Localized relaxation theory of circuits and its applications in electro-thermal analyses

In the high-performance IC design with increasing design complexity, it is a very important design content to efficiently analyze IC parameters. Thus, the electro-thermal (ET) analyses including power/ground (P/G) analysis and thermal analysis are hot topics in today’s IC research. Since ET analysis equation has a sparse, positive definite and strictly diagonally dominant coefficient-matrix, we prove that the ET analysis has the advantage of locality. Owing to this advantage, localized relaxation method is formally proposed, which has the same accuracy as the global relaxation done with the constraint of the same truncation error limitation. Based on the localized relaxation theory, an efficient and practical localized successive over-relaxation algorithm (LSOR2) is introduced and applied to solve the following three ET analysis problems. (1) Single-node statistical voltage analysis for over-IR-drop nodes in P/G networks; (2) single-node statistical temperature analysis for hot spots in 3D thermal analysis; (3) fast single open-defect analysis for P/G networks. A large amount of experimental data demonstrates that compared with the global successive over-relaxation (SOR) algorithm, LSOR2 can speed up 1–2 orders of magnitudes with the same accuracy in ET analyses.