Fast nonlinear dynamic compact thermal modeling with multiple heat sources in Ultra-Thin Chip Stacking Technology

This paper presents a model order reduction technique to extract nonlinear dynamic compact thermal models (DCTMs) with multiple (>1) heat sources. The approach leads to nonlinear DCTMs of small state-space dimensions that allow accurately reconstructing the whole temperature and heat flux fields induced by arbitrary power waveforms in complex structures composed by many materials with thermal conductivities showing different values and temperature dependences. The DCTM extraction requires a small fraction of the CPU time needed for the simulation of a power step response using the finite element method (FEM), while ensuring a very high level of precision. Therefore, the adoption of the proposed strategy is a viable alternative to FEM for performing efficient and accurate nonlinear transient analyses: the advantage of using a DCTM is magnified if many power input waveforms have to be considered for the same structure, since any further simulation of the extracted DCTM is nearly instantaneous. As a case study, the proposed technique is used to investigate the impact of two cooling solutions on the thermal behavior of a 3-die module in ultra-thin chip stacking technology.