Thermal Via Allocation for 3D ICs Considering Temporally and Spatially Variant Thermal Power

All existing methods for thermal-via allocation are based on a steady-state thermal analysis and may lead to excessive number of thermal vias. This paper develops an accurate and efficient thermal-via allocation considering temporally and spatially variant thermal-power. The transient temperature is calculated using macromodel by a structured and parameterized model reduction, which generates temperature sensitivity with respect to thermal-via via density. By defining a thermal-violation integral based on the transient temperature, a nonlinear optimization problem is formulated to allocate thermal-vias and minimize thermal violation integral. This optimization problem is transformed into a sequence of subproblems by Lagrangian relaxation, and each subproblem is solved by quadratic programming using sensitives from the macromodel. Experiments show that compared to the existing method using steady-state thermal analysis, our method is 126X faster to obtain the temperature profile, and reduces the number of thermal vias by 2.04X under the same temperature bound.