Boundary Integral Equation Methods For Three-dimensional Transmission Problems In Acoustic

Since the wave could propagate through different media in nature,it is necessary to establish some appropriate mathematical models to describe the phenomenon.In this the-sis,we mainly study the time-harmonic Helmholtz equations on acoustic scattering prob-lems in the three-dimensional case.There are many numerical methods to solve the trans-mission problems,including the finite element methods(FEM),finite difference method(FDM),finite volume method(FVM),boundary integral equation method(BIEM)and etc.,and in this thesis we mainly adopt BIEM to deal with the three-dimensional acoustic transmission problem.The BIEM has two main advantages.The first one is that we only need to discretize the boundary instead of the whole domain,while the second one is that we could reduce the dimension and meanwhile remain the high-order accuracy.Certainly it also owns some difficulties,such as how to handle the singularity of the integral kernel of the boundary integral operators.The thesis on three-dimensional acoustic transmission problems contains the two following parts.We first consider the exterior Dirichlet problem of the three-dimensional acoustic equations on the Lipschitz boundary.Utilizing the potential layer theory,the solution of the original equations could be represented by the single layer,and then we are able to obtain the boundary integral equation through the boundary condition.Based on this,we get a discrete system by using the Galerkin boundary element method.In turn by solving the linear equation group,the coefficients required could be obtained,and substituting them into the representation formulation of the original problem leads to the numerical results,as shown in the figure 3-2.Particularly,we introduce a special local coordinate system^[1,2]for the triangle boundary element,and thus the singularity of the boundary integral operators could be tackled effectively.Further we investigate the acoustic transmission problem on that an incident wave could penetrate the surface of an object in the second part.With the help of the Green's representations,we have the solution to the original problem,and then combining the boundary conditions leads to the coupled boundary integral equation system.We prove the existence and uniqueness of the weak solutions to the acoustic problem under the assumption of the Lipschitz boundary in the three-dimensional case.Finally a numerical example is illustrated to verify the effectiveness and the accuracy of the method.