Numerical Techniques for Integral Equations

Finding computationally efficient numerical techniques for simulation of three dimensional structures has been an important research topic in almost every engineering domain. Surprisingly, the most numerically intractable problem across these various disciplines can be reduced to the problem of solving a three-dimensional potential problem with a problem-specific Greens function. Application examples include electrostatic analysis of sensors and actuators; electromagnetic analysis of integrated circuit interconnect and packaging; detailed analysis of frequency response and loss in photonic devices; drag force analysis of micromachined structures; and potential flow based aircraft analysis. Over the last fifteen years, we have been developing fast methods for solving these problems and have developed widely used programs such as FastCap (capacitance), FastHenry (magnetoquasistatics), FastLap (general potential problems), FastImp (full wave impedence extraction),and FastStokes (fast fluid analysis). The most recent work is in developing higher order methods, methods that efficiently discretize curved geometries, methods that are more efficient for substrate problems, and methods for analyzing rough surfaces.

Although the boundary element method is a popular tool to solve the integral formulation of many three-dimensional potential problems, the method becomes slow when a large number of elements are needed to are used. This is because boundary-element methods lead to dense matrix problems which are typically solved with some form of Gaussian elimination. Generating dense systems implies that the required memory grows quadratically and required computational resources grows cubically with the number of unknowns needed to accurately discretize the problem.

Over the last fifteen years, a number of algorithms have been developed which can solve the dense systems associated with boundary-element methods in time and memory that grows only linearly with problem size. These fast algorithms typically combine iterative matrix solution methods with fast techniques for multiplying dense boundary-element matrices by vectors. Recent work in this area has been to develop precorrected-FFT techniques, which can work for general Greens functions, and Wavelet based techniques, which generate extremely effective preconditioners which accelerate iterative method convergence.