PhD Proposal: Fast and Accurate Boundary Element Methods in Three Dimensions

Talk
Ross Adelman
Time: 
04.28.2015 13:00 to 14:30
Location: 

AVW 3450

The Laplace equation in three dimensions is one of the most important partial differential equations (PDEs) in math and science. It is the governing equation for a very large number of physical phenomena, including electrostatics, magnetostatics, astrophysics, molecular dynamics, aerodynamics, and even in computer graphics. The purpose of my PhD research is to develop fast and accurate numerical solvers for the Laplace equation. In particular, I will focus on the Galerkin boundary element method (BEM).
First, I will develop new methods for implementing the Galerkin BEM. The Galerkin BEM is a powerful method for solving many PDEs, including the Laplace equation. When the boundary is discretized using triangular elements, constructing the system matrix requires computing double surface integrals over pairs of these triangles. Because the kernels being integrated are singular, these integrals can be dif?cult to compute, especially when the two triangles share a vertex, an edge, or are the same. We have developed a method for computing these integrals that completely avoids the singularity issue and only requires that completely regular integrals be computed explicitly.
Second, I will implement a baseline BEM in MATLAB. This is to verify that all the methods developed in the previous step work as advertised. Thus, accuracy is the most important aspect of this step. Various experiments will be performed to characterize the error behavior of the BEM. For example, several example problems will be constructed and solved to verify the accuracy.
Third, I will accelerate the BEM using the fast multipole method (FMM) and the GPU. The baseline BEM requires O(N^2) to store a problem and O(N^3) to solve the problem, where N is the number of elements. These heavy memory and computational demands effectively restrict the BEM to problems with fewer than 30,000 elements. However, the FMM can reduce the memory and computational complexity down to O(N), and using the GPU to accelerate the last step of the FMM can further increase performance. The FMM/GPU-accelerated BEM will be able to solve problems with millions of elements.
Lastly, when possible, I will release all code as freely available software.
Examining Committee:
Committee Chair: - Dr. Ramani Duraiswami
Dept’s Representative - Dr. Amitabh Varshney
Committee Member(s): - Dr. Howard Elman
- Dr. Nail Gumerov