Adaptive Transient Leakage-Aware Linearised Model for Thermal Analysis of 3-D ICs
Chao Zhanga, Milan Mihajlovićb and Vasilis F. Pavlidisc
School of Computer Science, The University of Manchester, Manchester, UK
aChao.Zhang@manchester.ac.uk
bmilan@cs.man.ac.uk
cpavlidis@cs.man.ac.uk
ABSTRACT
Physics-based models for thermal simulation that involve numerical solution of the heat equation are well placed to accurately capture the heterogeneity of materials and structures in modern 3-D integrated circuits (ICs). The introduction of nonlinear effects such as leakage power significantly improves their accuracy. However, this non-linearity increases considerably the complexity and computational time of the analysis. In this paper, we introduce a linearised thermal model by demonstrating that the weak temperature dependence of the specific heat and the thermal conductivity of IC related materials has only minor effect to computed temperature profiles. Thus, these parameters can be considered constant for the operating temperature ranges of modern ICs. The non-linearity in leakage power is approximated by a piecewise linear least square fit and the resulting model is linearised by exact Newton’s method. The method is applied to transient thermal analysis with adaptive time step selection, where we demonstrate the importance of applying Newton corrections to obtain the right time step size selection. The resulting method is typically 2-3× faster than a full non-linear method with a global relative error of less than 1%.
Keywords: Thermal analysis, Leakage power, Finite element method, Adaptive time integration.
