On the choice of the coupling parameter in boundary integral formulations of the exterior acoustic problem

Most commonly used boundary integral formulations of the exterior problem in time-harmonic acoustics, which are valid for all wave-numbers, involve a non-zero coupling parameter. In this paper we analyse the qualitative behaviour of the direct boundary integral formulation due to Burton and Miller. In particular we include a regularised formulation for the Neumann problem. By deriving the eigensystems of the operators we are able to obtain ′almost optimal′ values for the coupling parameter, as a function of the wave-number, so as to minimize the condition number of various formulations     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

Unified symmetric finite element and boundary integral variational coupling methods for linear fluid–structure interaction

This article concerns the development of energy-based variational formulations and their corresponding finite element–boundary element Rayleigh–Ritz approximations for solving the time-harmonic vibration and scattering problem of an inhomogeneous penetrable fluid or solid object immersed in a compressible, inviscid, homogeneous fluid. The resulting coupled finite element and boundary integral methods (FEM-BEM) have the following attractive features: (1) Separate direct and complementary variational principles lead naturally to several alternative structure variable and fluid variable methodologies. (2) The solution in the exterior region is represented by a combined single- and double-layer potential which ensures the validity of the methods for all wave numbers; even though this representation introduces hypersingular integrals, for actual computations the hypersingular operator may be rewritten in terms of single-layer potentials, which can be integrated by standard techniques. (3) Since the discretized equations for the interior region and for the boundary are derived from the first variation of bilinear functionals the resulting algebraic systems of equations are always symmetric. In addition, the transition conditions across the interface are natural. This allows one to approximate the solutions within the interior and exterior regions independently, without imposing any boundary constraints.     

Boundary Integral Equations for Mixed Boundary Value Problems in R3

Both exterior and interior mixed Dirichlet-Neumann problems in R³ for the scalar Helmholtz equation are solved via boundary integral equations. The integral equations are equivalent to the original problem in the sense that the traces of the weak seolution satisfy the integral equations, and, conversely, the solution of the integral equations inserted into Green's formula yields the solution of the mixed boundary value problem. The calculus of pseudodifferential operators is used to prove existence and regularity of the solution of the integral equations. The regularity results — obtained via Wiener-Hopf technique — show the explicit “edge” behavior of the solution near the submanifold which separates the Dirichlet boundary from the Neumann boundary.     

用Backus-Gilbert方法求解声波散射问题

利用位势理论将散射问题的外边界问题转化为第一类边界积分方程求解,再利用Backus-Gilbert方法给出了二维空间的数值结果,与Tikhonov正则化方法比较,虽然精度稍差一些,但是计算方法和计算机实现比较简单.     

Symmetric boundary integral formulations for Helmholtz transmission problems

In this work we propose and analyze numerical methods for the approximation of the solution of Helmholtz transmission problems in two or three dimensions. This kind of problems arises in many applications related to scattering of acoustic, thermal and electromagnetic waves. Formulations based on boundary integral methods are powerful tools to deal with transmission problems in unbounded media. Different formulations using boundary integral equations can be found in the literature. We propose here new symmetric formulations based on a paper by Martin Costabel and Ernst P. Stephan (1985), that uses the Calderón projector for the interior and exterior problems to develop closed expressions for the interior and exterior Dirichlet-to-Neumann operators. These operators are then matched to obtain an integral system that is equivalent to the Helmholtz transmission problem and uses Cauchy data on the transmission boundary as unknowns. We show how to simplify the aspect and analysis of the method by employing an additional mortar unknown with respect to the ones used in the original paper, writing it in an appropriate way to devise Krylov type iterations based on the separate Dirichlet-to-Neumann operators.     

An Interface Problem in Solid Mechanics with a Linear Elastic and a Hyperelastic Material

The three dimensional interface problem is considered with the homogeneous Lamé system in an unbounded exterior domain and some quasistatic nonlinear elastic material behavior in a bounded interior Lipschitz domain. The nonlinear material is of the Mooney-Rivlin type of polyconvex materials. We give a weak formulation of the interface problem based on minimizing the energy, and rewrite it in terms of boundary integral operators. Then, we prove existence of solutions.     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

Boundary integral method for scattering problems with cracks buried in a piecewise homogeneous medium

We consider the scattering of time‐harmonic acoustic plane waves by a crack buried in a piecewise homogeneous medium. The integral representation for a solution is obtained in the form of potentials by using Green's formula. The density in potentials satisfies the uniquely solvable Fredholm integral equation. Then we obtain the existence and uniqueness of the solution. Copyright © 2011 John Wiley & Sons, Ltd.     

Green's function-based integral approaches to linear transient boundary-value problems and their stability characteristics (I)

Two Green's function-based formulations are applied to the governing differential equation which describes unsteady heat or mass transport in an isotropic homogeneous 1-D domain. In this first part of a two series of papers, the linear form of the differential equation is addressed. The first formulation, herein denoted the quasi-steady Green element (QSGE) formulation, uses the Laplace differential operator as auxiliary equation to obtain the singular integral representation of the governing equation, while the second, denoted the transient Green element (TGE), uses the transient heat equation as auxiliary equation. The mathematical simplicity of the Green's function of the first formulation enhances the ease of solution of the integral equations and the resultant discrete equations. From the point of computational convenience, therefore, the first formulation is preferred. The stability characteristics of the two formulations are evaluated by examining how they propagate various Fourier harmonics in speed and amplitude. We found that both formulations correctly reproduce the theoretical speed of the harmonics, but fail to propagate the amplitude of the small harmonics correctly for Courant value of about unity. The QSGE formulation with difference weighting values between 0.67 and 0.75, and the TGE formulation provide optimal performance in numerical stability.     

L2-boundary integral equations for the robin problem

In this paper we study an integral equation method for the exterior Robin problem for the Helmholtz equation where the boundary condition is interpreted in the L²-sense. In particular, we derive a composite integral equation from Green's theorem which is uniquely solvable for all wave numbers.     

Asymptotic behaviour of solutions to semilinear wave equations with initial data of slow decay

Some useful and remarkable property are derived from a representation formula of a radially symmetric solution to the Cauchy problem for a homogeneous wave equation in odd space dimensions. These properties provide us with enough information to consider the semilinear case, namely, the associated integral equation with the problem will be considered on a weighted L∞-space. This formulation enables us to deal with the problem for slowly decaying initial data.     

Analytic solution of an exterior Neumann problem in a non‐convex domain

A new method, based on the Kelvin transformation and the Fokas integral method, is employed for solving analytically a potential problem in a non‐convex unbounded domain of ?², assuming the Neumann boundary condition. Taking advantage of the property of the Kelvin transformation to preserve harmonicity, we apply it to the present problem. In this way, the exterior potential problem is transformed to an equivalent one in the interior domain which is the Kelvin image of the original exterior one. An integral representation of the solution of the interior problem is obtained by employing the Kelvin inversion in ?² for the Neumann data and the ‘Neumann to Dirichlet’ map for the Dirichlet data. Applying next the ‘reverse’ Kelvin transformation, we finally obtain an integral representation of the solution of the original exterior Neumann problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Regularized Combined Field Integral Equations

Summary. Many boundary integral equations for exterior boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an L²()-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes.This article presents new combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods.     

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

A stream function-vorticity formulation coupled with boundary integrals for the two-dimensional exterior Stokes problem

In this work, we derive a stream function-vorticity variational formulation coupled with boundary integrals for the exterior Stokes problem in two dimensions, when the right-hand side has a bounded support. The stream function-vorticity formulation is expressed in a bounded region containing the support of the right-hand side, and the boundary conditions on the artificial boundary are obtained by an integral representation. We prove that this coupled formulation is equivalent to the original Stokes problem.     

Modified boundary integral formulations for the Helmholtz equation

In this paper we describe some modified regularized boundary integral equations to solve the exterior boundary value problem for the Helmholtz equation with either Dirichlet or Neumann boundary conditions. We formulate combined boundary integral equations which are uniquely solvable for all wave numbers even for Lipschitz boundaries Γ=∂Ω. This approach extends and unifies existing regularized combined boundary integral formulations.     

Boundary value problems in the theory of thermomicrostretch elastic solids

In this paper the linear theory of thermomicrostretch elastic solids is considered. The basic properties of wave numbers of the longitudinal and transverse plane waves are established.The existence of eigenfrequencies of the interior homogeneous boundary value problem (BVP) of steady oscillations is studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

The Behaviour of Elastic Fields and Boundary Integral Mellin Techniques Near Conical Points

For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.