Efficient full-chip thermal modeling and analysis

The ever-increasing power consumption and packaging density of integrated systems creates on-chip temperatures and gradients that can have a substantial impact on performance and reliability. While it is conceptually understood that a thermal equivalent circuit can be constructed to characterize the temperature gradients across the chip, direct and iterative solutions of the corresponding 3D equations are often intractable for a full-chip analysis. Multigrid accelerated iterative methods can be applied to solve the equivalent circuit problem that is provably symmetric positive definite; however, explicitly building the matrix problem is intractable for most full-chip problems. In This work we present a multigrid iterative approach for the full-chip thermal analysis which does not require explicit construction of the equivalent circuit matrix. We propose specific multigrid treatments to cope with the strong anisotropy of the full-chip thermal problem that is created by the vast difference in material thermal properties and chip geometries. Importantly, we demonstrate that only with careful thermal modeling assumptions and appropriate choices for grid hierarchy, multigrid operators and smoothing steps across grid points, can we accurately and efficiently analyze a full-chip thermal problem. Experimental results demonstrate the efficacy of the proposed multigrid methodology. Our prototyped thermal simulator is able to solve a steady-state problem with more than 10 million unknowns in 125 CPU seconds with a peak memory usage of 231 mega bytes.