Boundary Integral Equation Methods For Scattering Problems Of Several Wave Equations

The scattering problems of various wave equations,which arise from various applications,have attracted much attention of engineers and mathematicians for many years.This dissertation investigates the exterior Neumann boundary value problems of poroelastic wave in two and three dimensions and acoustic half-space problems,which are imposed in unbounded domain.Compared with the domain discretization methods,the boundary integral equation method possesses such advantages as only requiring discretization of boundaries and satisfying the radiation condition at infinity in a direct manner,which has advantages in solving such wave scattering problems.This dissertation consists of two parts.The first part is the numerical methods for the exterior Neumann boundary value problems in two-and three-dimensional poroelastic problems.The second part is the numerical methods for the acoustic scattering problems on a locally defected half-space in both two and three dimensions.For the poroelastic problems,we employ both the Burton-Miller formulation and the indirect method to construct two types of boundary integral equations for solving the exterior Neumann boundary value problems in two and three dimensions.These two kinds of boundary integral equations involve strongly-singular and hyper-singular boundary integral operators,which are well defined in the sense of Cauchy principal value and Hadamard finite part,respectively,and are difficult to be calculated directly.In addition,the eigenvalues of the hyper-singular integral operator accumulate at infinity.As a result,the solution of the integral equation by means of Krylov-subspace iterative solvers such as GMRES generally requires large numbers of iterations.To resolve the issues,on the one hand,with the help of Günter derivatives,Stokes formulations and the idea of integration-by-parts,the strongly-singular and hyper-singular integral operators are reformulated into compositions of weakly-singular integral operators and tangential-derivative operators,which allow us to prove the jump relations associated with the poroelastic layer potentials and boundary integral operators in a simple manner.On the other hand,relying on the Calderón relation and the investigated spectral properties of the boundary integral operators,we propose an analytical preconditioner,which can lead to the regularized integral equations of Burton-Miller formulation and indirect method.These two kinds of BIEs have good spectral properties,and the corresponding eigenvalues accumulate at three points,which are bounded away from zero and infinity.Compared with the original boundary integral equations,the numbers of iterations required for iterative solution of the regularized boundary integral equations are significantly decreased.Then,numerical examples of two-and three-dimensional poroelastic problems are presented to demonstrate the accuracy and efficiency of the proposed methodology by means of the Nystr(?)n method and a Chebyshev-based rectangular-polar solver,respectively.Considering the scattering problems in a locally perturbed half-space,based on the perfectly matched layer,this dissertation proposes a novel BIE solver for the acoustic scattering problems on a locally defected half-space in both two and three dimensions.The original scattering problem is reduced to a problem imposed in a bounded domain by the PML technique.Assuming the vanishing of the scattered field on the PML boundary,we derive BIEs on local defects only in terms of using the PML-transformed free-space Green’s function,and the four standard integral operators: single-layer,double-layer,transpose of double-layer,and hyper-singular operators.Then we derive two Fredholm equations of the second kind corresponding to the Dirichlet and Neumann boundary value problems.In addition,an extension of the high-order Chebyshev-based rectangular-polar singular-integration method to the current numerical implementation of all four boundary integral operators is introduced.The hyper-singular integral operator is transformed into combination of the weakly singular operators and the tangential derivatives.Numerical experiments for both two-and three-dimensional problems are carried out to demonstrate the accuracy and efficiency of the proposed solver.