Fast poisson solvers for thermal analysis

Abstract

Abstract-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.

Fast 3-D Thermal Simulation for Integrated Circuits With Domain Decomposition Method

“…A nonconformal discretization technique is then proposed, which discretizes the subdomains of heat spreader and heat sink with much coarser mesh, and thereby substantially reduces the solution time for them. With several 2-D and 3-D IC test cases, we demonstrate that the proposed DDM can be over 16× faster than other fast PCG algorithms [5], [8], [9], while using much less memory. And, the nonconformal discretization causes negligible error and is able to capture the hot spot accurately.…”

“…1). By approximating the whole thermal system with a single rectangular domain, substantial error (up to tens of degrees in temperature) may be introduced [4], [5]. Algorithms were recently proposed for the thermal simulation of 3-D ICs [6], [7], where the nonhomogeneous thermal conductivity due to through silicon vias (TSVs) was incorporated.…”

“…In [5], a preconditioned conjugate gradient (PCG) algorithm was proposed for the realistic pyramid IC model, which takes the solution of a larger and approximate rectangular domain by fast Poisson solver (FPS) as the preconditioner. It is called FPS-PCG algorithm, whose efficiency is however related to the Manuscript degree of irregularity in the realistic geometry.…”

“…We employ the relaxed nonoverlapping DDM, which divides the whole model into three rectangular subdomains, and enables fast convergence. For each subdomain, fast thermal simulator, like the FPS in [5], can be used to achieve the best efficiency. A nonconformal discretization technique is then proposed, which discretizes the subdomains of heat spreader and heat sink with much coarser mesh, and thereby substantially reduces the solution time for them.…”

Overcoming Variations in Nanometer-Scale Technologies

“…This may be leveraged to perform Green-function-based analysis [77], [78]. Moreover, the symmetry in thermal conductivities lends itself to the use of fast Poisson solver (FPS) methods [79], and a relationship between Green functions and FPS methods is demonstrated in [80]. 2) Optimization: Figure 6 shows a schematic of 3DIC with the associated packaging.…”