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 random-walk algorithm that solves the steady-state heat equation subject to Dirichlet boundary conditions. Our emphasis is the analysis of geometrically complex domains made up of piecewise-rectilinear boundaries and material interfaces. This work is principally motivated by the semiconductor industry, specifically, their aggressive development of so-called multichip module (MCM) technology. We give a mathematical derivation of the surface Green's function for Laplace's equation over a square region. From it, we obtain an infinite multiple-integral series expansion yielding temperature at any space point in the actual heat-equation problem domain. A stochastic floating random-walk algorithm is then deduced from the integral series expansion. To determine the volumetric thermal distribution within the domain, we introduce a unique linear, bilinear, and trigonometric splining procedure. A numerical-verification study employing two-dimensional finite-difference benchmark solutions has confirmed the accuracy of our algorithm and splining procedure.