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

The existing 3-D thermal-via allocation methods are based on the steady-state thermal analysis and may lead to excessive number of thermal vias. This paper develops an accurate and efficient thermal-via allocation considering the temporally and spatially variant thermal-power. The transient temperature is calculated by macromodel with a one-time structured and parameterized model reduction, which also generates temperature sensitivity with respect to thermal-via density. The proposed thermal-via allocation minimizes the time-integral of temperature violation, and is solved by a sequential quadratic programming algorithm with use of sensitivities from the macromodel. Compared to the existing method using the steady-state thermal analysis, our method in experiments is 126$\times$ faster to obtain temperature, and reduces the number of thermal vias by 2.04$\times$ under the same temperature bound.