Boundary Element Method For Seismic Wave Equation In Erequency-domain

Boundary Element Method(BEM) is a powerful method in solving partial equation.Since BEM divides only in boundary,in fact it will be dealt with lower dimensions. Reducing dimensions will reduce the unknowns of the algebra equations. It has advantages of less input data and more rapid compute speed. It is used to high dimension problem especially.The method of solving seismic wave equation in time-domain boundary element is direct which solve fundamental solutions of wave equations directly and boundary integral equation. But this equation has its shortage as followings: computing quantity of time-domain boundary element method will increase along with the increasing of time. The frequency-domain boundary element method of the former adopt the ordinary Fourier (double sides) transform, initial conditions cannot be reflected in Helmholtz equation.At present, BEM mainly apply to linear problem. There are few articles in studying nonlinear problem. In using frequency-domain FEM for wave equation, there are three difficulties as followings: The first is solving the fundamental solution of 2D Helmholtz equation. When λ— 0, the fundamental solution of 3D Helmholtz equation become the one of Laplace. As a result, these two problems have the results very similar results in using BEM. But in 2d Helmholtz equation there is not the relation , because when λ —> 0, there is not fundamental solution. The second is the BEM in complex domain. The whole domain will be divided into some even small domains in dealing uneven medium problem using BEM. This will increase the joint boundaries and bring on increasing the points inevitably, when there are many small domains, it no longer has the advantages of small computing. But if only not many domains, it has the advantages. The third is dealing with the general boundary condition problem and joint boundary. Boundary element method will solve a linear equations group finally which needs the linear relation of all unknown variables. But some boundary conditions, such as the absorbingconditions in this article, are not provided with linear property. This cannot apply the boundary conditions for the BEM. In the former articles about BEM, the boundary conditions were dealt with simply. The absorbing conditions in this article are mainly applied for FDM and applied for BEM directly.In the second chapter of this article, we introduce the single-side Fourier transform and discrete Fourier transform, single-side Fourier transform is same to the general one in theory basically. When u(x, y, z) = 0(t < 0), the single-side Fourier transform become the general one, and they have the same properties. The singe-side Fourier transform has the advantage of adding the initial conditions to the control equation directly. Discrete Fourier transform is dispersing the frequency after single-side Fourier transform. Frequency step and time step have the relation: Aw = -tStti ■ The third chapter introduce deducing fundamental solutions of the Helmholtz equation. The fundamental solutions of Helmholtz equation are not unique and this article gives two fundamental solutions. Using in spite of which fundamental solution, the solution of the equation is constant. The fundamental solution wants to be according to the calculating demand to select by examinations. The forth chapter introduce the frequency domain boundary element method. The single-side Fourier transform is firstly adopted to transform the wave equation from the time domain to the frequency domain. Discuss the deduct of boundary integral equation of the Helmholtz equation and its discretization and solving, the computation of coefficient matrix. This article put forward the general linear boundary conditions for the BEM. This condition is more universal then the third boundary condition. The last chapter studies the technique of dealing with the absorbing boundary conditions. About the problem that artificial absorbing condition can not be applied to BEM, this article put forward the linearize method for absorbing boundary condition. In this article,...