Interior Boundary Value Problems Research Articles

A periodic Green function for calculation of coloumbic latticepotentials

The modified Green function appropriate for solution of interior boundaryvalue problems of Laplace's equation in a three-dimensional rectangularparallelepiped, subject to periodic boundary conditions, is developed. Thisallows the determination of the potential due to an arbitrary continuouscharge distribution and its periodic replications in three dimensions.Summation of the eigenfunction expansion by application of thePoisson-Jacobi formula gives a Ewald sum, while application of the Poissonsummation formula results in a two-dimensional potential that is perturbedby a rapidly converging Fourier cosine series involving K0 Besselfunctions. The latter constitutes a generalization of formulae describedby Lekner. Numerical results show that the K0 expansion is more rapidlyconvergent than the Ewald sum, and could therefore substantially reduce thecomputational effort involved in the molecular simulation of ionic andpolar fluids. The Green function is also shown to be related to theasymptotic behaviour of lattice sums for the screened Coulomb potential, inthe limit as the screening constant tends to zero.

The Traction Problem in a Theory of Plane Strain Elasticity with Boundary Reinforcement

In a previous work, the authors used integral equation methods to solve interior and exterior mixed boundary value problems arising from a theory of plane strain elasticity with boundary reinforcement. In this paper, the authors generalize the results in their previous paper to account for the case when only traction is prescribed on the nonreinforced part of the boundary of the elastic body. This case is of particular interest, since the corresponding homogeneous systems of singular integral equations give rise to nontrivial solutions that affect the solvability of both the interior and exterior boundary value problems.

Numerical methods for flow in a porous medium with internal boundaries

Flow of fluids and transport of solutes in porous media are subjects of wide interest in several fields of applications: reservoir engineering, subsurface hydrology, chemical engineering, etc. In this paper we will study two-phase flow in a model consisting of two different types of sediments. Here, the absolute permeability, the relative permeabilities and the capillary pressure are discontinuous functions in space. This leads to interior boundary value problems at the interface between the sediments. The saturation Sw will be discontinuous or experience large gradients at the interface. A new solution procedure for such problems will be presented. The method combines the modified method of characteristics with a weak formulation where the basis functions are discontinuous at the interior boundary. The modified method of characteristics will provide a good first approximation for the jump in the discontinuous basis functions, which leads to a fast converging iterative solution scheme for the complete problem. The method has been implemented in a two-dimensional simulator, and results from numerical experiments will be presented.

Weak solutions of interior boundary-value problems for plates with transverse shear deformation

The existence, uniqueness, and continuous dependence on the data are investigated for the solutions of the equations of bending of plates with transverse shear deformation in a Sobolev space setting.

Elastic Boundary Conditions in the Theory of Plates

The direct boundary integral equation method is used to solve the interior and exterior boundary value problems with elastic conditions on the contour for plates with transverse shear deformation.

Semi‐inversion method for an accurate analysis of rectangular waveguide H plane angular discontinuities

The new efficient numerical‐analytical method of electromagnetic interior boundary value problem solving is presented with waveguide H plane angled bends as examples of implementation. The method essence consists in the analytical inversion of the main difference part of the initially derived ill‐posed matrix equation of the first kind. As a result, the problem is reduced to the solution of the second‐kind matrix equation with an invertible operator. The proposed method differs from other methods dealing with so‐called modified Wiener‐Hopf geometry structures. It is oriented to noncoordinate boundary value problems to which the ones of angular waveguide discontinuities are related. Detailed consideration of all parts of the numerical algorithm ensures its high efficiency. As application examples, the data for designing the matched truncated corners in a single‐mode waveguide and the results of comparison of the known high‐frequency heuristic estimations with the exact simulation of waveguide corners having a quasi‐optical mirror are presented.

A boundary element solution for a mode conversion study on the edge reflection of Lamb waves

The boundary element method, well known for bulk wave scattering, is extended to study the mode conversion phenomena of Lamb waves from a free edge. The elastodynamic interior boundary value problem is formulated as a hybrid boundary integral equation in conjunction with the normal mode expansion technique based on the Lamb wave dispersion equation. The present approach has the potential of easily handling the geometrical complexity of general guided wave scattering with improved computational efficiency due to the advantage of the boundary-type integral method. To check the accuracy of the boundary element program, vertical shear wave diffraction, due to a circular hole, is solved and compared with previous analytical solutions. Edge reflection factors for the multibackscattered modes in a steel plate are satisfied quite well with the principle of energy conservation. In the cases of A0, A1, and S0 incidence, the variations of the multireflection factors show similar tendencies to the existing results for glass. It is observed that the reflection of an incident wave becomes close to zero over a certain frequency range seen through energy interaction with other reflected modes, and increases again beyond this minimum point due to reverse mode conversion. The reflections of the higher symmetric incident modes, S1 and S2, are also investigated. It turns out that S1 mode is an unusual mode which is nearly unaffected by the mode conversion in the Lamb wave edge reflection.

Optimal Order Multigrid Methods for Solving Exterior Boundary Value Problems

The coupling of boundary elements and finite elements combines the advantage of boundary elements for treating domains extended to infinity and that of finite elements in treating the nonhomogeneity of equations and the complexity of domains. In the case of the Laplacian, by taking a circle or a sphere as the artificial coupling boundary, it is shown that the corresponding boundary integral equation can be solved without any cost and the coupled system is reduced to a simple finite element system. Two multigrid methods are proposed to solve this finite element linear system. Both methods are of optimal order and can be used to solve such finite element equations as efficiently as to solve those arising from interior boundary value problems. Numerical experiments are included to show the efficiency and advantages of the methods. An apparent significance of the methods is that the boundary elements appear neither in the discretization nor in the coding.

Transient sound radiation from impulsively accelerated bodies

This paper presents an integral formulation derived from the Kirchhoff integral theorem for predicting transient sound radiation from an impulsively accelerated body. The acoustic pressure is determined in an explicit form in the frequency domain, and subsequently converted to the time domain through Fourier transformations. This integral formulation can be extended to cases in which the body is subjected to an arbitrarily time-dependent excitation. The transient acoustic pressure is shown to be expressible as a convolution integral of the impulse response function to the time history of the surface velocity of the body. Since there is no relationship between the integral formulations for the exterior and interior regions, solutions for the radiated acoustic pressure in the exterior region are uniquem, even at frequencies corresponding to the eigenfrequencies of the related interior boundary value problem. Analytical results based on this integral formulation agree perfectly with those obtained using other methods for transient sound radiation from an explosively expanding sphere, impulsively accelerated rigid sphere, impulsively accelerated baffled, and unbaffled circular disks. Numerical implementations of the present integral formulation are expected to be more amenable to solution than directly solving the classical Kirchhoff integral formulation in which the frequency and time domains are fully coupled.

Transient sound radiation from impulsively accelerated bodies

This paper presents an integral formulation derived from the Kirchhoff integral theorem for predicting transient sound radiation from an impulsively accelerated body. The acoustic pressure is determined in an explicit form in the frequency domain, and subsequently converted to the time domain through Fourier transformations. This integral formulation can be extended to cases in which the body is subject to an arbitrarily time-dependent excitation. The transient acoustic pressure is shown to be expressible as a convolution integral of the impulse response function to the time history of the surface velocity of the body. Since there is no relationship between the integral formulations for the exterior and interior regions, solutions for the radiated acoustic pressure in the exterior region are unique, even at frequencies corresponding to the eigenfrequencies of the related interior boundary value problem. Analytical results based on this integral formulation agree perfectly with those obtained using other methods for transient sound radiation from an explosively expanding sphere, impulsively accelerated rigid sphere, impulsively accelerated baffled and unbaffled circular disks. [Work supported by ONR.]

Nonuniqueness of solutions to extended Kirchhoff integral formulations

This paper examines the nonuniqueness of solutions to an extended Kirchhoff integral formulation at certain discrete frequencies corresponding to the eigenfrequencies of the related interior boundary value problem. In particular, effects of surface motion and interaction between turbulence and vibrating surface in motion on nonuniqueness difficulties are investigated. It is shown that surface motion affects nonuniqueness difficulties in two ways: (1) it shifts the eigenfrequencies of the related interior boundary value problem and (2) it excites more eigenfrequencies at which nonuniqueness difficulties occur than the corresponding stationary case. The interaction between turbulence and vibrating surface in motion further shifts the eigenfrequencies. Although changes in these eigenfrequencies are small at low Mach numbers, they increase with the Mach number. It is also demonstrated that although an extended Kirchhoff integral equation fails to yield a unique solution at certain discrete frequencies, a combined extended surface Kirchhoff integral equation and an extended interior Kirchhoff integral equation always yield a unique solution for all frequencies. This is because there is only one set of solutions that satisfy both an extended surface Kirchhoff integral equation and an extended interior Kirchhoff integral equation.

Use of theR function method to calculate electrodynamic characteristics of waveguides with complex shapes

Modern radio physics has in recent years been characterized by rapid development of theory and engineering practice in the millimeter and submillimeter wave ranges. The dimensions of the systems, which are usually quite arbitrary in shape, are commensurate with or substantially in excess of the wavelength. This necessitates determination of the eigenvalues and eigenfunctions of regions with boundaries of complex form (waveguides, microstrip structures, resonators, etc.). One important problem is to lay the electrodynamic foundation for computer-aided design of microwave devices over a broad wavelength range [i]. A procedure for analytic description of the complex geometric shapes of real objects is especially important for development of the appropriate algorithm and their numerical implementation in computers. The theory of R functions, which is utilized in solving interior and exterior electrodynamic boundary value problems [2-11], is promising for this purpose.

LOW-FREQUENCY ACOUSTIC SCATTERING BY A HARD INVERSE-PROLATE SPHEROID

The solution is obtained via an appropriate use of the Kelvin inversion method and involves the solution of an interior boundary value problem, for a prolate spheroid with Robin conditions on its surface.

Convergence of the finite element method as applied to electromagnetic scattering problems in the presence of inhomogeneous media

Approximate radiation boundary conditions (BCs) make partial-differential-equation-based methods attractive for the solution of electromagnetic scattering problems, especially for geometrically complex inhomogenous targets. These BCs allow for an exterior boundary value problem (BVP) to be approximated by an interior BVP. The use of the second-order Bayliss-Turkel BC for two-dimensional inhomogeneous targets is considered. In numerical experiments the rate of convergence with respect to mesh size previously predicted continues to hold.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the equivalence of integral formulations for certain exterior and interior boundary value problems in mechanics

A procedure is given which relates various linear boundary value problems. It is shown that a relations exists between certain ‘exterior’ and ‘interior’ problems. For computational purposes an ‘interior’ problem is more desirable. The method is restricted by the requirement that the Neumann function, or its equivalent, be known for the problem being considered.

Boundary value problems of the Ginzburg–Landau equations

SynopsisFor a domain Ω in the Euclidean space Rd (d = 2, 3) existence of weak solutions for both interior and exterior Dirichlet boundary value problems of the Ginzburg-Landau equations are established without any restriction on the range of the coupling constant λ, thesize of Ω, or the boundary data. For the critical choice λ = 1, we prove the existence of confined multivortices in a bounded domain by a constructive monotone iteration method.

A generalized Fourier approximation in micropolar elasticity

Kupradze's method of generalized Fourier series is used to approximate solutions of interior boundary value problems of thin micropolar plates.

Plane interior boundary-value problems in microcontinuum fluid mechanics

This paper examines the fundamental question of existence and uniqueness of solutions of two plane interior boundary-value problems encountered in the field of microcontinuum fluid mechanics. These problems are analyzed using potential theoretic methods and with the aid of singular integral equations. The results obtained have their counterparts in the classical Navier-Stokes theory.

Some applications of a galerkin‐collocation method for boundary integral equations of the first kind

AbstractHere we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order ‐ 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B‐splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L2. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.

A penalty method for the approximate solution of stationary Maxwell equations

The paper deals with nonconforming finite element methods for the approximate solution of the interior boundary value problem for Maxwell equations in the time-harmonic case. The methods are based on penalization in the boundary conditions of total reflexion. Qualitative convergence results are obtained by a-priori estimates which are proven in the first part of this paper. The main object is to establish estimates for the global discretization error in various norms of the underlying spaces of approximating vector fields.