The Research on Making the Boundary Element Method Highly Accurate and Efficient

边界元法是求解偏微分方程的一种有效方法。本课题旨在提高该方法的精度和效率，并将其应用于求解反问题。在BEM中，每个元素都与所有其他元素相关，因此，随着元素数量(n)变大，生成和求解密集线性方程组的计算量（O (n^3)）和所需内存（O (n^2)）变得令人望而却步。为了克服这一问题，我们将快速多极子方法（FMM）和面板聚类方法应用于二维势问题、三维泊松问题以及二维和三维弹性静力学。该方法使用多极展开或泰勒展开，以便在要求的精度范围内近似和聚集相距很远的元素之间的效应，从而将计算和内存减少到接近O (n)。我们还将FMM应用于电子枪的三维边界元模拟，考虑了带电粒子的空间电荷效应。此外，我们开发了一种处理周期边界条件的方法，当使用fmm计算二维涡旋之间的力时。精确的积分计算我们发展了一种利用变量变换的自动数值积分方法，用于计算边界元法中在处理薄结构和间隙或计算非常靠近边界的场时出现的近奇异积分。我们将三维边界元和非线性优化应用于识别大脑中电流偶极子的问题，其中头部由三个具有不同电导率的区域建模。这是通过引入一个基于虚势的新目标函数而实现的，虚势是由于偶极子放置在无限均匀区域中。

The Boundary element Method (BEM) is a powerful method for solving partial differential equations. This research project is concerned with making the method more accurate and efficient, and on its application to inverse problems.1. Making the solution more efficientIn BEM,each element is related to all the other elements, so that the amount of computation (O (n^3)) and required memory (O (n^2)) for generating and solving the dense system of linear equations becomes prohibitive as the number elements (n) becomes large.In order to overcome this problem, we have applied the Fast Multipole Method (FMM) and the Panel Clustering Method to the 2-D potential problem, 3-D Poisson problem and 2-D and 3-D elastostatics. The methods use multipole expansions or Taylor expansions in order to approximate and cluster effects between elements far apart within the required accuracy, so that the computation and memory is reduced to nearly O (n).We have also applied the FMM to the three-dimensional boundary element simulation of the electron gun, taking the space charge effect of charged particles into account. Further, we developed a method for treating the periodic boundary condition when calculating the forces acting between vortices in 2-D using the FMM.2. Accurate computation of integralsWe have developed an automatic numerical integration method using variable transformations for the calculation of nearly singular integrals which appear in the boundary element method when treating thin structures and gaps or when calculating the field very near the boundary.3. Application to inverse problemsWe applied 3-D BEM and nonlinear optimization to the problem of identifying a current dipole in the brain, where the head is modelled by three regions with different conductivity. This was made possible by introducing a new object function based on the virtual potantial due to the dipole placed in the infinite homogeneous region.