Ph.D. Proposal Presentation by Jian Ding

Monday, March 17, 2003

(Dr. Wenjing Ye, advisor)

"Fast BEM Solutions for 3D Large Scale Problems"

The fast development in MEMS devices poses a great challenge in numerical modeling and simulation since the problems are usually coupled nonlinear problems and have complex 3-D geometry. The primary objective of this project is to develop fast boundary element method (BEM) solutions for 3-D large-scale problems with complex geometries, with focus on Slip Stokes and nonlinear problems. The fast slip Stokes solver, which has already been implemented, can solve for steady and oscillatory Stokes flow with slip boundary conditions. The computational cost of this solver is ?(nlogn) compared to ?(n2) required in existing methods. The approach for solving nonlinear problems using the BEM is to use a uniform 3-D grid (the same grid used in FastSlipStokes to accelerate the surface integrals) to evaluate the volume integrals resulted from the nonlinear terms in the integral formulation. This approach not only preserves the mesh advantage of the BEM, but also produces a fast algorithm since surface and volume integrals can be computed in one sweep on the same grid. The remaining of this Ph.D. work is on implementing this idea. A 3-D code based on this approach for nonlinear problems will be produced and tested on three problems, namely Poisson equation, Heat equation, and Navier-Stokes equations. Some related theoretical issues such as accuracy of the approximation will be studied as well.