beam with rigid bodies at the ends is addressed. Equations of motion are derived for two modal expansions of the beam deflection. The motion equations are cast in a first order form suitable for numerical integration. Selected FORTRAN programs are provided.

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.

Tidal Love Numbers of Neutron Stars

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

Hinderer, Tanja

For a variety of fully relativistic polytropic neutron star models we calculate the star's tidal Love number k{sub 2}. Most realistic equations of state for neutron stars can be approximated as a polytrope with an effective index n {approx} 0.5-1.0. The equilibrium stellar model is obtained by numerical integration of the Tolman-Oppenheimer-Volkhov equations. We calculate the linear l = 2 static perturbations to the Schwarzschild spacetime following the method of Thorne and Campolattaro. Combining the perturbed Einstein equations into a single second-order differential equation for the perturbation to the metric coefficient g{sub tt} and matching the exterior solution to themore » asymptotic expansion of the metric in the star's local asymptotic rest frame gives the Love number. Our results agree well with the Newtonian results in the weak field limit. The fully relativistic values differ from the Newtonian values by up to {approx}24%. The Love number is potentially measurable in gravitational wave signals from inspiralling binary neutron stars.« less

Exact solutions for postbuckling of a graded porous beam

NASA Astrophysics Data System (ADS)

Ma, L. S.; Ou, Z. Y.

2018-06-01

An exact, closed-form solution for the postbuckling responses of graded porous beams subjected to axially loading is obtained. It was assumed that the properties of the graded porous materials vary continuously through thickness of the beams, the equations governing the axial and transverse deformations are derived based on the classical beam theory and the physical neutral surface concept. The two equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformations. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for postbuckled deformation is obtained as a function of the applied load. The exact solutions explicitly describe the nonlinear equilibrium paths of the buckled beam and thus are able to provide insight into deformation problems. Based on the exact solutions obtained herein, the effects of various factors such as porosity distribution pattern, porosity coefficient and boundary conditions on postbuckling behavior of graded porous beams have been investigated.

Intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion.

PubMed

Thomas, Philipp; Matuschek, Hannes; Grima, Ramon

2012-01-01

The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen's system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA's performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circadian rhythms. The software iNA is freely available as executable binaries for Linux, MacOSX and Microsoft Windows, as well as the full source code under an open source license.

Intrinsic Noise Analyzer: A Software Package for the Exploration of Stochastic Biochemical Kinetics Using the System Size Expansion

PubMed Central

Grima, Ramon

2012-01-01

The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen’s system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA’s performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circadian rhythms. The software iNA is freely available as executable binaries for Linux, MacOSX and Microsoft Windows, as well as the full source code under an open source license. PMID:22723865

Lax Integrability and the Peakon Problem for the Modified Camassa-Holm Equation

NASA Astrophysics Data System (ADS)

Chang, Xiangke; Szmigielski, Jacek

2018-02-01

Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified Camassa-Holm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified Camassa-Holm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem, the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes' continued fractions and multi-point Padé approximations. We propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds, as well as Toda-like sorting property.

An improved two-dimensional depth-integrated flow equation for rough-walled fractures

NASA Astrophysics Data System (ADS)

Mallikamas, Wasin; Rajaram, Harihar

2010-08-01

We present the development of an improved 2-D flow equation for rough-walled fractures. Our improved equation accounts for the influence of midsurface tortuosity and the fact that the aperture normal to the midsurface is in general smaller than the vertical aperture. It thus improves upon the well-known Reynolds equation that is widely used for modeling flow in fractures. Unlike the Reynolds equation, our approach begins from the lubrication approximation applied in an inclined local coordinate system tangential to the fracture midsurface. The local flow equation thus obtained is rigorously transformed to an arbitrary global Cartesian coordinate system, invoking the concepts of covariant and contravariant transformations for vectors defined on surfaces. Unlike previously proposed improvements to the Reynolds equation, our improved flow equation accounts for tortuosity both along and perpendicular to a flow path. Our approach also leads to a well-defined anisotropic local transmissivity tensor relating the representations of the flux and head gradient vectors in a global Cartesian coordinate system. We show that the principal components of the transmissivity tensor and the orientation of its principal axes depend on the directional local midsurface slopes. In rough-walled fractures, the orientations of the principal axes of the local transmissivity tensor will vary from point to point. The local transmissivity tensor also incorporates the influence of the local normal aperture, which is uniquely defined at each point in the fracture. Our improved flow equation is a rigorous statement of mass conservation in any global Cartesian coordinate system. We present three examples of simple geometries to compare our flow equation to analytical solutions obtained using the exact Stokes equations: an inclined parallel plate, and circumferential and axial flows in an incomplete annulus. The effective transmissivities predicted by our flow equation agree very well with values obtained using the exact Stokes equations in all these cases. We discuss potential limitations of our depth-integrated equation, which include the neglect of convergence/divergence and the inaccuracies implicit in any depth-averaging process near sharp corners where the wall and midsurface curvatures are large.

Intermediate boundary conditions for LOD, ADI and approximate factorization methods

NASA Technical Reports Server (NTRS)

Leveque, R. J.

1985-01-01

A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second order accurate approximation to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The approximate factorization method is considered in slightly more generality.

Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials

NASA Astrophysics Data System (ADS)

Finster, Felix; Smoller, Joel

2010-09-01

A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by gluing together WKB and Airy solutions of corresponding one-dimensional Schrödinger equations. Our method is motivated by, and has applications to, the analysis of linear wave equations in the geometry of a rotating black hole.

Motion in a central field in the presence of a constant perturbing acceleration in a co-moving coordinate system

NASA Astrophysics Data System (ADS)

Sannikova, T. N.; Kholshevnikov, K. V.

2015-08-01

The motion of a point mass under the action of a gravitational force toward a central body and a perturbing acceleration P is considered. The magnitude of P is taken to be small compared to the main gravitational acceleration due to the central body, and the direction of P to be constant in a standard astronomical coordinate system with its origin at the central body and axes directed along the radius vector, the transversal, and the binormal. Consideration of a constant vector perturbing acceleration simplifies averaging of the Euler equations for the motion in osculating elements, making it straightforward to obtain evolutionary differential equations of motion in the mean elements, as was done earlier in a first small-parameter approximation. This paper is devoted to integration of the mean equations. The system is integratable by quadratures if at least one component of the perturbing acceleration is zero, and also if the orbit is initially circular. Moreover, all the quadratures can be expressed in terms of elementary functions and elliptical integrals of the first kind in Jacobi form. If all three components of P are non-zero, this problem reduces to a system of two first-order differential equations, which are apparently not integrable. Possible applications include the motion of natural and artificial satellites taking into account light pressure, the motion of a spacecraft with low thrust, and the motion of an asteroid subject to a thrust from an engine mounted on it or to a gravitational tractor designed, for example, to avoid a collision with Earth.

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....

Nonlinear Schroedinger Approximations for Partial Differential Equations with Quadratic and Quasilinear Terms

NASA Astrophysics Data System (ADS)

Cummings, Patrick

We consider the approximation of solutions of two complicated, physical systems via the nonlinear Schrodinger equation (NLS). In particular, we discuss the evolution of wave packets and long waves in two physical models. Due to the complicated nature of the equations governing many physical systems and the in-depth knowledge we have for solutions of the nonlinear Schrodinger equation, it is advantageous to use approximation results of this kind to model these physical systems. The approximations are simple enough that we can use them to understand the qualitative and quantitative behavior of the solutions, and by justifying them we can show that the behavior of the approximation captures the behavior of solutions to the original equation, at least for long, but finite time. We first consider a model of the water wave equations which can be approximated by wave packets using the NLS equation. We discuss a new proof that both simplifies and strengthens previous justification results of Schneider and Wayne. Rather than using analytic norms, as was done by Schneider and Wayne, we construct a modified energy functional so that the approximation holds for the full interval of existence of the approximate NLS solution as opposed to a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et al. We then consider the Klein-Gordon-Zakharov system and prove a long wave approximation result. In this case there is a non-trivial resonance that cannot be eliminated via a normal form transform. By combining the normal form transform for small Fourier modes and using analytic norms elsewhere, we can get a justification result on the order 1 over epsilon squared time scale.

An Efficient Algorithm for Perturbed Orbit Integration Combining Analytical Continuation and Modified Chebyshev Picard Iteration

NASA Astrophysics Data System (ADS)

Elgohary, T.; Kim, D.; Turner, J.; Junkins, J.

2014-09-01

Several methods exist for integrating the motion in high order gravity fields. Some recent methods use an approximate starting orbit, and an efficient method is needed for generating warm starts that account for specific low order gravity approximations. By introducing two scalar Lagrange-like invariants and employing Leibniz product rule, the perturbed motion is integrated by a novel recursive formulation. The Lagrange-like invariants allow exact arbitrary order time derivatives. Restricting attention to the perturbations due to the zonal harmonics J2 through J6, we illustrate an idea. The recursively generated vector-valued time derivatives for the trajectory are used to develop a continuation series-based solution for propagating position and velocity. Numerical comparisons indicate performance improvements of ~ 70X over existing explicit Runge-Kutta methods while maintaining mm accuracy for the orbit predictions. The Modified Chebyshev Picard Iteration (MCPI) is an iterative path approximation method to solve nonlinear ordinary differential equations. The MCPI utilizes Picard iteration with orthogonal Chebyshev polynomial basis functions to recursively update the states. The key advantages of the MCPI are as follows: 1) Large segments of a trajectory can be approximated by evaluating the forcing function at multiple nodes along the current approximation during each iteration. 2) It can readily handle general gravity perturbations as well as non-conservative forces. 3) Parallel applications are possible. The Picard sequence converges to the solution over large time intervals when the forces are continuous and differentiable. According to the accuracy of the starting solutions, however, the MCPI may require significant number of iterations and function evaluations compared to other integrators. In this work, we provide an efficient methodology to establish good starting solutions from the continuation series method; this warm start improves the performance of the MCPI significantly and will likely be useful for other applications where efficiently computed approximate orbit solutions are needed.

The reflection for dense plant canopies from the one-angle radiative transfer equation

NASA Technical Reports Server (NTRS)

Ganapol, B. D.; Lawless, James G. (Technical Monitor)

1994-01-01

An essential component of remote sensing of vegetation canopies from satellites is fundamental understanding. Since passive remote is driven by photons, the modeling of photon interactions with vegetation is a basic building block in that understanding. Several such photon transport models have been developed during the past two decades and continue to be developed. Different approaches have been followed including monte carlo, radiosity methods, geometric shadowing, and radiative transfer. In general, each approach has application for canopies with specific attributes. This presentation concerns the application of radiative transfer to dense vegetation canopies in which the soil does not participate. The approach taken here is novel in that a consistent theory for photon transport for non-rotationally invariant leaf scattering is developed in a canopy with a general leaf angle distribution (LAD). The theory is limited to the one-angle approximation (azimuthally averaged radiance) and is based on Chandrasekhar's analytical theory. While such a model is admittedly only approximate, it does fulfill a unique function in our search for understanding. In particular, the model is simple in its construct yet contains the essential features of canopy architecture that are mainly responsible for observed responses. Thus, this model will not only be a predictive tool but also an educational one. The mathematical setting is the radiative transfer equation in a dense (semiinfinite) canopy. The leaf scattering phase function is assumed to be Lambertian with different reflectance and transmittance. In addition, abaxial and adaxial differentiation is allowed which effectively destroys optical reciprocity. The analytical solution for the canopy BRDF is obtained by manipulation of the integral transport equation (a la Chandrasekhar) for a general LAD. With discretization of the. leaf angle, the resulting set of integral equations are solved iteratively including an acceleration procedure when the single scatter albedo is near one (in the NIR). Results will be compared to the LARS soybean canopy radiances as well as to broadleaf results from a recent Ames experiment.

Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry

NASA Astrophysics Data System (ADS)

Xu, Peiliang

2018-06-01

The numerical integration method has been routinely used by major institutions worldwide, for example, NASA Goddard Space Flight Center and German Research Center for Geosciences (GFZ), to produce global gravitational models from satellite tracking measurements of CHAMP and/or GRACE types. Such Earth's gravitational products have found widest possible multidisciplinary applications in Earth Sciences. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical and statistical point of view, satellite gravimetry from satellite tracking is essentially the problem of estimating unknown parameters in the Newton's nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in mathematics and statistics, chemistry and physics, and satellite gravimetry, is groundless, mathematically and physically. Given the Newton's nonlinear governing differential equations of satellite motion with unknown equation parameters and unknown initial conditions, we develop three methods to derive new local solutions around a nominal reference orbit, which are linked to measurements to estimate the unknown corrections to approximate values of the unknown parameters and the unknown initial conditions. Bearing in mind that satellite orbits can now be tracked almost continuously at unprecedented accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton's nonlinear governing differential equations of satellite motion for the next generation of global gravitational models. Since the solutions are global uniformly convergent, theoretically speaking, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements, leading to the production of global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, and recognizing the fact that satellite orbits are measured with random errors, we further reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton's governing differential equations as a condition adjustment model with unknown parameters, or equivalently, the weighted least squares estimation of unknown differential equation parameters with equality constraints, for the reconstruction of global high-precision, high-resolution gravitational models from modern (and future) satellite tracking measurements.

Projected regression method for solving Fredholm integral equations arising in the analytic continuation problem of quantum physics

NASA Astrophysics Data System (ADS)

Arsenault, Louis-François; Neuberg, Richard; Hannah, Lauren A.; Millis, Andrew J.

2017-11-01

We present a supervised machine learning approach to the inversion of Fredholm integrals of the first kind as they arise, for example, in the analytic continuation problem of quantum many-body physics. The approach provides a natural regularization for the ill-conditioned inverse of the Fredholm kernel, as well as an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. Applying machine learning to this database generates a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We suggest that the methodology will be similarly effective for other problems involving a formally ill-conditioned inversion of an integral operator, provided that the forward problem can be efficiently solved.

Yarkovsky-O'Keefe-Radzievskii-Paddack effect on tumbling objects

NASA Astrophysics Data System (ADS)

Breiter, S.; Rożek, A.; Vokrouhlický, D.

2011-11-01

A semi-analytical model of the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect on an asteroid spin in a non-principal axis rotation state is developed. The model describes the spin-state evolution in Deprit-Elipe variables, first-order averaged with respect to rotation and Keplerian orbital motion. Assuming zero conductivity, the YORP torque is represented by spherical harmonic series with vectorial coefficients, allowing us to use any degree and order of approximation. Within the quadrupole approximation of the illumination function we find the same first integrals involving rotational momentum, obliquity and dynamical inertia that were obtained by Cicaló & Scheeres. The integrals do not exist when higher degree terms of the illumination function are included, and then the asymptotic states known from Vokrouhlický et al. appear. This resolves an apparent contradiction between earlier results. Averaged equations of motion admit stable and unstable limit cycle solutions that were not previously detected. Non-averaged numerical integration by the Taylor series method for an exemplary shape of 3103 Eger is in good agreement with the semi-analytical theory.

A new method of imposing boundary conditions for hyperbolic equations

NASA Technical Reports Server (NTRS)

Funaro, D.; ative.

1987-01-01

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

A simplified computer program for the prediction of the linear stability behavior of liquid propellant combustors

NASA Technical Reports Server (NTRS)

Mitchell, C. E.; Eckert, K.

1979-01-01

A program for predicting the linear stability of liquid propellant rocket engines is presented. The underlying model assumptions and analytical steps necessary for understanding the program and its input and output are also given. The rocket engine is modeled as a right circular cylinder with an injector with a concentrated combustion zone, a nozzle, finite mean flow, and an acoustic admittance, or the sensitive time lag theory. The resulting partial differential equations are combined into two governing integral equations by the use of the Green's function method. These equations are solved using a successive approximation technique for the small amplitude (linear) case. The computational method used as well as the various user options available are discussed. Finally, a flow diagram, sample input and output for a typical application and a complete program listing for program MODULE are presented.

Vortex equations: Singularities, numerical solution, and axisymmetric vortex breakdown

NASA Technical Reports Server (NTRS)

Bossel, H. H.

1972-01-01

A method of weighted residuals for the computation of rotationally symmetric quasi-cylindrical viscous incompressible vortex flow is presented and used to compute a wide variety of vortex flows. The method approximates the axial velocity and circulation profiles by series of exponentials having (N + 1) and N free parameters, respectively. Formal integration results in a set of (2N + 1) ordinary differential equations for the free parameters. The governing equations are shown to have an infinite number of discrete singularities corresponding to critical values of the swirl parameters. The computations point to the controlling influence of the inner core flow on vortex behavior. They also confirm the existence of two particular critical swirl parameter values: one separates vortex flow which decays smoothly from vortex flow which eventually breaks down, and the second is the first singularity of the quasi-cylindrical system, at which point physical vortex breakdown is thought to occur.

Alien calculus and a Schwinger-Dyson equation: two-point function with a nonperturbative mass scale

NASA Astrophysics Data System (ADS)

Bellon, Marc P.; Clavier, Pierre J.

2018-02-01

Starting from the Schwinger-Dyson equation and the renormalization group equation for the massless Wess-Zumino model, we compute the dominant nonperturbative contributions to the anomalous dimension of the theory, which are related by alien calculus to singularities of the Borel transform on integer points. The sum of these dominant contributions has an analytic expression. When applied to the two-point function, this analysis gives a tame evolution in the deep euclidean domain at this approximation level, making doubtful the arguments on the triviality of the quantum field theory with positive β -function. On the other side, we have a singularity of the propagator for timelike momenta of the order of the renormalization group invariant scale of the theory, which has a nonperturbative relationship with the renormalization point of the theory. All these results do not seem to have an interpretation in terms of semiclassical analysis of a Feynman path integral.

Spacecraft self-contamination due to back-scattering of outgas products

NASA Technical Reports Server (NTRS)

Robertson, S. J.

1976-01-01

The back-scattering of outgas contamination near an orbiting spacecraft due to intermolecular collisions was analyzed. Analytical tools were developed for making reasonably accurate quantitative estimates of the outgas contamination return flux, given a knowledge of the pertinent spacecraft and orbit conditions. Two basic collision mechanisms were considered: (1) collisions involving only outgas molecules (self-scattering) and (2) collisions between outgas molecules and molecules in the ambient atmosphere (ambient-scattering). For simplicity, the geometry was idealized to a uniformly outgassing sphere and to a disk oriented normal to the freestream. The method of solution involved an integration of an approximation of the Boltzmann kinetic equation known as the BGK (or Krook) model equation. Results were obtained in the form of simple equations relating outgas return flux to spacecraft and orbit parameters. Results were compared with previous analyses based on more simplistic models of the collision processes.

Linearly exact parallel closures for slab geometry

NASA Astrophysics Data System (ADS)

Ji, Jeong-Young; Held, Eric D.; Jhang, Hogun

2013-08-01

Parallel closures are obtained by solving a linearized kinetic equation with a model collision operator using the Fourier transform method. The closures expressed in wave number space are exact for time-dependent linear problems to within the limits of the model collision operator. In the adiabatic, collisionless limit, an inverse Fourier transform is performed to obtain integral (nonlocal) parallel closures in real space; parallel heat flow and viscosity closures for density, temperature, and flow velocity equations replace Braginskii's parallel closure relations, and parallel flow velocity and heat flow closures for density and temperature equations replace Spitzer's parallel transport relations. It is verified that the closures reproduce the exact linear response function of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] for Landau damping given a temperature gradient. In contrast to their approximate closures where the vanishing viscosity coefficient numerically gives an exact response, our closures relate the heat flow and nonvanishing viscosity to temperature and flow velocity (gradients).

Low rank factorization of the Coulomb integrals for periodic coupled cluster theory.

PubMed

Hummel, Felix; Tsatsoulis, Theodoros; Grüneis, Andreas

2017-03-28

We study a tensor hypercontraction decomposition of the Coulomb integrals of periodic systems where the integrals are factorized into a contraction of six matrices of which only two are distinct. We find that the Coulomb integrals can be well approximated in this form already with small matrices compared to the number of real space grid points. The cost of computing the matrices scales as O(N 4 ) using a regularized form of the alternating least squares algorithm. The studied factorization of the Coulomb integrals can be exploited to reduce the scaling of the computational cost of expensive tensor contractions appearing in the amplitude equations of coupled cluster methods with respect to system size. We apply the developed methodologies to calculate the adsorption energy of a single water molecule on a hexagonal boron nitride monolayer in a plane wave basis set and periodic boundary conditions.

FELIX-1.0: A finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

NASA Astrophysics Data System (ADS)

Regnier, D.; Verrière, M.; Dubray, N.; Schunck, N.

2016-03-01

We describe the software package FELIX that solves the equations of the time-dependent generator coordinate method (TDGCM) in N-dimensions (N ≥ 1) under the Gaussian overlap approximation. The numerical resolution is based on the Galerkin finite element discretization of the collective space and the Crank-Nicolson scheme for time integration. The TDGCM solver is implemented entirely in C++. Several additional tools written in C++, Python or bash scripting language are also included for convenience. In this paper, the solver is tested with a series of benchmarks calculations. We also demonstrate the ability of our code to handle a realistic calculation of fission dynamics.

Applications of Laplace transform methods to airfoil motion and stability calculations

NASA Technical Reports Server (NTRS)

Edwards, J. W.

1979-01-01

This paper reviews the development of generalized unsteady aerodynamic theory and presents a derivation of the generalized Possio integral equation. Numerical calculations resolve questions concerning subsonic indicial lift functions and demonstrate the generation of Kutta waves at high values of reduced frequency, subsonic Mach number, or both. The use of rational function approximations of unsteady aerodynamic loads in aeroelastic stability calculations is reviewed, and a reformulation of the matrix Pade approximation technique is given. Numerical examples of flutter boundary calculations for a wing which is to be flight tested are given. Finally, a simplified aerodynamic model of transonic flow is used to study the stability of an airfoil exposed to supersonic and subsonic flow regions.

On recent advances and future research directions for computational fluid dynamics

NASA Technical Reports Server (NTRS)

Baker, A. J.; Soliman, M. O.; Manhardt, P. D.

1986-01-01

This paper highlights some recent accomplishments regarding CFD numerical algorithm constructions for generation of discrete approximate solutions to classes of Reynolds-averaged Navier-Stokes equations. Following an overview of turbulent closure modeling, and development of appropriate conservation law systems, a Taylor weak-statement semi-discrete approximate solution algorithm is developed. Various forms for completion to the final linear algebra statement are cited, as are a range of candidate numerical linear algebra solution procedures. This development sequence emphasizes the key building blocks of a CFD RNS algorithm, including solution trial and test spaces, integration procedure and added numerical stability mechanisms. A range of numerical results are discussed focusing on key topics guiding future research directions.

Modified equations of finite-size layered plates made of orthotropic material. Comparison of the results of numerical calculations with analytical solutions

NASA Astrophysics Data System (ADS)

Volchkov, Yu. M.

2017-09-01

This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the three-dimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of three-dimensional theory of elasticity, several approximations of each desired function (stresses and displacements) are used. The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of layered plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular layered plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3 %.

Properties of a soft-core model of methanol: An integral equation theory and computer simulation study

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

Huš, Matej; Urbic, Tomaz, E-mail: tomaz.urbic@fkkt.uni-lj.si; Munaò, Gianmarco

Thermodynamic and structural properties of a coarse-grained model of methanol are examined by Monte Carlo simulations and reference interaction site model (RISM) integral equation theory. Methanol particles are described as dimers formed from an apolar Lennard-Jones sphere, mimicking the methyl group, and a sphere with a core-softened potential as the hydroxyl group. Different closure approximations of the RISM theory are compared and discussed. The liquid structure of methanol is investigated by calculating site-site radial distribution functions and static structure factors for a wide range of temperatures and densities. Results obtained show a good agreement between RISM and Monte Carlo simulations.more » The phase behavior of methanol is investigated by employing different thermodynamic routes for the calculation of the RISM free energy, drawing gas-liquid coexistence curves that match the simulation data. Preliminary indications for a putative second critical point between two different liquid phases of methanol are also discussed.« less

Integral method for transient He II heat transfer in a semi-infinite domain

NASA Astrophysics Data System (ADS)

Baudouy, B.

2002-05-01

Integral methods are suited to solve a non-linear system of differential equations where the non-linearity can be found either in the differential equations or in the boundary conditions. Though they are approximate methods, they have proven to give simple solutions with acceptable accuracy for transient heat transfer in He II. Taking in account the temperature dependence of thermal properties, direct solutions are found without the need of adjusting a parameter. Previously, we have presented a solution for the clamped heat flux and in the present study this method is used to accommodate the clamped-temperature problem. In the case of constant thermal properties, this method yields results that are within a few percent of the exact solution for the heat flux at the axis origin. We applied this solution to analyze recovery from burnout and find an agreement within 10% at low heat flux, whereas at high heat flux the model deviates from the experimental data suggesting the need for a more refined thermal model.

Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry

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

Besse, Nicolas, E-mail: Nicolas.Besse@oca.eu; Institut Jean Lamour, UMR CNRS/UL 7198, Université de Lorraine, BP 70239 54506 Vandoeuvre-lès-Nancy Cedex; Coulette, David, E-mail: David.Coulette@ipcms.unistra.fr

2016-08-15

Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov–Poisson and Vlasov–Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to themore » VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, “Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry” (submitted)] and were found to be surprisingly close to those for the original gyrokinetic-Vlasov equations. The purpose of the present paper is to make these new ideas accessible to two readerships: applied mathematicians and plasma physicists.« less

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.

SALLY LEVEL II- COMPUTE AND INTEGRATE DISTURBANCE AMPLIFICATION RATES ON SWEPT AND TAPERED LAMINAR FLOW CONTROL WINGS WITH SUCTION

NASA Technical Reports Server (NTRS)

Srokowski, A. J.

1994-01-01

The computer program SALLY was developed to compute the incompressible linear stability characteristics and integrate the amplification rates of boundary layer disturbances on swept and tapered wings. For some wing designs, boundary layer disturbance can significantly alter the wing performance characteristics. This is particularly true for swept and tapered laminar flow control wings which incorporate suction to prevent boundary layer separation. SALLY should prove to be a useful tool in the analysis of these wing performance characteristics. The first step in calculating the disturbance amplification rates is to numerically solve the compressible laminar boundary-layer equation with suction for the swept and tapered wing. A two-point finite-difference method is used to solve the governing continuity, momentum, and energy equations. A similarity transformation is used to remove the wall normal velocity as a boundary condition and place it into the governing equations as a parameter. Thus the awkward nonlinear boundary condition is avoided. The resulting compressible boundary layer data is used by SALLY to compute the incompressible linear stability characteristics. The local disturbance growth is obtained from temporal stability theory and converted into a local growth rate for integration. The direction of the local group velocity is taken as the direction of integration. The amplification rate, or logarithmic disturbance amplitude ratio, is obtained by integration of the local disturbance growth over distance. The amplification rate serves as a measure of the growth of linear disturbances within the boundary layer and can serve as a guide in transition prediction. This program is written in FORTRAN IV and ASSEMBLER for batch execution and has been implemented on a CDC CYBER 70 series computer with a central memory requirement of approximately 67K (octal) of 60 bit words. SALLY was developed in 1979.

A comparison of integral equations and density functional theory versus Monte Carlo for hard dumbbells near a hard wall

NASA Astrophysics Data System (ADS)

Henderson, Douglas; Quintana, Jacqueline; Sokołowski, Stefan

1995-03-01

A comparison of Percus-Yevick-Pynn-Lado model theory and a density functional (DF) theory of nonuniform fluids of nonspherical particles is performed. The DF used is a new generalization of Tarazona's theory. The conclusion is that DF theory provides a preferable route to describe the system under consideration. Its accuracy can be improved with better approximation for the direct correlation function (DCF) for bulk system.

Segmented strings and the McMillan map

DOE PAGES

Gubser, Steven S.; Parikh, Sarthak; Witaszczyk, Przemek

2016-07-25

We present new exact solutions describing motions of closed segmented strings in AdS 3 in terms of elliptic functions. The existence of analytic expressions is due to the integrability of the classical equations of motion, which in our examples reduce to instances of the McMillan map. Here, we also obtain a discrete evolution rule for the motion in AdS 3 of arbitrary bound states of fundamental strings and D1-branes in the test approximation.

An Alternative Treatment of Heat Flow for Charge Transport in Semiconductor Devices (Postprint)

DTIC Science & Technology

2010-07-01

is tantamount to treating them as ideal gases. A three-dimensional ideal Fermi gas is spherically symmetric in momentum space, and its distribution in...the first mo- ment of the Boltzmann equation using the momentum relax- ation time and effective mass approximations.13 Neglecting any magnetic field and...where the integral is over all momentum vectors k, v is electron velocity, k is the momentum relaxation time, and kf denotes the gradient in momentum

The range of validity of the two-body approximation in models of terrestrial planet accumulation. II - Gravitational cross sections and runaway accretion

NASA Technical Reports Server (NTRS)

Wetherill, G. W.; Cox, L. P.

1985-01-01

The validity of the two-body approximation in calculating encounters between planetesimals has been evaluated as a function of the ratio of unperturbed planetesimal velocity (with respect to a circular orbit) to mutual escape velocity when their surfaces are in contact (V/V-sub-e). Impact rates as a function of this ratio are calculated to within about 20 percent by numerical integration of the equations of motion. It is found that when the ratio is greater than 0.4 the two-body approximation is a good one. Consequences of reducing the ratio to less than 0.02 are examined. Factors leading to an optimal size for growth of planetesimals from a swarm of given eccentricity and placing a limit on the extent of runaway accretion are derived.

MLFMA-accelerated Nyström method for ultrasonic scattering - Numerical results and experimental validation

NASA Astrophysics Data System (ADS)

Gurrala, Praveen; Downs, Andrew; Chen, Kun; Song, Jiming; Roberts, Ron

2018-04-01

Full wave scattering models for ultrasonic waves are necessary for the accurate prediction of voltage signals received from complex defects/flaws in practical nondestructive evaluation (NDE) measurements. We propose the high-order Nyström method accelerated by the multilevel fast multipole algorithm (MLFMA) as an improvement to the state-of-the-art full-wave scattering models that are based on boundary integral equations. We present numerical results demonstrating improvements in simulation time and memory requirement. Particularly, we demonstrate the need for higher order geom-etry and field approximation in modeling NDE measurements. Also, we illustrate the importance of full-wave scattering models using experimental pulse-echo data from a spherical inclusion in a solid, which cannot be modeled accurately by approximation-based scattering models such as the Kirchhoff approximation.

High Frequency Acoustic Propagation using Level Set Methods

DTIC Science & Technology

2007-01-01

solution of the high frequency approximation to the wave equation. Traditional solutions to the Eikonal equation in high frequency acoustics are...the Eikonal equation derived from the high frequency approximation to the wave equation, ucuH ∇±=∇ )(),( xx , with the nonnegative function c(x...For simplicity, we only consider the case ucuH ∇+=∇ )(),( xx . Two difficulties must be addressed when solving the Eikonal equation in a fixed

FAST TRACK COMMUNICATION Time-dependent exact solutions of the nonlinear Kompaneets equation

NASA Astrophysics Data System (ADS)

Ibragimov, N. H.

2010-12-01

Time-dependent exact solutions of the Kompaneets photon diffusion equation are obtained for several approximations of this equation. One of the approximations describes the case when the induced scattering is dominant. In this case, the Kompaneets equation has an additional symmetry which is used for constructing some exact solutions as group invariant solutions.

Semigroup theory and numerical approximation for equations in linear viscoelasticity

NASA Technical Reports Server (NTRS)

Fabiano, R. H.; Ito, K.

1990-01-01

A class of abstract integrodifferential equations used to model linear viscoelastic beams is investigated analytically, applying a Hilbert-space approach. The basic equation is rewritten as a Cauchy problem, and its well-posedness is demonstrated. Finite-dimensional subspaces of the state space and an estimate of the state operator are obtained; approximation schemes for the equations are constructed; and the convergence is proved using the Trotter-Kato theorem of linear semigroup theory. The actual convergence behavior of different approximations is demonstrated in numerical computations, and the results are presented in tables.

Heisenberg-Langevin versus quantum master equation

NASA Astrophysics Data System (ADS)

Boyanovsky, Daniel; Jasnow, David

2017-12-01

The quantum master equation is an important tool in the study of quantum open systems. It is often derived under a set of approximations, chief among them the Born (factorization) and Markov (neglect of memory effects) approximations. In this article we study the paradigmatic model of quantum Brownian motion of a harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic spectral density. We obtain analytically the exact solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the exact correlation functions to those obtained in the asymptotic long time limit with the quantum master equation in the Born approximation with and without the Markov approximation. In the latter case we implement a systematic derivative expansion that yields the exact asymptotic limit under the factorization approximation only. We find discrepancies that could be significant when the bandwidth of the bath Λ is much larger than the typical scales of the system. We study the exact interaction energy as a proxy for the correlations missed by the Born approximation and find that its dependence on Λ is similar to the discrepancy between the exact solution and that of the quantum master equation in the Born approximation. We quantify the regime of validity of the quantum master equation in the Born approximation with or without the Markov approximation in terms of the system's relaxation rate γ , its unrenormalized natural frequency Ω and Λ : γ /Ω ≪1 and also γ Λ /Ω2≪1 . The reliability of the Born approximation is discussed within the context of recent experimental settings and more general environments.

A simple molecular mechanics integrator in mixed rigid body and dihedral angle space

PubMed Central

Vitalis, Andreas; Pappu, Rohit V.

2014-01-01

We propose a numerical scheme to integrate equations of motion in a mixed space of rigid-body and dihedral angle coordinates. The focus of the presentation is biomolecular systems and the framework is applicable to polymers with tree-like topology. By approximating the effective mass matrix as diagonal and lumping all bias torques into the time dependencies of the diagonal elements, we take advantage of the formal decoupling of individual equations of motion. We impose energy conservation independently for every degree of freedom and this is used to derive a numerical integration scheme. The cost of all auxiliary operations is linear in the number of atoms. By coupling the scheme to one of two popular thermostats, we extend the method to sample constant temperature ensembles. We demonstrate that the integrator of choice yields satisfactory stability and is free of mass-metric tensor artifacts, which is expected by construction of the algorithm. Two fundamentally different systems, viz., liquid water and an α-helical peptide in a continuum solvent are used to establish the applicability of our method to a wide range of problems. The resultant constant temperature ensembles are shown to be thermodynamically accurate. The latter relies on detailed, quantitative comparisons to data from reference sampling schemes operating on exactly the same sets of degrees of freedom. PMID:25053299

Real time estimation of the heaving and pitching motions of a ship, using a Kalman filter

NASA Technical Reports Server (NTRS)

Triantafyllou, M.; Athans, M.

1981-01-01

In the present study the estimation of the heave and pitch motion of a ship is considered, using Kalman filtering techniques. A significant part of the study is devoted to constructing appropriate models for the sea and the ship. The governing equations are obtained from hydrodynamic considerations in the form of linear differential equations with frequency dependent coefficients. In addition, nonminimum phase characteristics are obtained due to the spatial integration of the water wave forces. The resulting transfer matrix function is irrational and nonminimum phase. The conditions for a finite-dimensional approximation are considered and the impact of the various parameters is assessed. A numerical application is considered for a DD-963 destroyer.

Single-step methods for predicting orbital motion considering its periodic components

NASA Astrophysics Data System (ADS)

Lavrov, K. N.

1989-01-01

Modern numerical methods for integration of ordinary differential equations can provide accurate and universal solutions to celestial mechanics problems. The implicit single sequence algorithms of Everhart and multiple step computational schemes using a priori information on periodic components can be combined to construct implicit single sequence algorithms which combine their advantages. The construction and analysis of the properties of such algorithms are studied, utilizing trigonometric approximation of the solutions of differential equations containing periodic components. The algorithms require 10 percent more machine memory than the Everhart algorithms, but are twice as fast, and yield short term predictions valid for five to ten orbits with good accuracy and five to six times faster than algorithms using other methods.

Non-LTE line formation in a magnetic field. I. Noncoherent scattering and true absorption

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

Domke, H.; Staude, J.

1973-08-01

The formation of a Zeeman-multiplet by noncoherent scattering and true absorption in a Milne-- Eddington atmosphere is considered assuming a homogeneous magnetic field and complete depolarization of the atomic line levels. The transfer equation for the Stokes parameters is transformed into a scalar integral equation of the Wiener-- Hopf type which is solved by Sobolev's method in closed form. The influence of the magnetic field on the mean scattering number in an infinite medium is discussed. The solution of the line formation problem is obtained for a Planckian source fruction. This solution may be simplified by making the ''finite fieldmore » approximation'', which should be sufficiently accurate for practical purposes. (auth)« less

Course 4: Density Functional Theory, Methods, Techniques, and Applications

NASA Astrophysics Data System (ADS)

Chrétien, S.; Salahub, D. R.

Contents 1 Introduction 2 Density functional theory 2.1 Hohenberg and Kohn theorems 2.2 Levy's constrained search 2.3 Kohn-Sham method 3 Density matrices and pair correlation functions 4 Adiabatic connection or coupling strength integration 5 Comparing and constrasting KS-DFT and HF-CI 6 Preparing new functionals 7 Approximate exchange and correlation functionals 7.1 The Local Spin Density Approximation (LSDA) 7.2 Gradient Expansion Approximation (GEA) 7.3 Generalized Gradient Approximation (GGA) 7.4 meta-Generalized Gradient Approximation (meta-GGA) 7.5 Hybrid functionals 7.6 The Optimized Effective Potential method (OEP) 7.7 Comparison between various approximate functionals 8 LAP correlation functional 9 Solving the Kohn-Sham equations 9.1 The Kohn-Sham orbitals 9.2 Coulomb potential 9.3 Exchange-correlation potential 9.4 Core potential 9.5 Other choices and sources of error 9.6 Functionality 10 Applications 10.1 Ab initio molecular dynamics for an alanine dipeptide model 10.2 Transition metal clusters: The ecstasy, and the agony... 10.3 The conversion of acetylene to benzene on Fe clusters 11 Conclusions

Consistent Yokoya-Chen Approximation to Beamstrahlung(LCC-0010)

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

Peskin, M

2004-04-22

I reconsider the Yokoya-Chen approximate evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical approximations for the electron and photon energy spectra. I compare these approximations to previous ones, and to simulation data.I reconsider the Yokoya-Chen approximate evolution equation for beamstrahlung and modify it slightly to generate simple, consistent analytical approximations for the electron and photon energy spectra. I compare these approximations to previous ones, and to simulation data.

Synoptic, Global Mhd Model For The Solar Corona

NASA Astrophysics Data System (ADS)

Cohen, Ofer; Sokolov, I. V.; Roussev, I. I.; Gombosi, T. I.

2007-05-01

The common techniques for mimic the solar corona heating and the solar wind acceleration in global MHD models are as follow. 1) Additional terms in the momentum and energy equations derived from the WKB approximation for the Alfv’en wave turbulence; 2) some empirical heat source in the energy equation; 3) a non-uniform distribution of the polytropic index, γ, used in the energy equation. In our model, we choose the latter approach. However, in order to get a more realistic distribution of γ, we use the empirical Wang-Sheeley-Arge (WSA) model to constrain the MHD solution. The WSA model provides the distribution of the asymptotic solar wind speed from the potential field approximation; therefore it also provides the distribution of the kinetic energy. Assuming that far from the Sun the total energy is dominated by the energy of the bulk motion and assuming the conservation of the Bernoulli integral, we can trace the total energy along a magnetic field line to the solar surface. On the surface the gravity is known and the kinetic energy is negligible. Therefore, we can get the surface distribution of γ as a function of the final speed originating from this point. By interpolation γ to spherically uniform value on the source surface, we use this spatial distribution of γ in the energy equation to obtain a self-consistent, steady state MHD solution for the solar corona. We present the model result for different Carrington Rotations.

Linear stability analysis of collective neutrino oscillations without spurious modes

NASA Astrophysics Data System (ADS)

Morinaga, Taiki; Yamada, Shoichi

2018-01-01

Collective neutrino oscillations are induced by the presence of neutrinos themselves. As such, they are intrinsically nonlinear phenomena and are much more complex than linear counterparts such as the vacuum or Mikheyev-Smirnov-Wolfenstein oscillations. They obey integro-differential equations, for which it is also very challenging to obtain numerical solutions. If one focuses on the onset of collective oscillations, on the other hand, the equations can be linearized and the technique of linear analysis can be employed. Unfortunately, however, it is well known that such an analysis, when applied with discretizations of continuous angular distributions, suffers from the appearance of so-called spurious modes: unphysical eigenmodes of the discretized linear equations. In this paper, we analyze in detail the origin of these unphysical modes and present a simple solution to this annoying problem. We find that the spurious modes originate from the artificial production of pole singularities instead of a branch cut on the Riemann surface by the discretizations. The branching point singularities on the Riemann surface for the original nondiscretized equations can be recovered by approximating the angular distributions with polynomials and then performing the integrals analytically. We demonstrate for some examples that this simple prescription does remove the spurious modes. We also propose an even simpler method: a piecewise linear approximation to the angular distribution. It is shown that the same methodology is applicable to the multienergy case as well as to the dispersion relation approach that was proposed very recently.

Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures

NASA Technical Reports Server (NTRS)

Gibson, J. S.; Adamian, A.

1988-01-01

An approximation theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that approximate closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The approximation scheme approximates the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

Seismic waves in heterogeneous material: subcell resolution of the discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Castro, Cristóbal E.; Käser, Martin; Brietzke, Gilbert B.

2010-07-01

We present an important extension of the arbitrary high-order discontinuous Galerkin (DG) finite-element method to model 2-D elastic wave propagation in highly heterogeneous material. In this new approach we include space-variable coefficients to describe smooth or discontinuous material variations inside each element using the same numerical approximation strategy as for the velocity-stress variables in the formulation of the elastic wave equation. The combination of the DG method with a time integration scheme based on the solution of arbitrary accuracy derivatives Riemann problems still provides an explicit, one-step scheme which achieves arbitrary high-order accuracy in space and time. Compared to previous formulations the new scheme contains two additional terms in the form of volume integrals. We show that the increasing computational cost per element can be overcompensated due to the improved material representation inside each element as coarser meshes can be used which reduces the total number of elements and therefore computational time to reach a desired error level. We confirm the accuracy of the proposed scheme performing convergence tests and several numerical experiments considering smooth and highly heterogeneous material. As the approximation of the velocity and stress variables in the wave equation and of the material properties in the model can be chosen independently, we investigate the influence of the polynomial material representation on the accuracy of the synthetic seismograms with respect to computational cost. Moreover, we study the behaviour of the new method on strong material discontinuities, in the case where the mesh is not aligned with such a material interface. In this case second-order linear material approximation seems to be the best choice, with higher-order intra-cell approximation leading to potential instable behaviour. For all test cases we validate our solution against the well-established standard fourth-order finite difference and spectral element method.

Effective use of surface-water management to control saltwater intrusion

NASA Astrophysics Data System (ADS)

Hughes, J. D.; White, J.

2012-12-01

The Biscayne aquifer in southeast Florida is susceptible to saltwater intrusion and inundation from rising sea-level as a result of high groundwater withdrawal rates and low topographic relief. Groundwater levels in the Biscayne aquifer are managed by an extensive canal system that is designed to control flooding, supply recharge to municipal well fields, and control saltwater intrusion. We present results from an integrated surface-water/groundwater model of a portion of the Biscayne aquifer to evaluate the ability of the existing managed surface-water control network to control saltwater intrusion. Surface-water stage and flow are simulated using a hydrodynamic model that solves the diffusive-wave approximation of the depth-integrated shallow surface-water equations. Variable-density groundwater flow and fluid density are solved using the Oberbeck--Boussinesq approximation of the three-dimensional variable-density groundwater flow equation and a sharp interface approximation, respectively. The surface-water and variable-density groundwater domains are implicitly coupled during each Picard iteration. The Biscayne aquifer is discretized into a multi-layer model having a 500-m square horizontal grid spacing. All primary and secondary surface-water features in the active model domain are discretized into segments using the 500-m square horizontal grid. A 15-year period of time is simulated and the model includes 66 operable surface-water control structures, 127 municipal production wells, and spatially-distributed daily internal and external hydrologic stresses. Numerical results indicate that the existing surface-water system can be effectively used in many locations to control saltwater intrusion in the Biscayne aquifer resulting from increases in groundwater withdrawals or sea-level rise expected to occur over the next 25 years. In other locations, numerical results indicate surface-water control structures and/or operations may need to be modified to control saltwater intrusion.

Probability Density Functions of Observed Rainfall in Montana

NASA Technical Reports Server (NTRS)

Larsen, Scott D.; Johnson, L. Ronald; Smith, Paul L.

1995-01-01

The question of whether a rain rate probability density function (PDF) can vary uniformly between precipitation events is examined. Image analysis on large samples of radar echoes is possible because of advances in technology. The data provided by such an analysis easily allow development of radar reflectivity factors (and by extension rain rate) distribution. Finding a PDF becomes a matter of finding a function that describes the curve approximating the resulting distributions. Ideally, one PDF would exist for all cases; or many PDF's that have the same functional form with only systematic variations in parameters (such as size or shape) exist. Satisfying either of theses cases will, validate the theoretical basis of the Area Time Integral (ATI). Using the method of moments and Elderton's curve selection criteria, the Pearson Type 1 equation was identified as a potential fit for 89 percent of the observed distributions. Further analysis indicates that the Type 1 curve does approximate the shape of the distributions but quantitatively does not produce a great fit. Using the method of moments and Elderton's curve selection criteria, the Pearson Type 1 equation was identified as a potential fit for 89% of the observed distributions. Further analysis indicates that the Type 1 curve does approximate the shape of the distributions but quantitatively does not produce a great fit.

Infrared-optical transmission and reflection measurements on loose powders

NASA Astrophysics Data System (ADS)

Kuhn, J.; Korder, S.; Arduini-Schuster, M. C.; Caps, R.; Fricke, J.

1993-09-01

A method is described to determine quantitatively the infrared-optical properties of loose powder beds via directional-hemispherical transmission and reflection measurements. Instead of the integration of the powders into a potassium bromide (KBr) or a paraffin oil matrix, which would drastically alter the scattering behavior, the powders are placed onto supporting layers of polyethylene (PE) and KBr. A commercial spectrometer is supplemented by an external optics, which enables measurements on horizontally arranged samples. For data evaluation we use a solution of the equation of radiative transfer in the 3-flux approximation under boundary conditions adapted to the PE or KBr/powder system. A comparison with Kubelka-Munk's theory and Schuster's 2-flux approximation is performed, which shows that 3-flux approximation yields results closest to the exact solution. Equations are developed, which correct transmission and reflection of the samples for the influence of the supporting layer and calculate the specific extinction and the albedo of the powder and thus enables us to separate scattering and absorption part of the extinction spectrum. Measurements on TiO2 powder are presented, which show the influence of preparation techniques and data evaluation with different methods to obtain the albedo. The specific extinction of various TiO2 powders is presented.

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

Study of the hard-disk system at high densities: the fluid-hexatic phase transition.

PubMed

Mier-Y-Terán, Luis; Machorro-Martínez, Brian Ignacio; Chapela, Gustavo A; Del Río, Fernando

2018-06-21

Integral equations of uniform fluids have been considered unable to predict any characteristic feature of the fluid-solid phase transition, including the shoulder that arises in the second peak of the fluid-phase radial distribution function, RDF, of hard-core systems obtained by computer simulations, at fluid densities very close to the structural two-step phase transition. This reasoning is based on the results of traditional integral approximations, like Percus-Yevick, PY, which does not show such a shoulder in hard-core systems, neither in two nor three dimensions. In this work, we present results of three Ansätze, based on the PY theory, that were proposed to remedy the lack of PY analytical solutions in two dimensions. This comparative study shows that one of those Ansätze does develop a shoulder in the second peak of the RDF at densities very close to the phase transition, qualitatively describing this feature. Since the shoulder grows into a peak at still higher densities, this integral equation approach predicts the appearance of an orientational order characteristic of the hexatic phase in a continuous fluid-hexatic phase transition.

A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach

NASA Astrophysics Data System (ADS)

Wu, Dongmei; Wang, Zhongcheng

2006-03-01

According to Mickens [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an approximation to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an approximation to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force Bcos(ωx), to find a periodic solution when the fundamental frequency is identical to ω, the corresponding Fourier series can be written as y˜(x)=∑n=1m acos[(2n-1)ωx]. How to calculate the coefficients of the Fourier series efficiently with a computer program is still an open problem. For HB method, by substituting approximation y˜(x) into force equation, expanding the resulting expression into a trigonometric series, then letting the coefficients of the resulting lowest-order harmonic be zero, one can obtain approximate coefficients of approximation y˜(x) [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563]. But for nonlinear differential equations such as Duffing equation, it is very difficult to construct higher-order analytical approximations, because the HB method requires solving a set of algebraic equations for a large number of unknowns with very complex nonlinearities. To overcome the difficulty, forty years ago, Urabe derived a computational method for Duffing equation based on Galerkin procedure [M. Urabe, A. Reiter, Numerical computation of nonlinear forced oscillations by Galerkin's procedure, J. Math. Anal. Appl. 14 (1966) 107-140]. Dooren obtained an approximate solution of the Duffing oscillator with a special set of parameters by using Urabe's method [R. van Dooren, Stabilization of Cowell's classic finite difference method for numerical integration, J. Comput. Phys. 16 (1974) 186-192]. In this paper, in the frame of the general HB method, we present a new iteration algorithm to calculate the coefficients of the Fourier series. By using this new method, the iteration procedure starts with a(x)cos(ωx)+b(x)sin(ωx), and the accuracy may be improved gradually by determining new coefficients a,a,… will be produced automatically in an one-by-one manner. In all the stage of calculation, we need only to solve a cubic equation. Using this new algorithm, we develop a Mathematica program, which demonstrates following main advantages over the previous HB method: (1) it avoids solving a set of associate nonlinear equations; (2) it is easier to be implemented into a computer program, and produces a highly accurate solution with analytical expression efficiently. It is interesting to find that, generally, for a given set of parameters, a nonlinear Duffing equation can have three independent oscillation modes. For some sets of the parameters, it can have two modes with complex displacement and one with real displacement. But in some cases, it can have three modes, all of them having real displacement. Therefore, we can divide the parameters into two classes, according to the solution property: there is only one mode with real displacement and there are three modes with real displacement. This program should be useful to study the dynamically periodic behavior of a Duffing oscillator and can provide an approximate analytical solution with high-accuracy for testing the error behavior of newly developed numerical methods with a wide range of parameters. Program summaryTitle of program:AnalyDuffing.nb Catalogue identifier:ADWR_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWR_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:none Computer for which the program is designed and others on which it has been tested:the program has been designed for a microcomputer and been tested on the microcomputer. Computers:IBM PC Installations:the address(es) of your computer(s) Operating systems under which the program has been tested:Windows XP Programming language used:Software Mathematica 4.2, 5.0 and 5.1 No. of lines in distributed program, including test data, etc.:23 663 No. of bytes in distributed program, including test data, etc.:152 321 Distribution format:tar.gz Memory required to execute with typical data:51 712 Bytes No. of bits in a word: No. of processors used:1 Has the code been vectorized?:no Peripherals used:no Program Library subprograms used:no Nature of physical problem:To find an approximate solution with analytical expressions for the undamped nonlinear Duffing equation with periodic driving force when the fundamental frequency is identical to the driving force. Method of solution:In the frame of the general HB method, by using a new iteration algorithm to calculate the coefficients of the Fourier series, we can obtain an approximate analytical solution with high-accuracy efficiently. Restrictions on the complexity of the problem:For problems, which have a large driving frequency, the convergence may be a little slow, because more iterative times are needed. Typical running time:several seconds Unusual features of the program:For an undamped Duffing equation, it can provide all the solutions or the oscillation modes with real displacement for any interesting parameters, for the required accuracy, efficiently. The program can be used to study the dynamically periodic behavior of a nonlinear oscillator, and can provide a high-accurate approximate analytical solution for developing high-accurate numerical method.

FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

NASA Astrophysics Data System (ADS)

Regnier, D.; Dubray, N.; Verrière, M.; Schunck, N.

2018-04-01

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this paper, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different types of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank-Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. We emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents).

The closure approximation in the hierarchy equations.

NASA Technical Reports Server (NTRS)

Adomian, G.

1971-01-01

The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the approximation in widely used hierarchy methods and the ?self-consistent field' approximation in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.