Fast poisson solvers for thermal analysis

Accurate and efficient thermal analysis for a VLSI chip is crucial, both for sign-off reliability verification and for design-time circuit optimization. To determine an accurate temperature profile, it is important to simulate a die together with its thermalmounts: this requires solving Poisson's equation on a nonrectangular 3D domain. This article presents a class of eigendecomposition- based Fast Poisson Solvers (FPS) for chip-level thermal analysis. We start with a solver that solves a rectangular 3D domain with mixed boundary conditions in O(N· logN) time, where N is the dimension of the finite difference matrix. Then we reveal, for the first time in the literature, a strong relation between fast Poisson solvers and Green-function-based methods. Finally, we propose an FPS method that leverages the preconditioned conjugate gradient method to solve nonrectangular 3D domains efficiently. We demonstrate this approach on thermal analysis of an industrial microprocessor, showing accurate results verified by a commercial tool, and that it solves a system of dimension 4.54e6 in only 13 conjugate gradient iterations, with a runtime of 65 seconds, a 15X speedup over the popular ICCG solver.