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Showing papers on "Boundary value problem published in 1992"


Journal ArticleDOI
TL;DR: In this article, a boundary condition formulation for the Navier-Stokes equations is proposed, which is compatible with non-disjoint algorithms applicable to direct simulations of turbulent flows.

3,214 citations


Book
01 Feb 1992
TL;DR: In this paper, Green's functions in domains bounded by a solid surface are used to define boundary integral methods based on the Stream Function, and numerical solutions of the integral equations are provided.
Abstract: 1. Preliminaries 2. Green's Functions and the Boundary Integral Equation 3. Green's Functions in domains bounded by a solid surface 4. Generalized boundary integral methods 5. Interfacial motion 6. Boundary integral methods based on the Stream Function 7. Discrete representation of a boundary 8. Numerical solution of the integral equations.

1,632 citations


Book
01 Mar 1992
TL;DR: In this paper, the authors studied the solutions of a boundary problem near corner edges and vertices and highlighted the singular solutions which carry the main physical information and which are given in their most explicit form to help potential users.
Abstract: This book studies the solutions of a boundary problem near corner edges and vertices. The exposition is introductory and self-contained. It focuses on real-life problems considered in the actual geometry met in the applcations. The book highlights the singular solutions which carry the main physical information and which are given in their most explicit form to help potential users.

1,103 citations


Journal ArticleDOI
TL;DR: In this paper, the yield strength not only depends on an equivalent plastic strain measure (hardening parameter), but also on the Laplacian thereof, and the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity.
Abstract: A plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure (hardening parameter), but also on the Laplacian thereof. The consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity. To properly solve the set of non-linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements. For appropriate boundary conditions this formulation can also be derived from a variational principle. Accordingly, the theory is complete

924 citations


Journal ArticleDOI
TL;DR: In this paper, a global method of generalised differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation.
Abstract: A global method of generalised differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.

807 citations


Book
01 Feb 1992
TL;DR: Green's Functions Heat Flux and Temperature Differential Energy Equation Boundary and Initial Conditions Integral Energy Equations Dirac Delta Function Steady Heat Conduction in One Dimension GF in the Infinite One-Dimensional Body Temperature in an Infinite 1-Ddimensional Body Two Interpretations of Green's Functions Temperature in Semi-Infinite Bodies Flat Plates Properties Common to Transient Green's Function Heterogeneous Bodies Anisotropic Bodies Transformations Non-Fourier Heat Fluid Flow in Ducts as mentioned in this paper Non-fourier heat Conduction Number
Abstract: Introduction to Green's Functions Heat Flux and Temperature Differential Energy Equation Boundary and Initial Conditions Integral Energy Equation Dirac Delta Function Steady Heat Conduction in One Dimension GF in the Infinite One-Dimensional Body Temperature in an Infinite One-Dimensional Body Two Interpretations of Green's Functions Temperature in Semi-Infinite Bodies Flat Plates Properties Common to Transient Green's Functions Heterogeneous Bodies Anisotropic Bodies Transformations Non-Fourier Heat Conduction Numbering System in Heat Conduction Geometry and Boundary Condition Numbering System Boundary Condition Modifiers Initial Temperature Distribution Interface Descriptors Numbering System for g(x, t) Examples of Numbering System Advantages of Numbering System Derivation of the Green's Function Solution Equation Derivation of the One-Dimensional Green's Function Solution Equation General Form of the Green's Function Solution Equation Alternative Green's Function Solution Equation Fin Term m2T Steady Heat Conduction Moving Solids Methods for Obtaining Green's Functions Method of Images Laplace Transform Method Method Of Separation of Variables Product Solution for Transient GF Method of Eigenfunction Expansions Steady Green's Functions Improvement of Convergence and Intrinsic Verification Identifying Convergence Problems Strategies to Improve Series Convergence Intrinsic Verification Rectangular Coordinates One-Dimensional Green's Functions Solution Equation Semi-Infinite One-Dimensional Bodies Flat Plates: Small-Cotime Green's Functions Flat Plates: Large-Cotime Green's Functions Flat Plates: The Nonhomogeneous Boundary Two-Dimensional Rectangular Bodies Two-Dimensional Semi-Infinite Bodies Steady State Cylindrical Coordinates Relations for Radial Heat Flow Infinite Body Separation of Variables for Radial Heat Flow Long Solid Cylinder Hollow Cylinder Infinite Body with a Circular Hole Thin Shells, T = T (phi, t) Limiting Cases for 2D and 3D Geometries Cylinders with T = T (r, z, t ) Disk Heat Source on a Semi-Infinite Body Bodies with T = T (r, phi, t ) Steady State Radial Heat Flow in Spherical Coordinates Green's Function Equation for Radial Spherical Heat Flow Infinite Body Separation of Variables for Radial Heat Flow in Spheres Temperature in Solid Spheres Temperature in Hollow Spheres Temperature in an Infinite Region Outside a Spherical Cavity Steady State Steady-Periodic Heat Conduction Steady-Periodic Relations One-Dimensional GF One-Dimensional Temperature Layered Bodies Two- and Three-Dimensional Cartesian Bodies Two-Dimensional Bodies in Cylindrical Coordinates Cylinder with T = T (r, phi, z,omega) Galerkin-Based Green's Functions and Solutions Green's Functions and Green's Function Solution Method Alternative form of the Green's Function Solution Basis Functions and Simple Matrix Operations Fins and Fin Effect Conclusions Applications of the Galerkin-Based Green's Functions Basis Functions in some Complex Geometries Heterogeneous Solids Steady-State Conduction Fluid Flow in Ducts Conclusion Unsteady Surface Element Method Duhamel's Theorem and Green's Function Method Unsteady Surface Element Formulations Approximate Analytical Solution (Single Element) Examples Problems References Appendices Index

770 citations




Book
01 Aug 1992
TL;DR: The DtN method for time-harmonic wave problems was proposed in this paper, where it was applied to beam and shell problems, as well as to time dependent problems.
Abstract: 1. Introduction and overview. 2. Boundary integral and boundary element methods. 3. Artificial boundary conditions and NRBCs. 4. Local non-reflecting boundary conditions. 5. Nonlocal non-reflecting boundary conditions. 6. Special numerical procedures for unbounded and large domains. Part II. 7. The DtN method. 8. Computational aspects of the DtN method. 9. Application of the DtN method to beam and shell problems. 10. The DtN method for time-harmonic waves. 11. The DtN method for time dependent problems. Appendix: The finite element method. References. Index.

497 citations


Journal ArticleDOI
TL;DR: In this article, a new boundary condition was proposed for beam propagation calculations that passes outgoing radiation freely with minimum reflection coefficient (as low as 3*10/sup -8/).
Abstract: A new boundary condition is presented for use in beam propagation calculations that passes outgoing radiation freely with minimum reflection coefficient (as low as 3*10/sup -8/). In conjunction with a standard Crank-Nicholson finite difference scheme, the assumption that the radiation field behaves as a complex exponential near the boundary is shown to result in a specific transparent boundary condition algorithm. In contrast to the commonly used absorber method, this algorithm contains no adjustable parameters, and is thus problem independent. It is shown to be accurate and robust for both two- and three-dimensional problems. >

392 citations


Book
17 Mar 1992
TL;DR: In this paper, the authors introduce the concept of Partial Differential Equations (PDE) as a way of separating variables in a partial differential equation and the Dirichlet condition.
Abstract: Chapter 1: Where PDEs Come From 1.1 What is a Partial Differential Equation? 1.2 First-Order Linear Equations 1.3 Flows, Vibrations, and Diffusions 1.4 Initial and Boundary Conditions 1.5 Well-Posed Problems 1.6 Types of Second-Order Equations Chapter 2: Waves and Diffusions 2.1 The Wave Equation 2.2 Causality and Energy 2.3 The Diffusion Equation 2.4 Diffusion on the Whole Line 2.5 Comparison of Waves and Diffusions Chapter 3: Reflections and Sources 3.1 Diffusion on the Half-Line 3.2 Reflections of Waves 3.3 Diffusion with a Source 3.4 Waves with a Source 3.5 Diffusion Revisited Chapter 4: Boundary Problems 4.1 Separation of Variables, The Dirichlet Condition 4.2 The Neumann Condition 4.3 The Robin Condition Chapter 5: Fourier Series 5.1 The Coefficients 5.2 Even, Odd, Periodic, and Complex Functions 5.3 Orthogonality and the General Fourier Series 5.4 Completeness 5.5 Completeness and the Gibbs Phenomenon 5.6 Inhomogeneous Boundary Conditions Chapter 6: Harmonic Functions 6.1 Laplace's Equation 6.2 Rectangles and Cubes 6.3 Poisson's Formula 6.4 Circles, Wedges, and Annuli Chapter 7: Green's Identities and Green's Functions 7.1 Green's First Identity 7.2 Green's Second Identity 7.3 Green's Functions 7.4 Half-Space and Sphere Chapter 8: Computation of Solutions 8.1 Opportunities and Dangers 8.2 Approximations of Diffusions 8.3 Approximations of Waves 8.4 Approximations of Laplace's Equation 8.5 Finite Element Method Chapter 9: Waves in Space 9.1 Energy and Causality 9.2 The Wave Equation in Space-Time 9.3 Rays, Singularities, and Sources 9.4 The Diffusion and Schrodinger Equations 9.5 The Hydrogen Atom Chapter 10: Boundaries in the Plane and in Space 10.1 Fourier's Method, Revisited 10.2 Vibrations of a Drumhead 10.3 Solid Vibrations in a Ball 10.4 Nodes 10.5 Bessel Functions 10.6 Legendre Functions 10.7 Angular Momentum in Quantum Mechanics Chapter 11: General Eigenvalue Problems 11.1 The Eigenvalues Are Minima of the Potential Energy 11.2 Computation of Eigenvalues 11.3 Completeness 11.4 Symmetric Differential Operators 11.5 Completeness and Separation of Variables 11.6 Asymptotics of the Eigenvalues Chapter 12: Distributions and Transforms 12.1 Distributions 12.2 Green's Functions, Revisited 12.3 Fourier Transforms 12.4 Source Functions 12.5 Laplace Transform Techniques Chapter 13: PDE Problems for Physics 13.1 Electromagnetism 13.2 Fluids and Acoustics 13.3 Scattering 13.4 Continuous Spectrum 13.5 Equations of Elementary Particles Chapter 14: Nonlinear PDEs 14.1 Shock Waves 14.2 Solitions 14.3 Calculus of Variations 14.4 Bifurcation Theory 14.5 Water Waves Appendix A.1 Continuous and Differentiable Functions A.2 Infinite Sets of Functions A.3 Differentiation and Integration A.4 Differential Equations A.5 The Gamma Function References Answers and Hints to Selected Exercises Index

03 Jan 1992
TL;DR: In this paper, the shape-from-shading problem is solved by linearization of the reflectance map about the current estimate of the surface orientation at each picture cell, which can find an exact solution of a given shape from shading problem even though a regularizing term is included.
Abstract: The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented. The robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell. The new scheme can find an exact solution of a given shape-from-shading problem even though a regularizing term is included. This is a reflection of the fact that shape-from-shading problems are {\it not} ill-posed when boundary conditions are available or when the image contains singular points.

Book
30 Sep 1992
TL;DR: The finite element method is the most effective method for the solution of composite laminates as discussed by the authors, but it is limited to simple geometries because of the difficulty in constructing the approximation functions for complicated geometrie.
Abstract: The partial differential equations governing composite laminates (see Section 2.4) of arbitrary geometries and boundary conditions cannot be solved in closed form. Analytical solutions of plate theories are available (see Reddy [1–5]) mostly for rectangular plates with all edges simply supported (i.e., the Navier solutions) or with two opposite edges simply supported and the remaining edges having arbitrary boundary conditions (i.e., the Levy solutions). The Rayleigh-Ritz and Galerkin methods can also be used to determine approximate analytical solutions, but they too are limited to simple geometries because of the difficulty in constructing the approximation functions for complicated geometries. The use of numerical methods facilitates the solution of these equations for problems of practical importance. Among the numerical methods available for the solution of differential equations defined over arbitrary domains, the finite element method is the most effective method. A brief introduction to the finite element method is presented in Section 3.2.

Journal ArticleDOI
TL;DR: In this paper, a new method is proposed for the calculation of the microcanonical cumulative reaction probability via flux autocorrelation relations, circumventing the need to compute the state-to-state dynamics.
Abstract: A new method is suggested for the calculation of the microcanonical cumulative reaction probability via flux autocorrelation relations. The Hamiltonian and the flux operators are computed in a discrete variable representation (DVR) and a well‐behaved representation for the Green’s operator, G(E+), is obtained by imposing absorbing boundary conditions (ABC). Applications to a one‐dimensional‐model problem and to the collinear H+H2 reaction show that the DVR‐ABC scheme provides a very efficient method for the direct calculation of the microcanonical probability, circumventing the need to compute the state‐to‐state dynamics. Our results indicate that the cumulative reaction probability can be calculated to a high accuracy using a rather small number of DVR points, confined to the vicinity of the transition state. Only limited information regarding the potential‐energy surface is therefore required, suggesting that this method would be applicable also to higher dimensionality problems, for which the complete potential surface is often unknown.

Journal ArticleDOI
TL;DR: In this article, the authors obtained existence and uniqueness theorems for the boundary value problem (1.1 t( 1.2) ) under natural conditions on f using degree-theoretic arguments.

Journal ArticleDOI
TL;DR: In this article, a methodology to improve the quality of the finite element calculations in the regions of unacceptable errors has been developed, where the superimposed regions can be of arbitrary shape, unlimited by the problem geometry, boundary conditions and the underlying mesh topography.


Journal ArticleDOI
TL;DR: In this article, a finite-difference vector beam propagation method (FD-VBPM) was proposed for two-dimensional waveguide structures and evaluated by calculating attenuation coefficients and the percentage errors of the propagation constants of the TE and TM modes of a step-index slab waveguide.
Abstract: The newly developed finite-difference vector beam propagation method (FD-VBPM) is analyzed and assessed for application to two-dimensional waveguide structures. The general formulations for the FD-VBPM are derived from the vector wave equations for the electric fields. The stability criteria, the numerical dissipation, and the dispersion of the finite-difference schemes are analyzed by applying the von Neumann method. Important issues regarding the implementation, such as the choice of reference refractive index, the application of numerical boundary conditions, and the use of numerical solution schemes, are discussed. The FD-VBPM is assessed by calculating the attenuation coefficients and the percentage errors of the propagation constants of the TE and TM modes of a step-index slab waveguide. Several salient features of the FD-VBPM are illustrated. >

Journal ArticleDOI
TL;DR: In this article, the hydrodynamic boundary condition at the interface between a porous and a plain medium is examined by direct simulation of the two-dimensional flow field near the interface of a porous medium made of cylinders.

Journal ArticleDOI
TL;DR: In this article, the authors present a numerical method for computing the motion of complex solid/liquid boundaries in crystal growth, which includes physical effects such as crystalline anisotropy, surface tension, molecular kinetics and undercooling.

Journal ArticleDOI
TL;DR: In this article, a generalized variational principle is used to formulate the equation of motion, taking into account the interlaminar stress concentration at the crack-tips, which is accomplished by introducing a "crack function" into the beam's compatibility relations.
Abstract: Free vibration of laminated composite beams is studied. The effect of interply delaminations on natural frequencies and mode shapes is evaluated both analytically and experimentally. The equation of motion and associated boundary conditions are derived for the free vibration of a composite beam with a delamination of arbitrary size and location. A generalized variational principle is used to formulate the equation of motion, taking into account the interlaminar stress concentration at the crack-tips. This is accomplished by introducing a 'crack function' into the beam's compatibility relations. This function has its maximum value at the crack tip and decays exponentially in the longitudinal direction. The rate of exponential decay is determined by a least-square fit with the experimental results. The effect of coupling between longitudinal vibration and bending vibration is considered in the present study. This coupling effect is found to significantly affect the natural frequencies and mode shapes of the delaminated beam.

Journal ArticleDOI
TL;DR: In this paper, the authors studied numerical methods for the one-dimensional heat equation with a singular forcing term, where the delta function was replaced by a discrete approximation, and the resulting equation was solved by a Crank-Nicolson method on a uniform grid.
Abstract: Numerical methods are studied for the one-dimensional heat equation with a singular forcing term, $u_t = u_{xx} + c(t)\delta (x - \alpha (t)).$ The delta function $\delta (x)$ is replaced by a discrete approximation $d_h (x)$ and the resulting equation is solved by a Crank–Nicolson method on a uniform grid. The accuracy of this method is analyzed for various choices of $d_h $. The case where $c(t)$ is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied. These problems serve as a model for the immersed boundary method of Peskin for incompressible flow problems in irregular regions. Some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions.

Journal ArticleDOI
TL;DR: In this article, a new K-tau model was proposed for near-wall turbulent flows, which is based on the K-omega model with the same asymptotic behavior.
Abstract: A variety of two-equation turbulence models,including several versions of the K-epsilon model as well as the K-omega model, are analyzed critically for near wall turbulent flows from a theoretical and computational standpoint. It is shown that the K-epsilon model has two major problems associated with it: the lack of natural boundary conditions for the dissipation rate and the appearance of higher-order correlations in the balance of terms for the dissipation rate at the wall. In so far as the former problem is concerned, either physically inconsistent boundary conditions have been used or the boundary conditions for the dissipation rate have been tied to higher-order derivatives of the turbulent kinetic energy which leads to numerical stiffness. The K-omega model can alleviate these problems since the asymptotic behavior of omega is known in more detail and since its near wall balance involves only exact viscous terms. However, the modeled form of the omega equation that is used in the literature is incomplete-an exact viscous term is missing which causes the model to behave in an asymptotically inconsistent manner. By including this viscous term and by introducing new wall damping functions with improved asymptotic behavior, a new K-tau model (where tau is identical with 1/omega is turbulent time scale) is developed. It is demonstrated that this new model is computationally robust and yields improved predictions for turbulent boundary layers.

Journal ArticleDOI
TL;DR: The wave equation subject to Dirichlet boundary conditions has a bound state in an infinite tube of constant cross section in any number of dimensions, provided that the tube is not exactly straight.
Abstract: The wave equation subject to Dirichlet boundary conditions has a bound state in an infinite tube of constant cross section in any number of dimensions, provided that the tube is not exactly straight. We prove this result, develop Green's-function methods to find the energy eigenvalue, and solve some simple cases. We discuss the implications for quantum systems and electromagnetic waveguides.

Journal ArticleDOI
TL;DR: In this paper, the Galerkin/least-squares method with DtN boundary conditions is designed to exhibit superior behavior for problems of acoustics, providing accurate solutions with relatively low mesh resolution and allowing numerical damping of unresolved waves.
Abstract: Finite element methods are constructed for the reduced wave equation in unbounded domains. Exterior boundary conditions for a computational problem are derived from an exact relation between the solution and its derivatives on an artificial boundary by the DtN method, precluding singular behavior in finite element models. Galerkin and Galerkin/least-squares finite element methods are presented. Model problems of radiation with inhomogeneous Neumann boundary conditions in plane and spherical configurations are employed to design and evaluate the numerical methods in the entire range of propagation and decay. The Galerkin/least-squares method with DtN boundary conditions is designed to exhibit superior behavior for problems of acoustics, providing accurate solutions with relatively low mesh resolution and allowing numerical damping of unresolved waves. General convergence results guarantee the good performance of Galerkin/least-squares methods on all configurations of practical interest. Numerical tests validate these conclusions.

Journal ArticleDOI
TL;DR: In this article, the authors extend Sonoda's proof of this result to conformal field theories defined on surfaces with boundaries and introduce four additional sewing constraints: three on the halfplane and one on the cylinder.

Journal ArticleDOI
TL;DR: In this article, the temperature changes that are possible in inhomogeneous low-density astrophysical plasmas were investigated for a variety of boundary distribution functions that occur in astrophysics, with emphasis placed on the spatial changes in temperature and their correlations with those of the density caused by time-independent, but spatially varying, conservative potentials.
Abstract: The temperature changes that are possible in inhomogeneous low-density astrophysical plasmas were investigated for a variety of boundary distribution functions that occur in astrophysics, with emphasis placed on the spatial changes in temperature and their correlations with those of the density caused by time-independent, but spatially varying, conservative potentials. It is proven that decelerating forces produce equilibrium temperatures that are anticorrelated with densities, provided that the boundary condition is non-Maxwellian, and the proof is extended analytically for a generalized Lorentzian distribution, showing that they obey a polytrope relation with the value of gamma between 0 and l.

Journal ArticleDOI
TL;DR: In this paper, a new outflow boundary condition, called the f?ee boundary condition is introduced and tested for two flow and heat transfer model problems, which is equivalent to extending the validity of the weak form of the governing equations to the synthetic outflow instead of replacing them there with unknown essential or natural boundary conditions.
Abstract: SUMMARY Boundary conditions come from Nature. Therefore these conditions exist at natural boundaries. Often, owing to limitations in computing power and means, large domains are truncated and confined between artificial synthetic boundaries. Then the required boundary conditions there cannot be provided naturally and there is a need to fabricate them by intuition, experience, asymptotic behaviour and numerical experimentation. In this work several kinds of outflow boundary conditions, including essential, natural and free boundar conditions, are evaluated for two flow and heat transfer model problems. A new outflow boundary condition, called hereafter the f?ee boundary condition, is introduced and tested. This free boundary condition is equivalent to extending the validity of the weak form of the governing equations to the synthetic outflow instead of replacing them there with unknown essential or natural boundary conditions. In the limit of zero Reynolds number the free boundary condition minimizes the energy functional among all possible choices of outflow boundary conditions. A review of results from applications of the same boundary conditions to several other flow situations is also presented and discussed.

Journal ArticleDOI
TL;DR: In this article, the authors demonstrate that there is an inherent mass balance error present in the operator-splitting algorithm for problems involving continuous mass influx boundary conditions, and they also present a variant of the normal operator splitting algorithm in which the order of solving the advection-dispersion and reaction operators is reversed at each time step.
Abstract: An operator-splitting approach is often used for the numerical solution of advection-dispersion-reaction problems. Operationally, this approach advances the solution over a single time step in two stages, one involving the solution of the nonreactive advection-dispersion equation and the other the solution of the reaction equations. The first stage is usually solved with a finite difference, finite element, or related technique, while the second stage is normally solved with an ordinary differential equation integrator. The only generally published guidelines on numerical accuracy suggest that the discretization errors associated with each stage must be small in order to achieve high accuracy of the overall solution. However, in this note we demonstrate that there is an inherent mass balance error present in the operator-splitting algorithm for problems involving continuous mass influx boundary conditions. The mass balance error does not exist for instantaneous mass input problems. These conclusions are based upon analysis of a simple first-order decay problem for which each stage of the calculation can be performed analytically (i.e., without discretization error). For this linear decay problem we find that the product of the first-order decay coefficient (k) times Δt must be less than approximately 0.1 in order for the mass balance error to be less than 5%. We also present a variant of the normal operator-splitting algorithm in which the order of solving the advection-dispersion and reaction operators is reversed at each time step. This modification reduces the mass balance error by more than a factor of 10 for a wide range of kΔt values.

Journal ArticleDOI
TL;DR: In this paper, a technique called superabsorption is proposed for improving absorbing boundary conditions in finite-difference time-domain methods, which can be applied to every known absorbing boundary condition and greatly reduces the numerical error caused by the boundary reflection.
Abstract: The authors propose a technique, which they call superabsorption, for improving absorbing boundary conditions in finite-difference time-domain methods. This method can be applied to every known absorbing boundary condition and greatly reduces the numerical error caused by the boundary reflection. The principle and analysis of the superabsorption method are presented. Numerical tests indicating the improvements obtained on many absorbing boundary conditions are reported. >