THEORY OF A FLOATING RANDOM-WALK ALGORITHM FOR SOLVING THE STEADY-STATE HEAT-EQUATION IN COMPLEX, MATERIALLY INHOMOGENEOUS RECTILINEAR DOMAINS

We present the theory and preliminary numerical results for a new rand om-walk algorithm that solves the steady-state heat equation subject t o Dirichlet boundary conditions. Our emphasis is the analysis of geome trically complex domains made up of piecewise-rectilinear boundaries a nd material interfaces. This work is principally motivated by the semi conductor industry, specifically, their aggressive development of so-c alled multichip module (MCM) technology. We give a mathematical deriva tion of the surface Green's function for Laplace's equation over a squ are region. From it, we obtain an infinite multiple-integral series ex pansion yielding temperature at any space point in the actual heat-equ ation problem domain. A stochastic floating random-walk algorithm is t hen deduced from the integral series expansion. To determine the volum etric thermal distribution within the domain, we introduce a unique li near, bilinear, and trigonometric splining procedure. A numerical-veri fication study employing two-dimensional finite-difference benchmark s olutions has confirmed the accuracy of our algorithm and splining proc edure.