Method of images for the fast calculation of temperature distributions in packaged VLSI chips

Thermal aware routing and placement algorithms are important in industry. Currently, there are reasonably fast Green\´s function based algorithms that calculate the temperature distribution in a chip made from a stack of different materials. However, the layers are all assumed to have the same size, thus neglecting the important fact that the thermal mounts which are placed underneath the chip can be significantly larger than the chip itself. In an earlier publication, we showed that the image blurring technique can be used to calculate quickly temperature distribution in realistic packages. For this method to be effective, temperature distribution for several point heat sources at the center and at the corner and edges of the chip should be calculated using finite element analysis (FEA) or measured. In addition, more accurate results require correction by a weighting function that will need several FEA simulations. In this paper, we introduce the convolution by images that take the symmetry of the thermal boundary conditions into account. Thus with only "two" finite element simulations, the steady-state temperature distribution for an arbitrary complex power dissipation profile in a packaged chip can be calculated. Several simulation results are presented. It is shown that the power blurring technique together with the method of images can reproduce the temperature profile with an error less than 0.5%.