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

边界元方法是求解偏微分方程组的一种强有力的方法。本研究项目致力于使该方法更加准确和高效，并将其应用于反问题1。为了提高求解效率，在边界元中，每个单元都与所有其他单元相关，因此，随着单元个数(N)的增加，生成和求解稠密线性方程组所需的计算量(O(n^3))和所需的内存(O(n^2))变得令人望而却步。为了克服这个问题，我们将快速多极子方法(FMM)和面板聚类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。该方法利用多极展开式或泰勒展开式，在所要求的精度范围内对相距较远的单元之间的效应进行近似和聚集，从而将计算量和存储空间降至接近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，我们发展了一种处理周期边界条件的方法，当使用FMM计算二维涡间作用力时。积分的精确计算我们发展了一种使用变量变换的自动数值积分方法，用于计算在处理薄结构和缝隙时或在计算非常接近边界的场时出现在边界元方法中的近奇异积分。逆问题的应用我们将三维边界元和非线性优化应用于识别大脑中的电流偶极子的问题，其中头部由三个具有不同电导率的区域建模。这是通过引入一个新的目标函数来实现的，该目标函数基于由于偶极子放置在无限均匀区域中而产生的虚势。

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.