2‐matrix‐based finite element linear solver for fast transient thermal analysis of high‐performance ICs

In this article, we propose ²‐based finite element (FE) solver for transient thermal analysis of high‐performance integrated circuits (ICs). ²‐matrix is a special subclass of hierarchical matrix or ‐matrix, which was shown to provide a data‐sparse way to approximate the matrices and their inverses with almost linear space and time complexities. In this work, we show that ²‐based mathematical framework can also be applied to FE‐based transient analysis of thermal parabolic partial differential equations. We show how the thermal matrix can be approximated by ²‐representations with controlled error. Then, we demonstrate that both storage and time complexities of the new solver are bounded by , where N is the matrix size. The method can be applied to any thermal structures for both steady and transient analysis. The numerical results from 3D ICs demonstrate the linear scalability of the proposed method in terms of both memory footprint and CPU time. The comparison with existing product‐quality LU solvers, CSPARSE and UMFPACK, on a number of 3D IC thermal matrices, shows that the new method is much more memory efficient than these methods, which however prevents the demonstration of the potential speedup of the proposed method over those methods. Copyright © 2014 John Wiley & Sons, Ltd.