Direct finite-element-based solver for 3D-IC thermal analysis via H-matrix representation

We propose, for the first time, the use of hierarchical matrix (H-matrix) in the efficient finite-element-based (FE-based) direct solver implementation for both steady and transient thermal analyses of three-dimensional integrated circuits (3D ICs). H-matrix was shown to provide a data-sparse way to approximate the matrices and their inverses with almost linear space and time complexities. We show this is also true for FE-based transient analysis of thermal parabolic partial differential equations (PDEs). Specifically, we show that the stiffness matrix from a FE-based steady and transient thermal analysis can be represented by H-matrix without any approximation, and its inverse and Cholesky factors can be evaluated by H-matrix with controlled accuracy. We then show that the memory and time complexities of the solver are bounded by O(k₁ N log N) and O(k₁²N log² N), respectively, for very large scale thermal systems, where k is a small quantity determined by accuracy requirements and N is the number of unknowns in the system. Numerical results validate and demonstrate the effectiveness of the proposed method in terms of predicted theoretical scalability.