A numerical method for solving the 3D unsteady incompressible Navier Stokes equations in curvilinear domains with complex immersed boundaries

NASA Astrophysics Data System (ADS)

Ge, Liang; Sotiropoulos, Fotis

2007-08-01

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457-492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

Numerical solution of system of boundary value problems using B-spline with free parameter

NASA Astrophysics Data System (ADS)

Gupta, Yogesh

2017-01-01

This paper deals with method of B-spline solution for a system of boundary value problems. The differential equations are useful in various fields of science and engineering. Some interesting real life problems involve more than one unknown function. These result in system of simultaneous differential equations. Such systems have been applied to many problems in mathematics, physics, engineering etc. In present paper, B-spline and B-spline with free parameter methods for the solution of a linear system of second-order boundary value problems are presented. The methods utilize the values of cubic B-spline and its derivatives at nodal points together with the equations of the given system and boundary conditions, ensuing into the linear matrix equation.

Solution of the Orr-Sommerfeld equation for the Blausius boundary-layer documentation of program ORRBL and a test case

NASA Technical Reports Server (NTRS)

Biringen, S.; Danabasoglu, G.

1988-01-01

A Chebyshev matrix collocation method is outlined for the solution of the Orr-Sommerfeld equation for the Blausius boundary layer. User information is provided for FORTRAN program ORRBL which solves the equation by the QR method.

A fourth-order box method for solving the boundary layer equations

NASA Technical Reports Server (NTRS)

Wornom, S. F.

1977-01-01

A fourth order box method for calculating high accuracy numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations is presented. The method is the natural extension of the second order Keller Box scheme to fourth order and is demonstrated with application to the incompressible, laminar and turbulent boundary layer equations. Numerical results for high accuracy test cases show the method to be significantly faster than other higher order and second order methods.

Boundary particle method for Laplace transformed time fractional diffusion equations

NASA Astrophysics Data System (ADS)

Fu, Zhuo-Jia; Chen, Wen; Yang, Hai-Tian

2013-02-01

This paper develops a novel boundary meshless approach, Laplace transformed boundary particle method (LTBPM), for numerical modeling of time fractional diffusion equations. It implements Laplace transform technique to obtain the corresponding time-independent inhomogeneous equation in Laplace space and then employs a truly boundary-only meshless boundary particle method (BPM) to solve this Laplace-transformed problem. Unlike the other boundary discretization methods, the BPM does not require any inner nodes, since the recursive composite multiple reciprocity technique (RC-MRM) is used to convert the inhomogeneous problem into the higher-order homogeneous problem. Finally, the Stehfest numerical inverse Laplace transform (NILT) is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding BPM solutions. In comparison with finite difference discretization, the LTBPM introduces Laplace transform and Stehfest NILT algorithm to deal with time fractional derivative term, which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability. Consequently, it can effectively simulate long time-history fractional diffusion systems. Error analysis and numerical experiments demonstrate that the present LTBPM is highly accurate and computationally efficient for 2D and 3D time fractional diffusion equations.

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

NASA Astrophysics Data System (ADS)

Shirvany, Yazdan; Hayati, Mohsen; Moradian, Rostam

2008-12-01

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.

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

Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics

NASA Astrophysics Data System (ADS)

Rangan, Aaditya V.; Cai, David; Tao, Louis

2007-02-01

Recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The moment equations arising from the kinetic theory are a system of (1 + 1)-dimensional nonlinear partial differential equations (PDE) on a bounded domain with nonlinear boundary conditions. The PDEs themselves are self-consistently specified by parameters which are functions of the boundary values of the solution. The moment equations can be stiff in space and time. Numerical methods are presented here for efficiently and accurately solving these moment equations. The essential ingredients in our numerical methods include: (i) the system is discretized in time with an implicit Euler method within a spectral deferred correction framework, therefore, the PDEs of the kinetic theory are reduced to a sequence, in time, of boundary value problems (BVPs) with nonlinear boundary conditions; (ii) a set of auxiliary parameters is introduced to recast the original BVP with nonlinear boundary conditions as BVPs with linear boundary conditions - with additional algebraic constraints on the auxiliary parameters; (iii) a careful combination of two Newton's iterates for the nonlinear BVP with linear boundary condition, interlaced with a Newton's iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self-consistent parameters. It is shown that a simple fixed-point iteration can only achieve a linear convergence for the self-consistent parameters. The practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples. It is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse-grained dynamical properties of integrate-and-fire neuronal networks.

Eigenvalue sensitivity analysis of planar frames with variable joint and support locations

NASA Technical Reports Server (NTRS)

Chuang, Ching H.; Hou, Gene J. W.

1991-01-01

Two sensitivity equations are derived in this study based upon the continuum approach for eigenvalue sensitivity analysis of planar frame structures with variable joint and support locations. A variational form of an eigenvalue equation is first derived in which all of the quantities are expressed in the local coordinate system attached to each member. Material derivative of this variational equation is then sought to account for changes in member's length and orientation resulting form the perturbation of joint and support locations. Finally, eigenvalue sensitivity equations are formulated in either domain quantities (by the domain method) or boundary quantities (by the boundary method). It is concluded that the sensitivity equation derived by the boundary method is more efficient in computation but less accurate than that of the domain method. Nevertheless, both of them in terms of computational efficiency are superior to the conventional direct differentiation method and the finite difference method.

Investigation of viscous/inviscid interaction in transonic flow over airfoils with suction

NASA Technical Reports Server (NTRS)

Vemuru, C. S.; Tiwari, S. N.

1988-01-01

The viscous/inviscid interaction over transonic airfoils with and without suction is studied. The streamline angle at the edge of the boundary layer is used to couple the viscous and inviscid flows. The potential flow equations are solved for the inviscid flow field. In the shock region, the Euler equations are solved using the method of integral relations. For this, the potential flow solution is used as the initial and boundary conditions. An integral method is used to solve the laminar boundary-layer equations. Since both methods are integral methods, a continuous interaction is allowed between the outer inviscid flow region and the inner viscous flow region. To avoid the Goldstein singularity near the separation point the laminar boundary-layer equations are derived in an inverse form to obtain solution for the flows with small separations. The displacement thickness distribution is specified instead of the usual pressure distribution to solve the boundry-layer equations. The Euler equations are solved for the inviscid flow using the finite volume technique and the coupling is achieved by a surface transpiration model. A method is developed to apply a minimum amount of suction that is required to have an attached flow on the airfoil. The turbulent boundary layer equations are derived using the bi-logarithmic wall law for mass transfer. The results are found to be in good agreement with available experimental data and with the results of other computational methods.

A general integral form of the boundary-layer equation for incompressible flow with an application to the calculation of the separation point of turbulent boundary layers

NASA Technical Reports Server (NTRS)

Tetervin, Neal; Lin, Chia Chiao

1951-01-01

A general integral form of the boundary-layer equation, valid for either laminar or turbulent incompressible boundary-layer flow, is derived. By using the experimental finding that all velocity profiles of the turbulent boundary layer form essentially a single-parameter family, the general equation is changed to an equation for the space rate of change of the velocity-profile shape parameter. The lack of precise knowledge concerning the surface shear and the distribution of the shearing stress across turbulent boundary layers prevented the attainment of a reliable method for calculating the behavior of turbulent boundary layers.

Some problems of the calculation of three-dimensional boundary layer flows on general configurations

NASA Technical Reports Server (NTRS)

Cebeci, T.; Kaups, K.; Mosinskis, G. J.; Rehn, J. A.

1973-01-01

An accurate solution of the three-dimensional boundary layer equations over general configurations such as those encountered in aircraft and space shuttle design requires a very efficient, fast, and accurate numerical method with suitable turbulence models for the Reynolds stresses. The efficiency, speed, and accuracy of a three-dimensional numerical method together with the turbulence models for the Reynolds stresses are examined. The numerical method is the implicit two-point finite difference approach (Box Method) developed by Keller and applied to the boundary layer equations by Keller and Cebeci. In addition, a study of some of the problems that may arise in the solution of these equations for three-dimensional boundary layer flows over general configurations.

Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

NASA Astrophysics Data System (ADS)

Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

2017-04-01

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

A stable penalty method for the compressible Navier-Stokes equations. 1: Open boundary conditions

NASA Technical Reports Server (NTRS)

Hesthaven, J. S.; Gottlieb, D.

1994-01-01

The purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible Navier-Stokes equations in three spatial dimensions. The treatment uses the conservation form of the Navier-Stokes equations and utilizes linearization and localization at the boundaries based on these variables. The proposed boundary conditions are applied through a penalty procedure, thus ensuring correct behavior of the scheme as the Reynolds number tends to infinity. The versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder.

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.

Theoretical study of the incompressible Navier-Stokes equations by the least-squares method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Loh, Ching Y.; Povinelli, Louis A.

1994-01-01

Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.

Convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment

NASA Technical Reports Server (NTRS)

Funaro, Daniele; Gottlieb, David

1989-01-01

A new method of imposing boundary conditions in the pseudospectral approximation of hyperbolic systems of equations is proposed. It is suggested to collocate the equations, not only at the inner grid points, but also at the boundary points and use the boundary conditions as penalty terms. In the pseudo-spectral Legrendre method with the new boundary treatment, a stability analysis for the case of a constant coefficient hyperbolic system is presented and error estimates are derived.

Program for the solution of multipoint boundary value problems of quasilinear differential equations

NASA Technical Reports Server (NTRS)

1973-01-01

Linear equations are solved by a method of superposition of solutions of a sequence of initial value problems. For nonlinear equations and/or boundary conditions, the solution is iterative and in each iteration a problem like the linear case is solved. A simple Taylor series expansion is used for the linearization of both nonlinear equations and nonlinear boundary conditions. The perturbation method of solution is used in preference to quasilinearization because of programming ease, and smaller storage requirements; and experiments indicate that the desired convergence properties exist although no proof or convergence is given.

Solution of Poisson's Equation with Global, Local and Nonlocal Boundary Conditions

ERIC Educational Resources Information Center

Aliev, Nihan; Jahanshahi, Mohammad

2002-01-01

Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the…

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min-max optimal control problems with uncertainty

NASA Astrophysics Data System (ADS)

Nikooeinejad, Z.; Delavarkhalafi, A.; Heydari, M.

2018-03-01

The difficulty of solving the min-max optimal control problems (M-MOCPs) with uncertainty using generalised Euler-Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler-Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.

UXO Discrimination in Cases with Overlapping Signatures

DTIC Science & Technology

2007-03-07

13. APPENDIX B: HFE -BIEM ..........................................................................................................290 - 7...First principals numerical solutions developed were a Hybrid Finite Element – Boundary Integral Equation Method ( HFE -BIEM) body of revolution (BOR...attacks, namely the Method of Auxiliary Sources (MAS) and the Hybrid Finite Element – Boundary Integral Equation Method ( HFE -BIEM). These work

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.

Filter method without boundary-value condition for simultaneous calculation of eigenfunction and eigenvalue of a stationary Schrödinger equation on a grid.

PubMed

Nurhuda, M; Rouf, A

2017-09-01

The paper presents a method for simultaneous computation of eigenfunction and eigenvalue of the stationary Schrödinger equation on a grid, without imposing boundary-value condition. The method is based on the filter operator, which selects the eigenfunction from wave packet at the rate comparable to δ function. The efficacy and reliability of the method are demonstrated by comparing the simulation results with analytical or numerical solutions obtained by using other methods for various boundary-value conditions. It is found that the method is robust, accurate, and reliable. Further prospect of filter method for simulation of the Schrödinger equation in higher-dimensional space will also be highlighted.

Integral method for the calculation of three-dimensional, laminar and turbulent boundary layers

NASA Technical Reports Server (NTRS)

Stock, H. W.

1978-01-01

The method for turbulent flows is a further development of an existing method; profile families with two parameters and a lag entrainment method replace the simple entrainment method and power profiles with one parameter. The method for laminar flows is a new development. Moment of momentum equations were used for the solution of the problem, the profile families were derived from similar solutions of boundary layer equations. Laminar and turbulent flows at the wings were calculated. The influence of wing tapering on the boundary layer development was shown. The turbulent boundary layer for a revolution ellipsoid is calculated for 0 deg and 10 deg incidence angles.

Solution of Grad-Shafranov equation by the method of fundamental solutions

NASA Astrophysics Data System (ADS)

Nath, D.; Kalra, M. S.; Kalra

2014-06-01

In this paper we have used the Method of Fundamental Solutions (MFS) to solve the Grad-Shafranov (GS) equation for the axisymmetric equilibria of tokamak plasmas with monomial sources. These monomials are the individual terms appearing on the right-hand side of the GS equation if one expands the nonlinear terms into polynomials. Unlike the Boundary Element Method (BEM), the MFS does not involve any singular integrals and is a meshless boundary-alone method. Its basic idea is to create a fictitious boundary around the actual physical boundary of the computational domain. This automatically removes the involvement of singular integrals. The results obtained by the MFS match well with the earlier results obtained using the BEM. The method is also applied to Solov'ev profiles and it is found that the results are in good agreement with analytical results.

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

NASA Astrophysics Data System (ADS)

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

2018-04-01

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

A method of boundary equations for unsteady hyperbolic problems in 3D

NASA Astrophysics Data System (ADS)

Petropavlovsky, S.; Tsynkov, S.; Turkel, E.

2018-07-01

We consider interior and exterior initial boundary value problems for the three-dimensional wave (d'Alembert) equation. First, we reduce a given problem to an equivalent operator equation with respect to unknown sources defined only at the boundary of the original domain. In doing so, the Huygens' principle enables us to obtain the operator equation in a form that involves only finite and non-increasing pre-history of the solution in time. Next, we discretize the resulting boundary equation and solve it efficiently by the method of difference potentials (MDP). The overall numerical algorithm handles boundaries of general shape using regular structured grids with no deterioration of accuracy. For long simulation times it offers sub-linear complexity with respect to the grid dimension, i.e., is asymptotically cheaper than the cost of a typical explicit scheme. In addition, our algorithm allows one to share the computational cost between multiple similar problems. On multi-processor (multi-core) platforms, it benefits from what can be considered an effective parallelization in time.

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

NASA Technical Reports Server (NTRS)

Nakagawa, Y.

1981-01-01

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

Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Soh, Woo Y.

1992-01-01

A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.

Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method

NASA Astrophysics Data System (ADS)

Nakamura, Gen; Wang, Haibing

2017-05-01

Consider the problem of reconstructing unknown Robin inclusions inside a heat conductor from boundary measurements. This problem arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the Robin inclusion and gave its rigorous mathematical justification. This method is non-iterative and based on the characterization of the solution to the so-called Neumann- to-Dirichlet map gap equation. In this paper, we give a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we clarify the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function associated with an initial-boundary value problem for the heat equation inside the Robin inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map and explains what is the input for the linear sampling method. Assuming that the Neumann-to-Dirichlet map gap equation has a unique solution, we also show the convergence of our method for noisy measurements. Second, we give the numerical implementation of the reconstruction method for two-dimensional spatial domains. The measurements for our inverse problem are simulated by solving the forward problem via the boundary integral equation method. Numerical results are presented to illustrate the efficiency and stability of the proposed method. By using a finite sequence of transient input over a time interval, we propose a new sampling method over the time interval by single measurement which is most likely to be practical.

Solution of the surface Euler equations for accurate three-dimensional boundary-layer analysis of aerodynamic configurations

NASA Technical Reports Server (NTRS)

Iyer, V.; Harris, J. E.

1987-01-01

The three-dimensional boundary-layer equations in the limit as the normal coordinate tends to infinity are called the surface Euler equations. The present paper describes an accurate method for generating edge conditions for three-dimensional boundary-layer codes using these equations. The inviscid pressure distribution is first interpolated to the boundary-layer grid. The surface Euler equations are then solved with this pressure field and a prescribed set of initial and boundary conditions to yield the velocities along the two surface coordinate directions. Results for typical wing and fuselage geometries are presented. The smoothness and accuracy of the edge conditions obtained are found to be superior to the conventional interpolation procedures.

An integral equation method for calculating sound field diffracted by a rigid barrier on an impedance ground.

PubMed

Zhao, Sipei; Qiu, Xiaojun; Cheng, Jianchun

2015-09-01

This paper proposes a different method for calculating a sound field diffracted by a rigid barrier based on the integral equation method, where a virtual boundary is assumed above the rigid barrier to divide the whole space into two subspaces. Based on the Kirchhoff-Helmholtz equation, the sound field in each subspace is determined with the source inside and the boundary conditions on the surface, and then the diffracted sound field is obtained by using the continuation conditions on the virtual boundary. Simulations are carried out to verify the feasibility of the proposed method. Compared to the MacDonald method and other existing methods, the proposed method is a rigorous solution for whole space and is also much easier to understand.

Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary.

PubMed

Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan

2016-01-01

Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations.

Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary

PubMed Central

Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan

2016-01-01

Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations. PMID:27031232

Development of monitoring system of helium leakage from canister

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

Toriu, D.; Ushijima, S.; Takeda, H.

2013-07-01

This paper presents a computational method for the helium leakage from a canister. The governing equations for compressible fluids consist of mass conservation equation in Eulerian description, momentum equations and energy equation. The numerical procedures are divided into three phases, advection, diffusion and acoustic phases, and the equations of compressible fluids are discretized with a finite volume method. Thus, the mass conservation law is sufficiently satisfied in the calculation region. In particular, our computational method enables us to predict the change of the temperature distributions around the canister boundaries by calculating the governing equations for the compressible gas flows, whichmore » are leaked out from a slight crack on the canister boundary. In order to confirm the validity of our method, it was applied to the basic problem, 2-dimensional natural convection flows in a rectangular cavity. As a result, it was shown that the naturally convected flows can be reasonably simulated by our method. Furthermore, numerical experiments were conducted for the helium leakage from canister and we derived a close relationship between the inner pressure and the boundary temperature distributions.« less

Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals.

PubMed

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel Antonio; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Marin-Hernandez, Antonio; Herrera-May, Agustin Leobardo; Diaz-Sanchez, Alejandro; Huerta-Chua, Jesus

2014-01-01

In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. 34L30.

Boundary regularized integral equation formulation of the Helmholtz equation in acoustics.

PubMed

Sun, Qiang; Klaseboer, Evert; Khoo, Boo-Cheong; Chan, Derek Y C

2015-01-01

A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals.

Boundary regularized integral equation formulation of the Helmholtz equation in acoustics

PubMed Central

Sun, Qiang; Klaseboer, Evert; Khoo, Boo-Cheong; Chan, Derek Y. C.

2015-01-01

A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals. PMID:26064591

Critical study of higher order numerical methods for solving the boundary-layer equations

NASA Technical Reports Server (NTRS)

Wornom, S. F.

1978-01-01

A fourth order box method is presented for calculating numerical solutions to parabolic, partial differential equations in two variables or ordinary differential equations. The method, which is the natural extension of the second order box scheme to fourth order, was demonstrated with application to the incompressible, laminar and turbulent, boundary layer equations. The efficiency of the present method is compared with two point and three point higher order methods, namely, the Keller box scheme with Richardson extrapolation, the method of deferred corrections, a three point spline method, and a modified finite element method. For equivalent accuracy, numerical results show the present method to be more efficient than higher order methods for both laminar and turbulent flows.

Computation of Three-Dimensional Boundary Layers Including Separation

DTIC Science & Technology

1987-02-01

As demonstrated by the 1968 and 1980 -1981 STANFORD Conferences, integral methods remain a valuable engineering tool to calculate the effects of...has been given by WHITFIELD, 1980 , which is valid over the whole thickness of the boundary layer. Another method to generate a velocity profiles...boundary layer equations and inviscid equations. A very clear presentation of the problem is given for example by VELOMAN, 1980 . 6.3. Three-dimensional

Computation of turbulent boundary layers on curved surfaces, 1 June 1975 - 31 January 1976

NASA Technical Reports Server (NTRS)

Wilcox, D. C.; Chambers, T. L.

1976-01-01

An accurate method was developed for predicting effects of streamline curvature and coordinate system rotation on turbulent boundary layers. A new two-equation model of turbulence was developed which serves as the basis of the study. In developing the new model, physical reasoning is combined with singular perturbation methods to develop a rational, physically-based set of equations which are, on the one hand, as accurate as mixing-length theory for equilibrium boundary layers and, on the other hand, suitable for computing effects of curvature and rotation. The equations are solved numerically for several boundary layer flows over plane and curved surfaces. For incompressible boundary layers, results of the computations are generally within 10% of corresponding experimental data. Somewhat larger discrepancies are noted for compressible applications.

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.

A Curved, Elastostatic Boundary Element for Plane Anisotropic Structures

NASA Technical Reports Server (NTRS)

Smeltzer, Stanley S.; Klang, Eric C.

2001-01-01

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

Topics in spectral methods

NASA Technical Reports Server (NTRS)

Gottlieb, D.; Turkel, E.

1985-01-01

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

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.

A symmetric Trefftz-DG formulation based on a local boundary element method for the solution of the Helmholtz equation

NASA Astrophysics Data System (ADS)

Barucq, H.; Bendali, A.; Fares, M.; Mattesi, V.; Tordeux, S.

2017-02-01

A general symmetric Trefftz Discontinuous Galerkin method is built for solving the Helmholtz equation with piecewise constant coefficients. The construction of the corresponding local solutions to the Helmholtz equation is based on a boundary element method. A series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters, and more importantly its outstanding ability to reduce the instabilities known as the "pollution effect" in the literature on numerical simulations of long-range wave propagation.

Calculation methods for compressible turbulent boundary layers, 1976

NASA Technical Reports Server (NTRS)

Bushnell, D. M.; Cary, A. M., Jr.; Harris, J. E.

1977-01-01

Equations and closure methods for compressible turbulent boundary layers are discussed. Flow phenomena peculiar to calculation of these boundary layers were considered, along with calculations of three dimensional compressible turbulent boundary layers. Procedures for ascertaining nonsimilar two and three dimensional compressible turbulent boundary layers were appended, including finite difference, finite element, and mass-weighted residual methods.

Analysis of the incomplete Galerkin method for modelling of smoothly-irregular transition between planar waveguides

NASA Astrophysics Data System (ADS)

Divakov, D.; Sevastianov, L.; Nikolaev, N.

2017-01-01

The paper deals with a numerical solution of the problem of waveguide propagation of polarized light in smoothly-irregular transition between closed regular waveguides using the incomplete Galerkin method. This method consists in replacement of variables in the problem of reduction of the Helmholtz equation to the system of differential equations by the Kantorovich method and in formulation of the boundary conditions for the resulting system. The formulation of the boundary problem for the ODE system is realized in computer algebra system Maple. The stated boundary problem is solved using Maples libraries of numerical methods.

A Chebyshev matrix method for spatial modes of the Orr-Sommerfeld equation

NASA Technical Reports Server (NTRS)

Danabasoglu, G.; Biringen, S.

1989-01-01

The Chebyshev matrix collocation method is applied to obtain the spatial modes of the Orr-Sommerfeld equation for Poiseuille flow and the Blausius boundary layer. The problem is linearized by the companion matrix technique for semi-infinite domain using a mapping transformation. The method can be easily adapted to problems with different boundary conditions requiring different transformations.

On the Boussinesq-Burgers equations driven by dynamic boundary conditions

NASA Astrophysics Data System (ADS)

Zhu, Neng; Liu, Zhengrong; Zhao, Kun

2018-02-01

We study the qualitative behavior of the Boussinesq-Burgers equations on a finite interval subject to the Dirichlet type dynamic boundary conditions. Assuming H1 ×H2 initial data which are compatible with boundary conditions and utilizing energy methods, we show that under appropriate conditions on the dynamic boundary data, there exist unique global-in-time solutions to the initial-boundary value problem, and the solutions converge to the boundary data as time goes to infinity, regardless of the magnitude of the initial data.

On solving wave equations on fixed bounded intervals involving Robin boundary conditions with time-dependent coefficients

NASA Astrophysics Data System (ADS)

van Horssen, Wim T.; Wang, Yandong; Cao, Guohua

2018-06-01

In this paper, it is shown how characteristic coordinates, or equivalently how the well-known formula of d'Alembert, can be used to solve initial-boundary value problems for wave equations on fixed, bounded intervals involving Robin type of boundary conditions with time-dependent coefficients. A Robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space-derivative on a boundary of the interval. Analytical methods, such as the method of separation of variables (SOV) or the Laplace transform method, are not applicable to those types of problems. The obtained analytical results by applying the proposed method, are in complete agreement with those obtained by using the numerical, finite difference method. For problems with time-independent coefficients in the Robin boundary condition(s), the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables, or by the finite difference method.

On the wall-normal velocity of the compressible boundary-layer equations

NASA Technical Reports Server (NTRS)

Pruett, C. David

1991-01-01

Numerical methods for the compressible boundary-layer equations are facilitated by transformation from the physical (x,y) plane to a computational (xi,eta) plane in which the evolution of the flow is 'slow' in the time-like xi direction. The commonly used Levy-Lees transformation results in a computationally well-behaved problem for a wide class of non-similar boundary-layer flows, but it complicates interpretation of the solution in physical space. Specifically, the transformation is inherently nonlinear, and the physical wall-normal velocity is transformed out of the problem and is not readily recovered. In light of recent research which shows mean-flow non-parallelism to significantly influence the stability of high-speed compressible flows, the contribution of the wall-normal velocity in the analysis of stability should not be routinely neglected. Conventional methods extract the wall-normal velocity in physical space from the continuity equation, using finite-difference techniques and interpolation procedures. The present spectrally-accurate method extracts the wall-normal velocity directly from the transformation itself, without interpolation, leaving the continuity equation free as a check on the quality of the solution. The present method for recovering wall-normal velocity, when used in conjunction with a highly-accurate spectral collocation method for solving the compressible boundary-layer equations, results in a discrete solution which is extraordinarily smooth and accurate, and which satisfies the continuity equation nearly to machine precision. These qualities make the method well suited to the computation of the non-parallel mean flows needed by spatial direct numerical simulations (DNS) and parabolized stability equation (PSE) approaches to the analysis of stability.

Adjoint Method and Predictive Control for 1-D Flow in NASA Ames 11-Foot Transonic Wind Tunnel

NASA Technical Reports Server (NTRS)

Nguyen, Nhan; Ardema, Mark

2006-01-01

This paper describes a modeling method and a new optimal control approach to investigate a Mach number control problem for the NASA Ames 11-Foot Transonic Wind Tunnel. The flow in the wind tunnel is modeled by the 1-D unsteady Euler equations whose boundary conditions prescribe a controlling action by a compressor. The boundary control inputs to the compressor are in turn controlled by a drive motor system and an inlet guide vane system whose dynamics are modeled by ordinary differential equations. The resulting Euler equations are thus coupled to the ordinary differential equations via the boundary conditions. Optimality conditions are established by an adjoint method and are used to develop a model predictive linear-quadratic optimal control for regulating the Mach number due to a test model disturbance during a continuous pitch

Boundary and Interface Conditions for High Order Finite Difference Methods Applied to the Euler and Navier-Strokes Equations

NASA Technical Reports Server (NTRS)

Nordstrom, Jan; Carpenter, Mark H.

1998-01-01

Boundary and interface conditions for high order finite difference methods applied to the constant coefficient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.

Transport synthetic acceleration with opposing reflecting boundary conditions

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

Zika, M.R.; Adams, M.L.

2000-02-01

The transport synthetic acceleration (TSA) scheme is extended to problems with opposing reflecting boundary conditions. This synthetic method employs a simplified transport operator as its low-order approximation. A procedure is developed that allows the use of the conjugate gradient (CG) method to solve the resulting low-order system of equations. Several well-known transport iteration algorithms are cast in a linear algebraic form to show their equivalence to standard iterative techniques. Source iteration in the presence of opposing reflecting boundary conditions is shown to be equivalent to a (poorly) preconditioned stationary Richardson iteration, with the preconditioner defined by the method of iteratingmore » on the incident fluxes on the reflecting boundaries. The TSA method (and any synthetic method) amounts to a further preconditioning of the Richardson iteration. The presence of opposing reflecting boundary conditions requires special consideration when developing a procedure to realize the CG method for the proposed system of equations. The CG iteration may be applied only to symmetric positive definite matrices; this condition requires the algebraic elimination of the boundary angular corrections from the low-order equations. As a consequence of this elimination, evaluating the action of the resulting matrix on an arbitrary vector involves two transport sweeps and a transmission iteration. Results of applying the acceleration scheme to a simple test problem are presented.« less

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

NASA Technical Reports Server (NTRS)

Meade, Andrew James, Jr.

1989-01-01

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

An implicit time-marching method for the three-dimensional Navier-Stokes equations of contravariant velocity components

NASA Astrophysics Data System (ADS)

Daiguji, Hisaaki; Yamamoto, Satoru

1988-12-01

The implicit time-marching finite-difference method for solving the three-dimensional compressible Euler equations developed by the authors is extended to the Navier-Stokes equations. The distinctive features of this method are to make use of momentum equations of contravariant velocities instead of physical boundaries, and to be able to treat the periodic boundary condition for the three-dimensional impeller flow easily. These equations can be solved by using the same techniques as the Euler equations, such as the delta-form approximate factorization, diagonalization and upstreaming. In addition to them, a simplified total variation diminishing scheme by the authors is applied to the present method in order to capture strong shock waves clearly. Finally, the computed results of the three-dimensional flow through a transonic compressor rotor with tip clearance are shown.

Group solution for unsteady free-convection flow from a vertical moving plate subjected to constant heat flux

NASA Astrophysics Data System (ADS)

Kassem, M.

2006-03-01

The problem of heat and mass transfer in an unsteady free-convection flow over a continuous moving vertical sheet in an ambient fluid is investigated for constant heat flux using the group theoretical method. The nonlinear coupled partial differential equation governing the flow and the boundary conditions are transformed to a system of ordinary differential equations with appropriate boundary conditions. The obtained ordinary differential equations are solved numerically using the shooting method. The effect of Prandlt number on the velocity and temperature of the boundary-layer is plotted in curves. A comparison with previous work is presented.

Boundary-integral methods in elasticity and plasticity. [solutions of boundary value problems

NASA Technical Reports Server (NTRS)

Mendelson, A.

1973-01-01

Recently developed methods that use boundary-integral equations applied to elastic and elastoplastic boundary value problems are reviewed. Direct, indirect, and semidirect methods using potential functions, stress functions, and displacement functions are described. Examples of the use of these methods for torsion problems, plane problems, and three-dimensional problems are given. It is concluded that the boundary-integral methods represent a powerful tool for the solution of elastic and elastoplastic problems.

A fast numerical solution of scattering by a cylinder: Spectral method for the boundary integral equations

NASA Technical Reports Server (NTRS)

Hu, Fang Q.

1994-01-01

It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converges slowly for high frequency waves. In this paper, we present a fast number solution for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of the integral kernels are removed and dealt with accurately. The proposed method preserves the spectral accuracy and is shown to have an exponential rate of convergence. Aspects of efficient implementation using FFT are discussed. Moreover, the boundary integral equations of combined single and double-layer representation are used in the present paper. This ensures the uniqueness of the numerical solution for the scattering problem at all frequencies. Although a strongly singular kernel is encountered for the Neumann boundary conditions, we show that the hypersingularity can be handled easily in the spectral method. Numerical examples that demonstrate the validity of the method are also presented.

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

NASA Astrophysics Data System (ADS)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

The Application of a Boundary Integral Equation Method to the Prediction of Ducted Fan Engine Noise

NASA Technical Reports Server (NTRS)

Dunn, M. H.; Tweed, J.; Farassat, F.

1999-01-01

The prediction of ducted fan engine noise using a boundary integral equation method (BIEM) is considered. Governing equations for the BIEM are based on linearized acoustics and describe the scattering of incident sound by a thin, finite-length cylindrical duct in the presence of a uniform axial inflow. A classical boundary value problem (BVP) is derived that includes an axisymmetric, locally reacting liner on the duct interior. Using potential theory, the BVP is recast as a system of hypersingular boundary integral equations with subsidiary conditions. We describe the integral equation derivation and solution procedure in detail. The development of the computationally efficient ducted fan noise prediction program TBIEM3D, which implements the BIEM, and its utility in conducting parametric noise reduction studies are discussed. Unlike prediction methods based on spinning mode eigenfunction expansions, the BIEM does not require the decomposition of the interior acoustic field into its radial and axial components which, for the liner case, avoids the solution of a difficult complex eigenvalue problem. Numerical spectral studies are presented to illustrate the nexus between the eigenfunction expansion representation and BIEM results. We demonstrate BIEM liner capability by examining radiation patterns for several cases of practical interest.

Semi-discrete Galerkin solution of the compressible boundary-layer equations with viscous-inviscid interaction

NASA Technical Reports Server (NTRS)

Day, Brad A.; Meade, Andrew J., Jr.

1993-01-01

A semi-discrete Galerkin (SDG) method is under development to model attached, turbulent, and compressible boundary layers for transonic airfoil analysis problems. For the boundary-layer formulation the method models the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby providing high resolution near the wall and permitting the use of a uniform finite element grid which automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past RAE 2822 and NACA 0012 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack.

Boundary value problems for multi-term fractional differential equations

NASA Astrophysics Data System (ADS)

Daftardar-Gejji, Varsha; Bhalekar, Sachin

2008-09-01

Multi-term fractional diffusion-wave equation along with the homogeneous/non-homogeneous boundary conditions has been solved using the method of separation of variables. It is observed that, unlike in the one term case, solution of multi-term fractional diffusion-wave equation is not necessarily non-negative, and hence does not represent anomalous diffusion of any kind.

Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications

NASA Technical Reports Server (NTRS)

Ibrahim, A. H.; Hou, G. J.-W.; Tiwari, S. N. (Principal Investigator)

1996-01-01

Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics

NASA Technical Reports Server (NTRS)

Bayliss, A.; Goldstein, C. I.; Turkel, E.

1984-01-01

The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner.

Diffuse interface immersed boundary method for multi-fluid flows with arbitrarily moving rigid bodies

NASA Astrophysics Data System (ADS)

Patel, Jitendra Kumar; Natarajan, Ganesh

2018-05-01

We present an interpolation-free diffuse interface immersed boundary method for multiphase flows with moving bodies. A single fluid formalism using the volume-of-fluid approach is adopted to handle multiple immiscible fluids which are distinguished using the volume fractions, while the rigid bodies are tracked using an analogous volume-of-solid approach that solves for the solid fractions. The solution to the fluid flow equations are carried out using a finite volume-immersed boundary method, with the latter based on a diffuse interface philosophy. In the present work, we assume that the solids are filled with a "virtual" fluid with density and viscosity equal to the largest among all fluids in the domain. The solids are assumed to be rigid and their motion is solved using Newton's second law of motion. The immersed boundary methodology constructs a modified momentum equation that reduces to the Navier-Stokes equations in the fully fluid region and recovers the no-slip boundary condition inside the solids. An implicit incremental fractional-step methodology in conjunction with a novel hybrid staggered/non-staggered approach is employed, wherein a single equation for normal momentum at the cell faces is solved everywhere in the domain, independent of the number of spatial dimensions. The scalars are all solved for at the cell centres, with the transport equations for solid and fluid volume fractions solved using a high-resolution scheme. The pressure is determined everywhere in the domain (including inside the solids) using a variable coefficient Poisson equation. The solution to momentum, pressure, solid and fluid volume fraction equations everywhere in the domain circumvents the issue of pressure and velocity interpolation, which is a source of spurious oscillations in sharp interface immersed boundary methods. A well-balanced algorithm with consistent mass/momentum transport ensures robust simulations of high density ratio flows with strong body forces. The proposed diffuse interface immersed boundary method is shown to be discretely mass-preserving while being temporally second-order accurate and exhibits nominal second-order accuracy in space. We examine the efficacy of the proposed approach through extensive numerical experiments involving one or more fluids and solids, that include two-particle sedimentation in homogeneous and stratified environment. The results from the numerical simulations show that the proposed methodology results in reduced spurious force oscillations in case of moving bodies while accurately resolving complex flow phenomena in multiphase flows with moving solids. These studies demonstrate that the proposed diffuse interface immersed boundary method, which could be related to a class of penalisation approaches, is a robust and promising alternative to computationally expensive conformal moving mesh algorithms as well as the class of sharp interface immersed boundary methods for multibody problems in multi-phase flows.

Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains

NASA Astrophysics Data System (ADS)

Ji, Songsong; Yang, Yibo; Pang, Gang; Antoine, Xavier

2018-01-01

The aim of this paper is to design some accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations in rectangular domains. The Laplace transform in time and discrete Fourier transform in space are applied to get Green's functions of the semi-discretized equations in unbounded domains with single-source. An algorithm is given to compute these Green's functions accurately through some recurrence relations. Furthermore, the finite-difference method is used to discretize the reduced problem with accurate boundary conditions. Numerical simulations are presented to illustrate the accuracy of our method in the case of the linear Schrödinger and heat equations. It is shown that the reflection at the corners is correctly eliminated.

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

NASA Astrophysics Data System (ADS)

Chen, Leilei; Zheng, Changjun; Chen, Haibo

2013-09-01

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

A numerical method for computing unsteady 2-D boundary layer flows

NASA Technical Reports Server (NTRS)

Krainer, Andreas

1988-01-01

A numerical method for computing unsteady two-dimensional boundary layers in incompressible laminar and turbulent flows is described and applied to a single airfoil changing its incidence angle in time. The solution procedure adopts a first order panel method with a simple wake model to solve for the inviscid part of the flow, and an implicit finite difference method for the viscous part of the flow. Both procedures integrate in time in a step-by-step fashion, in the course of which each step involves the solution of the elliptic Laplace equation and the solution of the parabolic boundary layer equations. The Reynolds shear stress term of the boundary layer equations is modeled by an algebraic eddy viscosity closure. The location of transition is predicted by an empirical data correlation originating from Michel. Since transition and turbulence modeling are key factors in the prediction of viscous flows, their accuracy will be of dominant influence to the overall results.

Numerical Treatment of Degenerate Diffusion Equations via Feller's Boundary Classification, and Applications

NASA Technical Reports Server (NTRS)

Cacio, Emanuela; Cohn, Stephen E.; Spigler, Renato

2011-01-01

A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration.

Accurate Projection Methods for the Incompressible Navier–Stokes Equations

DOE PAGES

Brown, David L.; Cortez, Ricardo; Minion, Michael L.

2001-04-10

This paper considers the accuracy of projection method approximations to the initial–boundary-value problem for the incompressible Navier–Stokes equations. The issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second-order methodology a decade and a half ago. It has been observed that while the velocity can be reliably computed to second-order accuracy in time and space, the pressure is typically only first-order accurate in the L ∞-norm. Here, we identify the source of this problem in the interplay of the global pressure-update formula with the numerical boundary conditions and presentsmore » an improved projection algorithm which is fully second-order accurate, as demonstrated by a normal mode analysis and numerical experiments. In addition, a numerical method based on a gauge variable formulation of the incompressible Navier–Stokes equations, which provides another option for obtaining fully second-order convergence in both velocity and pressure, is discussed. The connection between the boundary conditions for projection methods and the gauge method is explained in detail.« less

Numerical solution of the two-dimensional time-dependent incompressible Euler equations

NASA Technical Reports Server (NTRS)

Whitfield, David L.; Taylor, Lafayette K.

1994-01-01

A numerical method is presented for solving the artificial compressibility form of the 2D time-dependent incompressible Euler equations. The approach is based on using an approximate Riemann solver for the cell face numerical flux of a finite volume discretization. Characteristic variable boundary conditions are developed and presented for all boundaries and in-flow out-flow situations. The system of algebraic equations is solved using the discretized Newton-relaxation (DNR) implicit method. Numerical results are presented for both steady and unsteady flow.

Computation of transonic viscous-inviscid interacting flow

NASA Technical Reports Server (NTRS)

Whitfield, D. L.; Thomas, J. L.; Jameson, A.; Schmidt, W.

1983-01-01

Transonic viscous-inviscid interaction is considered using the Euler and inverse compressible turbulent boundary-layer equations. Certain improvements in the inverse boundary-layer method are mentioned, along with experiences in using various Runge-Kutta schemes to solve the Euler equations. Numerical conditions imposed on the Euler equations at a surface for viscous-inviscid interaction using the method of equivalent sources are developed, and numerical solutions are presented and compared with experimental data to illustrate essential points. Previously announced in STAR N83-17829

Boundary conditions for the solution of the three-dimensional Poisson equation in open metallic enclosures

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

Biswas, Debabrata; Singh, Gaurav; Kumar, Raghwendra

2015-09-15

Numerical solution of the Poisson equation in metallic enclosures, open at one or more ends, is important in many practical situations, such as high power microwave or photo-cathode devices. It requires imposition of a suitable boundary condition at the open end. In this paper, methods for solving the Poisson equation are investigated for various charge densities and aspect ratios of the open ends. It is found that a mixture of second order and third order local asymptotic boundary conditions is best suited for large aspect ratios, while a proposed non-local matching method, based on the solution of the Laplace equation,more » scores well when the aspect ratio is near unity for all charge density variations, including ones where the centre of charge is close to an open end or the charge density is non-localized. The two methods complement each other and can be used in electrostatic calculations where the computational domain needs to be terminated at the open boundaries of the metallic enclosure.« less

A multilevel correction adaptive finite element method for Kohn-Sham equation

NASA Astrophysics Data System (ADS)

Hu, Guanghui; Xie, Hehu; Xu, Fei

2018-02-01

In this paper, an adaptive finite element method is proposed for solving Kohn-Sham equation with the multilevel correction technique. In the method, the Kohn-Sham equation is solved on a fixed and appropriately coarse mesh with the finite element method in which the finite element space is kept improving by solving the derived boundary value problems on a series of adaptively and successively refined meshes. A main feature of the method is that solving large scale Kohn-Sham system is avoided effectively, and solving the derived boundary value problems can be handled efficiently by classical methods such as the multigrid method. Hence, the significant acceleration can be obtained on solving Kohn-Sham equation with the proposed multilevel correction technique. The performance of the method is examined by a variety of numerical experiments.

A spectrally accurate boundary-layer code for infinite swept wings

NASA Technical Reports Server (NTRS)

Pruett, C. David

1994-01-01

This report documents the development, validation, and application of a spectrally accurate boundary-layer code, WINGBL2, which has been designed specifically for use in stability analyses of swept-wing configurations. Currently, we consider only the quasi-three-dimensional case of an infinitely long wing of constant cross section. The effects of streamwise curvature, streamwise pressure gradient, and wall suction and/or blowing are taken into account in the governing equations and boundary conditions. The boundary-layer equations are formulated both for the attachment-line flow and for the evolving boundary layer. The boundary-layer equations are solved by marching in the direction perpendicular to the leading edge, for which high-order (up to fifth) backward differencing techniques are used. In the wall-normal direction, a spectral collocation method, based upon Chebyshev polynomial approximations, is exploited. The accuracy, efficiency, and user-friendliness of WINGBL2 make it well suited for applications to linear stability theory, parabolized stability equation methodology, direct numerical simulation, and large-eddy simulation. The method is validated against existing schemes for three test cases, including incompressible swept Hiemenz flow and Mach 2.4 flow over an airfoil swept at 70 deg to the free stream.

Nonequilibrium viscous flow over Jovian entry probes at high altitudes

NASA Technical Reports Server (NTRS)

Kumar, A.; Szema, K. Y.; Tiwari, S. N.

1979-01-01

The viscous chemical nonequilibrium flow around a Jovian entry body is investigated at high altitudes using two different methods. First method is only for the stagnation region and integrates the full Navier-Stokes equations from the body surface to the freestream. The second method uses viscous shock layer equations between the body surface and the shock. Due to low Reynolds numbers, both methods use surface slip boundary conditions and the second method also uses shock slip boundary conditions. The results of the two methods are compared at the stagnation point. It is found that the entire shock layer is under chemical nonequilibrium at higher altitudes and that the slip boundary conditions are important at these altitudes.

A computer program for calculating laminar and turbulent boundary layers for two-dimensional time-dependent flows

NASA Technical Reports Server (NTRS)

Cebeci, T.; Carr, L. W.

1978-01-01

A computer program is described which provides solutions of two dimensional equations appropriate to laminar and turbulent boundary layers for boundary conditions with an external flow which fluctuates in magnitude. The program is based on the numerical solution of the governing boundary layer equations by an efficient two point finite difference method. An eddy viscosity formulation was used to model the Reynolds shear stress term. The main features of the method are briefly described and instructions for the computer program with a listing are provided. Sample calculations to demonstrate its usage and capabilities for laminar and turbulent unsteady boundary layers with an external flow which fluctuated in magnitude are presented.

Similarity solution of the Boussinesq equation

NASA Astrophysics Data System (ADS)

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

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

A numerical investigation of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet via rational Chebyshev functions

NASA Astrophysics Data System (ADS)

Parand, Kourosh; Mahdi Moayeri, Mohammad; Latifi, Sobhan; Delkhosh, Mehdi

2017-07-01

In this paper, a spectral method based on the four kinds of rational Chebyshev functions is proposed to approximate the solution of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet. First, by using the quasilinearization method (QLM), the model which is a nonlinear ordinary differential equation is converted to a sequence of linear ordinary differential equations (ODEs). By applying the proposed method on the ODEs in each iteration, the equations are converted to a system of linear algebraic equations. The results indicate the high accuracy and convergence of our method. Moreover, the effects of the Eyring-Powell fluid material parameters are discussed.

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution.

PubMed

Lu, Benzhuo; Zhou, Y C; Huber, Gary A; Bond, Stephen D; Holst, Michael J; McCammon, J Andrew

2007-10-07

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.

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

Schlanderer, Stefan C., E-mail: stefan.schlanderer@unimelb.edu.au; Weymouth, Gabriel D., E-mail: G.D.Weymouth@soton.ac.uk; Sandberg, Richard D., E-mail: richard.sandberg@unimelb.edu.au

This paper introduces a virtual boundary method for compressible viscous fluid flow that is capable of accurately representing moving bodies in flow and aeroacoustic simulations. The method is the compressible extension of the boundary data immersion method (BDIM, Maertens & Weymouth (2015), ). The BDIM equations for the compressible Navier–Stokes equations are derived and the accuracy of the method for the hydrodynamic representation of solid bodies is demonstrated with challenging test cases, including a fully turbulent boundary layer flow and a supersonic instability wave. In addition we show that the compressible BDIM is able to accurately represent noise radiation frommore » moving bodies and flow induced noise generation without any penalty in allowable time step.« less

Boundary conditions for the solution of compressible Navier-Stokes equations by an implicit factored method

NASA Technical Reports Server (NTRS)

Shih, T. I.-P.; Smith, G. E.; Springer, G. S.; Rimon, Y.

1983-01-01

A method is presented for formulating the boundary conditions in implicit finite-difference form needed for obtaining solutions to the compressible Navier-Stokes equations by the Beam and Warming implicit factored method. The usefulness of the method was demonstrated (a) by establishing the boundary conditions applicable to the analysis of the flow inside an axisymmetric piston-cylinder configuration and (b) by calculating velocities and mass fractions inside the cylinder for different geometries and different operating conditions. Stability, selection of time step and grid sizes, and computer time requirements are discussed in reference to the piston-cylinder problem analyzed.

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.

Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary

NASA Astrophysics Data System (ADS)

Albanese, Guglielmo; Rigoli, Marco

2017-12-01

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary (M , ∂ M , 〈 , 〉) and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the so called Conformal Method.

A preconditioned formulation of the Cauchy-Riemann equations

NASA Technical Reports Server (NTRS)

Phillips, T. N.

1983-01-01

A preconditioning of the Cauchy-Riemann equations which results in a second-order system is described. This system is shown to have a unique solution if the boundary conditions are chosen carefully. This choice of boundary condition enables the solution of the first-order system to be retrieved. A numerical solution of the preconditioned equations is obtained by the multigrid method.

Nozzle Flow with Vibrational Nonequilibrium. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Landry, John Gary

1995-01-01

Flow of nitrogen gas through a converging-diverging nozzle is simulated. The flow is modeled using the Navier-Stokes equations that have been modified for vibrational nonequilibrium. The energy equation is replaced by two equations. One equation accounts for energy effects due to the translational and rotational degrees of freedom, and the other accounts for the affects due to the vibrational degree of freedom. The energy equations are coupled by a relaxation time which measures the time required for the vibrational energy component to equilibrate with the translational and rotational energy components. An improved relaxation time is used in this thesis. The equations are solved numerically using the Steger-Warming flux vector splitting method and the Implicit MacCormack method. The results show that uniform flow is produced outside of the boundary layer. Nonequilibrium exists in both the converging and diverging nozzle sections. The boundary layer region is characterized by a marked increase in translational-rotational temperature. The vibrational temperature remains frozen downstream of the nozzle, except in the boundary layer.

Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.

2005-09-01

We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.

The method of lines in analyzing solids containing cracks

NASA Technical Reports Server (NTRS)

Gyekenyesi, John P.

1990-01-01

A semi-numerical method is reviewed for solving a set of coupled partial differential equations subject to mixed and possibly coupled boundary conditions. The line method of analysis is applied to the Navier-Cauchy equations of elastic and elastoplastic equilibrium to calculate the displacement distributions in various, simple geometry bodies containing cracks. The application of this method to the appropriate field equations leads to coupled sets of simultaneous ordinary differential equations whose solutions are obtained along sets of lines in a discretized region. When decoupling of the equations and their boundary conditions is not possible, the use of a successive approximation procedure permits the analytical solution of the resulting ordinary differential equations. The use of this method is illustrated by reviewing and presenting selected solutions of mixed boundary value problems in three dimensional fracture mechanics. These solutions are of great importance in fracture toughness testing, where accurate stress and displacement distributions are required for the calculation of certain fracture parameters. Computations obtained for typical flawed specimens include that for elastic as well as elastoplastic response. Problems in both Cartesian and cylindrical coordinate systems are included. Results are summarized for a finite geometry rectangular bar with a central through-the-thickness or rectangular surface crack under remote uniaxial tension. In addition, stress and displacement distributions are reviewed for finite circular bars with embedded penny-shaped cracks, and rods with external annular or ring cracks under opening mode tension. The results obtained show that the method of lines presents a systematic approach to the solution of some three-dimensional mechanics problems with arbitrary boundary conditions. The advantage of this method over other numerical solutions is that good results are obtained even from the use of a relatively coarse grid.

Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations

NASA Technical Reports Server (NTRS)

Ibrahim, A. H.; Tiwari, S. N.; Smith, R. E.

1997-01-01

Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Solving Fluid Structure Interaction Problems with an Immersed Boundary Method

NASA Technical Reports Server (NTRS)

Barad, Michael F.; Brehm, Christoph; Kiris, Cetin C.

2016-01-01

An immersed boundary method for the compressible Navier-Stokes equations can be used for moving boundary problems as well as fully coupled fluid-structure interaction is presented. The underlying Cartesian immersed boundary method of the Launch Ascent and Vehicle Aerodynamics (LAVA) framework, based on the locally stabilized immersed boundary method previously presented by the authors, is extended to account for unsteady boundary motion and coupled to linear and geometrically nonlinear structural finite element solvers. The approach is validated for moving boundary problems with prescribed body motion and fully coupled fluid structure interaction problems. Keywords: Immersed Boundary Method, Higher-Order Finite Difference Method, Fluid Structure Interaction.

The dual boundary element formulation for elastoplastic fracture mechanics

NASA Astrophysics Data System (ADS)

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

1993-08-01

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

A New Formulation of Time Domain Boundary Integral Equation for Acoustic Wave Scattering in the Presence of a Uniform Mean Flow

NASA Technical Reports Server (NTRS)

Hu, Fang; Pizzo, Michelle E.; Nark, Douglas M.

2017-01-01

It has been well-known that under the assumption of a constant uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation, in both the time domain and the frequency domain. Compared with solving partial differential equations, numerical methods based on the boundary integral equation have the advantage of a reduced spatial dimension and, hence, requiring only a surface mesh. However, the constant uniform mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the uniform mean flow. In this paper, we argue that the proper boundary condition for the acoustic wave should not have its normal velocity be zero everywhere on the solid surfaces, as has been applied in the literature. A careful study of the acoustic energy conservation equation is presented that shows such a boundary condition in fact leads to erroneous source or sink points on solid surfaces not aligned with the mean flow. A new solid wall boundary condition is proposed that conserves the acoustic energy and a new time domain boundary integral equation is derived. In addition to conserving the acoustic energy, another significant advantage of the new equation is that it is considerably simpler than previous formulations. In particular, tangential derivatives of the solution on the solid surfaces are no longer needed in the new formulation, which greatly simplifies numerical implementation. Furthermore, stabilization of the new integral equation by Burton-Miller type reformulation is presented. The stability of the new formulation is studied theoretically as well as numerically by an eigenvalue analysis. Numerical solutions are also presented that demonstrate the stability of the new formulation.

Stability of mixing layers

NASA Technical Reports Server (NTRS)

Tam, Christopher; Krothapalli, A

1993-01-01

The research program for the first year of this project (see the original research proposal) consists of developing an explicit marching scheme for solving the parabolized stability equations (PSE). Performing mathematical analysis of the computational algorithm including numerical stability analysis and the determination of the proper boundary conditions needed at the boundary of the computation domain are implicit in the task. Before one can solve the parabolized stability equations for high-speed mixing layers, the mean flow must first be found. In the past, instability analysis of high-speed mixing layer has mostly been performed on mean flow profiles calculated by the boundary layer equations. In carrying out this project, it is believed that the boundary layer equations might not give an accurate enough nonparallel, nonlinear mean flow needed for parabolized stability analysis. A more accurate mean flow can, however, be found by solving the parabolized Navier-Stokes equations. The advantage of the parabolized Navier-Stokes equations is that its accuracy is consistent with the PSE method. Furthermore, the method of solution is similar. Hence, the major part of the effort of the work of this year has been devoted to the development of an explicit numerical marching scheme for the solution of the Parabolized Navier-Stokes equation as applied to the high-seed mixing layer problem.

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

PubMed Central

Macklin, Paul

2011-01-01

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth. PMID:21331304

Improved Boundary Conditions for Cell-centered Difference Schemes

NASA Technical Reports Server (NTRS)

VanderWijngaart, Rob F.; Klopfer, Goetz H.; Chancellor, Marisa K. (Technical Monitor)

1997-01-01

Cell-centered finite-volume (CCFV) schemes have certain attractive properties for the solution of the equations governing compressible fluid flow. Among others, they provide a natural vehicle for specifying flux conditions at the boundaries of the physical domain. Unfortunately, they lead to slow convergence for numerical programs utilizing them. In this report a method for investigating and improving the convergence of CCFV schemes is presented, which focuses on the effect of the numerical boundary conditions. The key to the method is the computation of the spectral radius of the iteration matrix of the entire demoralized system of equations, not just of the interior point scheme or the boundary conditions.

A general decay result of a viscoelastic equation with past history and boundary feedback

NASA Astrophysics Data System (ADS)

Messaoudi, Salim A.; Al-Gharabli, Mohammad M.

2015-08-01

In this paper, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and in the presence of infinite memory term. In the domain as well as on a part of the boundary, we use the multiplier method and some properties of the convex functions to prove an explicit and general decay result.

Modeling boundary measurements of scattered light using the corrected diffusion approximation

PubMed Central

Lehtikangas, Ossi; Tarvainen, Tanja; Kim, Arnold D.

2012-01-01

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion approximation models boundary measurements substantially better than the standard diffusion approximation in comparison to numerical solutions of the radiative transport equation. PMID:22435102

Shooting method for solution of boundary-layer flows with massive blowing

NASA Technical Reports Server (NTRS)

Liu, T.-M.; Nachtsheim, P. R.

1973-01-01

A modified, bidirectional shooting method is presented for solving boundary-layer equations under conditions of massive blowing. Unlike the conventional shooting method, which is unstable when the blowing rate increases, the proposed method avoids the unstable direction and is capable of solving complex boundary-layer problems involving mass and energy balance on the surface.

An interacting boundary layer model for cascades

NASA Technical Reports Server (NTRS)

Davis, R. T.; Rothmayer, A. P.

1983-01-01

A laminar, incompressible interacting boundary layer model is developed for two-dimensional cascades. In the limit of large cascade spacing these equations reduce to the interacting boundary layer equations for a single body immersed in an infinite stream. A fully implicit numerical method is used to solve the governing equations, and is found to be at least as efficient as the same technique applied to the single body problem. Solutions are then presented for a cascade of finite flat plates and a cascade of finite sine-waves, with cusped leading and trailing edges.

Initial-boundary value problems associated with the Ablowitz-Ladik system

NASA Astrophysics Data System (ADS)

Xia, Baoqiang; Fokas, A. S.

2018-02-01

We employ the Ablowitz-Ladik system as an illustrative example in order to demonstrate how to analyze initial-boundary value problems for integrable nonlinear differential-difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schrödinger and integrable discrete modified Korteweg-de Vries equations in terms of the solutions of appropriate matrix Riemann-Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations and the process of eliminating of the unknown boundary values.

A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions

NASA Astrophysics Data System (ADS)

Liska, Sebastian; Colonius, Tim

2017-02-01

A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original (Peskin's) immersed boundary method. Unlike Peskin's method, boundary forces are regarded as Lagrange multipliers that are used to satisfy the no-slip condition. The incompressible Navier-Stokes equations are discretized on an unbounded staggered Cartesian grid and are solved in a finite number of operations using lattice Green's function techniques. These techniques are used to automatically enforce the natural free-space boundary conditions and to implement a novel block-wise adaptive grid that significantly reduces the run-time cost of solutions by limiting operations to grid cells in the immediate vicinity and near-wake region of the immersed surface. These techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half-explicit Runge-Kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation, incompressibility constraint, and no-slip constraint. Linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation-free nested projection technique. The algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems, e.g. discrete Poisson problems, that are compatible with recent parallel fast multipole methods for difference equations. Numerical experiments on low-aspect-ratio flat plates and spheres at Reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation.

A fast Cauchy-Riemann solver. [differential equation solution for boundary conditions by finite difference approximation

NASA Technical Reports Server (NTRS)

Ghil, M.; Balgovind, R.

1979-01-01

The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.

Probabilistic boundary element method

NASA Technical Reports Server (NTRS)

Cruse, T. A.; Raveendra, S. T.

1989-01-01

The purpose of the Probabilistic Structural Analysis Method (PSAM) project is to develop structural analysis capabilities for the design analysis of advanced space propulsion system hardware. The boundary element method (BEM) is used as the basis of the Probabilistic Advanced Analysis Methods (PADAM) which is discussed. The probabilistic BEM code (PBEM) is used to obtain the structural response and sensitivity results to a set of random variables. As such, PBEM performs analogous to other structural analysis codes such as finite elements in the PSAM system. For linear problems, unlike the finite element method (FEM), the BEM governing equations are written at the boundary of the body only, thus, the method eliminates the need to model the volume of the body. However, for general body force problems, a direct condensation of the governing equations to the boundary of the body is not possible and therefore volume modeling is generally required.

A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition

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

Gerritsma, Marc; Bochev, Pavel

Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximatedmore » by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.« less

Three-dimensional boundary layer calculation by a characteristic method

NASA Technical Reports Server (NTRS)

Houdeville, R.

1992-01-01

A numerical method for solving the three-dimensional boundary layer equations for bodies of arbitrary shape is presented. In laminar flows, the application domain extends from incompressible to hypersonic flows with the assumption of chemical equilibrium. For turbulent boundary layers, the application domain is limited by the validity of the mixing length model used. In order to respect the hyperbolic nature of the equations reduced to first order partial derivative terms, the momentum equations are discretized along the local streamlines using of the osculator tangent plane at each node of the body fitted coordinate system. With this original approach, it is possible to overcome the use of the generalized coordinates, and therefore, it is not necessary to impose an extra hypothesis about the regularity of the mesh in which the boundary conditions are given. By doing so, it is possible to limit, and sometimes to suppress, the pre-treatment of the data coming from an inviscid calculation. Although the proposed scheme is only semi-implicit, the method remains numerically very efficient.

Spatiotemporal motion boundary detection and motion boundary velocity estimation for tracking moving objects with a moving camera: a level sets PDEs approach with concurrent camera motion compensation.

PubMed

Feghali, Rosario; Mitiche, Amar

2004-11-01

The purpose of this study is to investigate a method of tracking moving objects with a moving camera. This method estimates simultaneously the motion induced by camera movement. The problem is formulated as a Bayesian motion-based partitioning problem in the spatiotemporal domain of the image quence. An energy functional is derived from the Bayesian formulation. The Euler-Lagrange descent equations determine imultaneously an estimate of the image motion field induced by camera motion and an estimate of the spatiotemporal motion undary surface. The Euler-Lagrange equation corresponding to the surface is expressed as a level-set partial differential equation for topology independence and numerically stable implementation. The method can be initialized simply and can track multiple objects with nonsimultaneous motions. Velocities on motion boundaries can be estimated from geometrical properties of the motion boundary. Several examples of experimental verification are given using synthetic and real-image sequences.

A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition

DOE PAGES

Gerritsma, Marc; Bochev, Pavel

2016-03-22

Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximatedmore » by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.« less

A simplified analytic form for generation of axisymmetric plasma boundaries

DOE PAGES

Luce, Timothy C.

2017-02-23

An improved method has been formulated for generating analytic boundary shapes as input for axisymmetric MHD equilibria. This method uses the family of superellipses as the basis function, as previously introduced. The improvements are a simplified notation, reduction of the number of simultaneous nonlinear equations to be solved, and the realization that not all combinations of input parameters admit a solution to the nonlinear constraint equations. The method tests for the existence of a self-consistent solution and, when no solution exists, it uses a deterministic method to find a nearby solution. As a result, examples of generation of boundaries, includingmore » tests with an equilibrium solver, are given.« less

A simplified analytic form for generation of axisymmetric plasma boundaries

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

Luce, Timothy C.

An improved method has been formulated for generating analytic boundary shapes as input for axisymmetric MHD equilibria. This method uses the family of superellipses as the basis function, as previously introduced. The improvements are a simplified notation, reduction of the number of simultaneous nonlinear equations to be solved, and the realization that not all combinations of input parameters admit a solution to the nonlinear constraint equations. The method tests for the existence of a self-consistent solution and, when no solution exists, it uses a deterministic method to find a nearby solution. As a result, examples of generation of boundaries, includingmore » tests with an equilibrium solver, are given.« less

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.

PAN AIR: A Computer Program for Predicting Subsonic or Supersonic Linear Potential Flows About Arbitrary Configurations Using a Higher Order Panel Method. Volume 1; Theory Document (Version 1.1)

NASA Technical Reports Server (NTRS)

Magnus, Alfred E.; Epton, Michael A.

1981-01-01

An outline of the derivation of the differential equation governing linear subsonic and supersonic potential flow is given. The use of Green's Theorem to obtain an integral equation over the boundary surface is discussed. The engineering techniques incorporated in the PAN AIR (Panel Aerodynamics) program (a discretization method which solves the integral equation for arbitrary first order boundary conditions) are then discussed in detail. Items discussed include the construction of the compressibility transformations, splining techniques, imposition of the boundary conditions, influence coefficient computation (including the concept of the finite part of an integral), computation of pressure coefficients, and computation of forces and moments.

A study of the viscous and nonadiabatic flow in radial turbines

NASA Technical Reports Server (NTRS)

Khalil, I.; Tabakoff, W.

1981-01-01

A method for analyzing the viscous nonadiabatic flow within turbomachine rotors is presented. The field analysis is based upon the numerical integration of the incompressible Navier-Stokes equations together with the energy equation over the rotors blade-to-blade stream channels. The numerical code used to solve the governing equations employs a nonorthogonal boundary fitted coordinate system that suits the most complicated blade geometries. Effects of turbulence are modeled with two equations; one expressing the development of the turbulence kinetic energy and the other its dissipation rate. The method of analysis is applied to a radial inflow turbine. The solution obtained indicates the severity of the complex interaction mechanism that occurs between different flow regimes (i.e., boundary layers, recirculating eddies, separation zones, etc.). Comparison with nonviscous flow solutions tend to justify strongly the inadequacy of using the latter with standard boundary layer techniques to obtain viscous flow details within turbomachine rotors. Capabilities and limitations of the present method of analysis are discussed.

Discrete exterior calculus approach for discretizing Maxwell's equations on face-centered cubic grids for FDTD

NASA Astrophysics Data System (ADS)

Salmasi, Mahbod; Potter, Michael

2018-07-01

Maxwell's equations are discretized on a Face-Centered Cubic (FCC) lattice instead of a simple cubic as an alternative to the standard Yee method for improvements in numerical dispersion characteristics and grid isotropy of the method. Explicit update equations and numerical dispersion expressions, and the stability criteria are derived. Also, several tools available to the standard Yee method such as PEC/PMC boundary conditions, absorbing boundary conditions, and scattered field formulation are extended to this method as well. A comparison between the FCC and the Yee formulations is made, showing that the FCC method exhibits better dispersion compared to its Yee counterpart. Simulations are provided to demonstrate both the accuracy and grid isotropy improvement of the method.

The optimal modified variational iteration method for the Lane-Emden equations with Neumann and Robin boundary conditions

NASA Astrophysics Data System (ADS)

Singh, Randhir; Das, Nilima; Kumar, Jitendra

2017-06-01

An effective analytical technique is proposed for the solution of the Lane-Emden equations. The proposed technique is based on the variational iteration method (VIM) and the convergence control parameter h . In order to avoid solving a sequence of nonlinear algebraic or complicated integrals for the derivation of unknown constant, the boundary conditions are used before designing the recursive scheme for solution. The series solutions are found which converges rapidly to the exact solution. Convergence analysis and error bounds are discussed. Accuracy, applicability of the method is examined by solving three singular problems: i) nonlinear Poisson-Boltzmann equation, ii) distribution of heat sources in the human head, iii) second-kind Lane-Emden equation.

A superlinear convergence estimate for an iterative method for the biharmonic equation

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

Horn, M.A.

In [CDH] a method for the solution of boundary value problems for the biharmonic equation using conformal mapping was investigated. The method is an implementation of the classical method of Muskhelishvili. In [CDH] it was shown, using the Hankel structure, that the linear system in [Musk] is the discretization of the identify plus a compact operator, and therefore the conjugate gradient method will converge superlinearly. The purpose of this paper is to give an estimate of the superlinear convergence in the case when the boundary curve is in a Hoelder class.

Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method

NASA Astrophysics Data System (ADS)

Sohaib, Muhammad; Haq, Sirajul; Mukhtar, Safyan; Khan, Imad

2018-03-01

An efficient method is proposed to approximate sixth order boundary value problems. The proposed method is based on Legendre wavelet in which Legendre polynomial is used. The mechanism of the method is to use collocation points that converts the differential equation into a system of algebraic equations. For validation two test problems are discussed. The results obtained from proposed method are quite accurate, also close to exact solution, and other different methods. The proposed method is computationally more effective and leads to more accurate results as compared to other methods from literature.

An Immersed Boundary Method for Solving the Compressible Navier-Stokes Equations with Fluid Structure Interaction

NASA Technical Reports Server (NTRS)

Brehm, Christoph; Barad, Michael F.; Kiris, Cetin C.

2016-01-01

An immersed boundary method for the compressible Navier-Stokes equation and the additional infrastructure that is needed to solve moving boundary problems and fully coupled fluid-structure interaction is described. All the methods described in this paper were implemented in NASA's LAVA solver framework. The underlying immersed boundary method is based on the locally stabilized immersed boundary method that was previously introduced by the authors. In the present paper this method is extended to account for all aspects that are involved for fluid structure interaction simulations, such as fast geometry queries and stencil computations, the treatment of freshly cleared cells, and the coupling of the computational fluid dynamics solver with a linear structural finite element method. The current approach is validated for moving boundary problems with prescribed body motion and fully coupled fluid structure interaction problems in 2D and 3D. As part of the validation procedure, results from the second AIAA aeroelastic prediction workshop are also presented. The current paper is regarded as a proof of concept study, while more advanced methods for fluid structure interaction are currently being investigated, such as geometric and material nonlinearities, and advanced coupling approaches.

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.

Active flow control insight gained from a modified integral boundary layer equation

NASA Astrophysics Data System (ADS)

Seifert, Avraham

2016-11-01

Active Flow Control (AFC) can alter the development of boundary layers with applications (e.g., reducing drag by separation delay or separating the boundary layers and enhancing vortex shedding to increase drag). Historically, significant effects of steady AFC methods were observed. Unsteady actuation is significantly more efficient than steady. Full-scale AFC tests were conducted with varying levels of success. While clearly relevant to industry, AFC implementation relies on expert knowledge with proven intuition and or costly and lengthy computational efforts. This situation hinders the use of AFC while simple, quick and reliable design method is absent. An updated form of the unsteady integral boundary layer (UIBL) equations, that include AFC terms (unsteady wall transpiration and body forces) can be used to assist in AFC analysis and design. With these equations and given a family of suitable velocity profiles, the momentum thickness can be calculated and matched with an outer, potential flow solution in 2D and 3D manner to create an AFC design tool, parallel to proven tools for airfoil design. Limiting cases of the UIBL equation can be used to analyze candidate AFC concepts in terms of their capability to modify the boundary layers development and system performance.

On shifted Jacobi spectral method for high-order multi-point boundary value problems

NASA Astrophysics Data System (ADS)

Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.

2012-10-01

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss-Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.

The analytical solution for drug delivery system with nonhomogeneous moving boundary condition

NASA Astrophysics Data System (ADS)

Saudi, Muhamad Hakimi; Mahali, Shalela Mohd; Harun, Fatimah Noor

2017-08-01

This paper discusses the development and the analytical solution of a mathematical model based on drug release system from a swelling delivery device. The mathematical model is represented by a one-dimensional advection-diffusion equation with nonhomogeneous moving boundary condition. The solution procedures consist of three major steps. Firstly, the application of steady state solution method, which is used to transform the nonhomogeneous moving boundary condition to homogeneous boundary condition. Secondly, the application of the Landau transformation technique that gives a significant impact in removing the advection term in the system of equation and transforming the moving boundary condition to a fixed boundary condition. Thirdly, the used of separation of variables method to find the analytical solution for the resulted initial boundary value problem. The results show that the swelling rate of delivery device and drug release rate is influenced by value of growth factor r.

Calculation of Water Entry Problem for Free-falling Bodies Using a Developed Cartesian Cut Cell Mesh

NASA Astrophysics Data System (ADS)

Wenhua, Wang; Yanying, Wang

2010-05-01

This paper describes the development of free surface capturing method on Cartesian cut cell mesh to water entry problem for free-falling bodies with body-fluid interaction. The incompressible Euler equations for a variable density fluid system are presented as governing equations and the free surface is treated as a contact discontinuity by using free surface capturing method. In order to be convenient for dealing with the problem with moving body boundary, the Cartesian cut cell technique is adopted for generating the boundary-fitted mesh around body edge by cutting solid regions out of a background Cartesian mesh. Based on this mesh system, governing equations are discretized by finite volume method, and at each cell edge inviscid flux is evaluated by means of Roe's approximate Riemann solver. Furthermore, for unsteady calculation in time domain, a time accurate solution is achieved by a dual time-stepping technique with artificial compressibility method. For the body-fluid interaction, the projection method of momentum equations and exact Riemann solution are applied in the calculation of fluid pressure on the solid boundary. Finally, the method is validated by test case of water entry for free-falling bodies.

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

NASA Astrophysics Data System (ADS)

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

2014-08-01

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

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.

Influence of grain boundaries on the distribution of components in binary alloys

NASA Astrophysics Data System (ADS)

L'vov, P. E.; Svetukhin, V. V.

2017-12-01

Based on the free-energy density functional method (the Cahn-Hilliard equation), a phenomenological model that describes the influence of grain boundaries on the distribution of components in binary alloys has been developed. The model is built on the assumption of the difference between the interaction parameters of solid solution components in the bulk and at the grain boundary. The difference scheme based on the spectral method is proposed to solve the Cahn-Hilliard equation with interaction parameters depending on coordinates. Depending on the ratio between the interaction parameters in the bulk and at the grain boundary, temperature, and alloy composition, the model can give rise to different types of distribution of a dissolved component, namely, either depletion or enrichment of the grain-boundary area, preferential grainboundary precipitation, competitive precipitation in the bulk and at the grain boundary, etc.

A numerical approach to finding general stationary vacuum black holes

NASA Astrophysics Data System (ADS)

Adam, Alexander; Kitchen, Sam; Wiseman, Toby

2012-08-01

The Harmonic Einstein equation is the vacuum Einstein equation supplemented by a gauge fixing term which we take to be that of DeTurck. For static black holes analytically continued to Riemannian manifolds without boundary at the horizon, this equation has previously been shown to be elliptic, and Ricci flow and Newton’s method provide good numerical algorithms to solve it. Here we extend these techniques to the arbitrary cohomogeneity stationary case which must be treated in Lorentzian signature. For stationary spacetimes with globally timelike Killing vector the Harmonic Einstein equation is elliptic. In the presence of horizons and ergo-regions it is less obviously so. Motivated by the Rigidity theorem we study a class of stationary black hole spacetimes which is general enough to include many interesting higher dimensional solutions. We argue the Harmonic Einstein equation consistently truncates to this class of spacetimes giving an elliptic problem. The Killing horizons and axes of rotational symmetry are boundaries for this problem and we determine boundary conditions there. As a simple example we numerically construct 4D rotating black holes in a cavity using Anderson’s boundary conditions. We demonstrate both Newton’s method and Ricci flow to find these Lorentzian solutions.

Similarity solutions for unsteady free-convection flow from a continuous moving vertical surface

NASA Astrophysics Data System (ADS)

Abd-El-Malek, Mina B.; Kassem, Magda M.; Mekky, Mohammad L.

2004-03-01

The transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid. The thermal boundary layer induced within a vertical semi-infinite layer of Boussinseq fluid by a constant heated bounding plate. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The obtained ordinary differential equations are solved analytically for the temperature and numerically for the velocity using the shooting method. Effect of Prandtl number on the thermal boundary-layer and velocity boundary-layer are studied and plotted in curves.

User's Manual for FEM-BEM Method. 1.0

NASA Technical Reports Server (NTRS)

Butler, Theresa; Deshpande, M. D. (Technical Monitor)

2002-01-01

A user's manual for using FORTRAN code to perform electromagnetic analysis of arbitrarily shaped material cylinders using a hybrid method that combines the finite element method (FEM) and the boundary element method (BEM). In this method, the material cylinder is enclosed by a fictitious boundary and the Maxwell's equations are solved by FEM inside the boundary and by BEM outside the boundary. The electromagnetic scattering on several arbitrarily shaped material cylinders using this FORTRAN code is computed to as examples.

AN IMMERSED BOUNDARY METHOD FOR COMPLEX INCOMPRESSIBLE FLOWS

EPA Science Inventory

An immersed boundary method for time-dependant, three- dimensional, incompressible flows is presented in this paper. The incompressible Navier-Stokes equations are discretized using a low-diffusion flux splitting method for the inviscid fluxes and a second order central differenc...

On the laminar-turbulent transition in the boundary layer of streamwise corner

NASA Astrophysics Data System (ADS)

Kirilovskiy, S. V.; Boiko, A. V.; Poplavskaya, T. V.

2017-10-01

The work is aimed at developing methods of numerical simulation of incompressible non-symmetric flow in streamwise corner by solving the Navier-Stokes equations with ANSYS Fluent and the self-similar equations of boundary-layer type. A comparison of the computations with each other and experimental data is provided.

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.

Space-time domain solutions of the wave equation by a non-singular boundary integral method and Fourier transform.

PubMed

Klaseboer, Evert; Sepehrirahnama, Shahrokh; Chan, Derek Y C

2017-08-01

The general space-time evolution of the scattering of an incident acoustic plane wave pulse by an arbitrary configuration of targets is treated by employing a recently developed non-singular boundary integral method to solve the Helmholtz equation in the frequency domain from which the space-time solution of the wave equation is obtained using the fast Fourier transform. The non-singular boundary integral solution can enforce the radiation boundary condition at infinity exactly and can account for multiple scattering effects at all spacings between scatterers without adverse effects on the numerical precision. More generally, the absence of singular kernels in the non-singular integral equation confers high numerical stability and precision for smaller numbers of degrees of freedom. The use of fast Fourier transform to obtain the time dependence is not constrained to discrete time steps and is particularly efficient for studying the response to different incident pulses by the same configuration of scatterers. The precision that can be attained using a smaller number of Fourier components is also quantified.

Development of a coupled level set and immersed boundary method for predicting dam break flows

NASA Astrophysics Data System (ADS)

Yu, C. H.; Sheu, Tony W. H.

2017-12-01

Dam-break flow over an immersed stationary object is investigated using a coupled level set (LS)/immersed boundary (IB) method developed in Cartesian grids. This approach adopts an improved interface preserving level set method which includes three solution steps and the differential-based interpolation immersed boundary method to treat fluid-fluid and solid-fluid interfaces, respectively. In the first step of this level set method, the level set function ϕ is advected by a pure advection equation. The intermediate step is performed to obtain a new level set value through a new smoothed Heaviside function. In the final solution step, a mass correction term is added to the re-initialization equation to ensure the new level set is a distance function and to conserve the mass bounded by the interface. For accurately calculating the level set value, the four-point upwinding combined compact difference (UCCD) scheme with three-point boundary combined compact difference scheme is applied to approximate the first-order derivative term shown in the level set equation. For the immersed boundary method, application of the artificial momentum forcing term at points in cells consisting of both fluid and solid allows an imposition of velocity condition to account for the presence of solid object. The incompressible Navier-Stokes solutions are calculated using the projection method. Numerical results show that the coupled LS/IB method can not only predict interface accurately but also preserve the mass conservation excellently for the dam-break flow.

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.

A boundary integral approach to the scattering of nonplanar acoustic waves by rigid bodies

NASA Technical Reports Server (NTRS)

Gallman, Judith M.; Myers, M. K.; Farassat, F.

1990-01-01

The acoustic scattering of an incident wave by a rigid body can be described by a singular Fredholm integral equation of the second kind. This equation is derived by solving the wave equation using generalized function theory, Green's function for the wave equation in unbounded space, and the acoustic boundary condition for a perfectly rigid body. This paper will discuss the derivation of the wave equation, its reformulation as a boundary integral equation, and the solution of the integral equation by the Galerkin method. The accuracy of the Galerkin method can be assessed by applying the technique outlined in the paper to reproduce the known pressure fields that are due to various point sources. From the analysis of these simpler cases, the accuracy of the Galerkin solution can be inferred for the scattered pressure field caused by the incidence of a dipole field on a rigid sphere. The solution by the Galerkin technique can then be applied to such problems as a dipole model of a propeller whose pressure field is incident on a rigid cylinder. This is the groundwork for modeling the scattering of rotating blade noise by airplane fuselages.

Application of the Finite Element Method to Rotary Wing Aeroelasticity

NASA Technical Reports Server (NTRS)

Straub, F. K.; Friedmann, P. P.

1982-01-01

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method.

Theory of viscous transonic flow over airfoils at high Reynolds number

NASA Technical Reports Server (NTRS)

Melnik, R. E.; Chow, R.; Mead, H. R.

1977-01-01

This paper considers viscous flows with unseparated turbulent boundary layers over two-dimensional airfoils at transonic speeds. Conventional theoretical methods are based on boundary layer formulations which do not account for the effect of the curved wake and static pressure variations across the boundary layer in the trailing edge region. In this investigation an extended viscous theory is developed that accounts for both effects. The theory is based on a rational analysis of the strong turbulent interaction at airfoil trailing edges. The method of matched asymptotic expansions is employed to develop formal series solutions of the full Reynolds equations in the limit of Reynolds numbers tending to infinity. Procedures are developed for combining the local trailing edge solution with numerical methods for solving the full potential flow and boundary layer equations. Theoretical results indicate that conventional boundary layer methods account for only about 50% of the viscous effect on lift, the remaining contribution arising from wake curvature and normal pressure gradient effects.

Comparison of artificial absorbing boundaries for acoustic wave equation modelling

NASA Astrophysics Data System (ADS)

Gao, Yingjie; Song, Hanjie; Zhang, Jinhai; Yao, Zhenxing

2017-12-01

Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.

Application of the implicit MacCormack scheme to the PNS equations

NASA Technical Reports Server (NTRS)

Lawrence, S. L.; Tannehill, J. C.; Chaussee, D. S.

1983-01-01

The two-dimensional parabolized Navier-Stokes equations are solved using MacCormack's (1981) implicit finite-difference scheme. It is shown that this method for solving the parabolized Navier-Stokes equations does not require the inversion of block tridiagonal systems of algebraic equations and allows the original explicit scheme to be employed in those regions where implicit treatment is not needed. The finite-difference algorithm is discussed and the computational results for two laminar test cases are presented. Results obtained using this method for the case of a flat plate boundary layer are compared with those obtained using the conventional Beam-Warming scheme, as well as those obtained from a boundary layer code. The computed results for a more severe test of the method, the hypersonic flow past a 15 deg compression corner, are found to compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.

The complex variable boundary element method: Applications in determining approximative boundaries

USGS Publications Warehouse

Hromadka, T.V.

1984-01-01

The complex variable boundary element method (CVBEM) is used to determine approximation functions for boundary value problems of the Laplace equation such as occurs in potential theory. By determining an approximative boundary upon which the CVBEM approximator matches the desired constant (level curves) boundary conditions, the CVBEM is found to provide the exact solution throughout the interior of the transformed problem domain. Thus, the acceptability of the CVBEM approximation is determined by the closeness-of-fit of the approximative boundary to the study problem boundary. ?? 1984.

Calculation of unsteady transonic flows with mild separation by viscous-inviscid interaction

NASA Technical Reports Server (NTRS)

Howlett, James T.

1992-01-01

This paper presents a method for calculating viscous effects in two- and three-dimensional unsteady transonic flow fields. An integral boundary-layer method for turbulent viscous flow is coupled with the transonic small-disturbance potential equation in a quasi-steady manner. The viscous effects are modeled with Green's lag-entrainment equations for attached flow and an inverse boundary-layer method for flows that involve mild separation. The boundary-layer method is used stripwise to approximate three-dimensional effects. Applications are given for two-dimensional airfoils, aileron buzz, and a wing planform. Comparisons with inviscid calculations, other viscous calculation methods, and experimental data are presented. The results demonstrate that the present technique can economically and accurately calculate unsteady transonic flow fields that have viscous-inviscid interactions with mild flow separation.

Versions of the collocation and least squares method for solving biharmonic equations in non-canonical domains

NASA Astrophysics Data System (ADS)

Belyaev, V. A.; Shapeev, V. P.

2017-10-01

New versions of the collocations and least squares method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for the biharmonic equation in non-canonical domains. The solution of the biharmonic equation is used for simulating the stress-strain state of an isotropic plate under the action of transverse load. The differential problem is projected into a space of fourth-degree polynomials by the CLS method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLS method are implemented on the grids which are constructed in two different ways. It is shown that the approximate solution of problems converges with high order. Thus it matches with high accuracy with the analytical solution of the test problems in the case of known solution in the numerical experiments on the convergence of the solution of various problems on a sequence of grids.

Method of adiabatic modes in studying problems of smoothly irregular open waveguide structures

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

Sevastianov, L. A., E-mail: sevast@sci.pfu.edu.ru; Egorov, A. A.; Sevastyanov, A. L.

2013-02-15

Basic steps in developing an original method of adiabatic modes that makes it possible to solve the direct and inverse problems of simulating and designing three-dimensional multilayered smoothly irregular open waveguide structures are described. A new element in the method is that an approximate solution of Maxwell's equations is made to obey 'inclined' boundary conditions at the interfaces between themedia being considered. These boundary conditions take into account the obliqueness of planes tangent to nonplanar boundaries between the media and lead to new equations for coupled vector quasiwaveguide hybrid adiabatic modes. Solutions of these equations describe the phenomenon of 'entanglement'more » of two linear polarizations of an irregular multilayered waveguide, the appearance of a new mode in an entangled state, and the effect of rotation of the polarization plane of quasiwaveguide modes. The efficiency of the method is demonstrated by considering the example of numerically simulating a thin-film generalized waveguide Lueneburg lens.« less

Numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains.

PubMed

Li, Hongwei; Guo, Yue

2017-12-01

The numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains is considered by applying the artificial boundary method in this paper. In order to design the local absorbing boundary conditions for the coupled nonlinear Schrödinger equations, we generalize the unified approach previously proposed [J. Zhang et al., Phys. Rev. E 78, 026709 (2008)PLEEE81539-375510.1103/PhysRevE.78.026709]. Based on the methodology underlying the unified approach, the original problem is split into two parts, linear and nonlinear terms, and we then achieve a one-way operator to approximate the linear term to make the wave out-going, and finally we combine the one-way operator with the nonlinear term to derive the local absorbing boundary conditions. Then we reduce the original problem into an initial boundary value problem on the bounded domain, which can be solved by the finite difference method. The stability of the reduced problem is also analyzed by introducing some auxiliary variables. Ample numerical examples are presented to verify the accuracy and effectiveness of our proposed method.

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

PubMed

Yang, S A

2002-10-01

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

Numerical generation of two-dimensional grids by the use of Poisson equations with grid control at boundaries

NASA Technical Reports Server (NTRS)

Sorenson, R. L.; Steger, J. L.

1980-01-01

A method for generating boundary-fitted, curvilinear, two dimensional grids by the use of the Poisson equations is presented. Grids of C-type and O-type were made about airfoils and other shapes, with circular, rectangular, cascade-type, and other outer boundary shapes. Both viscous and inviscid spacings were used. In all cases, two important types of grid control can be exercised at both inner and outer boundaries. First is arbitrary control of the distances between the boundaries and the adjacent lines of the same coordinate family, i.e., stand-off distances. Second is arbitrary control of the angles with which lines of the opposite coordinate family intersect the boundaries. Thus, both grid cell size (or aspect ratio) and grid cell skewness are controlled at boundaries. Reasonable cell size and shape are ensured even in cases wherein extreme boundary shapes would tend to cause skewness or poorly controlled grid spacing. An inherent feature of the Poisson equations is that lines in the interior of the grid smoothly connect the boundary points (the grid mapping functions are second order differentiable).

Relaxation of the accelerating-gas boundary layer to the test-gas boundary layer on a flat plate in an expansion tube

NASA Technical Reports Server (NTRS)

Gupta, R. N.; Trimpi, R. L.

1973-01-01

An analytic investigation of the relaxation of the accelerating-gas boundary layer to the test-gas boundary layer over a flat plate mounted in an expansion tube has been conducted. In this treatment, nitrogen has been considered as the test gas and helium as the accelerating gas. The problem is analyzed in two conically similar limits: (1) when the time lag between the arrival of the shock and the interface at the leading edge of the plate is very large, and (2) when this time lag is negligible. The transient laminar boundary-layer equations of a perfect binary-gas mixture are taken as the flow governing equations. These coupled equations have been solved numerically by Gauss-Seidel line-relaxation method. The results predict the transient behavior as well as the time required for an all-helium accelerating-gas boundary layer to relax to an all-nitrogen boundary layer.

Model Predictive Optimal Control of a Time-Delay Distributed-Parameter Systems

NASA Technical Reports Server (NTRS)

Nguyen, Nhan

2006-01-01

This paper presents an optimal control method for a class of distributed-parameter systems governed by first order, quasilinear hyperbolic partial differential equations that arise in many physical systems. Such systems are characterized by time delays since information is transported from one state to another by wave propagation. A general closed-loop hyperbolic transport model is controlled by a boundary control embedded in a periodic boundary condition. The boundary control is subject to a nonlinear differential equation constraint that models actuator dynamics of the system. The hyperbolic equation is thus coupled with the ordinary differential equation via the boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to implement a model predictive control design for a wind tunnel to eliminate a transport delay effect that causes a poor Mach number regulation.

Unstructured Cartesian refinement with sharp interface immersed boundary method for 3D unsteady incompressible flows

NASA Astrophysics Data System (ADS)

Angelidis, Dionysios; Chawdhary, Saurabh; Sotiropoulos, Fotis

2016-11-01

A novel numerical method is developed for solving the 3D, unsteady, incompressible Navier-Stokes equations on locally refined fully unstructured Cartesian grids in domains with arbitrarily complex immersed boundaries. Owing to the utilization of the fractional step method on an unstructured Cartesian hybrid staggered/non-staggered grid layout, flux mismatch and pressure discontinuity issues are avoided and the divergence free constraint is inherently satisfied to machine zero. Auxiliary/hanging nodes are used to facilitate the discretization of the governing equations. The second-order accuracy of the solver is ensured by using multi-dimension Lagrange interpolation operators and appropriate differencing schemes at the interface of regions with different levels of refinement. The sharp interface immersed boundary method is augmented with local near-boundary refinement to handle arbitrarily complex boundaries. The discrete momentum equation is solved with the matrix free Newton-Krylov method and the Krylov-subspace method is employed to solve the Poisson equation. The second-order accuracy of the proposed method on unstructured Cartesian grids is demonstrated by solving the Poisson equation with a known analytical solution. A number of three-dimensional laminar flow simulations of increasing complexity illustrate the ability of the method to handle flows across a range of Reynolds numbers and flow regimes. Laminar steady and unsteady flows past a sphere and the oblique vortex shedding from a circular cylinder mounted between two end walls demonstrate the accuracy, the efficiency and the smooth transition of scales and coherent structures across refinement levels. Large-eddy simulation (LES) past a miniature wind turbine rotor, parameterized using the actuator line approach, indicates the ability of the fully unstructured solver to simulate complex turbulent flows. Finally, a geometry resolving LES of turbulent flow past a complete hydrokinetic turbine illustrates the potential of the method to simulate turbulent flows past geometrically complex bodies on locally refined meshes. In all the cases, the results are found to be in very good agreement with published data and savings in computational resources are achieved.

A finite element: Boundary integral method for electromagnetic scattering. Ph.D. Thesis Technical Report, Feb. - Sep. 1992

NASA Technical Reports Server (NTRS)

Collins, J. D.; Volakis, John L.

1992-01-01

A method that combines the finite element and boundary integral techniques for the numerical solution of electromagnetic scattering problems is presented. The finite element method is well known for requiring a low order storage and for its capability to model inhomogeneous structures. Of particular emphasis in this work is the reduction of the storage requirement by terminating the finite element mesh on a boundary in a fashion which renders the boundary integrals in convolutional form. The fast Fourier transform is then used to evaluate these integrals in a conjugate gradient solver, without a need to generate the actual matrix. This method has a marked advantage over traditional integral equation approaches with respect to the storage requirement of highly inhomogeneous structures. Rectangular, circular, and ogival mesh termination boundaries are examined for two-dimensional scattering. In the case of axially symmetric structures, the boundary integral matrix storage is reduced by exploiting matrix symmetries and solving the resulting system via the conjugate gradient method. In each case several results are presented for various scatterers aimed at validating the method and providing an assessment of its capabilities. Important in methods incorporating boundary integral equations is the issue of internal resonance. A method is implemented for their removal, and is shown to be effective in the two-dimensional and three-dimensional applications.

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems

NASA Astrophysics Data System (ADS)

Zheng, Chang-Jun; Chen, Hai-Bo; Chen, Lei-Lei

2013-04-01

This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/plane-symmetric acoustic wave problems. The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only. Moreover, a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived, and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating, translating and saving the multipole/local expansion coefficients of the image domain. The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems. As for exterior acoustic problems, the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method. Details on the implementation of the present method are described, and numerical examples are given to demonstrate its accuracy and efficiency.

Numerical and experimental investigation on static electric charge model at stable cone-jet region

NASA Astrophysics Data System (ADS)

Hashemi, Ali Reza; Pishevar, Ahmad Reza; Valipouri, Afsaneh; Pǎrǎu, Emilian I.

2018-03-01

In a typical electro-spinning process, the steady stretching process of the jet beyond the Taylor cone has a significant effect on the dimensions of resulting nanofibers. Also, it sets up the conditions for the onset of the bending instability. The focus of this work is the modeling and simulation of the initial stable jet phase seen during the electro-spinning process. The perturbation method was applied to solve hydrodynamic equations, and the electrostatic equation was solved by a boundary integral method. These equations were coupled with the stress boundary conditions derived appropriate at the fluid-fluid interface. Perturbation equations were discretized by the second-order finite difference method, and the Newton method was implemented to solve the discretized nonlinear system. Also, the boundary element method was utilized to solve the electrostatic equation. In the theoretical study, the fluid is described as a leaky dielectric with charges only on the jet surface in dielectric air. In this study, electric charges were modeled as static. Comparison of numerical and experimental results shows that at low flow rates and high electric field, good agreement was achieved because of the superior importance of the charge transport by conduction rather than convection and charge concentration. In addition, the effect of unevenness of the electric field around the nozzle tip was experimentally studied through plate-plate geometry as well as point-plate geometry.

Self-adaptive difference method for the effective solution of computationally complex problems of boundary layer theory

NASA Technical Reports Server (NTRS)

Schoenauer, W.; Daeubler, H. G.; Glotz, G.; Gruening, J.

1986-01-01

An implicit difference procedure for the solution of equations for a chemically reacting hypersonic boundary layer is described. Difference forms of arbitrary error order in the x and y coordinate plane were used to derive estimates for discretization error. Computational complexity and time were minimized by the use of this difference method and the iteration of the nonlinear boundary layer equations was regulated by discretization error. Velocity and temperature profiles are presented for Mach 20.14 and Mach 18.5; variables are velocity profiles, temperature profiles, mass flow factor, Stanton number, and friction drag coefficient; three figures include numeric data.

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

NASA Technical Reports Server (NTRS)

Lehman, L. L.

1982-01-01

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

The Use of Kruskal-Newton Diagrams for Differential Equations

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

T. Fishaleck and R.B. White

2008-02-19

The method of Kruskal-Newton diagrams for the solution of differential equations with boundary layers is shown to provide rapid intuitive understanding of layer scaling and can result in the conceptual simplification of some problems. The method is illustrated using equations arising in the theory of pattern formation and in plasma physics.

Integral approximations to classical diffusion and smoothed particle hydrodynamics

DOE PAGES

Du, Qiang; Lehoucq, R. B.; Tartakovsky, A. M.

2014-12-31

The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary.more » The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.« less

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

NASA Astrophysics Data System (ADS)

Latyshev, A. V.; Gordeeva, N. M.

2017-09-01

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov-Boltzmann equation with the Bhatnagar-Gross-Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi-Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.

Evaluation of finite difference and FFT-based solutions of the transport of intensity equation.

PubMed

Zhang, Hongbo; Zhou, Wen-Jing; Liu, Ying; Leber, Donald; Banerjee, Partha; Basunia, Mahmudunnabi; Poon, Ting-Chung

2018-01-01

A finite difference method is proposed for solving the transport of intensity equation. Simulation results show that although slower than fast Fourier transform (FFT)-based methods, finite difference methods are able to reconstruct the phase with better accuracy due to relaxed assumptions for solving the transport of intensity equation relative to FFT methods. Finite difference methods are also more flexible than FFT methods in dealing with different boundary conditions.

Integral Equations in Computational Electromagnetics: Formulations, Properties and Isogeometric Analysis

NASA Astrophysics Data System (ADS)

Lovell, Amy Elizabeth

Computational electromagnetics (CEM) provides numerical methods to simulate electromagnetic waves interacting with its environment. Boundary integral equation (BIE) based methods, that solve the Maxwell's equations in the homogeneous or piecewise homogeneous medium, are both efficient and accurate, especially for scattering and radiation problems. Development and analysis electromagnetic BIEs has been a very active topic in CEM research. Indeed, there are still many open problems that need to be addressed or further studied. A short and important list includes (1) closed-form or quasi-analytical solutions to time-domain integral equations, (2) catastrophic cancellations at low frequencies, (3) ill-conditioning due to high mesh density, multi-scale discretization, and growing electrical size, and (4) lack of flexibility due to re-meshing when increasing number of forward numerical simulations are involved in the electromagnetic design process. This dissertation will address those several aspects of boundary integral equations in computational electromagnetics. The first contribution of the dissertation is to construct quasi-analytical solutions to time-dependent boundary integral equations using a direct approach. Direct inverse Fourier transform of the time-harmonic solutions is not stable due to the non-existence of the inverse Fourier transform of spherical Hankel functions. Using new addition theorems for the time-domain Green's function and dyadic Green's functions, time-domain integral equations governing transient scattering problems of spherical objects are solved directly and stably for the first time. Additional, the direct time-dependent solutions, together with the newly proposed time-domain dyadic Green's functions, can enrich the time-domain spherical multipole theory. The second contribution is to create a novel method of moments (MoM) framework to solve electromagnetic boundary integral equation on subdivision surfaces. The aim is to avoid the meshing and re-meshing stages to accelerate the design process when the geometry needs to be updated. Two schemes to construct basis functions on the subdivision surface have been explored. One is to use the div-conforming basis function, and the other one is to create a rigorous iso-geometric approach based on the subdivision basis function with better smoothness properties. This new framework provides us better accuracy, more stability and high flexibility. The third contribution is a new stable integral equation formulation to avoid catastrophic cancellations due to low-frequency breakdown or dense-mesh breakdown. Many of the conventional integral equations and their associated post-processing operations suffer from numerical catastrophic cancellations, which can lead to ill-conditioning of the linear systems or serious accuracy problems. Examples includes low-frequency breakdown and dense mesh breakdown. Another instability may come from nontrivial null spaces of involving integral operators that might be related with spurious resonance or topology breakdown. This dissertation presents several sets of new boundary integral equations and studies their analytical properties. The first proposed formulation leads to the scalar boundary integral equations where only scalar unknowns are involved. Besides the requirements of gaining more stability and better conditioning in the resulting linear systems, multi-physics simulation is another driving force for new formulations. Scalar and vector potentials (rather than electromagnetic field) based formulation have been studied for this purpose. Those new contributions focus on different stages of boundary integral equations in an almost independent manner, e.g. isogeometric analysis framework can be used to solve different boundary integral equations, and the time-dependent solutions to integral equations from different formulations can be achieved through the same methodology proposed.

Effect of boundary treatments on quantum transport current in the Green's function and Wigner distribution methods for a nano-scale DG-MOSFET

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

Jiang Haiyan; Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001; Cai Wei

2010-06-20

In this paper, we conduct a study of quantum transport models for a two-dimensional nano-size double gate (DG) MOSFET using two approaches: non-equilibrium Green's function (NEGF) and Wigner distribution. Both methods are implemented in the framework of the mode space methodology where the electron confinements below the gates are pre-calculated to produce subbands along the vertical direction of the device while the transport along the horizontal channel direction is described by either approach. Each approach handles the open quantum system along the transport direction in a different manner. The NEGF treats the open boundaries with boundary self-energy defined by amore » Dirichlet to Neumann mapping, which ensures non-reflection at the device boundaries for electron waves leaving the quantum device active region. On the other hand, the Wigner equation method imposes an inflow boundary treatment for the Wigner distribution, which in contrast ensures non-reflection at the boundaries for free electron waves entering the device active region. In both cases the space-charge effect is accounted for by a self-consistent coupling with a Poisson equation. Our goals are to study how the device boundaries are treated in both transport models affects the current calculations, and to investigate the performance of both approaches in modeling the DG-MOSFET. Numerical results show mostly consistent quantum transport characteristics of the DG-MOSFET using both methods, though with higher transport current for the Wigner equation method, and also provide the current-voltage (I-V) curve dependence on various physical parameters such as the gate voltage and the oxide thickness.« less

The semi-discrete Galerkin finite element modelling of compressible viscous flow past an airfoil

NASA Technical Reports Server (NTRS)

Meade, Andrew J., Jr.

1992-01-01

A method is developed to solve the two-dimensional, steady, compressible, turbulent boundary-layer equations and is coupled to an existing Euler solver for attached transonic airfoil analysis problems. The boundary-layer formulation utilizes the semi-discrete Galerkin (SDG) method to model the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby permitting the use of a uniform finite element grid which provides high resolution near the wall and automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes, through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past NACA 0012 and RAE 2822 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack. All results show good agreement with experiment, and the coupled code proved to be a computationally-efficient and accurate airfoil analysis tool.

Survey of the status of finite element methods for partial differential equations

NASA Technical Reports Server (NTRS)

Temam, Roger

1986-01-01

The finite element methods (FEM) have proved to be a powerful technique for the solution of boundary value problems associated with partial differential equations of either elliptic, parabolic, or hyperbolic type. They also have a good potential for utilization on parallel computers particularly in relation to the concept of domain decomposition. This report is intended as an introduction to the FEM for the nonspecialist. It contains a survey which is totally nonexhaustive, and it also contains as an illustration, a report on some new results concerning two specific applications, namely a free boundary fluid-structure interaction problem and the Euler equations for inviscid flows.

A boundary value approach for solving three-dimensional elliptic and hyperbolic partial differential equations.

PubMed

Biala, T A; Jator, S N

2015-01-01

In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.

Unsteady boundary layer flow over a sphere in a porous medium

NASA Astrophysics Data System (ADS)

Mohammad, Nurul Farahain; Waini, Iskandar; Kasim, Abdul Rahman Mohd; Majid, Nurazleen Abdul

2017-08-01

This study focuses on the problem of unsteady boundary layer flow over a sphere in a porous medium. The governing equations which consists of a system of dimensional partial differential equations is applied with dimensionless parameter in order to attain non-dimensional partial differential equations. Later, the similarity transformation is performed in order to attain nonsimilar governing equations. Afterwards, the nonsimilar governing equations are solved numerically by using the Keller-Box method in Octave programme. The effect of porosity parameter is examined on separation time, velocity profile and skin friction of the unsteady flow. The results attained are presented in the form of table and graph.

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

NASA Astrophysics Data System (ADS)

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

2016-01-01

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

The numerical-analytical implementation of the cross-sections method to the open waveguide transition of the "horn" type

NASA Astrophysics Data System (ADS)

Divakov, Dmitriy; Malykh, Mikhail; Sevastianov, Leonid; Sevastianov, Anton; Tiutiunnik, Anastasiia

2017-04-01

In the paper we construct a method for approximate solution of the waveguide problem for guided modes of an open irregular waveguide transition. The method is based on straightening of the curved waveguide boundaries by introducing new variables and applying the Kantorovich method to the problem formulated in the new variables to get a system of ordinary second-order differential equations. In the method, the boundary conditions are formulated by analogy with the partial radiation conditions in the similar problem for closed waveguide transitions. The method is implemented in the symbolic-numeric form using the Maple computer algebra system. The coefficient matrices of the system of differential equations and boundary conditions are calculated symbolically, and then the obtained boundary-value problem is solved numerically using the finite difference method. The chosen coordinate functions of Kantorovich expansions provide good conditionality of the coefficient matrices. The numerical experiment simulating the propagation of guided modes in the open waveguide transition confirms the validity of the method proposed to solve the problem.

The stationary sine-Gordon equation on metric graphs: Exact analytical solutions for simple topologies

NASA Astrophysics Data System (ADS)

Sabirov, K.; Rakhmanov, S.; Matrasulov, D.; Susanto, H.

2018-04-01

We consider the stationary sine-Gordon equation on metric graphs with simple topologies. Exact analytical solutions are obtained for different vertex boundary conditions. It is shown that the method can be extended for tree and other simple graph topologies. Applications of the obtained results to branched planar Josephson junctions and Josephson junctions with tricrystal boundaries are discussed.

High-order finite-volume solutions of the steady-state advection-diffusion equation with nonlinear Robin boundary conditions

NASA Astrophysics Data System (ADS)

Lin, Zhi; Zhang, Qinghai

2017-09-01

We propose high-order finite-volume schemes for numerically solving the steady-state advection-diffusion equation with nonlinear Robin boundary conditions. Although the original motivation comes from a mathematical model of blood clotting, the nonlinear boundary conditions may also apply to other scientific problems. The main contribution of this work is a generic algorithm for generating third-order, fourth-order, and even higher-order explicit ghost-filling formulas to enforce nonlinear Robin boundary conditions in multiple dimensions. Under the framework of finite volume methods, this appears to be the first algorithm of its kind. Numerical experiments on boundary value problems show that the proposed fourth-order formula can be much more accurate and efficient than a simple second-order formula. Furthermore, the proposed ghost-filling formulas may also be useful for solving other partial differential equations.

A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics

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

Shao, Yan-Lin, E-mail: yanlin.shao@dnvgl.com; Faltinsen, Odd M.

2014-10-01

We propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3D Laplace equation. The computational domain is discretized by overlapping cells. Within each cell, the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials, which are the elementary solutions of Laplace equation. By its definition, the method is named as Harmonic Polynomial Cell (HPC) method. The characteristics of the accuracy and efficiency of the HPC method are demonstrated by studying analytical cases. Comparisons will be made with some other existing boundary element based methods,more » e.g. Quadratic Boundary Element Method (QBEM) and the Fast Multipole Accelerated QBEM (FMA-QBEM) and a fourth order Finite Difference Method (FDM). To demonstrate the applications of the method, it is applied to some studies relevant for marine hydrodynamics. Sloshing in 3D rectangular tanks, a fully-nonlinear numerical wave tank, fully-nonlinear wave focusing on a semi-circular shoal, and the nonlinear wave diffraction of a bottom-mounted cylinder in regular waves are studied. The comparisons with the experimental results and other numerical results are all in satisfactory agreement, indicating that the present HPC method is a promising method in solving potential-flow problems. The underlying procedure of the HPC method could also be useful in other fields than marine hydrodynamics involved with solving Laplace equation.« less

Immersed boundary method for Boltzmann model kinetic equations

NASA Astrophysics Data System (ADS)

Pekardan, Cem; Chigullapalli, Sruti; Sun, Lin; Alexeenko, Alina

2012-11-01

Three different immersed boundary method formulations are presented for Boltzmann model kinetic equations such as Bhatnagar-Gross-Krook (BGK) and Ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) model equations. 1D unsteady IBM solution for a moving piston is compared with the DSMC results and 2D quasi-steady microscale gas damping solutions are verified by a conformal finite volume method solver. Transient analysis for a sinusoidally moving beam is also carried out for the different pressure conditions (1 atm, 0.1 atm and 0.01 atm) corresponding to Kn=0.05,0.5 and 5. Interrelaxation method (Method 2) is shown to provide a faster convergence as compared to the traditional interpolation scheme used in continuum IBM formulations. Unsteady damping in rarefied regime is characterized by a significant phase-lag which is not captured by quasi-steady approximations.

A Semianalytical Model for Pumping Tests in Finite Heterogeneous Confined Aquifers With Arbitrarily Shaped Boundary

NASA Astrophysics Data System (ADS)

Wang, Lei; Dai, Cheng; Xue, Liang

2018-04-01

This study presents a Laplace-transform-based boundary element method to model the groundwater flow in a heterogeneous confined finite aquifer with arbitrarily shaped boundaries. The boundary condition can be Dirichlet, Neumann or Robin-type. The derived solution is analytical since it is obtained through the Green's function method within the domain. However, the numerical approximation is required on the boundaries, which essentially renders it a semi-analytical solution. The proposed method can provide a general framework to derive solutions for zoned heterogeneous confined aquifers with arbitrarily shaped boundary. The requirement of the boundary element method presented here is that the Green function must exist for a specific PDE equation. In this study, the linear equations for the two-zone and three-zone confined aquifers with arbitrarily shaped boundary is established in Laplace space, and the solution can be obtained by using any linear solver. Stehfest inversion algorithm can be used to transform it back into time domain to obtain the transient solution. The presented solution is validated in the two-zone cases by reducing the arbitrarily shaped boundaries to circular ones and comparing it with the solution in Lin et al. (2016, https://doi.org/10.1016/j.jhydrol.2016.07.028). The effect of boundary shape and well location on dimensionless drawdown in two-zone aquifers is investigated. Finally the drawdown distribution in three-zone aquifers with arbitrarily shaped boundary for constant-rate tests (CRT) and flow rate distribution for constant-head tests (CHT) are analyzed.

Efficient algorithms for analyzing the singularly perturbed boundary value problems of fractional order

NASA Astrophysics Data System (ADS)

Sayevand, K.; Pichaghchi, K.

2018-04-01

In this paper, we were concerned with the description of the singularly perturbed boundary value problems in the scope of fractional calculus. We should mention that, one of the main methods used to solve these problems in classical calculus is the so-called matched asymptotic expansion method. However we shall note that, this was not achievable via the existing classical definitions of fractional derivative, because they do not obey the chain rule which one of the key elements of the matched asymptotic expansion method. In order to accommodate this method to fractional derivative, we employ a relatively new derivative so-called the local fractional derivative. Using the properties of local fractional derivative, we extend the matched asymptotic expansion method to the scope of fractional calculus and introduce a reliable new algorithm to develop approximate solutions of the singularly perturbed boundary value problems of fractional order. In the new method, the original problem is partitioned into inner and outer solution equations. The reduced equation is solved with suitable boundary conditions which provide the terminal boundary conditions for the boundary layer correction. The inner solution problem is next solved as a solvable boundary value problem. The width of the boundary layer is approximated using appropriate resemblance function. Some theoretical results are established and proved. Some illustrating examples are solved and the results are compared with those of matched asymptotic expansion method and homotopy analysis method to demonstrate the accuracy and efficiency of the method. It can be observed that, the proposed method approximates the exact solution very well not only in the boundary layer, but also away from the layer.

Nonlinear power flow feedback control for improved stability and performance of airfoil sections

DOEpatents

Wilson, David G.; Robinett, III, Rush D.

2013-09-03

A computer-implemented method of determining the pitch stability of an airfoil system, comprising using a computer to numerically integrate a differential equation of motion that includes terms describing PID controller action. In one model, the differential equation characterizes the time-dependent response of the airfoil's pitch angle, .alpha.. The computer model calculates limit-cycles of the model, which represent the stability boundaries of the airfoil system. Once the stability boundary is known, feedback control can be implemented, by using, for example, a PID controller to control a feedback actuator. The method allows the PID controller gain constants, K.sub.I, K.sub.p, and K.sub.d, to be optimized. This permits operation closer to the stability boundaries, while preventing the physical apparatus from unintentionally crossing the stability boundaries. Operating closer to the stability boundaries permits greater power efficiencies to be extracted from the airfoil system.

Effect of nose shape on three-dimensional stagnation region streamlines and heating rates

NASA Technical Reports Server (NTRS)

Hassan, Basil; Dejarnette, Fred R.; Zoby, E. V.

1991-01-01

A new method for calculating the three-dimensional inviscid surface streamlines and streamline metrics using Cartesian coordinates and time as the independent variable of integration has been developed. The technique calculates the streamline from a specified point on the body to a point near the stagnation point by using a prescribed pressure distribution in the Euler equations. The differential equations, which are singular at the stagnation point, are of the two point boundary value problem type. Laminar heating rates are calculated using the axisymmetric analog concept for three-dimensional boundary layers and approximate solutions to the axisymmetric boundary layer equations. Results for elliptic conic forebody geometries show that location of the point of maximum heating depends on the type of conic in the plane of symmetry and the angle of attack, and that this location is in general different from the stagnation point. The new method was found to give smooth predictions of heat transfer in the nose region where previous methods gave oscillatory results.

Application of ANNs approach for wave-like and heat-like equations

NASA Astrophysics Data System (ADS)

Jafarian, Ahmad; Baleanu, Dumitru

2017-12-01

Artificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.

A method for solution of the Euler-Bernoulli beam equation in flexible-link robotic systems

NASA Technical Reports Server (NTRS)

Tzes, Anthony P.; Yurkovich, Stephen; Langer, F. Dieter

1989-01-01

An efficient numerical method for solving the partial differential equation (PDE) governing the flexible manipulator control dynamics is presented. A finite-dimensional model of the equation is obtained through discretization in both time and space coordinates by using finite-difference approximations to the PDE. An expert program written in the Macsyma symbolic language is utilized in order to embed the boundary conditions into the program, accounting for a mass carried at the tip of the manipulator. The advantages of the proposed algorithm are many, including the ability to (1) include any distributed actuation term in the partial differential equation, (2) provide distributed sensing of the beam displacement, (3) easily modify the boundary conditions through an expert program, and (4) modify the structure for running under a multiprocessor environment.

A numerical solution of a singular boundary value problem arising in boundary layer theory.

PubMed

Hu, Jiancheng

2016-01-01

In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner-Skan equation. And the one-dimensional third-order boundary value problem on interval [Formula: see text] is equivalently transformed into a second-order boundary value problem on finite interval [Formula: see text]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.

Implicit level set algorithms for modelling hydraulic fracture propagation.

PubMed

Peirce, A

2016-10-13

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

Implicit level set algorithms for modelling hydraulic fracture propagation

PubMed Central

2016-01-01

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

Discretization of the induced-charge boundary integral equation.

PubMed

Bardhan, Jaydeep P; Eisenberg, Robert S; Gillespie, Dirk

2009-07-01

Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.

Discretization of the induced-charge boundary integral equation

NASA Astrophysics Data System (ADS)

Bardhan, Jaydeep P.; Eisenberg, Robert S.; Gillespie, Dirk

2009-07-01

Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.

Thermodynamic evaluation of transonic compressor rotors using the finite volume approach

NASA Technical Reports Server (NTRS)

Nicholson, S.; Moore, J.

1986-01-01

A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite volume, time marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier Stokes equations. The entire calculation is performed in the physical domain. The method is currently limited to the calculation of attached flows. The features of the current method can be summarized as follows. Control volumes are chosen so that smoothing of flow properties, typically required for stability, is now needed. Different time steps are used in the different governing equations to improve the convergence speed of the viscous calculations. A new pressure interpolation scheme is introduced which improves the shock capturing ability of the method. A multi-volume method for pressure changes in the boundary layer allows calculations which use very long and thin control volumes. A special discretization technique is also used to stabilize these calculations. A special formulation of the energy equation is used to provide improved transient behavior of solutions which use the full energy equation. The method is then compared with a wide variety of test cases. The freestream Mach numbers range from 0.075 to 2.8 in the calculations. Transonic viscous flow in a converging diverging nozzle is calculated with the method; the Mach number upstream of the shock is approximately 1.25. The agreement between the calculated and measured shock strength and total pressure losses is good. Essentially incompressible turbulent boundary layer flow in a adverse pressure gradient is calculated and the computed distribution of mean velocity and shear stress are in good agreement with the measurements. At the other end of the Mach number range, a flat plate turbulent boundary layer with a freestream Mach number of 2.8 is calculated using the full energy equation; the computed total temperature distribution and recovery factor agree well with the measurements when a variable Prandtl number is used through the boundary layer.

GRAPE- TWO-DIMENSIONAL GRIDS ABOUT AIRFOILS AND OTHER SHAPES BY THE USE OF POISSON'S EQUATION

NASA Technical Reports Server (NTRS)

Sorenson, R. L.

1994-01-01

The ability to treat arbitrary boundary shapes is one of the most desirable characteristics of a method for generating grids, including those about airfoils. In a grid used for computing aerodynamic flow over an airfoil, or any other body shape, the surface of the body is usually treated as an inner boundary and often cannot be easily represented as an analytic function. The GRAPE computer program was developed to incorporate a method for generating two-dimensional finite-difference grids about airfoils and other shapes by the use of the Poisson differential equation. GRAPE can be used with any boundary shape, even one specified by tabulated points and including a limited number of sharp corners. The GRAPE program has been developed to be numerically stable and computationally fast. GRAPE can provide the aerodynamic analyst with an efficient and consistent means of grid generation. The GRAPE procedure generates a grid between an inner and an outer boundary by utilizing an iterative procedure to solve the Poisson differential equation subject to geometrical restraints. In this method, the inhomogeneous terms of the equation are automatically chosen such that two important effects are imposed on the grid. The first effect is control of the spacing between mesh points along mesh lines intersecting the boundaries. The second effect is control of the angles with which mesh lines intersect the boundaries. Along with the iterative solution to Poisson's equation, a technique of coarse-fine sequencing is employed to accelerate numerical convergence. GRAPE program control cards and input data are entered via the NAMELIST feature. Each variable has a default value such that user supplied data is kept to a minimum. Basic input data consists of the boundary specification, mesh point spacings on the boundaries, and mesh line angles at the boundaries. Output consists of a dataset containing the grid data and, if requested, a plot of the generated mesh. The GRAPE program is written in FORTRAN IV for batch execution and has been implemented on a CDC 6000 series computer with a central memory requirement of approximately 135K (octal) of 60 bit words. For plotted output the commercially available DISSPLA graphics software package is required. The GRAPE program was developed in 1980.

Finite element methods and Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Cuvelier, C.; Segal, A.; van Steenhoven, A. A.

This book is devoted to two and three-dimensional FEM analysis of the Navier-Stokes (NS) equations describing one flow of a viscous incompressible fluid. Three different approaches to the NS equations are described: a direct method, a penalty method, and a method that constructs discrete solenoidal vector fields. Subjects of current research which are important from the industrial/technological viewpoint are considered, including capillary-free boundaries, nonisothermal flows, turbulence, and non-Newtonian fluids.

Numerical and experimental study on the steady cone-jet mode of electro-centrifugal spinning

NASA Astrophysics Data System (ADS)

Hashemi, Ali Reza; Pishevar, Ahmad Reza; Valipouri, Afsaneh; Pǎrǎu, Emilian I.

2018-01-01

This study focuses on a numerical investigation of an initial stable jet through the air-sealed electro-centrifugal spinning process, which is known as a viable method for the mass production of nanofibers. A liquid jet undergoing electric and centrifugal forces, as well as other forces, first travels in a stable trajectory and then goes through an unstable curled path to the collector. In numerical modeling, hydrodynamic equations have been solved using the perturbation method—and the boundary integral method has been implemented to efficiently solve the electric potential equation. Hydrodynamic equations have been coupled with the electric field using stress boundary conditions at the fluid-fluid interface. Perturbation equations were discretized by a second order finite difference method, and the Newton method was implemented to solve the discretized non-linear system. Also, the boundary element method was utilized to solve electrostatic equations. In the theoretical study, the fluid was described as a leaky dielectric with charges only on the surface of the jet traveling in dielectric air. The effect of the electric field induced around the nozzle tip on the jet instability and trajectory deviation was also experimentally studied through plate-plate geometry as well as point-plate geometry. It was numerically found that the centrifugal force prevails on electric force by increasing the rotational speed. Therefore, the alteration of the applied voltage does not significantly affect the jet thinning profile or the jet trajectory.

A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation.

PubMed

Gumerov, Nail A; Duraiswami, Ramani

2009-01-01

The development of a fast multipole method (FMM) accelerated iterative solution of the boundary element method (BEM) for the Helmholtz equations in three dimensions is described. The FMM for the Helmholtz equation is significantly different for problems with low and high kD (where k is the wavenumber and D the domain size), and for large problems the method must be switched between levels of the hierarchy. The BEM requires several approximate computations (numerical quadrature, approximations of the boundary shapes using elements), and these errors must be balanced against approximations introduced by the FMM and the convergence criterion for iterative solution. These different errors must all be chosen in a way that, on the one hand, excess work is not done and, on the other, that the error achieved by the overall computation is acceptable. Details of translation operators for low and high kD, choice of representations, and BEM quadrature schemes, all consistent with these approximations, are described. A novel preconditioner using a low accuracy FMM accelerated solver as a right preconditioner is also described. Results of the developed solvers for large boundary value problems with 0.0001 less, similarkD less, similar500 are presented and shown to perform close to theoretical expectations.

A Numerical Method for Incompressible Flow with Heat Transfer

NASA Technical Reports Server (NTRS)

Sa, Jong-Youb; Kwak, Dochan

1997-01-01

A numerical method for the convective heat transfer problem is developed for low speed flow at mild temperatures. A simplified energy equation is added to the incompressible Navier-Stokes formulation by using Boussinesq approximation to account for the buoyancy force. A pseudocompressibility method is used to solve the resulting set of equations for steady-state solutions in conjunction with an approximate factorization scheme. A Neumann-type pressure boundary condition is devised to account for the interaction between pressure and temperature terms, especially near a heated or cooled solid boundary. It is shown that the present method is capable of predicting the temperature field in an incompressible flow.

Three-dimensional zonal grids about arbitrary shapes by Poisson's equation

NASA Technical Reports Server (NTRS)

Sorenson, Reese L.

1988-01-01

A method for generating 3-D finite difference grids about or within arbitrary shapes is presented. The 3-D Poisson equations are solved numerically, with values for the inhomogeneous terms found automatically by the algorithm. Those inhomogeneous terms have the effect near boundaries of reducing cell skewness and imposing arbitrary cell height. The method allows the region of interest to be divided into zones (blocks), allowing the method to be applicable to almost any physical domain. A FORTRAN program called 3DGRAPE has been written to implement the algorithm. Lastly, a method for redistributing grid points along lines normal to boundaries will be described.

Unsteady boundary layer flow and heat transfer of a Casson fluid past an oscillating vertical plate with Newtonian heating.

PubMed

Hussanan, Abid; Zuki Salleh, Mohd; Tahar, Razman Mat; Khan, Ilyas

2014-01-01

In this paper, the heat transfer effect on the unsteady boundary layer flow of a Casson fluid past an infinite oscillating vertical plate with Newtonian heating is investigated. The governing equations are transformed to a systems of linear partial differential equations using appropriate non-dimensional variables. The resulting equations are solved analytically by using the Laplace transform method and the expressions for velocity and temperature are obtained. They satisfy all imposed initial and boundary conditions and reduce to some well-known solutions for Newtonian fluids. Numerical results for velocity, temperature, skin friction and Nusselt number are shown in various graphs and discussed for embedded flow parameters. It is found that velocity decreases as Casson parameters increases and thermal boundary layer thickness increases with increasing Newtonian heating parameter.

A Heuristic Fast Method to Solve the Nonlinear Schroedinger Equation in Fiber Bragg Gratings with Arbitrary Shape Input Pulse

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

Emami, F.; Hatami, M.; Keshavarz, A. R.

2009-08-13

Using a combination of Runge-Kutta and Jacobi iterative method, we could solve the nonlinear Schroedinger equation describing the pulse propagation in FBGs. By decomposing the electric field to forward and backward components in fiber Bragg grating and utilizing the Fourier series analysis technique, the boundary value problem of a set of coupled equations governing the pulse propagation in FBG changes to an initial condition coupled equations which can be solved by simple Runge-Kutta method.

Boundary-layer effects in composite laminates: Free-edge stress singularities, part 6

NASA Technical Reports Server (NTRS)

Wanag, S. S.; Choi, I.

1981-01-01

A rigorous mathematical model was obtained for the boundary-layer free-edge stress singularity in angleplied and crossplied fiber composite laminates. The solution was obtained using a method consisting of complex-variable stress function potentials and eigenfunction expansions. The required order of the boundary-layer stress singularity is determined by solving the transcendental characteristic equation obtained from the homogeneous solution of the partial differential equations. Numerical results obtained show that the boundary-layer stress singularity depends only upon material elastic constants and fiber orientation of the adjacent plies. For angleplied and crossplied laminates the order of the singularity is weak in general.

Large Scale Flutter Data for Design of Rotating Blades Using Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Guruswamy, Guru P.

2012-01-01

A procedure to compute flutter boundaries of rotating blades is presented; a) Navier-Stokes equations. b) Frequency domain method compatible with industry practice. Procedure is initially validated: a) Unsteady loads with flapping wing experiment. b) Flutter boundary with fixed wing experiment. Large scale flutter computation is demonstrated for rotating blade: a) Single job submission script. b) Flutter boundary in 24 hour wall clock time with 100 cores. c) Linearly scalable with number of cores. Tested with 1000 cores that produced data in 25 hrs for 10 flutter boundaries. Further wall-clock speed-up is possible by performing parallel computations within each case.

Bounded Error Schemes for the Wave Equation on Complex Domains

NASA Technical Reports Server (NTRS)

Abarbanel, Saul; Ditkowski, Adi; Yefet, Amir

1998-01-01

This paper considers the application of the method of boundary penalty terms ("SAT") to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell's equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g. the staggered Yee scheme) - we achieve a decrease of two orders of magnitude in the level of the L2-error.

A composite velocity procedure for the compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Khosla, P. K.; Rubin, S. G.

1982-01-01

A new boundary-layer relaxation procedure is presented. In the spirit of the theory of matched asymptotic expansions, a multiplicative composite of the appropriate velocity representations for the inviscid and viscous regions is prescribed. The resulting equations are structured so that far from the surface of the body the momentum equations lead to the Bernoulli relation for the pressure, while the continuity equation reduces to the familiar compressible potential equation. Close to the body surface, the governing equations and solution techniques are characteristic of those describing interacting boundary-layers; although, the full Navier-Stokes equations are considered here. Laminar flow calculations for the subsonic flow over an axisymmetric boattail simulator geometry are presented for a variety of Reynolds and Mach numbers. A strongly implicit solution method is applied for the coupled velocity components.

Personal computer study of finite-difference methods for the transonic small disturbance equation

NASA Technical Reports Server (NTRS)

Bland, Samuel R.

1989-01-01

Calculation of unsteady flow phenomena requires careful attention to the numerical treatment of the governing partial differential equations. The personal computer provides a convenient and useful tool for the development of meshes, algorithms, and boundary conditions needed to provide time accurate solution of these equations. The one-dimensional equation considered provides a suitable model for the study of wave propagation in the equations of transonic small disturbance potential flow. Numerical results for effects of mesh size, extent, and stretching, time step size, and choice of far-field boundary conditions are presented. Analysis of the discretized model problem supports these numerical results. Guidelines for suitable mesh and time step choices are given.

Connectivity as an alternative to boundary integral equations: Construction of bases

PubMed Central

Herrera, Ismael; Sabina, Federico J.

1978-01-01

In previous papers Herrera developed a theory of connectivity that is applicable to the problem of connecting solutions defined in different regions, which occurs when solving partial differential equations and many problems of mechanics. In this paper we explain how complete connectivity conditions can be used to replace boundary integral equations in many situations. We show that completeness is satisfied not only in steady-state problems such as potential, reduced wave equation and static and quasi-static elasticity, but also in time-dependent problems such as heat and wave equations and dynamical elasticity. A method to obtain bases of connectivity conditions, which are independent of the regions considered, is also presented. PMID:16592522

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

NASA Astrophysics Data System (ADS)

Sardesai, Chetan R.

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

High-Order Finite-Difference Schemes for Numerical Simulation of Hypersonic Boundary-Layer Transition

NASA Astrophysics Data System (ADS)

Zhong, Xiaolin

1998-08-01

Direct numerical simulation (DNS) has become a powerful tool in studying fundamental phenomena of laminar-turbulent transition of high-speed boundary layers. Previous DNS studies of supersonic and hypersonic boundary layer transition have been limited to perfect-gas flow over flat-plate boundary layers without shock waves. For hypersonic boundary layers over realistic blunt bodies, DNS studies of transition need to consider the effects of bow shocks, entropy layers, surface curvature, and finite-rate chemistry. It is necessary that numerical methods for such studies are robust and high-order accurate both in resolving wide ranges of flow time and length scales and in resolving the interaction between the bow shocks and flow disturbance waves. This paper presents a new high-order shock-fitting finite-difference method for the DNS of the stability and transition of hypersonic boundary layers over blunt bodies with strong bow shocks and with (or without) thermo-chemical nonequilibrium. The proposed method includes a set of new upwind high-order finite-difference schemes which are stable and are less dissipative than a straightforward upwind scheme using an upwind-bias grid stencil, a high-order shock-fitting formulation, and third-order semi-implicit Runge-Kutta schemes for temporal discretization of stiff reacting flow equations. The accuracy and stability of the new schemes are validated by numerical experiments of the linear wave equation and nonlinear Navier-Stokes equations. The algorithm is then applied to the DNS of the receptivity of hypersonic boundary layers over a parabolic leading edge to freestream acoustic disturbances.

On the accurate long-time solution of the wave equation in exterior domains: Asymptotic expansions and corrected boundary conditions

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas; Hariharan, S. I.; Maccamy, R. C.

1993-01-01

We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.

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

NASA Astrophysics Data System (ADS)

Grigor’ev, Yu

2018-04-01

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

Computation of Nonlinear Backscattering Using a High-Order Numerical Method

NASA Technical Reports Server (NTRS)

Fibich, G.; Ilan, B.; Tsynkov, S.

2001-01-01

The nonlinear Schrodinger equation (NLS) is the standard model for propagation of intense laser beams in Kerr media. The NLS is derived from the nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. In this study we use a fourth-order finite-difference method supplemented by special two-way artificial boundary conditions (ABCs) to solve the NLH as a boundary value problem. Our numerical methodology allows for a direct comparison of the NLH and NLS models and for an accurate quantitative assessment of the backscattered signal.

A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation

DOE PAGES

Devendran, Dharshi; Graves, Daniel; Johansen, Hans; ...

2017-05-08

In this paper, we present a fourth-order algorithm to solve Poisson's equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second-order algorithm. We also discuss in depth strategies for retaining higher-order accuracy in the presence of nonsmooth geometries.

A fourth-order Cartesian grid embeddedboundary method for Poisson’s equation

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

Devendran, Dharshi; Graves, Daniel; Johansen, Hans

In this paper, we present a fourth-order algorithm to solve Poisson's equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second-order algorithm. We also discuss in depth strategies for retaining higher-order accuracy in the presence of nonsmooth geometries.

The Reduction of Ducted Fan Engine Noise Via A Boundary Integral Equation Method

NASA Technical Reports Server (NTRS)

Tweed, J.; Dunn, M.

1997-01-01

The development of a Boundary Integral Equation Method (BIEM) for the prediction of ducted fan engine noise is discussed. The method is motivated by the need for an efficient and versatile computational tool to assist in parametric noise reduction studies. In this research, the work in reference 1 was extended to include passive noise control treatment on the duct interior. The BEM considers the scattering of incident sound generated by spinning point thrust dipoles in a uniform flow field by a thin cylindrical duct. The acoustic field is written as a superposition of spinning modes. Modal coefficients of acoustic pressure are calculated term by term. The BEM theoretical framework is based on Helmholtz potential theory. A boundary value problem is converted to a boundary integral equation formulation with unknown single and double layer densities on the duct wall. After solving for the unknown densities, the acoustic field is easily calculated. The main feature of the BIEM is the ability to compute any portion of the sound field without the need to compute the entire field. Other noise prediction methods such as CFD and Finite Element methods lack this property. Additional BIEM attributes include versatility, ease of use, rapid noise predictions, coupling of propagation and radiation both forward and aft, implementable on midrange personal computers, and valid over a wide range of frequencies.

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.

Radiation boundary condition and anisotropy correction for finite difference solutions of the Helmholtz equation

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.; Webb, Jay C.

1994-01-01

In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The effectiveness of the correction factor in providing improvements to the computed solution is demonstrated in this paper.

Numerical boundary condition procedures and multigrid methods; Proceedings of the Symposium, NASA Ames Research Center, Moffett Field, CA, October 19-22, 1981

NASA Technical Reports Server (NTRS)

1982-01-01

Papers presented in this volume provide an overview of recent work on numerical boundary condition procedures and multigrid methods. The topics discussed include implicit boundary conditions for the solution of the parabolized Navier-Stokes equations for supersonic flows; far field boundary conditions for compressible flows; and influence of boundary approximations and conditions on finite-difference solutions. Papers are also presented on fully implicit shock tracking and on the stability of two-dimensional hyperbolic initial boundary value problems for explicit and implicit schemes.

Sensitivity Analysis of Hydraulic Head to Locations of Model Boundaries

DOE PAGES

Lu, Zhiming

2018-01-30

Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g. hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, wemore » derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. Finally, the approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.« less

Sensitivity Analysis of Hydraulic Head to Locations of Model Boundaries

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

Lu, Zhiming

Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g. hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, wemore » derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. Finally, the approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.« less

A collocation-shooting method for solving fractional boundary value problems

NASA Astrophysics Data System (ADS)

Al-Mdallal, Qasem M.; Syam, Muhammed I.; Anwar, M. N.

2010-12-01

In this paper, we discuss the numerical solution of special class of fractional boundary value problems of order 2. The method of solution is based on a conjugating collocation and spline analysis combined with shooting method. A theoretical analysis about the existence and uniqueness of exact solution for the present class is proven. Two examples involving Bagley-Torvik equation subject to boundary conditions are also presented; numerical results illustrate the accuracy of the present scheme.

An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition

NASA Astrophysics Data System (ADS)

Azarnavid, Babak; Parand, Kourosh; Abbasbandy, Saeid

2018-06-01

This article discusses an iterative reproducing kernel method with respect to its effectiveness and capability of solving a fourth-order boundary value problem with nonlinear boundary conditions modeling beams on elastic foundations. Since there is no method of obtaining reproducing kernel which satisfies nonlinear boundary conditions, the standard reproducing kernel methods cannot be used directly to solve boundary value problems with nonlinear boundary conditions as there is no knowledge about the existence and uniqueness of the solution. The aim of this paper is, therefore, to construct an iterative method by the use of a combination of reproducing kernel Hilbert space method and a shooting-like technique to solve the mentioned problems. Error estimation for reproducing kernel Hilbert space methods for nonlinear boundary value problems have yet to be discussed in the literature. In this paper, we present error estimation for the reproducing kernel method to solve nonlinear boundary value problems probably for the first time. Some numerical results are given out to demonstrate the applicability of the method.

Existence and Regularity Results for the Inviscid Primitive Equations with Lateral Periodicity

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

Hamouda, Makram, E-mail: mahamoud@indiana.edu; Jung, Chang-Yeol, E-mail: changyeoljung@gmail.com; Temam, Roger, E-mail: temam@indiana.edu

2016-06-15

The article is devoted to prove the existence and regularity of the solutions of the 3D inviscid Linearized Primitive Equations (LPEs) in a channel with lateral periodicity. This was assumed in a previous work (Hamouda et al. in Discret Contin Dyn Syst Ser S 6(2):401–422, 2013) which is concerned with the boundary layers generated by the corresponding viscous problem. Although the equations under investigation here are of hyperbolic type, the standard methods do not apply because of the specificity of the hyperbolic system. A set of non-local boundary conditions for the inviscid LPEs has to be imposed at the lateralmore » boundary of the channel making thus the system well-posed.« less

Falkner-Skan Boundary Layer Flow of a Sisko Fluid

NASA Astrophysics Data System (ADS)

Khan, Masood; Shahzad, Azeem

2012-09-01

In this paper, we investigate the steady boundary layer flow of a non-Newtonian fluid, represented by a Sisko fluid, over a wedge in a moving fluid. The equations of motion are derived for boundary layer flow of an incompressible Sisko fluid using appropriate similarity variables. The governing equations are reduced to a single third-order highly nonlinear ordinary differential equation in the dimensionless stream function, which is then solved analytically using the homotopy analysis method. Some important parameters have been discussed by this study, which include the power law index n, the material parameter A, the wedge shape factor b, and the skin friction coefficient Cf. A comprehensive study is made between the results of the Sisko and the power-law fluids.

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

NASA Astrophysics Data System (ADS)

Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohammed A.

2014-09-01

In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach

Inverse random source scattering for the Helmholtz equation in inhomogeneous media

NASA Astrophysics Data System (ADS)

Li, Ming; Chen, Chuchu; Li, Peijun

2018-01-01

This paper is concerned with an inverse random source scattering problem in an inhomogeneous background medium. The wave propagation is modeled by the stochastic Helmholtz equation with the source driven by additive white noise. The goal is to reconstruct the statistical properties of the random source such as the mean and variance from the boundary measurement of the radiated random wave field at multiple frequencies. Both the direct and inverse problems are considered. We show that the direct problem has a unique mild solution by a constructive proof. For the inverse problem, we derive Fredholm integral equations, which connect the boundary measurement of the radiated wave field with the unknown source function. A regularized block Kaczmarz method is developed to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

Review of literature on the finite-element solution of the equations of two-dimensional surface-water flow in the horizontal plane

USGS Publications Warehouse

Lee, Jonathan K.; Froehlich, David C.

1987-01-01

Published literature on the application of the finite-element method to solving the equations of two-dimensional surface-water flow in the horizontal plane is reviewed in this report. The finite-element method is ideally suited to modeling two-dimensional flow over complex topography with spatially variable resistance. A two-dimensional finite-element surface-water flow model with depth and vertically averaged velocity components as dependent variables allows the user great flexibility in defining geometric features such as the boundaries of a water body, channels, islands, dikes, and embankments. The following topics are reviewed in this report: alternative formulations of the equations of two-dimensional surface-water flow in the horizontal plane; basic concepts of the finite-element method; discretization of the flow domain and representation of the dependent flow variables; treatment of boundary conditions; discretization of the time domain; methods for modeling bottom, surface, and lateral stresses; approaches to solving systems of nonlinear equations; techniques for solving systems of linear equations; finite-element alternatives to Galerkin's method of weighted residuals; techniques of model validation; and preparation of model input data. References are listed in the final chapter.

Near Continuum Velocity and Temperature Coupled Compressible Boundary Layer Flow over a Flat Plate

NASA Astrophysics Data System (ADS)

He, Xin; Cai, Chunpei

2017-04-01

The problem of a compressible gas flows over a flat plate with the velocity-slip and temperature-jump boundary conditions are being studied. The standard single- shooting method is applied to obtain the exact solutions for velocity and temperature profiles when the momentum and energy equations are weakly coupled. A double-shooting method is applied if these two equations are closely coupled. If the temperature affects the velocity directly, more significant velocity slip happens at locations closer to the plate's leading edge, and inflections on the velocity profiles appear, indicating flows may become unstable. As a consequence, the temperature-jump and velocity-slip boundary conditions may trigger earlier flow transitions from a laminar to a turbulent flow state.

A computer program to generate two-dimensional grids about airfoils and other shapes by the use of Poisson's equation

NASA Technical Reports Server (NTRS)

Sorenson, R. L.

1980-01-01

A method for generating two dimensional finite difference grids about airfoils and other shapes by the use of the Poisson differential equation is developed. The inhomogeneous terms are automatically chosen such that two important effects are imposed on the grid at both the inner and outer boundaries. The first effect is control of the spacing between mesh points along mesh lines intersecting the boundaries. The second effect is control of the angles with which mesh lines intersect the boundaries. A FORTRAN computer program has been written to use this method. A description of the program, a discussion of the control parameters, and a set of sample cases are included.

Solving Differential Equations Using Modified Picard Iteration

ERIC Educational Resources Information Center

Robin, W. A.

2010-01-01

Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

A Novel Method for Modeling Neumann and Robin Boundary Conditions in Smoothed Particle Hydrodynamics

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

Ryan, Emily M.; Tartakovsky, Alexandre M.; Amon, Cristina

2010-08-26

In this paper we present an improved method for handling Neumann or Robin boundary conditions in smoothed particle hydrodynamics. The Neumann and Robin boundary conditions are common to many physical problems (such as heat/mass transfer), and can prove challenging to model in volumetric modeling techniques such as smoothed particle hydrodynamics (SPH). A new SPH method for diffusion type equations subject to Neumann or Robin boundary conditions is proposed. The new method is based on the continuum surface force model [1] and allows an efficient implementation of the Neumann and Robin boundary conditions in the SPH method for geometrically complex boundaries.more » The paper discusses the details of the method and the criteria needed to apply the model. The model is used to simulate diffusion and surface reactions and its accuracy is demonstrated through test cases for boundary conditions describing different surface reactions.« less

Similar solutions for viscous hypersonic flow over a slender three-fourths-power body of revolution

NASA Technical Reports Server (NTRS)

Lin, Chin-Shun

1987-01-01

For hypersonic flow with a shock wave, there is a similar solution consistent throughout the viscous and inviscid layers along a very slender three-fourths-power body of revolution The strong pressure interaction problem can then be treated by the method of similarity. Numerical calculations are performed in the viscous region with the edge pressure distribution known from the inviscid similar solutions. The compressible laminar boundary-layer equations are transformed into a system of ordinary differential equations. The resulting two-point boundary value problem is then solved by the Runge-Kutta method with a modified Newton's method for the corresponding boundary conditions. The effects of wall temperature, mass bleeding, and body transverse curvature are investigated. The induced pressure, displacement thickness, skin friction, and heat transfer due to the previously mentioned parameters are estimated and analyzed.

Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modelling

NASA Astrophysics Data System (ADS)

Gao, Hongwei; Zhang, Jianfeng

2008-09-01

The perfectly matched layer (PML) absorbing boundary condition is incorporated into an irregular-grid elastic-wave modelling scheme, thus resulting in an irregular-grid PML method. We develop the irregular-grid PML method using the local coordinate system based PML splitting equations and integral formulation of the PML equations. The irregular-grid PML method is implemented under a discretization of triangular grid cells, which has the ability to absorb incident waves in arbitrary directions. This allows the PML absorbing layer to be imposed along arbitrary geometrical boundaries. As a result, the computational domain can be constructed with smaller nodes, for instance, to represent the 2-D half-space by a semi-circle rather than a rectangle. By using a smooth artificial boundary, the irregular-grid PML method can also avoid the special treatments to the corners, which lead to complex computer implementations in the conventional PML method. We implement the irregular-grid PML method in both 2-D elastic isotropic and anisotropic media. The numerical simulations of a VTI lamb's problem, wave propagation in an isotropic elastic medium with curved surface and in a TTI medium demonstrate the good behaviour of the irregular-grid PML method.

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

NASA Technical Reports Server (NTRS)

Thomas, P. D.

1979-01-01

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

An improved viscid/inviscid interaction procedure for transonic flow over airfoils

NASA Technical Reports Server (NTRS)

Melnik, R. E.; Chow, R. R.; Mead, H. R.; Jameson, A.

1985-01-01

A new interacting boundary layer approach for computing the viscous transonic flow over airfoils is described. The theory includes a complete treatment of viscous interaction effects induced by the wake and accounts for normal pressure gradient effects across the boundary layer near trailing edges. The method is based on systematic expansions of the full Reynolds equation of turbulent flow in the limit of Reynolds numbers, Reynolds infinity. Procedures are developed for incorporating the local trailing edge solution into the numerical solution of the coupled full potential and integral boundary layer equations. Although the theory is strictly applicable to airfoils with cusped or nearly cusped trailing edges and to turbulent boundary layers that remain fully attached to the airfoil surface, the method was successfully applied to more general airfoils and to flows with small separation zones. Comparisons of theoretical solutions with wind tunnel data indicate the present method can accurately predict the section characteristics of airfoils including the absolute levels of drag.

Enriched reproducing kernel particle method for fractional advection-diffusion equation

NASA Astrophysics Data System (ADS)

Ying, Yuping; Lian, Yanping; Tang, Shaoqiang; Liu, Wing Kam

2018-06-01

The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection-diffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

The accurate solution of Poisson's equation by expansion in Chebyshev polynomials

NASA Technical Reports Server (NTRS)

Haidvogel, D. B.; Zang, T.

1979-01-01

A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.

Double Diffusive Magnetohydrodynamic (MHD) Mixed Convective Slip Flow along a Radiating Moving Vertical Flat Plate with Convective Boundary Condition

PubMed Central

Rashidi, Mohammad M.; Kavyani, Neda; Abelman, Shirley; Uddin, Mohammed J.; Freidoonimehr, Navid

2014-01-01

In this study combined heat and mass transfer by mixed convective flow along a moving vertical flat plate with hydrodynamic slip and thermal convective boundary condition is investigated. Using similarity variables, the governing nonlinear partial differential equations are converted into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved using a semi-numerical/analytical method called the differential transform method and results are compared with numerical results. Close agreement is found between the present method and the numerical method. Effects of the controlling parameters, including convective heat transfer, magnetic field, buoyancy ratio, hydrodynamic slip, mixed convective, Prandtl number and Schmidt number are investigated on the dimensionless velocity, temperature and concentration profiles. In addition effects of different parameters on the skin friction factor, , local Nusselt number, , and local Sherwood number are shown and explained through tables. PMID:25343360

An Examination of Higher-Order Treatments of Boundary Conditions in Split-Step Fourier Parabolic Equation Models

DTIC Science & Technology

2015-06-01

method provides improved agreement with a benchmark solution at longer ranges. 14. SUBJECT TERMS parabolic equation , Monterey Miami...elliptic Helmholtz wave equation dates back to mid-1940s, when Leontovich and Fock introduced the PE method to the problem of radio-wave propagation in...improvements in the solutions . B. PROBLEM STATEMENT The Monterey-Miami Parabolic Equation (MMPE) model was developed in the mid-1990s and since then has

Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions

NASA Astrophysics Data System (ADS)

Ding, Xiao-Li; Nieto, Juan J.

2017-11-01

In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

Partial differential equations of 3D boundary layer and their numerical solutions in turbomachinery

NASA Astrophysics Data System (ADS)

Zhang, Guoqing; Hua, Yaonan; Wu, Chung-Hua

1991-08-01

This paper studies the 3D boundary layer equations (3DBLE) and their numerical solutions in turbomachinery: (1) the general form of 3DBLE in turbomachines with rotational and curvature effects are derived under the semiorthogonal coordinate system, in which the normal pressure gradient is not equal to zero; (2) the method of solution of the 3DBLE is discussed; (3) the 3D boundary layers on the rotating blade surface, IGV endwall, rotor endwall (with a relatively moving boundary) are numerically solved, and the predicted data correlates well with the measured data; and (4) the comparison is made between the numerical results of 3DBLE with and without normal pressure gradient.

Boundary-Layer Receptivity and Integrated Transition Prediction

NASA Technical Reports Server (NTRS)

Chang, Chau-Lyan; Choudhari, Meelan

2005-01-01

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

Direct simulation of groundwater age

USGS Publications Warehouse

Goode, Daniel J.

1996-01-01

A new method is proposed to simulate groundwater age directly, by use of an advection-dispersion transport equation with a distributed zero-order source of unit (1) strength, corresponding to the rate of aging. The dependent variable in the governing equation is the mean age, a mass-weighted average age. The governing equation is derived from residence-time-distribution concepts for the case of steady flow. For the more general case of transient flow, a transient governing equation for age is derived from mass-conservation principles applied to conceptual “age mass.” The age mass is the product of the water mass and its age, and age mass is assumed to be conserved during mixing. Boundary conditions include zero age mass flux across all noflow and inflow boundaries and no age mass dispersive flux across outflow boundaries. For transient-flow conditions, the initial distribution of age must be known. The solution of the governing transport equation yields the spatial distribution of the mean groundwater age and includes diffusion, dispersion, mixing, and exchange processes that typically are considered only through tracer-specific solute transport simulation. Traditional methods have relied on advective transport to predict point values of groundwater travel time and age. The proposed method retains the simplicity and tracer-independence of advection-only models, but incorporates the effects of dispersion and mixing on volume-averaged age. Example simulations of age in two idealized regional aquifer systems, one homogeneous and the other layered, demonstrate the agreement between the proposed method and traditional particle-tracking approaches and illustrate use of the proposed method to determine the effects of diffusion, dispersion, and mixing on groundwater age.

Boundary control of elliptic solutions to enforce local constraints

NASA Astrophysics Data System (ADS)

Bal, G.; Courdurier, M.

We present a constructive method to devise boundary conditions for solutions of second-order elliptic equations so that these solutions satisfy specific qualitative properties such as: (i) the norm of the gradient of one solution is bounded from below by a positive constant in the vicinity of a finite number of prescribed points; (ii) the determinant of gradients of n solutions is bounded from below in the vicinity of a finite number of prescribed points. Such constructions find applications in recent hybrid medical imaging modalities. The methodology is based on starting from a controlled setting in which the constraints are satisfied and continuously modifying the coefficients in the second-order elliptic equation. The boundary condition is evolved by solving an ordinary differential equation (ODE) defined via appropriate optimality conditions. Unique continuations and standard regularity results for elliptic equations are used to show that the ODE admits a solution for sufficiently long times.

a Speculative Study on Negative-Dimensional Potential and Wave Problems by Implicit Calculus Modeling Approach

NASA Astrophysics Data System (ADS)

Chen, Wen; Wang, Fajie

Based on the implicit calculus equation modeling approach, this paper proposes a speculative concept of the potential and wave operators on negative dimensionality. Unlike the standard partial differential equation (PDE) modeling, the implicit calculus modeling approach does not require the explicit expression of the PDE governing equation. Instead the fundamental solution of physical problem is used to implicitly define the differential operator and to implement simulation in conjunction with the appropriate boundary conditions. In this study, we conjecture an extension of the fundamental solution of the standard Laplace and Helmholtz equations to negative dimensionality. And then by using the singular boundary method, a recent boundary discretization technique, we investigate the potential and wave problems using the fundamental solution on negative dimensionality. Numerical experiments reveal that the physics behaviors on negative dimensionality may differ on positive dimensionality. This speculative study might open an unexplored territory in research.

Friction-term response to boundary-condition type in flow models

USGS Publications Warehouse

Schaffranek, R.W.; Lai, C.

1996-01-01

The friction-slope term in the unsteady open-channel flow equations is examined using two numerical models based on different formulations of the governing equations and employing different solution methods. The purposes of the study are to analyze, evaluate, and demonstrate the behavior of the term in a set of controlled numerical experiments using varied types and combinations of boundary conditions. Results of numerical experiments illustrate that a given model can respond inconsistently for the identical resistance-coefficient value under different types and combinations of boundary conditions. Findings also demonstrate that two models employing different dependent variables and solution methods can respond similarly for the identical resistance-coefficient value under similar types and combinations of boundary conditions. Discussion of qualitative considerations and quantitative experimental results provides insight into the proper treatment, evaluation, and significance of the friction-slope term, thereby offering practical guidelines for model implementation and calibration.

Boundary condition at a two-phase interface in the lattice Boltzmann method for the convection-diffusion equation.

PubMed

Yoshida, Hiroaki; Kobayashi, Takayuki; Hayashi, Hidemitsu; Kinjo, Tomoyuki; Washizu, Hitoshi; Fukuzawa, Kenji

2014-07-01

A boundary scheme in the lattice Boltzmann method (LBM) for the convection-diffusion equation, which correctly realizes the internal boundary condition at the interface between two phases with different transport properties, is presented. The difficulty in satisfying the continuity of flux at the interface in a transient analysis, which is inherent in the conventional LBM, is overcome by modifying the collision operator and the streaming process of the LBM. An asymptotic analysis of the scheme is carried out in order to clarify the role played by the adjustable parameters involved in the scheme. As a result, the internal boundary condition is shown to be satisfied with second-order accuracy with respect to the lattice interval, if we assign appropriate values to the adjustable parameters. In addition, two specific problems are numerically analyzed, and comparison with the analytical solutions of the problems numerically validates the proposed scheme.

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.

Numerical modeling of crystal growth in Bridgman device

NASA Astrophysics Data System (ADS)

Vompe, Dmitry Aleksandrovich

1997-12-01

The standard model for the growth of a crystal from a pure substance or diluted binary mixture contains transport equations for heat and phase change conditions at the solidification front. A numerical method is constructed for simulations of crystal growth in a vertical Bridgman device. The method is based on a boundary fitting technique in which melted and solidified regions are mapped onto a fixed rectangular logical domain. The Alternating Directions scheme (ADI) is used to treat the diffusive terms implicitly, with explicit methods are used for the remaining terms in the mapped temperature equations with variable coefficients. The nonlinear equation for the solid/liquid interface motion is solved by the modified Euler technique. Results obtained from the calculations have been used to study the influence of various boundary conditions imposed on the sidewalls and the top and bottom of the ampoule. Conditions are identified that lead to a steadily growing crystal and results are compared with an asymptotic one- dimensional model. Criteria based on ampoule length and boundary conditions being derived and compared with a previously developed one-dimensional model. Various cases have been considered to determine conditions for maintaining a nearly flat interface. It was found that the interface amplitude can be decreased by a factor of 100 (even 1,000) by optimizing temperature boundary conditions.

On the existence of solutions to a one-dimensional degenerate nonlinear wave equation

NASA Astrophysics Data System (ADS)

Hu, Yanbo

2018-07-01

This paper is concerned with the degenerate initial-boundary value problem to the one-dimensional nonlinear wave equation utt =((1 + u) aux) x which arises in a number of various physical contexts. The global existence of smooth solutions to the degenerate problem was established under relaxed conditions on the initial-boundary data by the characteristic decomposition method. Moreover, we show that the solution is uniformly C 1 , α continuous up to the degenerate boundary and the degenerate curve is C 1 , α continuous for α ∈ (0 , min ⁡ a/1+a, 1/1+a).

Finite difference elastic wave modeling with an irregular free surface using ADER scheme

NASA Astrophysics Data System (ADS)

Almuhaidib, Abdulaziz M.; Nafi Toksöz, M.

2015-06-01

In numerical modeling of seismic wave propagation in the earth, we encounter two important issues: the free surface and the topography of the surface (i.e. irregularities). In this study, we develop a 2D finite difference solver for the elastic wave equation that combines a 4th- order ADER scheme (Arbitrary high-order accuracy using DERivatives), which is widely used in aeroacoustics, with the characteristic variable method at the free surface boundary. The idea is to treat the free surface boundary explicitly by using ghost values of the solution for points beyond the free surface to impose the physical boundary condition. The method is based on the velocity-stress formulation. The ultimate goal is to develop a numerical solver for the elastic wave equation that is stable, accurate and computationally efficient. The solver treats smooth arbitrary-shaped boundaries as simple plane boundaries. The computational cost added by treating the topography is negligible compared to flat free surface because only a small number of grid points near the boundary need to be computed. In the presence of topography, using 10 grid points per shortest shear-wavelength, the solver yields accurate results. Benchmark numerical tests using several complex models that are solved by our method and other independent accurate methods show an excellent agreement, confirming the validity of the method for modeling elastic waves with an irregular free surface.

Validation of three-dimensional incompressible spatial direct numerical simulation code: A comparison with linear stability and parabolic stability equation theories for boundary-layer transition on a flat plate

NASA Technical Reports Server (NTRS)

Joslin, Ronald D.; Streett, Craig L.; Chang, Chau-Lyan

1992-01-01

Spatially evolving instabilities in a boundary layer on a flat plate are computed by direct numerical simulation (DNS) of the incompressible Navier-Stokes equations. In a truncated physical domain, a nonstaggered mesh is used for the grid. A Chebyshev-collocation method is used normal to the wall; finite difference and compact difference methods are used in the streamwise direction; and a Fourier series is used in the spanwise direction. For time stepping, implicit Crank-Nicolson and explicit Runge-Kutta schemes are used to the time-splitting method. The influence-matrix technique is used to solve the pressure equation. At the outflow boundary, the buffer-domain technique is used to prevent convective wave reflection or upstream propagation of information from the boundary. Results of the DNS are compared with those from both linear stability theory (LST) and parabolized stability equation (PSE) theory. Computed disturbance amplitudes and phases are in very good agreement with those of LST (for small inflow disturbance amplitudes). A measure of the sensitivity of the inflow condition is demonstrated with both LST and PSE theory used to approximate inflows. Although the DNS numerics are very different than those of PSE theory, the results are in good agreement. A small discrepancy in the results that does occur is likely a result of the variation in PSE boundary condition treatment in the far field. Finally, a small-amplitude wave triad is forced at the inflow, and simulation results are compared with those of LST. Again, very good agreement is found between DNS and LST results for the 3-D simulations, the implication being that the disturbance amplitudes are sufficiently small that nonlinear interactions are negligible.

Divergence Boundary Conditions for Vector Helmholtz Equations with Divergence Constraints

NASA Technical Reports Server (NTRS)

Kangro, Urve; Nicolaides, Roy

1997-01-01

The idea of replacing a divergence constraint by a divergence boundary condition is investigated. The connections between the formulations are considered in detail. It is shown that the most common methods of using divergence boundary conditions do not always work properly. Necessary and sufficient conditions for the equivalence of the formulations are given.

Testing thermal gradient driving force for grain boundary migration using molecular dynamics simulations

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

Bai, Xian-Ming; Zhang, Yongfeng; Tonks, Michael R.

2015-02-01

Strong thermal gradients in low-thermal-conductivity ceramics may drive extended defects, such as grain boundaries and voids, to migrate in preferential directions. In this work, molecular dynamics simulations are conducted to study thermal gradient driven grain boundary migration and to verify a previously proposed thermal gradient driving force equation, using uranium dioxide as a model system. It is found that a thermal gradient drives grain boundaries to migrate up the gradient and the migration velocity increases under a constant gradient owing to the increase in mobility with temperature. Different grain boundaries migrate at very different rates due to their different intrinsicmore » mobilities. The extracted mobilities from the thermal gradient driven simulations are compared with those calculated from two other well-established methods and good agreement between the three different methods is found, demonstrating that the theoretical equation of the thermal gradient driving force is valid, although a correction of one input parameter should be made. The discrepancy in the grain boundary mobilities between modeling and experiments is also discussed.« less

Acceleration of incremental-pressure-correction incompressible flow computations using a coarse-grid projection method

NASA Astrophysics Data System (ADS)

Kashefi, Ali; Staples, Anne

2016-11-01

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping functions transfer data between the two grids. Here we propose a version of CGP for incompressible flow computations using incremental pressure correction methods, called IFEi-CGP (implicit-time-integration, finite-element, incremental coarse grid projection). Incremental pressure correction schemes solve Poisson's equation for an intermediate variable and not the pressure itself. This fact contributes to IFEi-CGP's efficiency in two ways. First, IFEi-CGP preserves the velocity field accuracy even for a high level of pressure field grid coarsening and thus significant speedup is achieved. Second, because incremental schemes reduce the errors that arise from boundaries with artificial homogenous Neumann conditions, CGP generates undamped flows for simulations with velocity Dirichlet boundary conditions. Comparisons of the data accuracy and CPU times for the incremental-CGP versus non-incremental-CGP computations are presented.

Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smooth particle hydrodynamics

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

Pan, Wenxiao; Daily, Michael D.; Baker, Nathan A.

2015-12-01

We demonstrate the accuracy and effectiveness of a Lagrangian particle-based method, smoothed particle hydrodynamics (SPH), to study diffusion in biomolecular systems by numerically solving the time-dependent Smoluchowski equation for continuum diffusion. The numerical method is first verified in simple systems and then applied to the calculation of ligand binding to an acetylcholinesterase monomer. Unlike previous studies, a reactive Robin boundary condition (BC), rather than the absolute absorbing (Dirichlet) boundary condition, is considered on the reactive boundaries. This new boundary condition treatment allows for the analysis of enzymes with "imperfect" reaction rates. Rates for inhibitor binding to mAChE are calculated atmore » various ionic strengths and compared with experiment and other numerical methods. We find that imposition of the Robin BC improves agreement between calculated and experimental reaction rates. Although this initial application focuses on a single monomer system, our new method provides a framework to explore broader applications of SPH in larger-scale biomolecular complexes by taking advantage of its Lagrangian particle-based nature.« less

Discrete transparent boundary conditions for the mixed KDV-BBM equation

NASA Astrophysics Data System (ADS)

Besse, Christophe; Noble, Pascal; Sanchez, David

2017-09-01

In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) and Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data.

Explicitly represented polygon wall boundary model for the explicit MPS method

NASA Astrophysics Data System (ADS)

Mitsume, Naoto; Yoshimura, Shinobu; Murotani, Kohei; Yamada, Tomonori

2015-05-01

This study presents an accurate and robust boundary model, the explicitly represented polygon (ERP) wall boundary model, to treat arbitrarily shaped wall boundaries in the explicit moving particle simulation (E-MPS) method, which is a mesh-free particle method for strong form partial differential equations. The ERP model expresses wall boundaries as polygons, which are explicitly represented without using the distance function. These are derived so that for viscous fluids, and with less computational cost, they satisfy the Neumann boundary condition for the pressure and the slip/no-slip condition on the wall surface. The proposed model is verified and validated by comparing computed results with the theoretical solution, results obtained by other models, and experimental results. Two simulations with complex boundary movements are conducted to demonstrate the applicability of the E-MPS method to the ERP model.

A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Scott, James R.

1994-01-01

This paper is concerned with the continued development of a new numerical method, the space-time solution element (STS) method, for solving conservation laws. The present work focuses on the two-dimensional, steady, incompressible Navier-Stokes equations. Using first an integral approach, and then a differential approach, the discrete flux conservation equations presented in a recent paper are rederived. Here a simpler method for determining the flux expressions at cell interfaces is given; a systematic and rigorous derivation of the conditions used to simulate the differential form of the governing conservation law(s) is provided; necessary and sufficient conditions for a discrete approximation to satisfy a conservation law in E2 are derived; and an estimate of the local truncation error is given. A specific scheme is then constructed for the solution of the thin airfoil boundary layer problem. Numerical results are presented which demonstrate the ability of the scheme to accurately resolve the developing boundary layer and wake regions using grids which are much coarser than those employed by other numerical methods. It is shown that ten cells in the cross-stream direction are sufficient to accurately resolve the developing airfoil boundary layer.

Accurate radiative transfer calculations for layered media.

PubMed

Selden, Adrian C

2016-07-01

Simple yet accurate results for radiative transfer in layered media with discontinuous refractive index are obtained by the method of K-integrals. These are certain weighted integrals applied to the angular intensity distribution at the refracting boundaries. The radiative intensity is expressed as the sum of the asymptotic angular intensity distribution valid in the depth of the scattering medium and a transient term valid near the boundary. Integrated boundary equations are obtained, yielding simple linear equations for the intensity coefficients, enabling the angular emission intensity and the diffuse reflectance (albedo) and transmittance of the scattering layer to be calculated without solving the radiative transfer equation directly. Examples are given of half-space, slab, interface, and double-layer calculations, and extensions to multilayer systems are indicated. The K-integral method is orders of magnitude more accurate than diffusion theory and can be applied to layered scattering media with a wide range of scattering albedos, with potential applications to biomedical and ocean optics.

Meshless method for solving fixed boundary problem of plasma equilibrium

NASA Astrophysics Data System (ADS)

Imazawa, Ryota; Kawano, Yasunori; Itami, Kiyoshi

2015-07-01

This study solves the Grad-Shafranov equation with a fixed plasma boundary by utilizing a meshless method for the first time. Previous studies have utilized a finite element method (FEM) to solve an equilibrium inside the fixed separatrix. In order to avoid difficulties of FEM (such as mesh problem, difficulty of coding, expensive calculation cost), this study focuses on the meshless methods, especially RBF-MFS and KANSA's method to solve the fixed boundary problem. The results showed that CPU time of the meshless methods was ten to one hundred times shorter than that of FEM to obtain the same accuracy.

Direct Discrete Method for Neutronic Calculations

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

Vosoughi, Naser; Akbar Salehi, Ali; Shahriari, Majid

The objective of this paper is to introduce a new direct method for neutronic calculations. This method which is named Direct Discrete Method, is simpler than the neutron Transport equation and also more compatible with physical meaning of problems. This method is based on physic of problem and with meshing of the desired geometry, writing the balance equation for each mesh intervals and with notice to the conjunction between these mesh intervals, produce the final discrete equations series without production of neutron transport differential equation and mandatory passing from differential equation bridge. We have produced neutron discrete equations for amore » cylindrical shape with two boundary conditions in one group energy. The correction of the results from this method are tested with MCNP-4B code execution. (authors)« less

Double slip effects of Magnetohydrodynamic (MHD) boundary layer flow over an exponentially stretching sheet with radiation, heat source and chemical reaction

NASA Astrophysics Data System (ADS)

Shaharuz Zaman, Azmanira; Aziz, Ahmad Sukri Abd; Ali, Zaileha Md

2017-09-01

The double slips effect on the magnetohydrodynamic boundary layer flow over an exponentially stretching sheet with suction/blowing, radiation, chemical reaction and heat source is presented in this analysis. By using the similarity transformation, the governing partial differential equations of momentum, energy and concentration are transformed into the non-linear ordinary equations. These equations are solved using Runge-Kutta-Fehlberg method with shooting technique in MAPLE software environment. The effects of the various parameter on the velocity, temperature and concentration profiles are graphically presented and discussed.

Solution of the modified Helmholtz equation in a triangular domain and an application to diffusion-limited coalescence.

PubMed

ben-Avraham, D; Fokas, A S

2001-07-01

A new transform method for solving boundary value problems for linear and integrable nonlinear partial differential equations recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q(xx)(x,y)+q(yy)(x,y)-4 beta(2)q(x,y)=0 in the triangular domain 0< or =x< or =L-y< or =L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A+A<==>A, in the segment (-L/2,L/2), with traps at the edges.

Turbulence modeling for hypersonic flows

NASA Technical Reports Server (NTRS)

Marvin, J. G.; Coakley, T. J.

1989-01-01

Turbulence modeling for high speed compressible flows is described and discussed. Starting with the compressible Navier-Stokes equations, methods of statistical averaging are described by means of which the Reynolds-averaged Navier-Stokes equations are developed. Unknown averages in these equations are approximated using various closure concepts. Zero-, one-, and two-equation eddy viscosity models, algebraic stress models and Reynolds stress transport models are discussed. Computations of supersonic and hypersonic flows obtained using several of the models are discussed and compared with experimental results. Specific examples include attached boundary layer flows, shock wave boundary layer interactions and compressible shear layers. From these examples, conclusions regarding the status of modeling and recommendations for future studies are discussed.

Large-eddy simulation of a turbulent mixing layer

NASA Technical Reports Server (NTRS)

Mansour, N. N.; Ferziger, J. H.; Reynolds, W. C.

1978-01-01

The three dimensional, time dependent (incompressible) vorticity equations were used to simulate numerically the decay of isotropic box turbulence and time developing mixing layers. The vorticity equations were spatially filtered to define the large scale turbulence field, and the subgrid scale turbulence was modeled. A general method was developed to show numerical conservation of momentum, vorticity, and energy. The terms that arise from filtering the equations were treated (for both periodic boundary conditions and no stress boundary conditions) in a fast and accurate way by using fast Fourier transforms. Use of vorticity as the principal variable is shown to produce results equivalent to those obtained by use of the primitive variable equations.

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.

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

NASA Astrophysics Data System (ADS)

Pipkins, Daniel Scott

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

Implicit solution of Navier-Stokes equations on staggered curvilinear grids using a Newton-Krylov method with a novel analytical Jacobian.

NASA Astrophysics Data System (ADS)

Borazjani, Iman; Asgharzadeh, Hafez

2015-11-01

Flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates with explicit and semi-implicit schemes. Implicit schemes can be used to overcome these restrictions. However, implementing implicit solver for nonlinear equations including Navier-Stokes is not straightforward. Newton-Krylov subspace methods (NKMs) are one of the most advanced iterative methods to solve non-linear equations such as implicit descritization of the Navier-Stokes equation. The efficiency of NKMs massively depends on the Jacobian formation method, e.g., automatic differentiation is very expensive, and matrix-free methods slow down as the mesh is refined. Analytical Jacobian is inexpensive method, but derivation of analytical Jacobian for Navier-Stokes equation on staggered grid is challenging. The NKM with a novel analytical Jacobian was developed and validated against Taylor-Green vortex and pulsatile flow in a 90 degree bend. The developed method successfully handled the complex geometries such as an intracranial aneurysm with multiple overset grids, and immersed boundaries. It is shown that the NKM with an analytical Jacobian is 3 to 25 times faster than the fixed-point implicit Runge-Kutta method, and more than 100 times faster than automatic differentiation depending on the grid (size) and the flow problem. The developed methods are fully parallelized with parallel efficiency of 80-90% on the problems tested.

Artificial Boundary Conditions Based on the Difference Potentials Method

NASA Technical Reports Server (NTRS)

Tsynkov, Semyon V.

1996-01-01

While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears to be most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems present a wide class of practically important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms. Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABC's) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computationally) expensive. The methods from the second group (local ABC's) are, as a rule, algorithmically simple, numerically cheap, and geometrically universal; however, they usually lack accuracy of computations. In this paper we first present a survey and provide a comparative assessment of different existing methods for constructing the ABC's. Then, we describe a relatively new ABC's technique of ours and review the corresponding results. This new technique, in our opinion, is currently one of the most promising in the field. It enables one to construct such ABC's that combine the advantages relevant to the two aforementioned classes of existing methods. Our approach is based on application of the difference potentials method attributable to V. S. Ryaben'kii. This approach allows us to obtain highly accurate ABC's in the form of certain (nonlocal) boundary operator equations. The operators involved are analogous to the pseudodifferential boundary projections first introduced by A. P. Calderon and then also studied by R. T. Seeley. The apparatus of the boundary pseudodifferential equations, which has formerly been used mostly in the qualitative theory of integral equations and PDE'S, is now effectively employed for developing numerical methods in the different fields of scientific computing.

A method for the computational modeling of the physics of heart murmurs

NASA Astrophysics Data System (ADS)

Seo, Jung Hee; Bakhshaee, Hani; Garreau, Guillaume; Zhu, Chi; Andreou, Andreas; Thompson, William R.; Mittal, Rajat

2017-05-01

A computational method for direct simulation of the generation and propagation of blood flow induced sounds is proposed. This computational hemoacoustic method is based on the immersed boundary approach and employs high-order finite difference methods to resolve wave propagation and scattering accurately. The current method employs a two-step, one-way coupled approach for the sound generation and its propagation through the tissue. The blood flow is simulated by solving the incompressible Navier-Stokes equations using the sharp-interface immersed boundary method, and the equations corresponding to the generation and propagation of the three-dimensional elastic wave corresponding to the murmur are resolved with a high-order, immersed boundary based, finite-difference methods in the time-domain. The proposed method is applied to a model problem of aortic stenosis murmur and the simulation results are verified and validated by comparing with known solutions as well as experimental measurements. The murmur propagation in a realistic model of a human thorax is also simulated by using the computational method. The roles of hemodynamics and elastic wave propagation on the murmur are discussed based on the simulation results.

Photonic band gap structure simulator

DOEpatents

Chen, Chiping; Shapiro, Michael A.; Smirnova, Evgenya I.; Temkin, Richard J.; Sirigiri, Jagadishwar R.

2006-10-03

A system and method for designing photonic band gap structures. The system and method provide a user with the capability to produce a model of a two-dimensional array of conductors corresponding to a unit cell. The model involves a linear equation. Boundary conditions representative of conditions at the boundary of the unit cell are applied to a solution of the Helmholtz equation defined for the unit cell. The linear equation can be approximated by a Hermitian matrix. An eigenvalue of the Helmholtz equation is calculated. One computation approach involves calculating finite differences. The model can include a symmetry element, such as a center of inversion, a rotation axis, and a mirror plane. A graphical user interface is provided for the user's convenience. A display is provided to display to a user the calculated eigenvalue, corresponding to a photonic energy level in the Brilloin zone of the unit cell.

Singularity Preserving Numerical Methods for Boundary Integral Equations

NASA Technical Reports Server (NTRS)

Kaneko, Hideaki (Principal Investigator)

1996-01-01

In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.

The F(N) method for the one-angle radiative transfer equation applied to plant canopies

NASA Technical Reports Server (NTRS)

Ganapol, B. D.; Myneni, R. B.

1992-01-01

The paper presents a semianalytical solution method, called the F(N) method, for the one-angle radiative transfer equation in slab geometry. The F(N) method is based on two integral equations specifying the intensities exiting the boundaries of the vegetation canopy; the solution is obtained through an expansion in a set of basis functions with expansion coefficients to be determined. The advantage of this method is that it avoids spatial truncation error entirely because it requires discretization only in the angular variable.

Unsteady MHD free convection flow of casson fluid over an inclined vertical plate embedded in a porous media

NASA Astrophysics Data System (ADS)

Manideep, P.; Raju, R. Srinivasa; Rao, T. Siva Nageswar; Reddy, G. Jithender

2018-05-01

This paper deals, an unsteady magnetohydrodynamic heat transfer natural convection flow of non-Newtonian Casson fluid over an inclined vertical plate embedded in a porous media with the presence of boundary conditions such as oscillating velocity, constant wall temperature. The governing dimensionless boundary layer partial differential equations are reduced to simultaneous algebraic linear equation for velocity, temperature of Casson fluid through finite element method. Those equations are solved by Thomas algorithm after imposing the boundary conditions through MATLAB for analyzing the behavior of Casson fluid velocity and temperature with various physical parameters. Also analyzed the local skin-friction and rate of heat transfer. Compared the present results with earlier reported studies, the results are comprehensively authenticated and robust FEM.

Corrected simulations for one-dimensional diffusion processes with naturally occurring boundaries.

PubMed

Shafiey, Hassan; Gan, Xinjun; Waxman, David

2017-11-01

To simulate a diffusion process, a usual approach is to discretize the time in the associated stochastic differential equation. This is the approach used in the Euler method. In the present work we consider a one-dimensional diffusion process where the terms occurring, within the stochastic differential equation, prevent the process entering a region. The outcome is a naturally occurring boundary (which may be absorbing or reflecting). A complication occurs in a simulation of this situation. The term involving a random variable, within the discretized stochastic differential equation, may take a trajectory across the boundary into a "forbidden region." The naive way of dealing with this problem, which we refer to as the "standard" approach, is simply to reset the trajectory to the boundary, based on the argument that crossing the boundary actually signifies achieving the boundary. In this work we show, within the framework of the Euler method, that such resetting introduces a spurious force into the original diffusion process. This force may have a significant influence on trajectories that come close to a boundary. We propose a corrected numerical scheme, for simulating one-dimensional diffusion processes with naturally occurring boundaries. This involves correcting the standard approach, so that an exact property of the diffusion process is precisely respected. As a consequence, the proposed scheme does not introduce a spurious force into the dynamics. We present numerical test cases, based on exactly soluble one-dimensional problems with one or two boundaries, which suggest that, for a given value of the discrete time step, the proposed scheme leads to substantially more accurate results than the standard approach. Alternatively, the standard approach needs considerably more computation time to obtain a comparable level of accuracy to the proposed scheme, because the standard approach requires a significantly smaller time step.

Corrected simulations for one-dimensional diffusion processes with naturally occurring boundaries

NASA Astrophysics Data System (ADS)

Shafiey, Hassan; Gan, Xinjun; Waxman, David

2017-11-01

To simulate a diffusion process, a usual approach is to discretize the time in the associated stochastic differential equation. This is the approach used in the Euler method. In the present work we consider a one-dimensional diffusion process where the terms occurring, within the stochastic differential equation, prevent the process entering a region. The outcome is a naturally occurring boundary (which may be absorbing or reflecting). A complication occurs in a simulation of this situation. The term involving a random variable, within the discretized stochastic differential equation, may take a trajectory across the boundary into a "forbidden region." The naive way of dealing with this problem, which we refer to as the "standard" approach, is simply to reset the trajectory to the boundary, based on the argument that crossing the boundary actually signifies achieving the boundary. In this work we show, within the framework of the Euler method, that such resetting introduces a spurious force into the original diffusion process. This force may have a significant influence on trajectories that come close to a boundary. We propose a corrected numerical scheme, for simulating one-dimensional diffusion processes with naturally occurring boundaries. This involves correcting the standard approach, so that an exact property of the diffusion process is precisely respected. As a consequence, the proposed scheme does not introduce a spurious force into the dynamics. We present numerical test cases, based on exactly soluble one-dimensional problems with one or two boundaries, which suggest that, for a given value of the discrete time step, the proposed scheme leads to substantially more accurate results than the standard approach. Alternatively, the standard approach needs considerably more computation time to obtain a comparable level of accuracy to the proposed scheme, because the standard approach requires a significantly smaller time step.

Application of the Discrete Regularization Method to the Inverse of the Chord Vibration Equation

NASA Astrophysics Data System (ADS)

Wang, Linjun; Han, Xu; Wei, Zhouchao

The inverse problem of the initial condition about the boundary value of the chord vibration equation is ill-posed. First, we transform it into a Fredholm integral equation. Second, we discretize it by the trapezoidal formula method, and then obtain a severely ill-conditioned linear equation, which is sensitive to the disturbance of the data. In addition, the tiny error of right data causes the huge concussion of the solution. We cannot obtain good results by the traditional method. In this paper, we solve this problem by the Tikhonov regularization method, and the numerical simulations demonstrate that this method is feasible and effective.

Completed Beltrami-Michell Formulation for Analyzing Radially Symmetrical Bodies

NASA Technical Reports Server (NTRS)

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

1994-01-01

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

Boundary transfer matrices and boundary quantum KZ equations

NASA Astrophysics Data System (ADS)

Vlaar, Bart

2015-07-01

A simple relation between inhomogeneous transfer matrices and boundary quantum Knizhnik-Zamolodchikov (KZ) equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus, the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin's boundary transfer matrices by merely imposing appropriate reflection equations, in particular without using the conditions of crossing symmetry and unitarity of the R-matrix.

Existence and stability of periodic solutions of quasi-linear Korteweg — de Vries equation

NASA Astrophysics Data System (ADS)

Glyzin, S. D.; Kolesov, A. Yu; Preobrazhenskaia, M. M.

2017-01-01

We consider the scalar nonlinear differential-difference equation with two delays, which models electrical activity of a neuron. Under some additional suppositions for this equation well known method of quasi-normal forms can be applied. Its essence lies in the formal normalization of the Poincare - Dulac obtaining quasi-normal form and the subsequent application of the theorems of conformity. In this case, the result of the application of quasi-normal forms is a countable system of differential-difference equations, which can be turned into a boundary value problem of the Korteweg - de Vries equation. The investigation of this boundary value problem allows us to draw a conclusion about the behaviour of the original equation. Namely, for a suitable choice of parameters in the framework of this equation is implemented buffer phenomenon consisting in the presence of the bifurcation mechanism for the birth of an arbitrarily large number of stable cycles.

Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations.

PubMed

Meffin, Hamish; Tahayori, Bahman; Grayden, David B; Burkitt, Anthony N

2012-12-01

Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace's equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite's membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.

Modeling of Structural-Acoustic Interaction Using Coupled FE/BE Method and Control of Interior Acoustic Pressure Using Piezoelectric Actuators

NASA Technical Reports Server (NTRS)

Mei, Chuh; Shi, Yacheng

1997-01-01

A coupled finite element (FE) and boundary element (BE) approach is presented to model full coupled structural/acoustic/piezoelectric systems. The dual reciprocity boundary element method is used so that the natural frequencies and mode shapes of the coupled system can be obtained, and to extend this approach to time dependent problems. The boundary element method is applied to interior acoustic domains, and the results are very accurate when compared with limited exact solutions. Structural-acoustic problems are then analyzed with the coupled finite element/boundary element method, where the finite element method models the structural domain and the boundary element method models the acoustic domain. Results for a system consisting of an isotropic panel and a cubic cavity are in good agreement with exact solutions and experiment data. The response of a composite panel backed cavity is then obtained. The results show that the mass and stiffness of piezoelectric layers have to be considered. The coupled finite element and boundary element equations are transformed into modal coordinates, which is more convenient for transient excitation. Several transient problems are solved based on this formulation. Two control designs, a linear quadratic regulator (LQR) and a feedforward controller, are applied to reduce the acoustic pressure inside the cavity based on the equations in modal coordinates. The results indicate that both controllers can reduce the interior acoustic pressure and the plate deflection.

Preconditioned conjugate residual methods for the solution of spectral equations

NASA Technical Reports Server (NTRS)

Wong, Y. S.; Zang, T. A.; Hussaini, M. Y.

1986-01-01

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Numerical simulation of axisymmetric turbulent flow in combustors and diffusors. Ph.D. Thesis. Final Report

NASA Technical Reports Server (NTRS)

Yung, Chain Nan

1988-01-01

A method for predicting turbulent flow in combustors and diffusers is developed. The Navier-Stokes equations, incorporating a turbulence kappa-epsilon model equation, were solved in a nonorthogonal curvilinear coordinate system. The solution applied the finite volume method to discretize the differential equations and utilized the SIMPLE algorithm iteratively to solve the differenced equations. A zonal grid method, wherein the flow field was divided into several subsections, was developed. This approach permitted different computational schemes to be used in the various zones. In addition, grid generation was made a more simple task. However, treatment of the zonal boundaries required special handling. Boundary overlap and interpolating techniques were used and an adjustment of the flow variables was required to assure conservation of mass, momentum and energy fluxes. The numerical accuracy was assessed using different finite differencing methods, i.e., hybrid, quadratic upwind and skew upwind, to represent the convection terms. Flows in different geometries of combustors and diffusers were simulated and results compared with experimental data and good agreement was obtained.

Boundary-element modelling of dynamics in external poroviscoelastic problems

NASA Astrophysics Data System (ADS)

Igumnov, L. A.; Litvinchuk, S. Yu; Ipatov, A. A.; Petrov, A. N.

2018-04-01

A problem of a spherical cavity in porous media is considered. Porous media are assumed to be isotropic poroelastic or isotropic poroviscoelastic. The poroviscoelastic formulation is treated as a combination of Biot’s theory of poroelasticity and elastic-viscoelastic correspondence principle. Such viscoelastic models as Kelvin–Voigt, Standard linear solid, and a model with weakly singular kernel are considered. Boundary field study is employed with the help of the boundary element method. The direct approach is applied. The numerical scheme is based on the collocation method, regularized boundary integral equation, and Radau stepped scheme.

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

One shot methods for optimal control of distributed parameter systems 1: Finite dimensional control

NASA Technical Reports Server (NTRS)

Taasan, Shlomo

1991-01-01

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.

High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities

NASA Astrophysics Data System (ADS)

Britt, Darrell Steven, Jr.

Problems of time-harmonic wave propagation arise in important fields of study such as geological surveying, radar detection/evasion, and aircraft design. These often involve highfrequency waves, which demand high-order methods to mitigate the dispersion error. We propose a high-order method for computing solutions to the variable-coefficient inhomogeneous Helmholtz equation in two dimensions on domains bounded by piecewise smooth curves of arbitrary shape with a finite number of boundary singularities at known locations. We utilize compact finite difference (FD) schemes on regular structured grids to achieve highorder accuracy due to their efficiency and simplicity, as well as the capability to approximate variable-coefficient differential operators. In this work, a 4th-order compact FD scheme for the variable-coefficient Helmholtz equation on a Cartesian grid in 2D is derived and tested. The well known limitation of finite differences is that they lose accuracy when the boundary curve does not coincide with the discretization grid, which is a severe restriction on the geometry of the computational domain. Therefore, the algorithm presented in this work combines high-order FD schemes with the method of difference potentials (DP), which retains the efficiency of FD while allowing for boundary shapes that are not aligned with the grid without sacrificing the accuracy of the FD scheme. Additionally, the theory of DP allows for the universal treatment of the boundary conditions. One of the significant contributions of this work is the development of an implementation that accommodates general boundary conditions (BCs). In particular, Robin BCs with discontinuous coefficients are studied, for which we introduce a piecewise parameterization of the boundary curve. Problems with discontinuities in the boundary data itself are also studied. We observe that the design convergence rate suffers whenever the solution loses regularity due to the boundary conditions. This is because the FD scheme is only consistent for classical solutions of the PDE. For this reason, we implement the method of singularity subtraction as a means for restoring the design accuracy of the scheme in the presence of singularities at the boundary. While this method is well studied for low order methods and for problems in which singularities arise from the geometry (e.g., corners), we adapt it to our high-order scheme for curved boundaries via a conformal mapping and show that it can also be used to restore accuracy when the singularity arises from the BCs rather than the geometry. Altogether, the proposed methodology for 2D boundary value problems is computationally efficient, easily handles a wide class of boundary conditions and boundary shapes that are not aligned with the discretization grid, and requires little modification for solving new problems.

A Study of Multigrid Preconditioners Using Eigensystem Analysis

NASA Technical Reports Server (NTRS)

Roberts, Thomas W.; Swanson, R. C.

2005-01-01

The convergence properties of numerical schemes for partial differential equations are studied by examining the eigensystem of the discrete operator. This method of analysis is very general, and allows the effects of boundary conditions and grid nonuniformities to be examined directly. Algorithms for the Laplace equation and a two equation model hyperbolic system are examined.

Finite element approximation of the radiative transport equation in a medium with piece-wise constant refractive index

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

Lehtikangas, O., E-mail: Ossi.Lehtikangas@uef.fi; Tarvainen, T.; Department of Computer Science, University College London, Gower Street, London WC1E 6BT

2015-02-01

The radiative transport equation can be used as a light transport model in a medium with scattering particles, such as biological tissues. In the radiative transport equation, the refractive index is assumed to be constant within the medium. However, in biomedical media, changes in the refractive index can occur between different tissue types. In this work, light propagation in a medium with piece-wise constant refractive index is considered. Light propagation in each sub-domain with a constant refractive index is modeled using the radiative transport equation and the equations are coupled using boundary conditions describing Fresnel reflection and refraction phenomena onmore » the interfaces between the sub-domains. The resulting coupled system of radiative transport equations is numerically solved using a finite element method. The approach is tested with simulations. The results show that this coupled system describes light propagation accurately through comparison with the Monte Carlo method. It is also shown that neglecting the internal changes of the refractive index can lead to erroneous boundary measurements of scattered light.« less

Heat transfer to the transpired turbulent boundary layer.

NASA Technical Reports Server (NTRS)

Kays, W. M.

1972-01-01

This paper contains a summarization of five years work on an investigation on heat transfer to the transpired turbulent boundary layer. Experimental results are presented for friction coefficient and Stanton number over a wide range of blowing and suction for the case of constant free-stream velocity, holding certain blowing parameters constant. The problem of the accelerated turbulent boundary layer with transpiration is considered, experimental data are presented and discussed, and theoretical models for solution of the momentum equation under these conditions are presented. Data on turbulent Prandtl number are presented so that solutions to the energy equation may be obtained. Some examples of boundary layer heat transfer and friction coefficient predictions are presented using one of the models discussed, employing a finite difference solution method.

Numerical techniques for the solution of the compressible Navier-Stokes equations and implementation of turbulence models. [separated turbulent boundary layer flow problems

NASA Technical Reports Server (NTRS)

Baldwin, B. S.; Maccormack, R. W.; Deiwert, G. S.

1975-01-01

The time-splitting explicit numerical method of MacCormack is applied to separated turbulent boundary layer flow problems. Modifications of this basic method are developed to counter difficulties associated with complicated geometry and severe numerical resolution requirements of turbulence model equations. The accuracy of solutions is investigated by comparison with exact solutions for several simple cases. Procedures are developed for modifying the basic method to improve the accuracy. Numerical solutions of high-Reynolds-number separated flows over an airfoil and shock-separated flows over a flat plate are obtained. A simple mixing length model of turbulence is used for the transonic flow past an airfoil. A nonorthogonal mesh of arbitrary configuration facilitates the description of the flow field. For the simpler geometry associated with the flat plate, a rectangular mesh is used, and solutions are obtained based on a two-equation differential model of turbulence.

Pentadiagonal alternating-direction-implicit finite-difference time-domain method for two-dimensional Schrödinger equation

NASA Astrophysics Data System (ADS)

Tay, Wei Choon; Tan, Eng Leong

2014-07-01

In this paper, we have proposed a pentadiagonal alternating-direction-implicit (Penta-ADI) finite-difference time-domain (FDTD) method for the two-dimensional Schrödinger equation. Through the separation of complex wave function into real and imaginary parts, a pentadiagonal system of equations for the ADI method is obtained, which results in our Penta-ADI method. The Penta-ADI method is further simplified into pentadiagonal fundamental ADI (Penta-FADI) method, which has matrix-operator-free right-hand-sides (RHS), leading to the simplest and most concise update equations. As the Penta-FADI method involves five stencils in the left-hand-sides (LHS) of the pentadiagonal update equations, special treatments that are required for the implementation of the Dirichlet's boundary conditions will be discussed. Using the Penta-FADI method, a significantly higher efficiency gain can be achieved over the conventional Tri-ADI method, which involves a tridiagonal system of equations.

Explicit finite-volume time-marching calculations of total temperature distributions in turbulent flow

NASA Technical Reports Server (NTRS)

Nicholson, Stephen; Moore, Joan G.; Moore, John

1987-01-01

A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite volume, time-marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier-Stokes equations. The entire calculation is performed in the physical domain. This paper investigates the introduction of a new formulation of the energy equation which gives improved transient behavior as the calculation converges. The effect of variable Prandtl number on the temperature distribution through the boundary layer is also investigated. A turbulent boundary layer in an adverse pressure gradient (M = 0.55) is used to demonstrate the improved transient temperature distribution obtained when the new formulation of the energy equation is used. A flat plate turbulent boundary layer with a supersonic free-stream Mach number of 2.8 is used to investigate the effect of Prandtl number on the distribution of properties through the boundary layer. The computed total temperature distribution and recovery factor agree well with the measurements when a variable Prandtl number is used through the boundary layer.

Explicit finite-volume time-marching calculations of total temperature distributions in turbulent flow

NASA Technical Reports Server (NTRS)

Nicholson, Stephen; Moore, Joan G.; Moore, John

1986-01-01

A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite volume, time-marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier-Stokes equations. The entire calculation is performed in the physical domain. This paper investigates the introduction of a new formulation of the energy equation which gives improved transient behavior as the calculation converges. The effect of variable Prandtl number on the temperature distribution through the boundary layer is also investigated. A turbulent boundary layer in an adverse pressure gradient (M = 0.55) is used to demonstrate the improved transient temperature distribution obtained when the new formulation of the energy equation is used. A flat plate turbulent boundary layer with a supersonic free-stream Mach number of 2.8 is used to investigate the effect of Prandtl number on the distribution of properties through the boundary layer. The computed total temperature distribution and recovery factor agree well with the measurements when a variable Prandtl number is used through the boundary layer.

Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations

NASA Astrophysics Data System (ADS)

Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, Robert A.

2014-03-01

A Jacobi-Gauss-Lobatto collocation (J-GL-C) method, used in combination with the implicit Runge-Kutta method of fourth order, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schrödinger equations (NLSE) with initial-boundary data in 1+1 dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C is employed for approximating the functional dependence on the spatial variable, using (N-1) nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of 2(N-1) first-order ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small.

Smoothed Particle Hydrodynamics and its applications for multiphase flow and reactive transport in porous media

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

Tartakovsky, Alexandre M.; Trask, Nathaniel; Pan, K.

2016-03-11

Smoothed Particle Hydrodynamics (SPH) is a Lagrangian method based on a meshless discretization of partial differential equations. In this review, we present SPH discretization of the Navier-Stokes and Advection-Diffusion-Reaction equations, implementation of various boundary conditions, and time integration of the SPH equations, and we discuss applications of the SPH method for modeling pore-scale multiphase flows and reactive transport in porous and fractured media.

Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography

NASA Astrophysics Data System (ADS)

Medl'a, Matej; Mikula, Karol; Čunderlík, Róbert; Macák, Marek

2018-01-01

The paper presents a numerical solution of the oblique derivative boundary value problem on and above the Earth's topography using the finite volume method (FVM). It introduces a novel method for constructing non-uniform hexahedron 3D grids above the Earth's surface. It is based on an evolution of a surface, which approximates the Earth's topography, by mean curvature. To obtain optimal shapes of non-uniform 3D grid, the proposed evolution is accompanied by a tangential redistribution of grid nodes. Afterwards, the Laplace equation is discretized using FVM developed for such a non-uniform grid. The oblique derivative boundary condition is treated as a stationary advection equation, and we derive a new upwind type discretization suitable for non-uniform 3D grids. The discretization of the Laplace equation together with the discretization of the oblique derivative boundary condition leads to a linear system of equations. The solution of this system gives the disturbing potential in the whole computational domain including the Earth's surface. Numerical experiments aim to show properties and demonstrate efficiency of the developed FVM approach. The first experiments study an experimental order of convergence of the method. Then, a reconstruction of the harmonic function on the Earth's topography, which is generated from the EGM2008 or EIGEN-6C4 global geopotential model, is presented. The obtained FVM solutions show that refining of the computational grid leads to more precise results. The last experiment deals with local gravity field modelling in Slovakia using terrestrial gravity data. The GNSS-levelling test shows accuracy of the obtained local quasigeoid model.

An implicit-iterative solution of the heat conduction equation with a radiation boundary condition

NASA Technical Reports Server (NTRS)

Williams, S. D.; Curry, D. M.

1977-01-01

For the problem of predicting one-dimensional heat transfer between conducting and radiating mediums by an implicit finite difference method, four different formulations were used to approximate the surface radiation boundary condition while retaining an implicit formulation for the interior temperature nodes. These formulations are an explicit boundary condition, a linearized boundary condition, an iterative boundary condition, and a semi-iterative boundary method. The results of these methods in predicting surface temperature on the space shuttle orbiter thermal protection system model under a variety of heating rates were compared. The iterative technique caused the surface temperature to be bounded at each step. While the linearized and explicit methods were generally more efficient, the iterative and semi-iterative techniques provided a realistic surface temperature response without requiring step size control techniques.

Numerical computations on one-dimensional inverse scattering problems

NASA Technical Reports Server (NTRS)

Dunn, M. H.; Hariharan, S. I.

1983-01-01

An approximate method to determine the index of refraction of a dielectric obstacle is presented. For simplicity one dimensional models of electromagnetic scattering are treated. The governing equations yield a second order boundary value problem, in which the index of refraction appears as a functional parameter. The availability of reflection coefficients yield two additional boundary conditions. The index of refraction by a k-th order spline which can be written as a linear combination of B-splines is approximated. For N distinct reflection coefficients, the resulting N boundary value problems yield a system of N nonlinear equations in N unknowns which are the coefficients of the B-splines.

Three-dimensional analysis of chevron-notched specimens by boundary integral method

NASA Technical Reports Server (NTRS)

Mendelson, A.; Ghosn, L.

1983-01-01

The chevron-notched short bar and short rod specimens was analyzed by the boundary integral equations method. This method makes use of boundary surface elements in obtaining the solution. The boundary integral models were composed of linear triangular and rectangular surface segments. Results were obtained for two specimens with width to thickness ratios of 1.45 and 2.00 and for different crack length to width ratios ranging from 0.4 to 0.7. Crack opening displacement and stress intensity factors determined from displacement calculations along the crack front and compliance calculations were compared with experimental values and with finite element analysis.

A well-posed and stable stochastic Galerkin formulation of the incompressible Navier–Stokes equations with random data

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

Pettersson, Per, E-mail: per.pettersson@uib.no; Nordström, Jan, E-mail: jan.nordstrom@liu.se; Doostan, Alireza, E-mail: alireza.doostan@colorado.edu

2016-02-01

We present a well-posed stochastic Galerkin formulation of the incompressible Navier–Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimatemore » for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined.« less

A transonic interactive boundary-layer theory for laminar and turbulent flow over swept wings

NASA Technical Reports Server (NTRS)

Woodson, Shawn H.; Dejarnette, Fred R.

1988-01-01

A 3-D laminar and turbulent boundary-layer method is developed for compressible flow over swept wings. The governing equations and curvature terms are derived in detail for a nonorthogonal, curvilinear coordinate system. Reynolds shear-stress terms are modeled by the Cebeci-Smith eddy-viscosity formulation. The governing equations are descretized using the second-order accurate, predictor-corrector finite-difference technique of Matsuno, which has the advantage that the crossflow difference formulas are formed independent of the sign of the crossflow velocity component. The method is coupled with a full potential wing/body inviscid code (FLO-30) and the inviscid-viscous interaction is performed by updating the original wing surface with the viscous displacement surface calculated by the boundary-layer code. The number of these global iterations ranged from five to twelve depending on Mach number, sweep angle, and angle of attack. Several test cases are computed by this method and the results are compared with another inviscid-viscous interaction method (TAWFIVE) and with experimental data.

Time-Accurate, Unstructured-Mesh Navier-Stokes Computations with the Space-Time CESE Method

NASA Technical Reports Server (NTRS)

Chang, Chau-Lyan

2006-01-01

Application of the newly emerged space-time conservation element solution element (CESE) method to compressible Navier-Stokes equations is studied. In contrast to Euler equations solvers, several issues such as boundary conditions, numerical dissipation, and grid stiffness warrant systematic investigations and validations. Non-reflecting boundary conditions applied at the truncated boundary are also investigated from the stand point of acoustic wave propagation. Validations of the numerical solutions are performed by comparing with exact solutions for steady-state as well as time-accurate viscous flow problems. The test cases cover a broad speed regime for problems ranging from acoustic wave propagation to 3D hypersonic configurations. Model problems pertinent to hypersonic configurations demonstrate the effectiveness of the CESE method in treating flows with shocks, unsteady waves, and separations. Good agreement with exact solutions suggests that the space-time CESE method provides a viable alternative for time-accurate Navier-Stokes calculations of a broad range of problems.

Singular boundary method for global gravity field modelling

NASA Astrophysics Data System (ADS)

Cunderlik, Robert

2014-05-01

The singular boundary method (SBM) and method of fundamental solutions (MFS) are meshless boundary collocation techniques that use the fundamental solution of a governing partial differential equation (e.g. the Laplace equation) as their basis functions. They have been developed to avoid singular numerical integration as well as mesh generation in the traditional boundary element method (BEM). SBM have been proposed to overcome a main drawback of MFS - its controversial fictitious boundary outside the domain. The key idea of SBM is to introduce a concept of the origin intensity factors that isolate singularities of the fundamental solution and its derivatives using some appropriate regularization techniques. Consequently, the source points can be placed directly on the real boundary and coincide with the collocation nodes. In this study we deal with SBM applied for high-resolution global gravity field modelling. The first numerical experiment presents a numerical solution to the fixed gravimetric boundary value problem. The achieved results are compared with the numerical solutions obtained by MFS or the direct BEM indicating efficiency of all methods. In the second numerical experiments, SBM is used to derive the geopotential and its first derivatives from the Tzz components of the gravity disturbing tensor observed by the GOCE satellite mission. A determination of the origin intensity factors allows to evaluate the disturbing potential and gravity disturbances directly on the Earth's surface where the source points are located. To achieve high-resolution numerical solutions, the large-scale parallel computations are performed on the cluster with 1TB of the distributed memory and an iterative elimination of far zones' contributions is applied.

Solution of time fractional Black-Scholes European option pricing equation arising in financial market

NASA Astrophysics Data System (ADS)

Ravi Kanth, A. S. V.; Aruna, K.

2016-12-01

In this paper, we present fractional differential transform method (FDTM) and modified fractional differential transform method (MFDTM) for the solution of time fractional Black-Scholes European option pricing equation. The method finds the solution without any discretization, transformation, or restrictive assumptions with the use of appropriate initial or boundary conditions. The efficiency and exactitude of the proposed methods are tested by means of three examples.

Dynamic analysis of a hollow cylinder subject to a dual traveling force imposed on its inner surface

NASA Astrophysics Data System (ADS)

Lee, Sooyoung; Seok, Jongwon

2015-03-01

The dynamic behavior of a hollow cylinder under a dual traveling force applied to the inner surface is investigated in this study. The cylinder is constrained at both the top and bottom surfaces not to move in the length direction but free in other directions. And a dual force travels at a constant velocity along the length direction on the inner surface of the hollow cylinder. The resulting governing field equations and the associated boundary conditions are ruled by the general Hooke's law. Due to the nature of the field equations, proper adjoint system of equations and biorthogonality conditions were derived in a precise and detailed manner. To solve these field equations in this study, the method of separation of variable is used and the method of Fro¨benius is employed for the differential equations in the radial direction. Using the field equations, the eigenanalyses on both the original and its adjoint system were performed with great care, which results in the eigenfunction sets of both systems. The biorthogonality conditions were applied to the field equations to obtain the discretized equation for each mode. Using the solutions of the discretized equations that account for the boundary forcing terms, the critical speed for a dual traveling force for each mode could be computed.

Embedding methods for the steady Euler equations

NASA Technical Reports Server (NTRS)

Chang, S. H.; Johnson, G. M.

1983-01-01

An approach to the numerical solution of the steady Euler equations is to embed the first-order Euler system in a second-order system and then to recapture the original solution by imposing additional boundary conditions. Initial development of this approach and computational experimentation with it were previously based on heuristic physical reasoning. This has led to the construction of a relaxation procedure for the solution of two-dimensional steady flow problems. The theoretical justification for the embedding approach is addressed. It is proven that, with the appropriate choice of embedding operator and additional boundary conditions, the solution to the embedded system is exactly the one to the original Euler equations. Hence, solving the embedded version of the Euler equations will not produce extraneous solutions.

The three-wave equation on the half-line

NASA Astrophysics Data System (ADS)

Xu, Jian; Fan, Engui

2014-01-01

The Fokas method is used to analyze the initial-boundary value problem for the three-wave equation p-{bi-bj}/{ai-aj}p+∑k ({bk-bj}/{ak-aj}-{bi-bk}/{ai-ak})pp=0, i,j,k=1,2,3, on the half-line. Assuming that the solution p(x,t) exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ.

Compact scheme for systems of equations applied to fundamental problems of mechanics of continua

NASA Technical Reports Server (NTRS)

Klimkowski, Jerzy Z.

1990-01-01

Compact scheme formulation was used in the treatment of boundary conditions for a system of coupled diffusion and Poisson equations. Models and practical solutions of specific engineering problems arising in solid mechanics, chemical engineering, heat transfer and fuid mechanics are described and analyzed for efficiency and accuracy. Only 2-D cases are discussed and a new method of numerical treatment of boundary conditions common in the fundamental problems of mechanics of continua is presented.

Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations

NASA Technical Reports Server (NTRS)

Jawerth, Bjoern; Sweldens, Wim

1993-01-01

We present ideas on how to use wavelets in the solution of boundary value ordinary differential equations. Rather than using classical wavelets, we adapt their construction so that they become (bi)orthogonal with respect to the inner product defined by the operator. The stiffness matrix in a Galerkin method then becomes diagonal and can thus be trivially inverted. We show how one can construct an O(N) algorithm for various constant and variable coefficient operators.

Numerical solution of the time fractional reaction-diffusion equation with a moving boundary

NASA Astrophysics Data System (ADS)

Zheng, Minling; Liu, Fawang; Liu, Qingxia; Burrage, Kevin; Simpson, Matthew J.

2017-06-01

A fractional reaction-diffusion model with a moving boundary is presented in this paper. An efficient numerical method is constructed to solve this moving boundary problem. Our method makes use of a finite difference approximation for the temporal discretization, and spectral approximation for the spatial discretization. The stability and convergence of the method is studied, and the errors of both the semi-discrete and fully-discrete schemes are derived. Numerical examples, motivated by problems from developmental biology, show a good agreement with the theoretical analysis and illustrate the efficiency of our method.

Effects of boundary layer refraction and fuselage scattering on fuselage surface noise from advanced turboprop propellers

NASA Technical Reports Server (NTRS)

Mcaninch, G. L.; Rawls, J. W., Jr.

1984-01-01

An acoustic disturbance's propagation through a boundary layer is discussed with a view to the analysis of the acoustic field generated by a propfan rotor incident to the fuselage of an aircraft. Applying the parallel flow assumption, the resulting partial differential equations are reduced to an ordinary acoustic pressure differential equation by means of the Fourier transform. The methods used for the solution of this equation include those of Frobenius and of analytic continuation; both yield exact solutions in series form. Two models of the aircraft fuselage-boundary layer system are considered, in the first of which the fuselage is replaced by a flat plate and the acoustic field is assumed to be two-dimensional, while in the second the fuselage is a cylinder in a fully three-dimensional acoustic field. It is shown that the boundary layer correction improves theory-data comparisons over simple application of a pressure-doubling rule at the fuselage.

A globally convergent and closed analytical solution of the Blasius equation with beneficial applications

NASA Astrophysics Data System (ADS)

Zheng, Jun; Han, Xinyue; Wang, ZhenTao; Li, Changfeng; Zhang, Jiazhong

2017-06-01

For about a century, people have been trying to seek for a globally convergent and closed analytical solution (CAS) of the Blasius Equation (BE). In this paper, we proposed a formally satisfied solution which could be parametrically expressed by two power series. Some analytical results of the laminar boundary layer of a flat plate, that were not analytically given in former studies, e.g. the thickness of the boundary layer and higher order derivatives, could be obtained based on the solution. Besides, the heat transfer in the laminar boundary layer of a flat plate with constant temperature could also be analytically formulated. Especially, the solution of the singular situation with Prandtl number Pr=0, which seems impossible to be analyzed in prior studies, could be given analytically. The method for finding the CAS of Blasius equation was also utilized in the problem of the boundary layer regulation through wall injection and slip velocity on the wall surface.

Uniqueness of boundary blow-up solutions on exterior domain of RN

NASA Astrophysics Data System (ADS)

Dong, Wei; Pang, Changci

2007-06-01

In this paper, we consider the existence and uniqueness of positive solutions of the degenerate logistic type elliptic equation where N[greater-or-equal, slanted]2, D[subset of]RN is a bounded domain with smooth boundary and a(x), b(x) are continuous functions on RN with b(x)[greater-or-equal, slanted]0, b(x)[not identical with]0. We show that under rather general conditions on a(x) and b(x) for large x, there exists a unique positive solution. Our results improve the corresponding ones in [W. Dong, Y. Du, Unbounded principal eigenfunctions and the logistic equation on RN, Bull. Austral. Math. Soc. 67 (2003) 413-427] and [Y. Du, L. Ma, Logistic type equations on RN by a squeezing method involving boundary blow-up solutions, J. London Math. Soc. (2) 64 (2001) 107-124].

The effect of convective boundary condition on MHD mixed convection boundary layer flow over an exponentially stretching vertical sheet

NASA Astrophysics Data System (ADS)

Isa, Siti Suzilliana Putri Mohamed; Arifin, Norihan Md.; Nazar, Roslinda; Bachok, Norfifah; Ali, Fadzilah Md

2017-12-01

A theoretical study that describes the magnetohydrodynamic mixed convection boundary layer flow with heat transfer over an exponentially stretching sheet with an exponential temperature distribution has been presented herein. This study is conducted in the presence of convective heat exchange at the surface and its surroundings. The system is controlled by viscous dissipation and internal heat generation effects. The governing nonlinear partial differential equations are converted into ordinary differential equations by a similarity transformation. The converted equations are then solved numerically using the shooting method. The results related to skin friction coefficient, local Nusselt number, velocity and temperature profiles are presented for several sets of values of the parameters. The effects of the governing parameters on the features of the flow and heat transfer are examined in detail in this study.

Moving charged particles in lattice Boltzmann-based electrokinetics

NASA Astrophysics Data System (ADS)

Kuron, Michael; Rempfer, Georg; Schornbaum, Florian; Bauer, Martin; Godenschwager, Christian; Holm, Christian; de Graaf, Joost

2016-12-01

The motion of ionic solutes and charged particles under the influence of an electric field and the ensuing hydrodynamic flow of the underlying solvent is ubiquitous in aqueous colloidal suspensions. The physics of such systems is described by a coupled set of differential equations, along with boundary conditions, collectively referred to as the electrokinetic equations. Capuani et al. [J. Chem. Phys. 121, 973 (2004)] introduced a lattice-based method for solving this system of equations, which builds upon the lattice Boltzmann algorithm for the simulation of hydrodynamic flow and exploits computational locality. However, thus far, a description of how to incorporate moving boundary conditions into the Capuani scheme has been lacking. Moving boundary conditions are needed to simulate multiple arbitrarily moving colloids. In this paper, we detail how to introduce such a particle coupling scheme, based on an analogue to the moving boundary method for the pure lattice Boltzmann solver. The key ingredients in our method are mass and charge conservation for the solute species and a partial-volume smoothing of the solute fluxes to minimize discretization artifacts. We demonstrate our algorithm's effectiveness by simulating the electrophoresis of charged spheres in an external field; for a single sphere we compare to the equivalent electro-osmotic (co-moving) problem. Our method's efficiency and ease of implementation should prove beneficial to future simulations of the dynamics in a wide range of complex nanoscopic and colloidal systems that were previously inaccessible to lattice-based continuum algorithms.

A mathematical model for simulating noise suppression of lined ejectors

NASA Technical Reports Server (NTRS)

Watson, Willie R.

1994-01-01

A mathematical model containing the essential features embodied in the noise suppression of lined ejectors is presented. Although some simplification of the physics is necessary to render the model mathematically tractable, the current model is the most versatile and technologically advanced at the current time. A system of linearized equations and the boundary conditions governing the sound field are derived starting from the equations of fluid dynamics. A nonreflecting boundary condition is developed. In view of the complex nature of the equations, a parametric study requires the use of numerical techniques and modern computers. A finite element algorithm that solves the differential equations coupled with the boundary condition is then introduced. The numerical method results in a matrix equation with several hundred thousand degrees of freedom that is solved efficiently on a supercomputer. The model is validated by comparing results either with exact solutions or with approximate solutions from other works. In each case, excellent correlations are obtained. The usefulness of the model as an optimization tool and the importance of variable impedance liners as a mechanism for achieving broadband suppression within a lined ejector are demonstrated.

Optimal analytic method for the nonlinear Hasegawa-Mima equation

NASA Astrophysics Data System (ADS)

Baxter, Mathew; Van Gorder, Robert A.; Vajravelu, Kuppalapalle

2014-05-01

The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10-15 are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.

Formulation of boundary conditions for the multigrid acceleration of the Euler and Navier Stokes equations

NASA Technical Reports Server (NTRS)

Jentink, Thomas Neil; Usab, William J., Jr.

1990-01-01

An explicit, Multigrid algorithm was written to solve the Euler and Navier-Stokes equations with special consideration given to the coarse mesh boundary conditions. These are formulated in a manner consistent with the interior solution, utilizing forcing terms to prevent coarse-mesh truncation error from affecting the fine-mesh solution. A 4-Stage Hybrid Runge-Kutta Scheme is used to advance the solution in time, and Multigrid convergence is further enhanced by using local time-stepping and implicit residual smoothing. Details of the algorithm are presented along with a description of Jameson's standard Multigrid method and a new approach to formulating the Multigrid equations.

Unsteady boundary layer rotating flow and heat transfer in a copper-water nanofluid over a shrinking sheet

NASA Astrophysics Data System (ADS)

Dzulkifli, Nor Fadhilah; Bachok, Norfifah; Yacob, Nor Azizah; Arifin, Norihan Md; Rosali, Haliza

2017-04-01

The study of unsteady three-dimensional boundary layer rotating flow with heat transfer in Copper-water nanofluid over a shrinking sheet is discussed. The governing equations in terms of partial differential equations are transformed to ordinary differential equations by introducing the appropriate similarity variables which are then solved numerically by a shooting method with Maple software. The numerical results of velocity gradient in x and y directions, skin friction coefficient and local Nusselt number as well as dual velocity and temperature profiles are shown graphically. The study revealed that dual solutions exist in certain range of s > 0.

A numerical scheme to solve unstable boundary value problems

NASA Technical Reports Server (NTRS)

Kalnay Derivas, E.

1975-01-01

A new iterative scheme for solving boundary value problems is presented. It consists of the introduction of an artificial time dependence into a modified version of the system of equations. Then explicit forward integrations in time are followed by explicit integrations backwards in time. The method converges under much more general conditions than schemes based in forward time integrations (false transient schemes). In particular it can attain a steady state solution of an elliptical system of equations even if the solution is unstable, in which case other iterative schemes fail to converge. The simplicity of its use makes it attractive for solving large systems of nonlinear equations.

A two-component Matched Interface and Boundary (MIB) regularization for charge singularity in implicit solvation

NASA Astrophysics Data System (ADS)

Geng, Weihua; Zhao, Shan

2017-12-01

We present a new Matched Interface and Boundary (MIB) regularization method for treating charge singularity in solvated biomolecules whose electrostatics are described by the Poisson-Boltzmann (PB) equation. In a regularization method, by decomposing the potential function into two or three components, the singular component can be analytically represented by the Green's function, while other components possess a higher regularity. Our new regularization combines the efficiency of two-component schemes with the accuracy of the three-component schemes. Based on this regularization, a new MIB finite difference algorithm is developed for solving both linear and nonlinear PB equations, where the nonlinearity is handled by using the inexact-Newton's method. Compared with the existing MIB PB solver based on a three-component regularization, the present algorithm is simpler to implement by circumventing the work to solve a boundary value Poisson equation inside the molecular interface and to compute related interface jump conditions numerically. Moreover, the new MIB algorithm becomes computationally less expensive, while maintains the same second order accuracy. This is numerically verified by calculating the electrostatic potential and solvation energy on the Kirkwood sphere on which the analytical solutions are available and on a series of proteins with various sizes.

A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives

NASA Astrophysics Data System (ADS)

Macías-Díaz, J. E.

2017-12-01

In this manuscript, we consider an initial-boundary-value problem governed by a (1 + 1)-dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics. The model considers the presence of a spatial Laplacian of fractional order which is defined in terms of Riesz fractional derivatives, as well as the inclusion of a generic continuously differentiable potential. It is known that the undamped regime has an associated positive energy functional, and we show here that it is preserved throughout time under suitable boundary conditions. To approximate the solutions of this model, we propose a finite-difference discretization based on fractional centered differences. Some discrete quantities are proposed in this work to estimate the energy functional, and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative. Moreover, we establish suitable computational constraints under which the discrete energy of the system is positive. The method is consistent of second order, and is both stable and convergent. The numerical simulations shown here illustrate the most important features of our numerical methodology.

Stability of hyperbolic-parabolic mixed type equations with partial boundary condition

NASA Astrophysics Data System (ADS)

Zhan, Huashui; Feng, Zhaosheng

2018-06-01

In this paper, we are concerned with the hyperbolic-parabolic mixed type equations with the non-homogeneous boundary condition. If it is degenerate on the boundary, the part of the boundary whose boundary value should be imposed, is determined by the entropy condition from the convection term. If there is no convection term in the equation, we show that the stability of solutions can be proved without any boundary condition. If the equation is completely degenerate, we show that the stability of solutions can be established just based on the partial boundary condition.

Double diffusive magnetohydrodynamic (MHD) mixed convective slip flow along a radiating moving vertical flat plate with convective boundary condition.

PubMed

Rashidi, Mohammad M; Kavyani, Neda; Abelman, Shirley; Uddin, Mohammed J; Freidoonimehr, Navid

2014-01-01

In this study combined heat and mass transfer by mixed convective flow along a moving vertical flat plate with hydrodynamic slip and thermal convective boundary condition is investigated. Using similarity variables, the governing nonlinear partial differential equations are converted into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved using a semi-numerical/analytical method called the differential transform method and results are compared with numerical results. Close agreement is found between the present method and the numerical method. Effects of the controlling parameters, including convective heat transfer, magnetic field, buoyancy ratio, hydrodynamic slip, mixed convective, Prandtl number and Schmidt number are investigated on the dimensionless velocity, temperature and concentration profiles. In addition effects of different parameters on the skin friction factor, [Formula: see text], local Nusselt number, [Formula: see text], and local Sherwood number [Formula: see text] are shown and explained through tables.

New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov–Galerkin method

PubMed Central

Doha, E.H.; Abd-Elhameed, W.M.; Youssri, Y.H.

2014-01-01

Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient. PMID:26425358

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

PubMed Central

Feischl, Michael; Gantner, Gregor; Praetorius, Dirk

2015-01-01

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

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

NASA Technical Reports Server (NTRS)

Ghosn, L. J.

1988-01-01

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

Practical calculation of laminar and turbulent bled-off boundary layers

NASA Technical Reports Server (NTRS)

Eppler, R.

1978-01-01

Bleed-off of boundary layer material is shown to be an effective means for reducing drag by conserving the laminar boundary layer and preventing separation of the turbulent boundary layer. The case in which the two effects of bleed-off overlap is examined. Empirical methods are extended to the case of bleed-off. Laminar and turbulent boundary layers are treated simultaneously and the approximation differential equations are solved without an uncertain error. The case without bleed-off is also treated.

Fourth order difference methods for hyperbolic IBVP's

NASA Technical Reports Server (NTRS)

Gustafsson, Bertil; Olsson, Pelle

1994-01-01

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

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

NASA Astrophysics Data System (ADS)

Bao, Gang; Xu, Liwei; Yin, Tao

2017-11-01

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

Nonequilibrium flows with smooth particle applied mechanics

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

Kum, Oyeon

1995-07-01

Smooth particle methods are relatively new methods for simulating solid and fluid flows through they have a 20-year history of solving complex hydrodynamic problems in astrophysics, such as colliding planets and stars, for which correct answers are unknown. The results presented in this thesis evaluate the adaptability or fitness of the method for typical hydrocode production problems. For finite hydrodynamic systems, boundary conditions are important. A reflective boundary condition with image particles is a good way to prevent a density anomaly at the boundary and to keep the fluxes continuous there. Boundary values of temperature and velocity can be separatelymore » controlled. The gradient algorithm, based on differentiating the smooth particle expression for (uρ) and (Tρ), does not show numerical instabilities for the stress tensor and heat flux vector quantities which require second derivatives in space when Fourier`s heat-flow law and Newton`s viscous force law are used. Smooth particle methods show an interesting parallel linking to them to molecular dynamics. For the inviscid Euler equation, with an isentropic ideal gas equation of state, the smooth particle algorithm generates trajectories isomorphic to those generated by molecular dynamics. The shear moduli were evaluated based on molecular dynamics calculations for the three weighting functions, B spline, Lucy, and Cusp functions. The accuracy and applicability of the methods were estimated by comparing a set of smooth particle Rayleigh-Benard problems, all in the laminar regime, to corresponding highly-accurate grid-based numerical solutions of continuum equations. Both transient and stationary smooth particle solutions reproduce the grid-based data with velocity errors on the order of 5%. The smooth particle method still provides robust solutions at high Rayleigh number where grid-based methods fails.« less

A Discrete Analysis of Non-reflecting Boundary Conditions for Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Hu, Fang Q.; Atkins, Harold L.

2003-01-01

We present a discrete analysis of non-reflecting boundary conditions for the discontinuous Galerkin method. The boundary conditions considered in this paper include the recently proposed Perfectly Matched Layer absorbing boundary condition for the linearized Euler equation and two non-reflecting boundary conditions based on the characteristic decomposition of the flux on the boundary. The analyses for the three boundary conditions are carried out in a unifled way. In each case, eigensolutions of the discrete system are obtained and applied to compute the numerical reflection coefficients of a specified out-going wave. The dependencies of the reflections at the boundary on the out-going wave angle and frequency as well as the mesh sizes arc? studied. Comparisons with direct numerical simulation results are also presented.

GRUMFOIL: A computer code for the viscous transonic flow over airfoils

NASA Technical Reports Server (NTRS)

Mead, H. R.; Melnik, R. E.

1985-01-01

A user's manual which describes the operation of the computer program, GRUMFOIL is presented. The program computes the viscous transonic flow over two dimensional airfoils using a boundary layer type viscid-inviscid interaction approach. The inviscid solution is obtained by a multigrid method for the full potential equation. The boundary layer solution is based on integral entrainment methods.

Preconditioned characteristic boundary conditions based on artificial compressibility method for solution of incompressible flows

NASA Astrophysics Data System (ADS)

Hejranfar, Kazem; Parseh, Kaveh

2017-09-01

The preconditioned characteristic boundary conditions based on the artificial compressibility (AC) method are implemented at artificial boundaries for the solution of two- and three-dimensional incompressible viscous flows in the generalized curvilinear coordinates. The compatibility equations and the corresponding characteristic variables (or the Riemann invariants) are mathematically derived and then applied as suitable boundary conditions in a high-order accurate incompressible flow solver. The spatial discretization of the resulting system of equations is carried out by the fourth-order compact finite-difference (FD) scheme. In the preconditioning applied here, the value of AC parameter in the flow field and also at the far-field boundary is automatically calculated based on the local flow conditions to enhance the robustness and performance of the solution algorithm. The code is fully parallelized using the Concurrency Runtime standard and Parallel Patterns Library (PPL) and its performance on a multi-core CPU is analyzed. The incompressible viscous flows around a 2-D circular cylinder, a 2-D NACA0012 airfoil and also a 3-D wavy cylinder are simulated and the accuracy and performance of the preconditioned characteristic boundary conditions applied at the far-field boundaries are evaluated in comparison to the simplified boundary conditions and the non-preconditioned characteristic boundary conditions. It is indicated that the preconditioned characteristic boundary conditions considerably improve the convergence rate of the solution of incompressible flows compared to the other boundary conditions and the computational costs are significantly decreased.

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.

Calculation of Moment Matrix Elements for Bilinear Quadrilaterals and Higher-Order Basis Functions

DTIC Science & Technology

2016-01-06

methods are known as boundary integral equation (BIE) methods and the present study falls into this category. The numerical solution of the BIE is...iterated integrals. The inner integral involves the product of the free-space Green’s function for the Helmholtz equation multiplied by an appropriate...Website: http://www.wipl-d.com/ 5. Y. Zhang and T. K. Sarkar, Parallel Solution of Integral Equation -Based EM Problems in the Frequency Domain. New

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

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Rahimian, Abtin; Zorin, Denis

2017-04-01

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

A Numerical Analysis of the Transient Response of an Ablation System Including Effects of Thermal Nonequilibrium, Mass Transfer and Chemical Kinetics. Ph.D Thesis - Virginia Polytechnic Inst. and State Univ.

NASA Technical Reports Server (NTRS)

Clark, R. K.

1972-01-01

The differential equations governing the transient response of a one-dimensional ablative thermal protection system undergoing stagnation ablation are derived. These equations are for thermal nonequilibrium effects between the pyrolysis gases and the char layer and kinetically controlled chemical reactions and mass transfer between the pyrolysis gases and the char layer. The boundary conditions are written for the particular case of stagnation heating with surface removal by oxidation or sublimation and pyrolysis of the uncharred layer occurring in a plane. The governing equations and boundary conditions are solved numerically using the modified implicit method (Crank-Nicolson method). Numerical results are compared with exact solutions for a number of simplified cases. The comparison is favorable in each instance.

An efficient solution technique for shockwave-boundary layer interactions with flow separation and slot suction effects

NASA Technical Reports Server (NTRS)

Edwards, Jack R.; Mcrae, D. Scott

1991-01-01

An efficient method for computing two-dimensional compressible Navier-Stokes flow fields is presented. The solution algorithm is a fully-implicit approximate factorization technique based on an unsymmetric line Gauss-Seidel splitting of the equation system Jacobian matrix. Convergence characteristics are improved by the addition of acceleration techniques based on Shamanskii's method for nonlinear equations and Broyden's quasi-Newton update. Characteristic-based differencing of the equations is provided by means of Van Leer's flux vector splitting. In this investigation, emphasis is placed on the fast and accurate computation of shock-wave-boundary layer interactions with and without slot suction effects. In the latter context, a set of numerical boundary conditions for simulating the transpiration flow in an open slot is devised. Both laminar and turbulent cases are considered, with turbulent closure provided by a modified Cebeci-Smith algebraic model. Comparisons with computational and experimental data sets are presented for a variety of interactions, and a fully-coupled simulation of a plenum chamber/inlet flowfield with shock interaction and suction is also shown and discussed.

Numerical methods for large-scale, time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Turkel, E.

1979-01-01

A survey of numerical methods for time dependent partial differential equations is presented. The emphasis is on practical applications to large scale problems. A discussion of new developments in high order methods and moving grids is given. The importance of boundary conditions is stressed for both internal and external flows. A description of implicit methods is presented including generalizations to multidimensions. Shocks, aerodynamics, meteorology, plasma physics and combustion applications are also briefly described.

A High-Order Immersed Boundary Method for Acoustic Wave Scattering and Low-Mach Number Flow-Induced Sound in Complex Geometries

PubMed Central

Seo, Jung Hee; Mittal, Rajat

2010-01-01

A new sharp-interface immersed boundary method based approach for the computation of low-Mach number flow-induced sound around complex geometries is described. The underlying approach is based on a hydrodynamic/acoustic splitting technique where the incompressible flow is first computed using a second-order accurate immersed boundary solver. This is followed by the computation of sound using the linearized perturbed compressible equations (LPCE). The primary contribution of the current work is the development of a versatile, high-order accurate immersed boundary method for solving the LPCE in complex domains. This new method applies the boundary condition on the immersed boundary to a high-order by combining the ghost-cell approach with a weighted least-squares error method based on a high-order approximating polynomial. The method is validated for canonical acoustic wave scattering and flow-induced noise problems. Applications of this technique to relatively complex cases of practical interest are also presented. PMID:21318129

A parallel overset-curvilinear-immersed boundary framework for simulating complex 3D incompressible flows

PubMed Central

Borazjani, Iman; Ge, Liang; Le, Trung; Sotiropoulos, Fotis

2013-01-01

We develop an overset-curvilinear immersed boundary (overset-CURVIB) method in a general non-inertial frame of reference to simulate a wide range of challenging biological flow problems. The method incorporates overset-curvilinear grids to efficiently handle multi-connected geometries and increase the resolution locally near immersed boundaries. Complex bodies undergoing arbitrarily large deformations may be embedded within the overset-curvilinear background grid and treated as sharp interfaces using the curvilinear immersed boundary (CURVIB) method (Ge and Sotiropoulos, Journal of Computational Physics, 2007). The incompressible flow equations are formulated in a general non-inertial frame of reference to enhance the overall versatility and efficiency of the numerical approach. Efficient search algorithms to identify areas requiring blanking, donor cells, and interpolation coefficients for constructing the boundary conditions at grid interfaces of the overset grid are developed and implemented using efficient parallel computing communication strategies to transfer information among sub-domains. The governing equations are discretized using a second-order accurate finite-volume approach and integrated in time via an efficient fractional-step method. Various strategies for ensuring globally conservative interpolation at grid interfaces suitable for incompressible flow fractional step methods are implemented and evaluated. The method is verified and validated against experimental data, and its capabilities are demonstrated by simulating the flow past multiple aquatic swimmers and the systolic flow in an anatomic left ventricle with a mechanical heart valve implanted in the aortic position. PMID:23833331

Clinical characterization of 2D pressure field in human left ventricles

NASA Astrophysics Data System (ADS)

Borja, Maria; Rossini, Lorenzo; Martinez-Legazpi, Pablo; Benito, Yolanda; Alhama, Marta; Yotti, Raquel; Perez Del Villar, Candelas; Gonzalez-Mansilla, Ana; Barrio, Alicia; Fernandez-Aviles, Francisco; Bermejo, Javier; Khan, Andrew; Del Alamo, Juan Carlos

2014-11-01

The evaluation of left ventricle (LV) function in the clinical setting remains a challenge. Pressure gradient is a reliable and reproducible indicator of the LV function. We obtain 2D relative pressure field in the LV using in-vivo measurements obtained by processing Doppler-echocardiography images of healthy and dilated hearts. Exploiting mass conservation, we solve the Poisson pressure equation (PPE) dropping the time derivatives and viscous terms. The flow acceleration appears only in the boundary conditions, making our method weakly sensible to the time resolution of in-vivo acquisitions. To ensure continuity with respect to the discrete operator and grid used, a potential flow correction is applied beforehand, which gives another Poisson equation. The new incompressible velocity field ensures that the compatibility equation for the PPE is satisfied. Both Poisson equations are efficiently solved on a Cartesian grid using a multi-grid method and immersed boundary for the LV wall. The whole process is computationally inexpensive and could play a diagnostic role in the clinical assessment of LV function.

Generating a Multiphase Equation of State with Swarm Intelligence

NASA Astrophysics Data System (ADS)

Cox, Geoffrey

2017-06-01

Hydrocode calculations require knowledge of the variation of pressure of a material with density and temperature, which is given by the equation of state. An accurate model needs to account for discontinuities in energy, density and properties of a material across a phase boundary. When generating a multiphase equation of state the modeller attempts to balance the agreement between the available data for compression, expansion and phase boundary location. However, this can prove difficult because minor adjustments in the equation of state for a single phase can have a large impact on the overall phase diagram. Recently, Cox and Christie described a method for combining statistical-mechanics-based condensed matter physics models with a stochastic analysis technique called particle swarm optimisation. The models produced show good agreement with experiment over a wide range of pressure-temperature space. This talk details the general implementation of this technique, shows example results, and describes the types of analysis that can be performed with this method.

Global boundary flattening transforms for acoustic propagation under rough sea surfaces.

PubMed

Oba, Roger M

2010-07-01

This paper introduces a conformal transform of an acoustic domain under a one-dimensional, rough sea surface onto a domain with a flat top. This non-perturbative transform can include many hundreds of wavelengths of the surface variation. The resulting two-dimensional, flat-topped domain allows direct application of any existing, acoustic propagation model of the Helmholtz or wave equation using transformed sound speeds. Such a transform-model combination applies where the surface particle velocity is much slower than sound speed, such that the boundary motion can be neglected. Once the acoustic field is computed, the bijective (one-to-one and onto) mapping permits the field interpolation in terms of the original coordinates. The Bergstrom method for inverse Riemann maps determines the transform by iterated solution of an integral equation for a surface matching term. Rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the Helmholtz equation.

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.

Calculation of three-dimensional compressible laminar and turbulent boundary layers. Calculation of three-dimensional compressible boundary layers on arbitrary wings

NASA Technical Reports Server (NTRS)

Cebeci, T.; Kaups, K.; Ramsey, J.; Moser, A.

1975-01-01

A very general method for calculating compressible three-dimensional laminar and turbulent boundary layers on arbitrary wings is described. The method utilizes a nonorthogonal coordinate system for the boundary-layer calculations and includes a geometry package that represents the wing analytically. In the calculations all the geometric parameters of the coordinate system are accounted for. The Reynolds shear-stress terms are modeled by an eddy-viscosity formulation developed by Cebeci. The governing equations are solved by a very efficient two-point finite-difference method used earlier by Keller and Cebeci for two-dimensional flows and later by Cebeci for three-dimensional flows.

A numerical scheme for singularly perturbed reaction-diffusion problems with a negative shift via numerov method

NASA Astrophysics Data System (ADS)

Dinesh Kumar, S.; Nageshwar Rao, R.; Pramod Chakravarthy, P.

2017-11-01

In this paper, we consider a boundary value problem for a singularly perturbed delay differential equation of reaction-diffusion type. We construct an exponentially fitted numerical method using Numerov finite difference scheme, which resolves not only the boundary layers but also the interior layers arising from the delay term. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

Isospectrals of non-uniform Rayleigh beams with respect to their uniform counterparts

PubMed Central

Ganguli, Ranjan

2018-01-01

In this paper, we look for non-uniform Rayleigh beams isospectral to a given uniform Rayleigh beam. Isospectral systems are those that have the same spectral properties, i.e. the same free vibration natural frequencies for a given boundary condition. A transformation is proposed that converts the fourth-order governing differential equation of non-uniform Rayleigh beam into a uniform Rayleigh beam. If the coefficients of the transformed equation match with those of the uniform beam equation, then the non-uniform beam is isospectral to the given uniform beam. The boundary-condition configuration should be preserved under this transformation. We present the constraints under which the boundary configurations will remain unchanged. Frequency equivalence of the non-uniform beams and the uniform beam is confirmed by the finite-element method. For the considered cases, examples of beams having a rectangular cross section are presented to show the application of our analysis. PMID:29515879

The SMM Model as a Boundary Value Problem Using the Discrete Diffusion Equation

NASA Technical Reports Server (NTRS)

Campbell, Joel

2007-01-01

A generalized single step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.

The SMM model as a boundary value problem using the discrete diffusion equation.

PubMed

Campbell, Joel

2007-12-01

A generalized single-step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.

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.

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.

Determining the Transference Number of H[superscript +](aq) by a Modified Moving Boundary Method: A Directed Study for the Undergraduate Physical Chemistry Laboratory

ERIC Educational Resources Information Center

Dabke, Rajeev B.; Gebeyehu, Zewdu; Padelford, Jonathan

2012-01-01

A directed study for the undergraduate physical chemistry laboratory for determining the transference number of H[superscript +](aq) using a modified moving boundary method is presented. The laboratory study combines Faraday's laws of electrolysis with mole ratios and the perfect gas equation. The volume of hydrogen gas produced at the cathode is…

Calculation of compressible boundary layer flow about airfoils by a finite element/finite difference method

NASA Technical Reports Server (NTRS)

Strong, Stuart L.; Meade, Andrew J., Jr.

1992-01-01

Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.

A Chebyshev Collocation Method for Moving Boundaries, Heat Transfer, and Convection During Directional Solidification

NASA Technical Reports Server (NTRS)

Zhang, Yiqiang; Alexander, J. I. D.; Ouazzani, J.

1994-01-01

Free and moving boundary problems require the simultaneous solution of unknown field variables and the boundaries of the domains on which these variables are defined. There are many technologically important processes that lead to moving boundary problems associated with fluid surfaces and solid-fluid boundaries. These include crystal growth, metal alloy and glass solidification, melting and name propagation. The directional solidification of semi-conductor crystals by the Bridgman-Stockbarger method is a typical example of such a complex process. A numerical model of this growth method must solve the appropriate heat, mass and momentum transfer equations and determine the location of the melt-solid interface. In this work, a Chebyshev pseudospectra collocation method is adapted to the problem of directional solidification. Implementation involves a solution algorithm that combines domain decomposition, finite-difference preconditioned conjugate minimum residual method and a Picard type iterative scheme.

On the Lagrangian description of unsteady boundary-layer separation. I - General theory

NASA Technical Reports Server (NTRS)

Van Dommelen, Leon L.; Cowley, Stephen J.

1990-01-01

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

On the Lagrangian description of unsteady boundary layer separation. Part 1: General theory

NASA Technical Reports Server (NTRS)

Vandommelen, Leon L.; Cowley, Stephen J.

1989-01-01

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer solutions.

PubMed

Cengizci, Süleyman; Atay, Mehmet Tarık; Eryılmaz, Aytekin

2016-01-01

This paper is concerned with two-point boundary value problems for singularly perturbed nonlinear ordinary differential equations. The case when the solution only has one boundary layer is examined. An efficient method so called Successive Complementary Expansion Method (SCEM) is used to obtain uniformly valid approximations to this kind of solutions. Four test problems are considered to check the efficiency and accuracy of the proposed method. The numerical results are found in good agreement with exact and existing solutions in literature. The results confirm that SCEM has a superiority over other existing methods in terms of easy-applicability and effectiveness.

Velocity boundary conditions for vorticity formulations of the incompressible Navier-Stokes equations

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

Kempka, S.N.; Strickland, J.H.; Glass, M.W.

1995-04-01

formulation to satisfy velocity boundary conditions for the vorticity form of the incompressible, viscous fluid momentum equations is presented. The tangential and normal components of the velocity boundary condition are satisfied simultaneously by creating vorticity adjacent to boundaries. The newly created vorticity is determined using a kinematical formulation which is a generalization of Helmholtz` decomposition of a vector field. Though it has not been generally recognized, these formulations resolve the over-specification issue associated with creating voracity to satisfy velocity boundary conditions. The generalized decomposition has not been widely used, apparently due to a lack of a useful physical interpretation. Anmore » analysis is presented which shows that the generalized decomposition has a relatively simple physical interpretation which facilitates its numerical implementation. The implementation of the generalized decomposition is discussed in detail. As an example the flow in a two-dimensional lid-driven cavity is simulated. The solution technique is based on a Lagrangian transport algorithm in the hydrocode ALEGRA. ALEGRA`s Lagrangian transport algorithm has been modified to solve the vorticity transport equation and the generalized decomposition, thus providing a new, accurate method to simulate incompressible flows. This numerical implementation and the new boundary condition formulation allow vorticity-based formulations to be used in a wider range of engineering problems.« less

Regularity estimates up to the boundary for elliptic systems of difference equations

NASA Technical Reports Server (NTRS)

Strikwerda, J. C.; Wade, B. A.; Bube, K. P.

1986-01-01

Regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations were proved. The regularity estimates, obtained for boundary fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudo-difference operators to treat systems with variable coefficients. The elliptic systems of difference equations and the boundary conditions which are considered are very general in form. The regularity of a regular elliptic system of difference equations was proved equivalent to the nonexistence of eigensolutions. The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundstrom (1972) and others for hyperbolic difference equations.

Completed Beltrami-Michell Formulation in Polar Coordinates

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Hopkins, Dale A.

2005-01-01

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

An Immersed Boundary-Lattice Boltzmann Method for Simulating Particulate Flows

NASA Astrophysics Data System (ADS)

Zhang, Baili; Cheng, Ming; Lou, Jing

2013-11-01

A two-dimensional momentum exchange-based immersed boundary-lattice Boltzmann method developed by X.D. Niu et al. (2006) has been extended in three-dimensions for solving fluid-particles interaction problems. This method combines the most desirable features of the lattice Boltzmann method and the immersed boundary method by using a regular Eulerian mesh for the flow domain and a Lagrangian mesh for the moving particles in the flow field. The non-slip boundary conditions for the fluid and the particles are enforced by adding a force density term into the lattice Boltzmann equation, and the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. This method preserves the advantages of lattice Boltzmann method in tracking a group of particles and, at the same time, provides an alternative approach to treat solid-fluid boundary conditions. Numerical validations show that the present method is very accurate and efficient. The present method will be further developed to simulate more complex problems with particle deformation, particle-bubble and particle-droplet interactions.

Elastic Green’s Function in Anisotropic Bimaterials Considering Interfacial Elasticity

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

Juan, Pierre -Alexandre; Dingreville, Remi

Here, the two-dimensional elastic Green’s function is calculated for a general anisotropic elastic bimaterial containing a line dislocation and a concentrated force while accounting for the interfacial structure by means of a generalized interfacial elasticity paradigm. The introduction of the interface elasticity model gives rise to boundary conditions that are effectively equivalent to those of a weakly bounded interface. The equations of elastic equilibrium are solved by complex variable techniques and the method of analytical continuation. The solution is decomposed into the sum of the Green’s function corresponding to the perfectly bonded interface and a perturbation term corresponding to themore » complex coupling nature between the interface structure and a line dislocation/concentrated force. Such construct can be implemented into the boundary integral equations and the boundary element method for analysis of nano-layered structures and epitaxial systems where the interface structure plays an important role.« less

Elastic Green’s Function in Anisotropic Bimaterials Considering Interfacial Elasticity

DOE PAGES

Juan, Pierre -Alexandre; Dingreville, Remi

2017-09-13

Here, the two-dimensional elastic Green’s function is calculated for a general anisotropic elastic bimaterial containing a line dislocation and a concentrated force while accounting for the interfacial structure by means of a generalized interfacial elasticity paradigm. The introduction of the interface elasticity model gives rise to boundary conditions that are effectively equivalent to those of a weakly bounded interface. The equations of elastic equilibrium are solved by complex variable techniques and the method of analytical continuation. The solution is decomposed into the sum of the Green’s function corresponding to the perfectly bonded interface and a perturbation term corresponding to themore » complex coupling nature between the interface structure and a line dislocation/concentrated force. Such construct can be implemented into the boundary integral equations and the boundary element method for analysis of nano-layered structures and epitaxial systems where the interface structure plays an important role.« less

Numerical simulation of separated flows. Ph.D. Thesis - Stanford Univ., Calif.

NASA Technical Reports Server (NTRS)

Spalart, P. R.; Leonard, A.; Baganoff, D.

1983-01-01

A new numerical method, based on the Vortex Method, for the simulation of two-dimensional separated flows, was developed and tested on a wide range of gases. The fluid is incompressible and the Reynolds number is high. A rigorous analytical basis for the representation of the Navier-Stokes equation in terms of the vorticity is used. An equation for the control of circulation around each body is included. An inviscid outer flow (computed by the Vortex Method) was coupled with a viscous boundary layer flow (computed by an Eulerian method). This version of the Vortex Method treats bodies of arbitrary shape, and accurately computes the pressure and shear stress at the solid boundary. These two quantities reflect the structure of the boundary layer. Several versions of the method are presented and applied to various problems, most of which have massive separation. Comparison of its results with other results, generally experimental, demonstrates the reliability and the general accuracy of the new method, with little dependence on empirical parameters. Many of the complex features of the flow past a circular cylinder, over a wide range of Reynolds numbers, are correctly reproduced.

New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.; Bassuony, M. A.

2013-03-01

This paper is concerned with spectral Galerkin algorithms for solving high even-order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions. The proposed algorithms are extended to solve two-dimensional high even-order differential equations. The key to the efficiency of these algorithms is to construct compact combinations of Chebyshev polynomials of the third and fourth kinds as basis functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithms, and some comparisons with some other methods are made.

Spectral methods in time for a class of parabolic partial differential equations

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

Ierley, G.; Spencer, B.; Worthing, R.

1992-09-01

In this paper, we introduce a fully spectral solution for the partial differential equation u[sub t] + uu[sub x] + vu[sub xx] + [mu]u[sub xxx] + [lambda]u[sub xxxx] = O. For periodic boundary conditions in space, the use of a Fourier expansion in x admits of a particularly efficient algorithm with respect to expansion of the time dependence in a Chebyshev series. Boundary conditions other than periodic may still be treated with reasonable, though lesser, efficiency. for all cases, very high accuracy is attainable at moderate computational cost relative to the expense of variable order finite difference methods in time.more » 14 refs., 9 figs.« less

Airfoil Design Using a Coupled Euler and Integral Boundary Layer Method with Adjoint Based Sensitivities

NASA Technical Reports Server (NTRS)

Edwards, S.; Reuther, J.; Chattot, J. J.

1997-01-01

The objective of this paper is to present a control theory approach for the design of airfoils in the presence of viscous compressible flows. A coupled system of the integral boundary layer and the Euler equations is solved to provide rapid flow simulations. An adjunct approach consistent with the complete coupled state equations is employed to obtain the sensitivities needed to drive a numerical optimization algorithm. Design to target pressure distribution is demonstrated on an RAE 2822 airfoil at transonic speed.

Computer simulation of solutions of polyharmonic equations in plane domain

NASA Astrophysics Data System (ADS)

Kazakova, A. O.

2018-05-01

A systematic study of plane problems of the theory of polyharmonic functions is presented. A method of reducing boundary problems for polyharmonic functions to the system of integral equations on the boundary of the domain is given and a numerical algorithm for simulation of solutions of this system is suggested. Particular attention is paid to the numerical solution of the main tasks when the values of the function and its derivatives are given. Test examples are considered that confirm the effectiveness and accuracy of the suggested algorithm.

Thermal boundary layer due to sudden heating of fluid

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

Kurkal, K.R.; Munukutla, S.

This paper proposes to solve computationally the heat-transfer problems (introduced by Munukutla and Venkataraman, 1988) related to a closed-cycle pulsed high-power laser flow loop. The continuity and the momentum equations as well as the unsteady energy equation are solved using the Keller-Box method. The solutions were compared with the steady-state solutions at large times, and the comparison was found to be excellent. Empirical formulas are proposed for calculating the time-dependent boundary-layer thickness and mass-heat transfer, that can be used by laser flow loop designers. 6 refs.

Thermal boundary layer due to sudden heating of fluid

NASA Astrophysics Data System (ADS)

Kurkal, K. R.; Munukutla, S.

1989-10-01

This paper proposes to solve computationally the heat-transfer problems (introduced by Munukutla and Venkataraman, 1988) related to a closed-cycle pulsed high-power laser flow loop. The continuity and the momentum equations as well as the unsteady energy equation are solved using the Keller-Box method. The solutions were compared with the steady-state solutions at large times, and the comparison was found to be excellent. Empirical formulas are proposed for calculating the time-dependent boundary-layer thickness and mass-heat transfer, that can be used by laser flow loop designers.

Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method

NASA Astrophysics Data System (ADS)

Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye

2018-04-01

The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.

FFT-based computation of the bioheat transfer equation for the HCC ultrasound surgery therapy modeling.

PubMed

Dillenseger, Jean-Louis; Esneault, Simon; Garnier, Carole

2008-01-01

This paper describes a modeling method of the tissue temperature evolution over time in hyperthermia. More precisely, this approach is used to simulate the hepatocellular carcinoma curative treatment by a percutaneous high intensity ultrasound surgery. The tissue temperature evolution over time is classically described by Pennes' bioheat transfer equation which is generally solved by a finite difference method. In this paper we will present a method where the bioheat transfer equation can be algebraically solved after a Fourier transformation over the space coordinates. The implementation and boundary conditions of this method will be shown and compared with the finite difference method.

The origin of spurious solutions in computational electromagnetics

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Wu, Jie; Povinelli, L. A.

1995-01-01

The origin of spurious solutions in computational electromagnetics, which violate the divergence equations, is deeply rooted in a misconception about the first-order Maxwell's equations and in an incorrect derivation and use of the curl-curl equations. The divergence equations must be always included in the first-order Maxwell's equations to maintain the ellipticity of the system in the space domain and to guarantee the uniqueness of the solution and/or the accuracy of the numerical solutions. The div-curl method and the least-squares method provide rigorous derivation of the equivalent second-order Maxwell's equations and their boundary conditions. The node-based least-squares finite element method (LSFEM) is recommended for solving the first-order full Maxwell equations directly. Examples of the numerical solutions by LSFEM for time-harmonic problems are given to demonstrate that the LSFEM is free of spurious solutions.

Computational aspects of unsteady flows

NASA Technical Reports Server (NTRS)

Cebeci, T.; Carr, L. W.; Khattab, A. A.; Schimke, S. M.

1985-01-01

The calculation of unsteady flows and the development of numerical methods for solving unsteady boundary layer equations and their application to the flows around important configurations such as oscillating airfoils are presented. A brief review of recent work is provided with emphasis on the need for numerical methods which can overcome possible problems associated with flow reversal and separation. The zig-zag and characteristic box schemes are described in this context, and when embodied in a method which permits interaction between solutions of inviscid and viscous equations, the characteristic box scheme is shown to avoid the singularity associated with boundary layer equations and prescribed pressure gradient. Calculations were performed for a cylinder started impulsively from rest and oscillating airfoils. The results are presented and discussed. It is conlcuded that turbulence models based on an algebraic specification of eddy viscosity can be adequate, that location of translation is important to the calculation of the location of flow separation and, therefore, to the overall lift of an oscillating airfoil.

Three dimensional flow field inside compressor rotor, including blade boundary layers

NASA Technical Reports Server (NTRS)

Galmes, J. M.; Pouagere, M.; Lakshminarayana, B.

1982-01-01

The Reynolds stress equation, pressure strain correlation, and dissipative terms and diffusion are discussed in relation to turbulence modelling using the Reynolds stress model. Algebraic modeling of Reynolds stresses and calculation of the boundary layer over an axial cylinder are examined with regards to the kinetic energy model for turbulence modelling. The numerical analysis of blade and hub wall boundary layers, and an experimental study of rotor blade boundary layer in an axial flow compressor rotor are discussed. The Patankar-Spalding numerical method for two dimensional boundary layers is included.

Boundary-fitted coordinate systems for numerical solution of partial differential equations - A review

NASA Technical Reports Server (NTRS)

Thompson, J. F.; Warsi, Z. U. A.; Mastin, C. W.

1982-01-01

A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.

Analytical study of Cattaneo-Christov heat flux model for a boundary layer flow of Oldroyd-B fluid

NASA Astrophysics Data System (ADS)

F, M. Abbasi; M, Mustafa; S, A. Shehzad; M, S. Alhuthali; T, Hayat

2016-01-01

We investigate the Cattaneo-Christov heat flux model for a two-dimensional laminar boundary layer flow of an incompressible Oldroyd-B fluid over a linearly stretching sheet. Mathematical formulation of the boundary layer problems is given. The nonlinear partial differential equations are converted into the ordinary differential equations using similarity transformations. The dimensionless velocity and temperature profiles are obtained through optimal homotopy analysis method (OHAM). The influences of the physical parameters on the velocity and the temperature are pointed out. The results show that the temperature and the thermal boundary layer thickness are smaller in the Cattaneo-Christov heat flux model than those in the Fourier’s law of heat conduction. Project supported by the Deanship of Scientific Research (DSR) King Abdulaziz University, Jeddah, Saudi Arabia (Grant No. 32-130-36-HiCi).

Boundary integral equation analysis for suspension of spheres in Stokes flow

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Veerapaneni, Shravan

2018-06-01

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

Numerical simulation of the control of the three-dimensional transition process in boundary layers

NASA Technical Reports Server (NTRS)

Kral, L. D.; Fasel, H. F.

1990-01-01

Surface heating techniques to control the three-dimensional laminar-turbulent transition process are numerically investigated for a water boundary layer. The Navier-Stokes and energy equations are solved using a fully implicit finite difference/spectral method. The spatially evolving boundary layer is simulated. Results of both passive and active methods of control are shown for small amplitude two-dimensional and three-dimensional disturbance waves. Control is also applied to the early stages of the secondary instability process using passive or active control techniques.

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

Cao, Yong; Chu, Yuchuan; He, Xiaoming

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.

The effect of Reynolds number and turbulence on airfoil aerodynamics at -90 degrees incidence

NASA Technical Reports Server (NTRS)

Stremel, Paul M.

1993-01-01

A method has been developed for calculating the viscous flow about airfoils in with and without deflected flaps at -90 deg incidence. This method provides for the solution of the unsteady incompressible Navier-Stokes equations by means of an implicit technique. The solution is calculated on a body-fitted computational mesh using a staggered grid method. The vorticity is defined at the node points, and the velocity components are defined at the mesh-cell sides. The staggered-grid orientation provides for accurate representation of vorticity at the node points and the continuity equation at the mesh-cell centers. The method provides for the direct solution of the flow field and satisfies the continuity equation to machine zero at each time-step. The method is evaluated in terms of its ability to predict two-dimensional flow about an airfoil at -90 degrees incidence for varying Reynolds number and different boundary layer models. A laminar and a turbulent boundary layer models. A laminar and a turbulent boundary layer model are considered in the evaluation of the method. The variation of the average loading and surface pressure distribution due to flap deflection, Reynolds number, and laminar or turbulent flow are presented and compared with experimental results. The comparisons indicate that the calculated drag and drag reduction caused by flap deflection and the calculated average surface pressure are in excellent agreement with the measured results at a similar Reynolds number.

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

NASA Technical Reports Server (NTRS)

Skollermo, G.

1979-01-01

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Robust iterative method for nonlinear Helmholtz equation

NASA Astrophysics Data System (ADS)

Yuan, Lijun; Lu, Ya Yan

2017-08-01

A new iterative method is developed for solving the two-dimensional nonlinear Helmholtz equation which governs polarized light in media with the optical Kerr nonlinearity. In the strongly nonlinear regime, the nonlinear Helmholtz equation could have multiple solutions related to phenomena such as optical bistability and symmetry breaking. The new method exhibits a much more robust convergence behavior than existing iterative methods, such as frozen-nonlinearity iteration, Newton's method and damped Newton's method, and it can be used to find solutions when good initial guesses are unavailable. Numerical results are presented for the scattering of light by a nonlinear circular cylinder based on the exact nonlocal boundary condition and a pseudospectral method in the polar coordinate system.

Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence

NASA Technical Reports Server (NTRS)

Sandham, N. D.; Yee, H. C.; Kwak, Dochan (Technical Monitor)

2000-01-01

A stable high order numerical scheme for direct numerical simulation (DNS) of shock-free compressible turbulence is presented. The method is applicable to general geometries. It contains no upwinding, artificial dissipation, or filtering. Instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points (interior scheme) as well as the boundary points (boundary scheme). An entropy splitting approach splits the inviscid flux derivatives into conservative and non-conservative portions. The spatial difference operators satisfy a summation by parts condition leading to a stable scheme (combined interior and boundary schemes) for the initial boundary value problem using a generalized energy estimate. A Laplacian formulation of the viscous and heat conduction terms on the right hand side of the Navier-Stokes equations is used to ensure that any tendency to odd-even decoupling associated with central schemes can be countered by the fluid viscosity. A special formulation of the continuity equation is used, based on similar arguments. The resulting methods are able to minimize spurious high frequency oscillation producing nonlinear instability associated with pure central schemes, especially for long time integration simulation such as DNS. For validation purposes, the methods are tested in a DNS of compressible turbulent plane channel flow at a friction Mach number of 0.1 where a very accurate turbulence data base exists. It is demonstrated that the methods are robust in terms of grid resolution, and in good agreement with incompressible channel data, as expected at this Mach number. Accurate turbulence statistics can be obtained with moderate grid sizes. Stability limits on the range of the splitting parameter are determined from numerical tests.

Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow

NASA Astrophysics Data System (ADS)

Henshaw, William D.; Schwendeman, Donald W.

2006-08-01

We consider the solution of the reactive and non-reactive Euler equations on two-dimensional domains that evolve in time. The domains are discretized using moving overlapping grids. In a typical grid construction, boundary-fitted grids are used to represent moving boundaries, and these grids overlap with stationary background Cartesian grids. Block-structured adaptive mesh refinement (AMR) is used to resolve fine-scale features in the flow such as shocks and detonations. Refinement grids are added to base-level grids according to an estimate of the error, and these refinement grids move with their corresponding base-level grids. The numerical approximation of the governing equations takes place in the parameter space of each component grid which is defined by a mapping from (fixed) parameter space to (moving) physical space. The mapped equations are solved numerically using a second-order extension of Godunov's method. The stiff source term in the reactive case is handled using a Runge-Kutta error-control scheme. We consider cases when the boundaries move according to a prescribed function of time and when the boundaries of embedded bodies move according to the surface stress exerted by the fluid. In the latter case, the Newton-Euler equations describe the motion of the center of mass of the each body and the rotation about it, and these equations are integrated numerically using a second-order predictor-corrector scheme. Numerical boundary conditions at slip walls are described, and numerical results are presented for both reactive and non-reactive flows that demonstrate the use and accuracy of the numerical approach.

A versatile embedded boundary adaptive mesh method for compressible flow in complex geometry

NASA Astrophysics Data System (ADS)

Al-Marouf, M.; Samtaney, R.

2017-05-01

We present an embedded ghost fluid method for numerical solutions of the compressible Navier Stokes (CNS) equations in arbitrary complex domains. A PDE multidimensional extrapolation approach is used to reconstruct the solution in the ghost fluid regions and imposing boundary conditions on the fluid-solid interface, coupled with a multi-dimensional algebraic interpolation for freshly cleared cells. The CNS equations are numerically solved by the second order multidimensional upwind method. Block-structured adaptive mesh refinement, implemented with the Chombo framework, is utilized to reduce the computational cost while keeping high resolution mesh around the embedded boundary and regions of high gradient solutions. The versatility of the method is demonstrated via several numerical examples, in both static and moving geometry, ranging from low Mach number nearly incompressible flows to supersonic flows. Our simulation results are extensively verified against other numerical results and validated against available experimental results where applicable. The significance and advantages of our implementation, which revolve around balancing between the solution accuracy and implementation difficulties, are briefly discussed as well.

A NURBS-enhanced finite volume solver for steady Euler equations

NASA Astrophysics Data System (ADS)

Meng, Xucheng; Hu, Guanghui

2018-04-01

In Hu and Yi (2016) [20], a non-oscillatory k-exact reconstruction method was proposed towards the high-order finite volume methods for steady Euler equations, which successfully demonstrated the high-order behavior in the simulations. However, the degeneracy of the numerical accuracy of the approximate solutions to problems with curved boundary can be observed obviously. In this paper, the issue is resolved by introducing the Non-Uniform Rational B-splines (NURBS) method, i.e., with given discrete description of the computational domain, an approximate NURBS curve is reconstructed to provide quality quadrature information along the curved boundary. The advantages of using NURBS include i). both the numerical accuracy of the approximate solutions and convergence rate of the numerical methods are improved simultaneously, and ii). the NURBS curve generation is independent of other modules of the numerical framework, which makes its application very flexible. It is also shown in the paper that by introducing more elements along the normal direction for the reconstruction patch of the boundary element, significant improvement in the convergence to steady state can be achieved. The numerical examples confirm the above features very well.

A fully Sinc-Galerkin method for Euler-Bernoulli beam models

NASA Technical Reports Server (NTRS)

Smith, R. C.; Bowers, K. L.; Lund, J.

1990-01-01

A fully Sinc-Galerkin method in both space and time is presented for fourth-order time-dependent partial differential equations with fixed and cantilever boundary conditions. The Sinc discretizations for the second-order temporal problem and the fourth-order spatial problems are presented. Alternate formulations for variable parameter fourth-order problems are given which prove to be especially useful when applying the forward techniques to parameter recovery problems. The discrete system which corresponds to the time-dependent partial differential equations of interest are then formulated. Computational issues are discussed and a robust and efficient algorithm for solving the resulting matrix system is outlined. Numerical results which highlight the method are given for problems with both analytic and singular solutions as well as fixed and cantilever boundary conditions.

A dynamical regularization algorithm for solving inverse source problems of elliptic partial differential equations

NASA Astrophysics Data System (ADS)

Zhang, Ye; Gong, Rongfang; Cheng, Xiaoliang; Gulliksson, Mårten

2018-06-01

This study considers the inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary data. The unknown source term is to be determined by additional boundary conditions. Unlike the existing methods found in the literature, which usually employ the first-order in time gradient-like system (such as the steepest descent methods) for numerically solving the regularized optimization problem with a fixed regularization parameter, we propose a novel method with a second-order in time dissipative gradient-like system and a dynamical selected regularization parameter. A damped symplectic scheme is proposed for the numerical solution. Theoretical analysis is given for both the continuous model and the numerical algorithm. Several numerical examples are provided to show the robustness of the proposed algorithm.

Boundary layers at the interface of two different shear flows

NASA Astrophysics Data System (ADS)

Weidman, Patrick D.; Wang, C. Y.

2018-05-01

We present solutions for the boundary layer between two uniform shear flows flowing in the same direction. In the upper layer, the flow has shear strength a, fluid density ρ1, and kinematic viscosity ν1, while the lower layer has shear strength b, fluid density ρ2, and kinematic viscosity ν2. Similarity transformations reduce the boundary-layer equations to a pair of ordinary differential equations governed by three dimensionless parameters: the shear strength ratio γ = b/a, the density ratio ρ = ρ2/ρ1, and the viscosity ratio ν = ν2/ν1. Further analysis shows that an affine transformation reduces this multi-parameter problem to a single ordinary differential equation which may be efficiently integrated as an initial-value problem. Solutions of the original boundary-value problem are shown to agree with the initial-value integrations, but additional dual and quadruple solutions are found using this method. We argue on physical grounds and through bifurcation analysis that these additional solutions are not tenable. The present problem is applicable to the trailing edge flow over a thin airfoil with camber.

On radiative heat transfer in stagnation point flow of MHD Carreau fluid over a stretched surface

NASA Astrophysics Data System (ADS)

Khan, Masood; Sardar, Humara; Mudassar Gulzar, M.

2018-03-01

This paper investigates the behavior of MHD stagnation point flow of Carreau fluid in the presence of infinite shear rate viscosity. Additionally heat transfer analysis in the existence of non-linear radiation with convective boundary condition is performed. Moreover effects of Joule heating is observed and mathematical analysis is presented in the presence of viscous dissipation. The suitable transformations are employed to alter the leading partial differential equations to a set of ordinary differential equations. The subsequent non-straight common ordinary differential equations are solved numerically by an effective numerical approach specifically Runge-Kutta Fehlberg method alongside shooting technique. It is found that the higher values of Hartmann number (M) correspond to thickening of the thermal and thinning of momentum boundary layer thickness. The analysis further reveals that the fluid velocity is diminished by increasing the viscosity ratio parameter (β∗) and opposite trend is observed for temperature profile for both hydrodynamic and hydromagnetic flows. In addition the momentum boundary layer thickness is increased with velocity ratio parameter (α) and opposite is true for thermal boundary layer thickness.

Dual Dynamically Orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions

NASA Astrophysics Data System (ADS)

Musharbash, Eleonora; Nobile, Fabio

2018-02-01

In this paper we propose a method for the strong imposition of random Dirichlet boundary conditions in the Dynamical Low Rank (DLR) approximation of parabolic PDEs and, in particular, incompressible Navier Stokes equations. We show that the DLR variational principle can be set in the constrained manifold of all S rank random fields with a prescribed value on the boundary, expressed in low rank format, with rank smaller then S. We characterize the tangent space to the constrained manifold by means of a Dual Dynamically Orthogonal (Dual DO) formulation, in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions, consistent with the original problem. The Dual DO formulation is also convenient to include the incompressibility constraint, when dealing with incompressible Navier Stokes equations. We show the performance of the proposed Dual DO approximation on two numerical test cases: the classical benchmark of a laminar flow around a cylinder with random inflow velocity, and a biomedical application for simulating blood flow in realistic carotid artery reconstructed from MRI data with random inflow conditions coming from Doppler measurements.

Unsteady transonic viscous-inviscid interaction using Euler and boundary-layer equations

NASA Technical Reports Server (NTRS)

Pirzadeh, Shahyar; Whitfield, Dave

1989-01-01

The Euler code is used extensively for computation of transonic unsteady aerodynamics. The boundary layer code solves the 3-D, compressible, unsteady, mean flow kinetic energy integral boundary layer equations in the direct mode. Inviscid-viscous coupling is handled using porosity boundary conditions. Some of the advantages and disadvantages of using the Euler and boundary layer equations for investigating unsteady viscous-inviscid interaction is examined.

A collocation--Galerkin finite element model of cardiac action potential propagation.

PubMed

Rogers, J M; McCulloch, A D

1994-08-01

A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, we observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques.

Exact semi-separation of variables in waveguides with non-planar boundaries

NASA Astrophysics Data System (ADS)

Athanassoulis, G. A.; Papoutsellis, Ch. E.

2017-05-01

Series expansions of unknown fields Φ =∑φn Zn in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions Zn are determined by solving local Sturm-Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to Zn cannot be compatible with the physical boundary conditions of Φ, leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in Athanassoulis & Belibassakis (Athanassoulis & Belibassakis 1999 J. Fluid Mech. 389, 275-301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly convergent representation of the field Φ, valid for any smooth, non-planar boundaries and any smooth enough Φ. This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.

Scattering Of Nonplanar Acoustic Waves

NASA Technical Reports Server (NTRS)

Gillman, Judith M.; Farassat, F.; Myers, M. K.

1995-01-01

Report presents theoretical study of scattering of nonplanar acoustic waves by rigid bodies. Study performed as part of effort to develop means of predicting scattering, from aircraft fuselages, of noise made by rotating blades. Basic approach was to model acoustic scattering by use of boundary integral equation to solve equation by the Galerkin method.

Prediction of Undsteady Flows in Turbomachinery Using the Linearized Euler Equations on Deforming Grids

NASA Technical Reports Server (NTRS)

Clark, William S.; Hall, Kenneth C.

1994-01-01

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.

A rigorous and simpler method of image charges

NASA Astrophysics Data System (ADS)

Ladera, C. L.; Donoso, G.

2016-07-01

The method of image charges relies on the proven uniqueness of the solution of the Laplace differential equation for an electrostatic potential which satisfies some specified boundary conditions. Granted by that uniqueness, the method of images is rightly described as nothing but shrewdly guessing which and where image charges are to be placed to solve the given electrostatics problem. Here we present an alternative image charges method that is based not on guessing but on rigorous and simpler theoretical grounds, namely the constant potential inside any conductor and the application of powerful geometric symmetries. The aforementioned required uniqueness and, more importantly, guessing are therefore both altogether dispensed with. Our two new theoretical fundaments also allow the image charges method to be introduced in earlier physics courses for engineering and sciences students, instead of its present and usual introduction in electromagnetic theory courses that demand familiarity with the Laplace differential equation and its boundary conditions.

Phase retrieval with the transport-of-intensity equation in an arbitrarily-shaped aperture by iterative discrete cosine transforms

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

Huang, Lei; Zuo, Chao; Idir, Mourad

A novel transport-of-intensity equation (TIE) based phase retrieval method is proposed with putting an arbitrarily-shaped aperture into the optical wavefield. In this arbitrarily-shaped aperture, the TIE can be solved under non-uniform illuminations and even non-homogeneous boundary conditions by iterative discrete cosine transforms with a phase compensation mechanism. Simulation with arbitrary phase, arbitrary aperture shape, and non-uniform intensity distribution verifies the effective compensation and high accuracy of the proposed method. Experiment is also carried out to check the feasibility of the proposed method in real measurement. Comparing to the existing methods, the proposed method is applicable for any types of phasemore » distribution under non-uniform illumination and non-homogeneous boundary conditions within an arbitrarily-shaped aperture, which enables the technique of TIE with hard aperture become a more flexible phase retrieval tool in practical measurements.« less

Phase retrieval with the transport-of-intensity equation in an arbitrarily-shaped aperture by iterative discrete cosine transforms

DOE PAGES

Huang, Lei; Zuo, Chao; Idir, Mourad; ...

2015-04-21

A novel transport-of-intensity equation (TIE) based phase retrieval method is proposed with putting an arbitrarily-shaped aperture into the optical wavefield. In this arbitrarily-shaped aperture, the TIE can be solved under non-uniform illuminations and even non-homogeneous boundary conditions by iterative discrete cosine transforms with a phase compensation mechanism. Simulation with arbitrary phase, arbitrary aperture shape, and non-uniform intensity distribution verifies the effective compensation and high accuracy of the proposed method. Experiment is also carried out to check the feasibility of the proposed method in real measurement. Comparing to the existing methods, the proposed method is applicable for any types of phasemore » distribution under non-uniform illumination and non-homogeneous boundary conditions within an arbitrarily-shaped aperture, which enables the technique of TIE with hard aperture become a more flexible phase retrieval tool in practical measurements.« less

Analysis of iced wings

NASA Technical Reports Server (NTRS)

Cebeci, T.; Chen, H. H.; Kaups, K.; Schimke, S.; Shin, J.

1992-01-01

A method for computing ice shapes along the leading edge of a wing and a method for predicting its aerodynamic performance degradation due to icing is described. Ice shapes are computed using an extension of the LEWICE code which was developed for airfoils. The aerodynamic properties of the iced wing are determined with an interactive scheme in which the solutions of the inviscid flow equations are obtained from a panel method and the solutions of the viscous flow equations are obtained from an inverse three-dimensional finite-difference boundary-layer method. A new interaction law is used to couple the inviscid and viscous flow solutions. The application of the LEWICE wing code to the calculation of ice shapes on a MS-317 swept wing shows good agreement with measurements. The interactive boundary-layer method is applied to a tapered ice wing in order to study the effect of icing on the aerodynamic properties of the wing at several angles of attack.

Acoustic coupled fluid-structure interactions using a unified fast multipole boundary element method.

PubMed

Wilkes, Daniel R; Duncan, Alec J

2015-04-01

This paper presents a numerical model for the acoustic coupled fluid-structure interaction (FSI) of a submerged finite elastic body using the fast multipole boundary element method (FMBEM). The Helmholtz and elastodynamic boundary integral equations (BIEs) are, respectively, employed to model the exterior fluid and interior solid domains, and the pressure and displacement unknowns are coupled between conforming meshes at the shared boundary interface to achieve the acoustic FSI. The low frequency FMBEM is applied to both BIEs to reduce the algorithmic complexity of the iterative solution from O(N(2)) to O(N(1.5)) operations per matrix-vector product for N boundary unknowns. Numerical examples are presented to demonstrate the algorithmic and memory complexity of the method, which are shown to be in good agreement with the theoretical estimates, while the solution accuracy is comparable to that achieved by a conventional finite element-boundary element FSI model.

Eigenmode Analysis of Boundary Conditions for One-Dimensional Preconditioned Euler Equations

NASA Technical Reports Server (NTRS)

Darmofal, David L.

1998-01-01

An analysis of the effect of local preconditioning on boundary conditions for the subsonic, one-dimensional Euler equations is presented. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective with preconditioning, and, at low Mach numbers, disturbances do not decay. Other boundary conditions are investigated which are non-reflective with preconditioning and numerical results are presented confirming the analysis.

Unstructured High-Order Galerkin-Temporal- Boundary Methods for the Klein-Gordon Equation with Non-Reflecting Boundary Conditions

DTIC Science & Technology

2010-06-01

9 C. Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . 11 1. Gravity Effects . . . . . . . . . . . . . . . . . . . . . . . . . 12 2...describe the high-order spectral element method used to discretize the problem in space (up to 16th order polynomials ) in Chapter IV. Chapter V discusses...inertial frame. Body forces are those acting on the fluid volume that are proportional to the mass. The body forces considered here are gravity and

Analysis of a diffuse interface model of multispecies tumor growth

NASA Astrophysics Data System (ADS)

Dai, Mimi; Feireisl, Eduard; Rocca, Elisabetta; Schimperna, Giulio; Schonbek, Maria E.

2017-04-01

We consider a diffuse interface model for tumor growth recently proposed in Chen et al (2014 Int. J. Numer. Methods Biomed. Eng. 30 726-54). In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn-Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity \\mathbf{u} satisfies \\mathbf{u}\\centerdot ν >0 , where ν is the outer normal to the boundary of the domain.

Vacuum stress energy density and its gravitational implications

NASA Astrophysics Data System (ADS)

Estrada, Ricardo; Fulling, Stephen A.; Kaplan, Lev; Kirsten, Klaus; Liu, Zhonghai; Milton, Kimball A.

2008-04-01

In nongravitational physics the local density of energy is often regarded as merely a bookkeeping device; only total energy has an experimental meaning—and it is only modulo a constant term. But in general relativity the local stress-energy tensor is the source term in Einstein's equation. In closed universes, and those with Kaluza-Klein dimensions, theoretical consistency demands that quantum vacuum energy should exist and have gravitational effects, although there are no boundary materials giving rise to that energy by van der Waals interactions. In the lab there are boundaries, and in general the energy density has a nonintegrable singularity as a boundary is approached (for idealized boundary conditions). As pointed out long ago by Candelas and Deutsch, in this situation there is doubt about the viability of the semiclassical Einstein equation. Our goal is to show that the divergences in the linearized Einstein equation can be renormalized to yield a plausible approximation to the finite theory that presumably exists for realistic boundary conditions. For a scalar field with Dirichlet or Neumann boundary conditions inside a rectangular parallelepiped, we have calculated by the method of images all components of the stress tensor, for all values of the conformal coupling parameter and an exponential ultraviolet cutoff parameter. The qualitative features of contributions from various classes of closed classical paths are noted. Then the Estrada-Kanwal distributional theory of asymptotics, particularly the moment expansion, is used to show that the linearized Einstein equation with the stress-energy near a plane boundary as source converges to a consistent theory when the cutoff is removed. This paper reports work in progress on a project combining researchers in Texas, Louisiana and Oklahoma. It is supported by NSF Grants PHY-0554849 and PHY-0554926.

Applying Boundary Conditions Using a Time-Dependent Lagrangian for Modeling Laser-Plasma Interactions

NASA Astrophysics Data System (ADS)

Reyes, Jonathan; Shadwick, B. A.

2016-10-01

Modeling the evolution of a short, intense laser pulse propagating through an underdense plasma is of particular interest in the physics of laser-plasma interactions. Numerical models are typically created by first discretizing the equations of motion and then imposing boundary conditions. Using the variational principle of Chen and Sudan, we spatially discretize the Lagrangian density to obtain discrete equations of motion and a discrete energy conservation law which is exactly satisfied regardless of the spatial grid resolution. Modifying the derived equations of motion (e.g., enforcing boundary conditions) generally ruins energy conservation. However, time-dependent terms can be added to the Lagrangian which force the equations of motion to have the desired boundary conditions. Although some foresight is needed to choose these time-dependent terms, this approach provides a mechanism for energy to exit the closed system while allowing the conservation law to account for the loss. An appropriate time discretization scheme is selected based on stability analysis and resolution requirements. We present results using this variational approach in a co-moving coordinate system and compare such results to those using traditional second-order methods. This work was supported by the U. S. Department of Energy under Contract No. DE-SC0008382 and by the National Science Foundation under Contract No. PHY- 1104683.

FDM study of ion exchange diffusion equation in glass

NASA Astrophysics Data System (ADS)

Zhou, Zigang; Yang, Yongjia; Wang, Qiang; Sun, Guangchun

2009-05-01

Ion-exchange technique in glass was developed to fabricate gradient refractive index optical devices. In this paper, the Finite Difference Method(FDM), which is used for the solution of ion-diffusion equation, is reported. This method transforms continual diffusion equation to separate difference equation. It unitizes the matrix of MATLAB program to solve the iteration process. The collation results under square boundary condition show that it gets a more accurate numerical solution. Compared to experiment data, the relative error is less than 0.2%. Furthermore, it has simply operation and kinds of output solutions. This method can provide better results for border-proliferation of the hexagonal and the channel devices too.

On the inclusion of mass source terms in a single-relaxation-time lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Aursjø, Olav; Jettestuen, Espen; Vinningland, Jan Ludvig; Hiorth, Aksel

2018-05-01

We present a lattice Boltzmann algorithm for incorporating a mass source in a fluid flow system. The proposed mass source/sink term, included in the lattice Boltzmann equation, maintains the Galilean invariance and the accuracy of the overall method, while introducing a mass source/sink term in the fluid dynamical equations. The method can, for instance, be used to inject or withdraw fluid from any preferred lattice node in a system. This suggests that injection and withdrawal of fluid does not have to be introduced through cumbersome, and sometimes less accurate, boundary conditions. The method also suggests that, through a chosen equation of state relating mass density to pressure, the proposed mass source term will render it possible to set a preferred pressure at any lattice node in a system. We demonstrate how this model handles injection and withdrawal of a fluid. And we show how it can be used to incorporate pressure boundaries. The accuracy of the algorithm is identified through a Chapman-Enskog expansion of the model and supported by the numerical simulations.

Three-dimensional unsteady Euler equations solutions on dynamic grids

NASA Technical Reports Server (NTRS)

Belk, D. M.; Janus, J. M.; Whitfield, D. L.

1985-01-01

A method is presented for solving the three-dimensional unsteady Euler equations on dynamic grids based on flux vector splitting. The equations are cast in curvilinear coordinates and a finite volume discretization is used for handling arbitrary geometries. The discretized equations are solved using an explicit upwind second-order predictor corrector scheme that is stable for a CFL of 2. Characteristic variable boundary conditions are developed and used for unsteady impermeable surfaces and for the far-field boundary. Dynamic-grid results are presented for an oscillating air-foil and for a store separating from a reflection plate. For the cases considered of stores separating from a reflection plate, the unsteady aerodynamic forces on the store are significantly different from forces obtained by steady-state aerodynamics with the body inclination angle changed to account for plunge velocity.

Initial-Boundary Value Problem for Two-Component Gerdjikov-Ivanov Equation with 3 × 3 Lax Pair on Half-Line

NASA Astrophysics Data System (ADS)

Zhu, Qiao-Zhen; Fan, En-Gui; Xu, Jian

2017-10-01

The Fokas unified method is used to analyze the initial-boundary value problem of two-component Gerdjikov-Ivanonv equation on the half-line. It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 × 3 Riemann-Hilbert problem. The Dirichlet to Neumann map is obtained through the global relation. Supported by grants from the National Science Foundation of China under Grant No. 11671095, National Science Foundation of China under Grant No. 11501365, Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant No 15YF1408100, and the Hujiang Foundation of China (B14005)

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

Kim, K.; Petersson, N. A.; Rodgers, A.

Acoustic waveform modeling is a computationally intensive task and full three-dimensional simulations are often impractical for some geophysical applications such as long-range wave propagation and high-frequency sound simulation. In this study, we develop a two-dimensional high-order accurate finite-difference code for acoustic wave modeling. We solve the linearized Euler equations by discretizing them with the sixth order accurate finite difference stencils away from the boundary and the third order summation-by-parts (SBP) closure near the boundary. Non-planar topographic boundary is resolved by formulating the governing equation in curvilinear coordinates following the interface. We verify the implementation of the algorithm by numerical examplesmore » and demonstrate the capability of the proposed method for practical acoustic wave propagation problems in the atmosphere.« less

Slip Boundary Conditions for the Compressible Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

Aoki, Kazuo; Baranger, Céline; Hattori, Masanari; Kosuge, Shingo; Martalò, Giorgio; Mathiaud, Julien; Mieussens, Luc

2017-11-01

The slip boundary conditions for the compressible Navier-Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman-Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar-Gross-Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.

Hypersonic three-dimensional nonequilibrium boundary-layer equations in generalized curvilinear coordinates

NASA Technical Reports Server (NTRS)

Lee, Jong-Hun

1993-01-01

The basic governing equations for the second-order three-dimensional hypersonic thermal and chemical nonequilibrium boundary layer are derived by means of an order-of-magnitude analysis. A two-temperature concept is implemented into the system of boundary-layer equations by simplifying the rather complicated general three-temperature thermal gas model. The equations are written in a surface-oriented non-orthogonal curvilinear coordinate system, where two curvilinear coordinates are non-orthogonial and a third coordinate is normal to the surface. The equations are described with minimum use of tensor expressions arising from the coordinate transformation, to avoid unnecessary confusion for readers. The set of equations obtained will be suitable for the development of a three-dimensional nonequilibrium boundary-layer code. Such a code could be used to determine economically the aerodynamic/aerothermodynamic loads to the surfaces of hypersonic vehicles with general configurations. In addition, the basic equations for three-dimensional stagnation flow, of which solution is required as an initial value for space-marching integration of the boundary-layer equations, are given along with the boundary conditions, the boundary-layer parameters, and the inner-outer layer matching procedure. Expressions for the chemical reaction rates and the thermodynamic and transport properties in the thermal nonequilibrium environment are explicitly given.

Solution of partial differential equations on vector and parallel computers

NASA Technical Reports Server (NTRS)

Ortega, J. M.; Voigt, R. G.

1985-01-01

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.

Numerical methods for incompressible viscous flows with engineering applications

NASA Technical Reports Server (NTRS)

Rose, M. E.; Ash, R. L.

1988-01-01

A numerical scheme has been developed to solve the incompressible, 3-D Navier-Stokes equations using velocity-vorticity variables. This report summarizes the development of the numerical approximation schemes for the divergence and curl of the velocity vector fields and the development of compact schemes for handling boundary and initial boundary value problems.

Vibration analysis of rotating functionally graded Timoshenko microbeam based on modified couple stress theory under different temperature distributions

NASA Astrophysics Data System (ADS)

Ghadiri, Majid; Shafiei, Navvab

2016-04-01

In this study, thermal vibration of rotary functionally graded Timoshenko microbeam has been analyzed based on modified couple stress theory considering temperature change in four types of temperature distribution on thermal environment. Material properties of FG microbeam are supposed to be temperature dependent and vary continuously along the thickness according to the power-law form. The axial forces are also included in the model as the thermal and true spatial variation due to the rotation. Governing equations and boundary conditions have been derived by employing Hamiltonian's principle. The differential quadrature method is employed to solve the governing equations for cantilever and propped cantilever boundary conditions. Validations are done by comparing available literatures and obtained results which indicate accuracy of applied method. Results represent effects of temperature changes, different boundary conditions, nondimensional angular velocity, length scale parameter, different boundary conditions, FG index and beam thickness on fundamental, second and third nondimensional frequencies. Results determine critical values of temperature changes and other essential parameters which can be applicable to design micromachines like micromotor and microturbine.

Finite Difference Time Marching in the Frequency Domain: A Parabolic Formulation for the Convective Wave Equation

NASA Technical Reports Server (NTRS)

Baumeister, K. J.; Kreider, K. L.

1996-01-01

An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Numerical Recovering of a Speed of Sound by the BC-Method in 3D

NASA Astrophysics Data System (ADS)

Pestov, Leonid; Bolgova, Victoria; Danilin, Alexandr

We develop the numerical algorithm for solving the inverse problem for the wave equation by the Boundary Control method. The problem, which we refer to as a forward one, is an initial boundary value problem for the wave equation with zero initial data in the bounded domain. The inverse problem is to find the speed of sound c(x) by the measurements of waves induced by a set of boundary sources. The time of observation is assumed to be greater then two acoustical radius of the domain. The numerical algorithm for sound reconstruction is based on two steps. The first one is to find a (sufficiently large) number of controls {f_j} (the basic control is defined by the position of the source and some time delay), which generates the same number of known harmonic functions, i.e. Δ {u_j}(.,T) = 0 , where {u_j} is the wave generated by the control {f_j} . After that the linear integral equation w.r.t. the speed of sound is obtained. The piecewise constant model of the speed is used. The result of numerical testing of 3-dimensional model is presented.

PAN AIR: A computer program for predicting subsonic or supersonic linear potential flows about arbitrary configurations using a higher order panel method. Volume 1: Theory document (version 3.0)

NASA Technical Reports Server (NTRS)

Epton, Michael A.; Magnus, Alfred E.

1990-01-01

An outline of the derivation of the differential equation governing linear subsonic and supersonic potential flow is given. The use of Green's Theorem to obtain an integral equation over the boundary surface is discussed. The engineering techniques incorporated in the Panel Aerodynamics (PAN AIR) program (a discretization method which solves the integral equation for arbitrary first order boundary conditions) are then discussed in detail. Items discussed include the construction of the compressibility transformation, splining techniques, imposition of the boundary conditions, influence coefficient computation (including the concept of the finite part of an integral), computation of pressure coefficients, and computation of forces and moments. Principal revisions to version 3.0 are the following: (1) appendices H and K more fully describe the Aerodynamic Influence Coefficient (AIC) construction; (2) appendix L now provides a complete description of the AIC solution process; (3) appendix P is new and discusses the theory for the new FDP module (which calculates streamlines and offbody points); and (4) numerous small corrections and revisions reflecting the MAG module rewrite.

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.

Controlling Wavebreaking in a Viscous Fluid Conduit

NASA Astrophysics Data System (ADS)

Anderson, Dalton; Maiden, Michelle; Hoefer, Mark

2015-11-01

This poster will present a new technique in the experimental investigation of dispersive hydrodynamics. In shallow water flows, internal ocean waves, superfluids, and optical media, wave breaking can be resolved by a dispersive shock wave (DSW). In this work, an experimental method to control the location of DSW formation (gradient catastrophe) is explained. The central idea is to convert an initial value problem (Riemann problem) into an equivalent boundary value problem. The system to which this technique is applied is a fluid conduit resulting from high viscosity contrast between a buoyant interior and heavier exterior fluid. The conduit cross-sectional area is modeled by a nonlinear, conservative, dispersive, third order partial differential equation. Using this model, the aim is to predict the breaking location of a DSW by controlling one boundary condition. An analytical expression for this boundary condition is derived by solving the dispersionless equation backward in time from the desired step via the method of characteristics. This is used in experiment to generate an injection rate profile for a high precision piston pump. This translates to the desired conduit shape. Varying the jump height and desired breaking location indicates good control of DSW formation. This result can be improved by deriving a conduit profile by numerical simulation of the full model equation. Controlling the breaking location of a DSW allows for the investigation of dynamics independent of the boundary. Support provided by NSF CAREER DMS-1255422 , NSF EXTREEMS.

Boundary-artifact-free phase retrieval with the transport of intensity equation II: applications to microlens characterization.

PubMed

Zuo, Chao; Chen, Qian; Li, Hongru; Qu, Weijuan; Asundi, Anand

2014-07-28

Boundary conditions play a crucial role in the solution of the transport of intensity equation (TIE). If not appropriately handled, they can create significant boundary artifacts across the reconstruction result. In a previous paper [Opt. Express 22, 9220 (2014)], we presented a new boundary-artifact-free TIE phase retrieval method with use of discrete cosine transform (DCT). Here we report its experimental investigations with applications to the micro-optics characterization. The experimental setup is based on a tunable lens based 4f system attached to a non-modified inverted bright-field microscope. We establish inhomogeneous Neumann boundary values by placing a rectangular aperture in the intermediate image plane of the microscope. Then the boundary values are applied to solve the TIE with our DCT-based TIE solver. Experimental results on microlenses highlight the importance of boundary conditions that often overlooked in simplified models, and confirm that our approach effectively avoid the boundary error even when objects are located at the image borders. It is further demonstrated that our technique is non-interferometric, accurate, fast, full-field, and flexible, rendering it a promising metrological tool for the micro-optics inspection.

The role of nonlinear critical layers in boundary layer transition

NASA Technical Reports Server (NTRS)

Goldstein, M.E.

1995-01-01

Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear spatially growing instability waves evolve downstream in nominally two-dimensional laminar boundary layers. The first nonlinear reaction takes place locally within a so-called 'critical layer', with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves - which may or may not be accompanied by an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy either a single integro-differential equation or a pair of integro-differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.

The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A careful study of the boundary error

NASA Technical Reports Server (NTRS)

Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul; Don, Wai-Sun

1993-01-01

The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) time advancement reduces the formal accuracy of the space-time method to first order locally, and second order globally, independently of the spatial operator. This counter intuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case: (1) impose the exact boundary condition only at the end of the complete RK cycle, (2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases, results in a scheme with much reduced CFL condition, rendering the RK scheme less attractive. The second method retains the same allowable time step as the periodic problem. However it is a general remedy only for the linear case. For non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.

Finite difference methods for transient signal propagation in stratified dispersive media

NASA Technical Reports Server (NTRS)

Lam, D. H.

1975-01-01

Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.

ZZ-Type a posteriori error estimators for adaptive boundary element methods on a curve☆

PubMed Central

Feischl, Michael; Führer, Thomas; Karkulik, Michael; Praetorius, Dirk

2014-01-01

In the context of the adaptive finite element method (FEM), ZZ-error estimators named after Zienkiewicz and Zhu (1987) [52] are mathematically well-established and widely used in practice. In this work, we propose and analyze ZZ-type error estimators for the adaptive boundary element method (BEM). We consider weakly singular and hyper-singular integral equations and prove, in particular, convergence of the related adaptive mesh-refining algorithms. Throughout, the theoretical findings are underlined by numerical experiments. PMID:24748725

Calculation of laminar heating rates on three-dimensional configurations using the axisymmetric analogue

NASA Technical Reports Server (NTRS)

Hamilton, H. H., II

1980-01-01

A theoretical method was developed for computing approximate laminar heating rates on three dimensional configurations at angle of attack. The method is based on the axisymmetric analogue which is used to reduce the three dimensional boundary layer equations along surface streamlines to an equivalent axisymmetric form by using the metric coefficient which describes streamline divergence (or convergence). The method was coupled with a three dimensional inviscid flow field program for computing surface streamline paths, metric coefficients, and boundary layer edge conditions.

Theoretical study of the tunnel-boundary lift interference due to slotted walls in the presence of the trailing-vortex system of a lifting model

NASA Technical Reports Server (NTRS)

Matthews, Clarence W

1955-01-01

The equations presented in this report give the interference on the trailing-vortex system of a uniformly loaded finite-span wing in a circular tunnel containing partly open and partly closed walls, with special reference to symmetrical arrangements of the open and closed portions. Methods are given for extending the equations to include tunnel shapes other than circular. The rectangular tunnel is used to demonstrate these methods. The equations are also extended to nonuniformly loaded wings.

Discontinuous Finite Element Quasidiffusion Methods

DOE PAGES

Anistratov, Dmitriy Yurievich; Warsa, James S.

2018-05-21

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

Discontinuous Finite Element Quasidiffusion Methods

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

Anistratov, Dmitriy Yurievich; Warsa, James S.

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

Exact solution of conductive heat transfer in cylindrical composite laminate

NASA Astrophysics Data System (ADS)

Kayhani, M. H.; Shariati, M.; Nourozi, M.; Karimi Demneh, M.

2009-11-01

This paper presents an exact solution for steady-state conduction heat transfer in cylindrical composite laminates. This laminate is cylindrical shape and in each lamina, fibers have been wound around the cylinder. In this article heat transfer in composite laminates is being investigated, by using separation of variables method and an analytical relation for temperature distribution in these laminates has been obtained under specific boundary conditions. Also Fourier coefficients in each layer obtain by solving set of equations that related to thermal boundary layer conditions at inside and outside of the cylinder also thermal continuity and heat flux continuity between each layer is considered. In this research LU factorization method has been used to solve the set of equations.

Numerical simulation of the vortical flow around a pitching airfoil

NASA Astrophysics Data System (ADS)

Fu, Xiang; Li, Gaohua; Wang, Fuxin

2017-04-01

In order to study the dynamic behaviors of the flapping wing, the vortical flow around a pitching NACA0012 airfoil is investigated. The unsteady flow field is obtained by a very efficient zonal procedure based on the velocity-vorticity formulation and the Reynolds number based on the chord length of the airfoil is set to 1 million. The zonal procedure divides up the whole computation domain in to three zones: potential flow zone, boundary layer zone and Navier-Stokes zone. Since the vorticity is absent in the potential flow zone, the vorticity transport equation needs only to be solved in the boundary layer zone and Navier-Stokes zone. Moreover, the boundary layer equations are solved in the boundary layer zone. This arrangement drastically reduces the computation time against the traditional numerical method. After the flow field computation, the evolution of the vortices around the airfoil is analyzed in detail.

Computation of three-dimensional compressible boundary layers to fourth-order accuracy on wings and fuselages

NASA Technical Reports Server (NTRS)

Iyer, Venkit

1990-01-01

A solution method, fourth-order accurate in the body-normal direction and second-order accurate in the stream surface directions, to solve the compressible 3-D boundary layer equations is presented. The transformation used, the discretization details, and the solution procedure are described. Ten validation cases of varying complexity are presented and results of calculation given. The results range from subsonic flow to supersonic flow and involve 2-D or 3-D geometries. Applications to laminar flow past wing and fuselage-type bodies are discussed. An interface procedure is used to solve the surface Euler equations with the inviscid flow pressure field as the input to assure accurate boundary conditions at the boundary layer edge. Complete details of the computer program used and information necessary to run each of the test cases are given in the Appendix.

Calculation of three-dimensional compressible laminar and turbulent boundary layers. An implicit finite-difference procedure for solving the three-dimensional compressible laminar, transitional, and turbulent boundary-layer equations

NASA Technical Reports Server (NTRS)

Harris, J. E.

1975-01-01

An implicit finite-difference procedure is presented for solving the compressible three-dimensional boundary-layer equations. The method is second-order accurate, unconditionally stable (conditional stability for reverse cross flow), and efficient from the viewpoint of computer storage and processing time. The Reynolds stress terms are modeled by (1) a single-layer mixing length model and (2) a two-layer eddy viscosity model. These models, although simple in concept, accurately predicted the equilibrium turbulent flow for the conditions considered. Numerical results are compared with experimental wall and profile data for a cone at an angle of attack larger than the cone semiapex angle. These comparisons clearly indicate that the numerical procedure and turbulence models accurately predict the experimental data with as few as 21 nodal points in the plane normal to the wall boundary.

A Quadtree-gridding LBM with Immersed Boundary for Two-dimension Viscous Flows

NASA Astrophysics Data System (ADS)

Yao, Jieke; Feng, Wenliang; Chen, Bin; Zhou, Wei; Cao, Shikun

2017-07-01

An un-uniform quadtree grids lattice Boltzmann method (LBM) with immersed boundary is presented in this paper. In overlapping for different level grids, temporal and spatial interpolation are necessary to ensure the continuity of physical quantity. In order to take advantage of the equation for temporal and spatial step in the same level grids, equal interval interpolation, which is simple to apply to any refined boundary grids in the LBM, is adopted in temporal and spatial aspects to obtain second-order accuracy. The velocity correction, which can guarantee more preferably no-slip boundary condition than the direct forcing method and the momentum exchange method in the traditional immersed-boundary LBM, is used for solid boundary to make the best of Cartesian grid. In present quadtree-gridding immersed-boundary LBM, large eddy simulation (LES) is adopted to simulate the flows over obstacle in higher Reynolds number (Re). The incompressible viscous flows over circular cylinder are carried out, and a great agreement is obtained.

Discretization of three-dimensional free surface flows and moving boundary problems via elliptic grid methods based on variational principles

NASA Astrophysics Data System (ADS)

Fraggedakis, D.; Papaioannou, J.; Dimakopoulos, Y.; Tsamopoulos, J.

2017-09-01

A new boundary-fitted technique to describe free surface and moving boundary problems is presented. We have extended the 2D elliptic grid generator developed by Dimakopoulos and Tsamopoulos (2003) [19] and further advanced by Chatzidai et al. (2009) [18] to 3D geometries. The set of equations arises from the fulfillment of the variational principles established by Brackbill and Saltzman (1982) [21], and refined by Christodoulou and Scriven (1992) [22]. These account for both smoothness and orthogonality of the grid lines of tessellated physical domains. The elliptic-grid equations are accompanied by new boundary constraints and conditions which are based either on the equidistribution of the nodes on boundary surfaces or on the existing 2D quasi-elliptic grid methodologies. The capabilities of the proposed algorithm are first demonstrated in tests with analytically described complex surfaces. The sequence in which these tests are presented is chosen to help the reader build up experience on the best choice of the elliptic grid parameters. Subsequently, the mesh equations are coupled with the Navier-Stokes equations, in order to reveal the full potential of the proposed methodology in free surface flows. More specifically, the problem of gas assisted injection in ducts of circular and square cross-sections is examined, where the fluid domain experiences extreme deformations. Finally, the flow-mesh solver is used to calculate the equilibrium shapes of static menisci in capillary tubes.

MHD Free Convective Boundary Layer Flow of a Nanofluid past a Flat Vertical Plate with Newtonian Heating Boundary Condition

PubMed Central

Uddin, Mohammed J.; Khan, Waqar A.; Ismail, Ahmed I.

2012-01-01

Steady two dimensional MHD laminar free convective boundary layer flows of an electrically conducting Newtonian nanofluid over a solid stationary vertical plate in a quiescent fluid taking into account the Newtonian heating boundary condition is investigated numerically. A magnetic field can be used to control the motion of an electrically conducting fluid in micro/nano scale systems used for transportation of fluid. The transport equations along with the boundary conditions are first converted into dimensionless form and then using linear group of transformations, the similarity governing equations are developed. The transformed equations are solved numerically using the Runge-Kutta-Fehlberg fourth-fifth order method with shooting technique. The effects of different controlling parameters, namely, Lewis number, Prandtl number, buoyancy ratio, thermophoresis, Brownian motion, magnetic field and Newtonian heating on the flow and heat transfer are investigated. The numerical results for the dimensionless axial velocity, temperature and nanoparticle volume fraction as well as the reduced Nusselt and Sherwood number have been presented graphically and discussed. It is found that the rate of heat and mass transfer increase as Newtonian heating parameter increases. The dimensionless velocity and temperature distributions increase with the increase of Newtonian heating parameter. The results of the reduced heat transfer rate is compared for convective heating boundary condition and found an excellent agreement. PMID:23166688

Examining Potential Boundary Bias Effects in Kernel Smoothing on Equating: An Introduction for the Adaptive and Epanechnikov Kernels.

PubMed

Cid, Jaime A; von Davier, Alina A

2015-05-01

Test equating is a method of making the test scores from different test forms of the same assessment comparable. In the equating process, an important step involves continuizing the discrete score distributions. In traditional observed-score equating, this step is achieved using linear interpolation (or an unscaled uniform kernel). In the kernel equating (KE) process, this continuization process involves Gaussian kernel smoothing. It has been suggested that the choice of bandwidth in kernel smoothing controls the trade-off between variance and bias. In the literature on estimating density functions using kernels, it has also been suggested that the weight of the kernel depends on the sample size, and therefore, the resulting continuous distribution exhibits bias at the endpoints, where the samples are usually smaller. The purpose of this article is (a) to explore the potential effects of atypical scores (spikes) at the extreme ends (high and low) on the KE method in distributions with different degrees of asymmetry using the randomly equivalent groups equating design (Study I), and (b) to introduce the Epanechnikov and adaptive kernels as potential alternative approaches to reducing boundary bias in smoothing (Study II). The beta-binomial model is used to simulate observed scores reflecting a range of different skewed shapes.

Direct numerical simulation of laminar-turbulent flow over a flat plate at hypersonic flow speeds

NASA Astrophysics Data System (ADS)

Egorov, I. V.; Novikov, A. V.

2016-06-01

A method for direct numerical simulation of a laminar-turbulent flow around bodies at hypersonic flow speeds is proposed. The simulation is performed by solving the full three-dimensional unsteady Navier-Stokes equations. The method of calculation is oriented to application of supercomputers and is based on implicit monotonic approximation schemes and a modified Newton-Raphson method for solving nonlinear difference equations. By this method, the development of three-dimensional perturbations in the boundary layer over a flat plate and in a near-wall flow in a compression corner is studied at the Mach numbers of the free-stream of M = 5.37. In addition to pulsation characteristic, distributions of the mean coefficients of the viscous flow in the transient section of the streamlined surface are obtained, which enables one to determine the beginning of the laminar-turbulent transition and estimate the characteristics of the turbulent flow in the boundary layer.

Least-Squares PN Formulation of the Transport Equation Using Self-Adjoint-Angular-Flux Consistent Boundary Conditions

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

Laboure, Vincent M.; Wang, Yaqi; DeHart, Mark D.

In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids [1] in the context of Continuous Finite Element Methods (CFEM).We first deriveweakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment [2], in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework [3] using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutionsmore » (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.« less

Least-Squares PN Formulation of the Transport Equation Using Self-Adjoint-Angular-Flux Consistent Boundary Conditions.

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

Vincent M. Laboure; Yaqi Wang; Mark D. DeHart

In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids in the context of Continuous Finite Element Methods (CFEM).We first deriveweakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment, in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and findmore » the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.« less

A computationally efficient modelling of laminar separation bubbles

NASA Technical Reports Server (NTRS)

Maughmer, Mark D.

1988-01-01

The goal of this research is to accurately predict the characteristics of the laminar separation bubble and its effects on airfoil performance. To this end, a model of the bubble is under development and will be incorporated in the analysis section of the Eppler and Somers program. As a first step in this direction, an existing bubble model was inserted into the program. It was decided to address the problem of the short bubble before attempting the prediction of the long bubble. In the second place, an integral boundary-layer method is believed more desirable than a finite difference approach. While these two methods achieve similar prediction accuracy, finite-difference methods tend to involve significantly longer computer run times than the integral methods. Finally, as the boundary-layer analysis in the Eppler and Somers program employs the momentum and kinetic energy integral equations, a short-bubble model compatible with these equations is most preferable.

Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip

PubMed Central

Filobello-Nino, U.; Vazquez-Leal, H.; Sarmiento-Reyes, A.; Benhammouda, B.; Jimenez-Fernandez, V. M.; Pereyra-Diaz, D.; Perez-Sesma, A.; Cervantes-Perez, J.; Huerta-Chua, J.; Sanchez-Orea, J.; Contreras-Hernandez, A. D.

2014-01-01

The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient. PMID:27433526

A numerical method for osmotic water flow and solute diffusion with deformable membrane boundaries in two spatial dimension

NASA Astrophysics Data System (ADS)

Yao, Lingxing; Mori, Yoichiro

2017-12-01

Osmotic forces and solute diffusion are increasingly seen as playing a fundamental role in cell movement. Here, we present a numerical method that allows for studying the interplay between diffusive, osmotic and mechanical effects. An osmotically active solute obeys a advection-diffusion equation in a region demarcated by a deformable membrane. The interfacial membrane allows transmembrane water flow which is determined by osmotic and mechanical pressure differences across the membrane. The numerical method is based on an immersed boundary method for fluid-structure interaction and a Cartesian grid embedded boundary method for the solute. We demonstrate our numerical algorithm with the test case of an osmotic engine, a recently proposed mechanism for cell propulsion.

Application of a soft computing technique in predicting the percentage of shear force carried by walls in a rectangular channel with non-homogeneous roughness.

PubMed

Khozani, Zohreh Sheikh; Bonakdari, Hossein; Zaji, Amir Hossein

2016-01-01

Two new soft computing models, namely genetic programming (GP) and genetic artificial algorithm (GAA) neural network (a combination of modified genetic algorithm and artificial neural network methods) were developed in order to predict the percentage of shear force in a rectangular channel with non-homogeneous roughness. The ability of these methods to estimate the percentage of shear force was investigated. Moreover, the independent parameters' effectiveness in predicting the percentage of shear force was determined using sensitivity analysis. According to the results, the GP model demonstrated superior performance to the GAA model. A comparison was also made between the GP program determined as the best model and five equations obtained in prior research. The GP model with the lowest error values (root mean square error ((RMSE) of 0.0515) had the best function compared with the other equations presented for rough and smooth channels as well as smooth ducts. The equation proposed for rectangular channels with rough boundaries (RMSE of 0.0642) outperformed the prior equations for smooth boundaries.

Application of different variants of the BEM in numerical modeling of bioheat transfer problems.

PubMed

Majchrzak, Ewa

2013-09-01

Heat transfer processes proceeding in the living organisms are described by the different mathematical models. In particular, the typical continuous model of bioheat transfer bases on the most popular Pennes equation, but the Cattaneo-Vernotte equation and the dual phase lag equation are also used. It should be pointed out that in parallel are also examined the vascular models, and then for the large blood vessels and tissue domain the energy equations are formulated separately. In the paper the different variants of the boundary element method as a tool of numerical solution of bioheat transfer problems are discussed. For the steady state problems and the vascular models the classical BEM algorithm and also the multiple reciprocity BEM are presented. For the transient problems connected with the heating of tissue, the various tissue models are considered for which the 1st scheme of the BEM, the BEM using discretization in time and the general BEM are applied. Examples of computations illustrate the possibilities of practical applications of boundary element method in the scope of bioheat transfer problems.

Dust transportation in bounday layers on complex areas

NASA Astrophysics Data System (ADS)

Karelsky, Kirill; Petrosyan, Arakel

2017-04-01

This presentation is aimed at creating and realization of new physical model of impurity transfer (solid particles and heavy gases) in areas with non-flat and/or nonstationary boundaries. The main idea of suggested method is to use non-viscous equations for solid particles transport modeling in the vicinity of complex boundary. In viscous atmosphere with as small as one likes coefficient of molecular viscosity, the non-slip boundary condition on solid surface must be observed. This postulates the reduction of velocity to zero at a solid surface. It is unconditionally in this case Prandtle hypothesis must be observed: for rather wide range of conditions in the surface neighboring layers energy dissipation of atmosphere flows is comparable by magnitude with manifestation of inertia forces. That is why according to Prandtle hypothesis in atmosphere movement characterizing by a high Reynolds number the boundary layer is forming near a planet surface, within which the required transition from zero velocities at the surface to magnitudes at the external boundary of the layer that are quite close to ones in ideal atmosphere flow. In that layer fast velocity gradients cause viscous effects to be comparable in magnitude with inertia forces influence. For conditions considered essential changes of hydrodynamic fields near solid boundary caused not only by nonslip condition but also by a various relief of surface: mountains, street canyons, individual buildings. Transport of solid particles, their ascent and precipitation also result in dramatic changes of meteorological fields. As dynamic processes of solid particles transfer accompanying the flow past of complex relief surface by wind flows is of our main interest we are to use equations of non-viscous hydrodynamic. We should put up with on the one hand idea of high wind gradients in the boundary layer and on the other hand disregard of molecular viscosity in two-phase atmosphere equations. We deal with describing high field gradients with the aid of scheme viscosity of numerical algorithm used to model near-surface phenomena. This idea is implemented in the model of ideal gas equations with variable equation of state describing particulates transportation within boundary layer with obstacles.

The Renormalization-Group Method in the Problem on Calculation of the Spectral Energy Density of Fluid Turbulence

NASA Astrophysics Data System (ADS)

Teodorovich, E. V.

2018-03-01

In order to find the shape of energy spectrum within the framework of the model of stationary homogeneous isotropic turbulence, the renormalization-group equations, which reflect the Markovian nature of the mechanism of energy transfer along the wavenumber spectrum, are used in addition to the dimensional considerations and the energy balance equation. For the spectrum, the formula depends on three parameters, namely, the wavenumber, which determines the upper boundary of the range of the turbulent energy production, the spectral flux through this boundary, and the fluid kinematic viscosity.

Noniterative three-dimensional grid generation using parabolic partial differential equations

NASA Technical Reports Server (NTRS)

Edwards, T. A.

1985-01-01

A new algorithm for generating three-dimensional grids has been developed and implemented which numerically solves a parabolic partial differential equation (PDE). The solution procedure marches outward in two coordinate directions, and requires inversion of a scalar tridiagonal system in the third. Source terms have been introduced to control the spacing and angle of grid lines near the grid boundaries, and to control the outer boundary point distribution. The method has been found to generate grids about 100 times faster than comparable grids generated via solution of elliptic PDEs, and produces smooth grids for finite-difference flow calculations.

High speed propeller acoustics and aerodynamics - A boundary element approach

NASA Technical Reports Server (NTRS)

Farassat, F.; Myers, M. K.; Dunn, M. H.

1989-01-01

The Boundary Element Method (BEM) is applied in this paper to the problems of acoustics and aerodynamics of high speed propellers. The underlying theory is described based on the linearized Ffowcs Williams-Hawkings equation. The surface pressure on the blade is assumed unknown in the aerodynamic problem. It is obtained by solving a singular integral equation. The acoustic problem is then solved by moving the field point inside the fluid medium and evaluating some surface and line integrals. Thus the BEM provides a powerful technique in calculation of high speed propeller aerodynamics and acoustics.

The high hall ventilation with the simplified simulation of the fan

NASA Astrophysics Data System (ADS)

Kyncl, Martin; Pelant, Jaroslav

2018-06-01

Here we work with the system of equations describing the non-stationary compressible turbulent multi-component flow in the gravitational field. We focus on the numerical simulation of the fan situated inside the high hall. The RANS equations are discretized with the use of the finite volume method. The original modification of the Riemann problem and its solution is used at the boundaries. The combination of specific boundary conditions is used for the simulation of the fan. The presented computational results are computed with own-developed code (C, FORTRAN, multiprocessor, unstructured meshes in general).

Numerical Solution of the Electron Transport Equation in the Upper Atmosphere

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

Woods, Mark Christopher; Holmes, Mark; Sailor, William C

A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplifieed problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other to at least two significant digits.

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.

Absorbing boundary conditions for second-order hyperbolic equations

NASA Technical Reports Server (NTRS)

Jiang, Hong; Wong, Yau Shu

1989-01-01

A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.

Development of a Perfectly Matched Layer Technique for a Discontinuous-Galerkin Spectral-Element Method

NASA Technical Reports Server (NTRS)

Garai, Anirban; Murman, Scott M.; Madavan, Nateri K.

2016-01-01

The numerical simulation of many aerodynamic non-periodic flows of practical interest involves discretized computational domains that often must be artificially truncated. Appropriate boundary conditions are required at these truncated domain boundaries, and ideally, these boundary conditions should be perfectly "absorbing" or "nonreflecting" so that they do not contaminate the flow field in the interior of the domain. The proper specification of these boundaries is critical to the stability, accuracy, convergence, and quality of the numerical solution, and has been the topic of considerable research. The need for accurate boundary specification has been underscored in recent years with efforts to apply higher-fidelity methods (DNS, LES) in conjunction with high-order low-dissipation numerical schemes to realistic flow configurations. One of the most popular choices for specifying these boundaries is the characteristics-based boundary condition where the linearized flow field at the boundaries are decomposed into characteristic waves using either one-dimensional Riemann or other multi-dimensional Riemann approximations. The values of incoming characteristics are then suitably modified. The incoming characteristics are specified at the in flow boundaries, and at the out flow boundaries the variation of the incoming characteristic is zeroed out to ensure no reflection. This, however, makes the problem ill-posed requiring the use of an ad-hoc parameter to allow small reflections that make the solution stable. Generally speaking, such boundary conditions work reasonably well when the characteristic flow direction is normal to the boundary, but reflects spurious energy otherwise. An alternative to the characteristic-based boundary condition is to add additional "buffer" regions to the main computational domain near the artificial boundaries, and solve a different set of equations in the buffer region in order to minimize acoustic reflections. One approach that has been used involves modeling the pressure fluctuations as acoustic waves propagating in the far-field relative to a single noise-source inside the buffer region. This approach treats vorticity-induced pressure fluctuations the same as acoustic waves. Another popular approach, often referred to as the "sponge layer," attempts to dampen the flow perturbations by introducing artificial dissipation in the buffer region. Although the artificial dissipation removes all perturbations inside the sponge layer, incoming waves are still reflected from the interface boundary between the computational domain and the sponge layer. The effect of these refkections can be somewhat mitigated by appropriately selecting the artificial dissipation strength and the extent of the sponge layer. One of the most promising variants on the buffer region approach is the Perfectly Matched Layer (PML) technique. The PML technique mitigates spurious reflections from boundaries and interfaces by dampening the perturbation modes inside the buffer region such that their eigenfunctions remain unchanged. The technique was first developed by Berenger for application to problems involving electromagnetic wave propagation. It was later extended to the linearized Euler, Euler and Navier-Stokes equations by Hu and his coauthors. The PML technique ensures the no-reflection property for all waves, irrespective of incidence angle, wavelength, and propagation direction. Although the technique requires the solution of a set of auxiliary equations, the computational overhead is easily justified since it allows smaller domain sizes and can provide better accuracy, stability, and convergence of the numerical solution. In this paper, the PML technique is developed in the context of a high-order spectral-element Discontinuous Galerkin (DG) method. The technique is compared to other approaches to treating the in flow and out flow boundary, such as those based on using characteristic boundary conditions and sponge layers. The superiority of the current PML technique over other approaches is demonstrated for a range of test cases, viz., acoustic pulse propagation, convective vortex, shear layer flow, and low-pressure turbine cascade flow. The paper is structured as follows. We first derive the PML equations from the non{linear Euler equations. A short description of the higher-order DG method used is then described. Preliminary results for the four test cases considered are then presented and discussed. Details regarding current work that will be included in the final paper are also provided.

Validation of a Node-Centered Wall Function Model for the Unstructured Flow Code FUN3D

NASA Technical Reports Server (NTRS)

Carlson, Jan-Renee; Vasta, Veer N.; White, Jeffery

2015-01-01

In this paper, the implementation of two wall function models in the Reynolds averaged Navier-Stokes (RANS) computational uid dynamics (CFD) code FUN3D is described. FUN3D is a node centered method for solving the three-dimensional Navier-Stokes equations on unstructured computational grids. The first wall function model, based on the work of Knopp et al., is used in conjunction with the one-equation turbulence model of Spalart-Allmaras. The second wall function model, also based on the work of Knopp, is used in conjunction with the two-equation k-! turbulence model of Menter. The wall function models compute the wall momentum and energy flux, which are used to weakly enforce the wall velocity and pressure flux boundary conditions in the mean flow momentum and energy equations. These wall conditions are implemented in an implicit form where the contribution of the wall function model to the Jacobian are also included. The boundary conditions of the turbulence transport equations are enforced explicitly (strongly) on all solid boundaries. The use of the wall function models is demonstrated on four test cases: a at plate boundary layer, a subsonic di user, a 2D airfoil, and a 3D semi-span wing. Where possible, different near-wall viscous spacing tactics are examined. Iterative residual convergence was obtained in most cases. Solution results are compared with theoretical and experimental data for several variations of grid spacing. In general, very good comparisons with data were achieved.

Numerical Boundary Conditions for Specular Reflection in a Level-Sets-Based Wavefront Propagation Method

DTIC Science & Technology

2012-12-01

acoustics One begins with Eikonal equation for the acoustic phase function S(t,x) as derived from the geometric acoustics (high frequency) approximation to...zb(x) is smooth and reasonably approximated as piecewise linear. The time domain ray (characteristic) equations for the Eikonal equation are ẋ(t)= c...travel time is affected, which is more physically relevant than global error in φ since it provides the phase information for the Eikonal equation (2.1

Numerical study of shock-wave/boundary layer interactions in premixed hydrogen-air hypersonic flows

NASA Technical Reports Server (NTRS)

Yungster, Shaye

1991-01-01

A computational study of shock wave/boundary layer interactions involving premixed combustible gases, and the resulting combustion processes is presented. The analysis is carried out using a new fully implicit, total variation diminishing (TVD) code developed for solving the fully coupled Reynolds-averaged Navier-Stokes equations and species continuity equations in an efficient manner. To accelerate the convergence of the basic iterative procedure, this code is combined with vector extrapolation methods. The chemical nonequilibrium processes are simulated by means of a finite-rate chemistry model for hydrogen-air combustion. Several validation test cases are presented and the results compared with experimental data or with other computational results. The code is then applied to study shock wave/boundary layer interactions in a ram accelerator configuration. Results indicate a new combustion mechanism in which a shock wave induces combustion in the boundary layer, which then propagates outwards and downstream. At higher Mach numbers, spontaneous ignition in part of the boundary layer is observed, which eventually extends along the entire boundary layer at still higher values of the Mach number.

Investigations on entropy layer along hypersonic hyperboloids using a defect boundary layer

NASA Technical Reports Server (NTRS)

Brazier, J. P.; Aupoix, B.; Cousteix, J.

1992-01-01

A defect approach coupled with matched asymptotic expansions is used to derive a new set of boundary layer equations. This method ensures a smooth matching of the boundary layer with the inviscid solution. These equations are solved to calculate boundary layers over hypersonic blunt bodies involving the entropy gradient effect. Systematic comparisons are made for both axisymmetric and plane flows in several cases with different Mach and Reynolds numbers. After a brief survey of the entropy layer characteristics, the defect boundary layer results are compared with standard boundary layer and full Navier-Stokes solutions. The entropy gradient effects are found to be more important in the axisymmetric case than in the plane one. The wall temperature has a great influence on the results through the displacement effect. Good predictions can be obtained with the defect approach over a cold wall in the nose region, with a first order solution. However, the defect approach gives less accurate results far from the nose on axisymmetric bodies because of the thinning of the entropy layer.

Numerical study of shock-wave/boundary layer interactions in premixed hydrogen-air hypersonic flows

NASA Technical Reports Server (NTRS)

Yungster, Shaye

1990-01-01

A computational study of shock wave/boundary layer interactions involving premixed combustible gases, and the resulting combustion processes is presented. The analysis is carried out using a new fully implicit, total variation diminishing (TVD) code developed for solving the fully coupled Reynolds-averaged Navier-Stokes equations and species continuity equations in an efficient manner. To accelerate the convergence of the basic iterative procedure, this code is combined with vector extrapolation methods. The chemical nonequilibrium processes are simulated by means of a finite-rate chemistry model for hydrogen-air combustion. Several validation test cases are presented and the results compared with experimental data or with other computational results. The code is then applied to study shock wave/boundary layer interactions in a ram accelerator configuration. Results indicate a new combustion mechanism in which a shock wave induces combustion in the boundary layer, which then propagates outwards and downstream. At higher Mach numbers, spontaneous ignition in part of the boundary layer is observed, which eventually extends along the entire boundary layer at still higher values of the Mach number.

Three-dimensional elliptic grid generation technique with application to turbomachinery cascades

NASA Technical Reports Server (NTRS)

Chen, S. C.; Schwab, J. R.

1988-01-01

Described is a numerical method for generating 3-D grids for turbomachinery computational fluid dynamic codes. The basic method is general and involves the solution of a quasi-linear elliptic partial differential equation via pointwise relaxation with a local relaxation factor. It allows specification of the grid point distribution on the boundary surfaces, the grid spacing off the boundary surfaces, and the grid orthogonality at the boundary surfaces. A geometry preprocessor constructs the grid point distributions on the boundary surfaces for general turbomachinery cascades. Representative results are shown for a C-grid and an H-grid for a turbine rotor. Two appendices serve as user's manuals for the basic solver and the geometry preprocessor.

Parallel computation using boundary elements in solid mechanics

NASA Technical Reports Server (NTRS)

Chien, L. S.; Sun, C. T.

1990-01-01

The inherent parallelism of the boundary element method is shown. The boundary element is formulated by assuming the linear variation of displacements and tractions within a line element. Moreover, MACSYMA symbolic program is employed to obtain the analytical results for influence coefficients. Three computational components are parallelized in this method to show the speedup and efficiency in computation. The global coefficient matrix is first formed concurrently. Then, the parallel Gaussian elimination solution scheme is applied to solve the resulting system of equations. Finally, and more importantly, the domain solutions of a given boundary value problem are calculated simultaneously. The linear speedups and high efficiencies are shown for solving a demonstrated problem on Sequent Symmetry S81 parallel computing system.

Elastic interactions of a fatigue crack with a micro-defect by the mixed boundary integral equation method

NASA Technical Reports Server (NTRS)

Lua, Yuan J.; Liu, Wing K.; Belytschko, Ted

1993-01-01

In this paper, the mixed boundary integral equation method is developed to study the elastic interactions of a fatigue crack and a micro-defect such as a void, a rigid inclusion or a transformation inclusion. The method of pseudo-tractions is employed to study the effect of a transformation inclusion. An enriched element which incorporates the mixed-mode stress intensity factors is applied to characterize the singularity at a moving crack tip. In order to evaluate the accuracy of the numerical procedure, the analysis of a crack emanating from a circular hole in a finite plate is performed and the results are compared with the available numerical solution. The effects of various micro-defects on the crack path and fatigue life are investigated. The results agree with the experimental observations.

Assessment of a 3-D boundary layer code to predict heat transfer and flow losses in a turbine

NASA Technical Reports Server (NTRS)

Anderson, O. L.

1984-01-01

Zonal concepts are utilized to delineate regions of application of three-dimensional boundary layer (DBL) theory. The zonal approach requires three distinct analyses. A modified version of the 3-DBL code named TABLET is used to analyze the boundary layer flow. This modified code solves the finite difference form of the compressible 3-DBL equations in a nonorthogonal surface coordinate system which includes coriolis forces produced by coordinate rotation. These equations are solved using an efficient, implicit, fully coupled finite difference procedure. The nonorthogonal surface coordinate system is calculated using a general analysis based on the transfinite mapping of Gordon which is valid for any arbitrary surface. Experimental data is used to determine the boundary layer edge conditions. The boundary layer edge conditions are determined by integrating the boundary layer edge equations, which are the Euler equations at the edge of the boundary layer, using the known experimental wall pressure distribution. Starting solutions along the inflow boundaries are estimated by solving the appropriate limiting form of the 3-DBL equations.

Higher-order continuation for the determination of robot workspace boundaries

NASA Astrophysics Data System (ADS)

Hentz, Gauthier; Charpentier, Isabelle; Renaud, Pierre

2016-02-01

In the medical and surgical fields, robotics may be of great interest for safer and more accurate procedures. Space constraints for a robotic assistant are however strict. Therefore, roboticists study non-conventional mechanisms with advantageous size/workspace ratios. The determination of mechanism workspace, and primarily its boundaries, is thus of major importance. This Note builds on boundary equation definition, continuation and automatic differentiation to propose a general, accurate, fast and automated method for the determination of mechanism workspace. The method is illustrated with a planar RRR mechanism and a three-dimensional Orthoglide parallel mechanism.

Effect of thermal radiation and chemical reaction on non-Newtonian fluid through a vertically stretching porous plate with uniform suction

NASA Astrophysics Data System (ADS)

Khan, Zeeshan; Khan, Ilyas; Ullah, Murad; Tlili, I.

2018-06-01

In this work, we discuss the unsteady flow of non-Newtonian fluid with the properties of heat source/sink in the presence of thermal radiation moving through a binary mixture embedded in a porous medium. The basic equations of motion including continuity, momentum, energy and concentration are simplified and solved analytically by using Homotopy Analysis Method (HAM). The energy and concentration fields are coupled with Dankohler and Schmidt numbers. By applying suitable transformation, the coupled nonlinear partial differential equations are converted to couple ordinary differential equations. The effect of physical parameters involved in the solutions of velocity, temperature and concentration profiles are discussed by assign numerical values and results obtained shows that the velocity, temperature and concentration profiles are influenced appreciably by the radiation parameter, Prandtl number, suction/injection parameter, reaction order index, solutal Grashof number and the thermal Grashof. It is observed that the non-Newtonian parameter H leads to an increase in the boundary layer thickness. It was established that the Prandtl number decreases thee thermal boundary layer thickness which helps in maintaining system temperature of the fluid flow. It is observed that the temperature profiles higher for heat source parameter and lower for heat sink parameter throughout the boundary layer. Fromm this simulation it is analyzed that an increase in the Schmidt number decreases the concentration boundary layer thickness. Additionally, for the sake of comparison numerical method (ND-Solve) and Adomian Decomposition Method are also applied and good agreement is found.

Generalized Lagrangian Jacobi Gauss collocation method for solving unsteady isothermal gas through a micro-nano porous medium

NASA Astrophysics Data System (ADS)

Parand, Kourosh; Latifi, Sobhan; Delkhosh, Mehdi; Moayeri, Mohammad M.

2018-01-01

In the present paper, a new method based on the Generalized Lagrangian Jacobi Gauss (GLJG) collocation method is proposed. The nonlinear Kidder equation, which explains unsteady isothermal gas through a micro-nano porous medium, is a second-order two-point boundary value ordinary differential equation on the unbounded interval [0, ∞). Firstly, using the quasilinearization method, the equation is converted to a sequence of linear ordinary differential equations. Then, by using the GLJG collocation method, the problem is reduced to solving a system of algebraic equations. It must be mentioned that this equation is solved without domain truncation and variable changing. A comparison with some numerical solutions made and the obtained results indicate that the presented solution is highly accurate. The important value of the initial slope, y'(0), is obtained as -1.191790649719421734122828603800159364 for η = 0.5. Comparing to the best result obtained so far, it is accurate up to 36 decimal places.

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

NASA Technical Reports Server (NTRS)

Avis, L. M.

1976-01-01

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

Numerical solutions of the Navier-Stokes equations for the supersonic laminar flow over a two-dimensional compression corner

NASA Technical Reports Server (NTRS)

Carter, J. E.

1972-01-01

Numerical solutions have been obtained for the supersonic, laminar flow over a two-dimensional compression corner. These solutions were obtained as steady-state solutions to the unsteady Navier-Stokes equations using the finite difference method of Brailovskaya, which has second-order accuracy in the spatial coordinates. Good agreement was obtained between the computed results and wall pressure distributions measured experimentally for Mach numbers of 4 and 6.06, and respective Reynolds numbers, based on free-stream conditions and the distance from the leading edge to the corner. In those calculations, as well as in others, sufficient resolution was obtained to show the streamline pattern in the separation bubble. Upstream boundary conditions to the compression corner flow were provided by numerically solving the unsteady Navier-Stokes equations for the flat plate flow field, beginning at the leading edge. The compression corner flow field was enclosed by a computational boundary with the unknown boundary conditions supplied by extrapolation from internally computed points.

A level-set method for two-phase flows with moving contact line and insoluble surfactant

NASA Astrophysics Data System (ADS)

Xu, Jian-Jun; Ren, Weiqing

2014-04-01

A level-set method for two-phase flows with moving contact line and insoluble surfactant is presented. The mathematical model consists of the Navier-Stokes equation for the flow field, a convection-diffusion equation for the surfactant concentration, together with the Navier boundary condition and a condition for the dynamic contact angle derived by Ren et al. (2010) [37]. The numerical method is based on the level-set continuum surface force method for two-phase flows with surfactant developed by Xu et al. (2012) [54] with some cautious treatment for the boundary conditions. The numerical method consists of three components: a flow solver for the velocity field, a solver for the surfactant concentration, and a solver for the level-set function. In the flow solver, the surface force is dealt with using the continuum surface force model. The unbalanced Young stress at the moving contact line is incorporated into the Navier boundary condition. A convergence study of the numerical method and a parametric study are presented. The influence of surfactant on the dynamics of the moving contact line is illustrated using examples. The capability of the level-set method to handle complex geometries is demonstrated by simulating a pendant drop detaching from a wall under gravity.

Truly self-consistent solution of Kohn-Sham equations for extended systems with inhomogeneous electron gas

NASA Astrophysics Data System (ADS)

Shul'man, A. Ya; Posvyanskii, D. V.

2014-05-01

The density functional approach in the Kohn-Sham approximation is widely used to study properties of many-electron systems. Due to the nonlinearity of the Kohn-Sham equations, the general self-consistent solution method for infinite systems involves iterations with alternate solutions of the Poisson and Schrödinger equations. One of problems with such an approach is that the charge distribution, updated by solving the Schrodinger equation, may be incompatible with the boundary conditions of the Poisson equation for Coulomb potential. The resulting instability or divergence manifests itself most appreciably in the case of infinitely extended systems because the corresponding boundary-value problem becomes singular. In this work the stable iterative scheme for solving the Kohn-Sham equations for infinite systems with inhomogeneous electron gas is described based on eliminating the long-range character of the Coulomb interaction, which causes the tight coupling of the charge distribution with the boundary conditions. This algorithm has been previously successfully implemented in the calculation of work function and surface energy of simple metals in the jellium model. Here it is used to calculate the energy spectrum of quasi-two-dimensional electron gas in the accumulation layer at the semiconductor surface n-InAs. The electrons in such a structure occupy states that belong to both discrete and continuous parts of the energy spectrum. This causes the problems of convergence in the usually used approaches, which do not exist in our case. Because of the narrow bandgap of InAs, it is necessary to take the nonparabolicity of the conduction band into account; this is done by means of a new effective mass method. The calculated quasi-two-dimensional energy bands correspond well to experimental data measured by the angle resolved photoelectron spectroscopy technique.

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

NASA Astrophysics Data System (ADS)

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

2018-04-01

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

Boundary Conditions for Infinite Conservation Laws

NASA Astrophysics Data System (ADS)

Rosenhaus, V.; Bruzón, M. S.; Gandarias, M. L.

2016-12-01

Regular soliton equations (KdV, sine-Gordon, NLS) are known to possess infinite sets of local conservation laws. Some other classes of nonlinear PDE possess infinite-dimensional symmetries parametrized by arbitrary functions of independent or dependent variables; among them are Zabolotskaya-Khokhlov, Kadomtsev-Petviashvili, Davey-Stewartson equations and Born-Infeld equation. Boundary conditions were shown to play an important role for the existence of local conservation laws associated with infinite-dimensional symmetries. In this paper, we analyze boundary conditions for the infinite conserved densities of regular soliton equations: KdV, potential KdV, Sine-Gordon equation, and nonlinear Schrödinger equation, and compare them with boundary conditions for the conserved densities obtained from infinite-dimensional symmetries with arbitrary functions of independent and dependent variables.

Efficient Fluid Dynamic Design Optimization Using Cartesian Grids

NASA Technical Reports Server (NTRS)

Dadone, A.; Grossman, B.; Sellers, Bill (Technical Monitor)

2004-01-01

This report is subdivided in three parts. The first one reviews a new approach to the computation of inviscid flows using Cartesian grid methods. The crux of the method is the curvature-corrected symmetry technique (CCST) developed by the present authors for body-fitted grids. The method introduces ghost cells near the boundaries whose values are developed from an assumed flow-field model in vicinity of the wall consisting of a vortex flow, which satisfies the normal momentum equation and the non-penetration condition. The CCST boundary condition was shown to be substantially more accurate than traditional boundary condition approaches. This improved boundary condition is adapted to a Cartesian mesh formulation, which we call the Ghost Body-Cell Method (GBCM). In this approach, all cell centers exterior to the body are computed with fluxes at the four surrounding cell edges. There is no need for special treatment corresponding to cut cells which complicate other Cartesian mesh methods.

Prediction of submarine scattered noise by the acoustic analogy

NASA Astrophysics Data System (ADS)

Testa, C.; Greco, L.

2018-07-01

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

Providing the physical basis of SCS curve number method and its proportionality relationship from Richards' equation

NASA Astrophysics Data System (ADS)

Hooshyar, M.; Wang, D.

2016-12-01

The empirical proportionality relationship, which indicates that the ratio of cumulative surface runoff and infiltration to their corresponding potentials are equal, is the basis of the extensively used Soil Conservation Service Curve Number (SCS-CN) method. The objective of this paper is to provide the physical basis of the SCS-CN method and its proportionality hypothesis from the infiltration excess runoff generation perspective. To achieve this purpose, an analytical solution of Richards' equation is derived for ponded infiltration in shallow water table environment under the following boundary conditions: 1) the soil is saturated at the land surface; and 2) there is a no-flux boundary which moves downward. The solution is established based on the assumptions of negligible gravitational effect, constant soil water diffusivity, and hydrostatic soil moisture profile between the no-flux boundary and water table. Based on the derived analytical solution, the proportionality hypothesis is a reasonable approximation for rainfall partitioning at the early stage of ponded infiltration in areas with a shallow water table for coarse textured soils.

An analytical solution of Richards' equation providing the physical basis of SCS curve number method and its proportionality relationship

NASA Astrophysics Data System (ADS)

Hooshyar, Milad; Wang, Dingbao

2016-08-01

The empirical proportionality relationship, which indicates that the ratio of cumulative surface runoff and infiltration to their corresponding potentials are equal, is the basis of the extensively used Soil Conservation Service Curve Number (SCS-CN) method. The objective of this paper is to provide the physical basis of the SCS-CN method and its proportionality hypothesis from the infiltration excess runoff generation perspective. To achieve this purpose, an analytical solution of Richards' equation is derived for ponded infiltration in shallow water table environment under the following boundary conditions: (1) the soil is saturated at the land surface; and (2) there is a no-flux boundary which moves downward. The solution is established based on the assumptions of negligible gravitational effect, constant soil water diffusivity, and hydrostatic soil moisture profile between the no-flux boundary and water table. Based on the derived analytical solution, the proportionality hypothesis is a reasonable approximation for rainfall partitioning at the early stage of ponded infiltration in areas with a shallow water table for coarse textured soils.

Flow and Turbulence Modeling and Computation of Shock Buffet Onset for Conventional and Supercritical Airfoils

NASA Technical Reports Server (NTRS)

Bartels, Robert E.

1998-01-01

Flow and turbulence models applied to the problem of shock buffet onset are studied. The accuracy of the interactive boundary layer and the thin-layer Navier-Stokes equations solved with recent upwind techniques using similar transport field equation turbulence models is assessed for standard steady test cases, including conditions having significant shock separation. The two methods are found to compare well in the shock buffet onset region of a supercritical airfoil that involves strong trailing-edge separation. A computational analysis using the interactive-boundary layer has revealed a Reynolds scaling effect in the shock buffet onset of the supercritical airfoil, which compares well with experiment. The methods are next applied to a conventional airfoil. Steady shock-separated computations of the conventional airfoil with the two methods compare well with experiment. Although the interactive boundary layer computations in the shock buffet region compare well with experiment for the conventional airfoil, the thin-layer Navier-Stokes computations do not. These findings are discussed in connection with possible mechanisms important in the onset of shock buffet and the constraints imposed by current numerical modeling techniques.

The boundary element method applied to 3D magneto-electro-elastic dynamic problems

NASA Astrophysics Data System (ADS)

Igumnov, L. A.; Markov, I. P.; Kuznetsov, Iu A.

2017-11-01

Due to the coupling properties, the magneto-electro-elastic materials possess a wide number of applications. They exhibit general anisotropic behaviour. Three-dimensional transient analyses of magneto-electro-elastic solids can hardly be found in the literature. 3D direct boundary element formulation based on the weakly-singular boundary integral equations in Laplace domain is presented in this work for solving dynamic linear magneto-electro-elastic problems. Integral expressions of the three-dimensional fundamental solutions are employed. Spatial discretization is based on a collocation method with mixed boundary elements. Convolution quadrature method is used as a numerical inverse Laplace transform scheme to obtain time domain solutions. Numerical examples are provided to illustrate the capability of the proposed approach to treat highly dynamic problems.

Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition

DOE PAGES

Cao, Yong; Chu, Yuchuan; He, Xiaoming; ...

2013-01-01

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.

Meshless Method with Operator Splitting Technique for Transient Nonlinear Bioheat Transfer in Two-Dimensional Skin Tissues

PubMed Central

Zhang, Ze-Wei; Wang, Hui; Qin, Qing-Hua

2015-01-01

A meshless numerical scheme combining the operator splitting method (OSM), the radial basis function (RBF) interpolation, and the method of fundamental solutions (MFS) is developed for solving transient nonlinear bioheat problems in two-dimensional (2D) skin tissues. In the numerical scheme, the nonlinearity caused by linear and exponential relationships of temperature-dependent blood perfusion rate (TDBPR) is taken into consideration. In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step. Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation. Finally, the full fields consisting of the particular and homogeneous solutions are enforced to fit the NHGE at interpolation points and the boundary conditions at boundary collocations for determining unknowns at each time step. The proposed method is verified by comparison of other methods. Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue. PMID:25603180