| This thesis describes the development of a new computational approach for rapid and accurate evaluation of the three-dimensional potential field and its gradient. Efficient evaluation of the potential field is an essential requirement for simulation of large ensembles of particles in many applications including astrophysics, plasma physics, fluid dynamics, molecular dynamics, VLSI systems, and micro-electro-mechanical systems. Current methods use multipole expansion of spherical harmonics for the potential field, which is computationally expensive in terms of running time and memory requirements when a high degree of accuracy is desired. The mathematical background for the approach proposed in this thesis stems from an exponential integral representation of Green's function and an approximation to the integral using Gauss quadratures. The translations are simple in structure, error-free, and independent of the approximation, which enables the overall accuracy and computational performance to be controlled externally via the approximation. In addition, the gradient of the potential can be readily retrieved as a by-product of the computational process. More importantly, the memory requirement is independent of the desired degree of accuracy. The technique presented here opens new possibilities for efficient distributed computing and parallel processing of large-scale simulation of particle systems.; The research described in this thesis makes the following key contributions: (i) A randomized algorithm is devised for generating exponential expansion of the Green's function. Given an intended error, this probabilistic-type scheme constructs the Gauss quadratures for the expansions with sizes as small as possible. It makes use of the available Gauss-Legendre quadratures and does not require solving non-linear equations. (ii) Employing the exponential expansion, a theory is developed for efficient evaluation of the potential field and its gradient. The main motivation is to provide an alternative to the current methods for N-body problems involving millions of particles. (iii) A new formulation is proposed for the charge density problem. It yields integral equations of the second kind, for which efficient numerical algorithms are available. The resultant discretization matrices have improved conditioning and the convergence rate is faster. (iv) A simulator is designed and implemented for numerically extracting capacitance of multi-layered dielectric multi-conductor systems. It is used for studying capacitance associated with amorphous silicon thin-film transistors and imaging arrays. |
