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

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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.