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 thermal mounts: this requires solving Poisson's equation on a non-rectangular 3D domain. This paper presents a class of eigendecomposition- based fast Poisson solvers (FPS) for chiplevel thermal analysis. We start with a solver that solves a rectangular 3D domain with mixed boundary conditions in O(NlogN) 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-functionbased methods. Finally, we propose an FPS method that leverages the preconditioned conjugate gradient method to solve non-rectangular 3D domains efficiently. We demonstrate that this approach solves a system of dimension 5.33e6 in only 11 Conjugate Gradient iterations, with a runtime of 171 seconds, a 6X speedup over the popular ICCG solver.