Research And Application Of Adaptive Time Step Finite Element Method Based On Domain Decomposition In Thermal Simulation Of Chips

Modern integrated circuits are developing in the direction of ultra-large scale,the feature size of the chip is constantly approaching the theoretical limit,and the number of stacked layers is also rising.Due to the limited size of the chip,the heat production rate per unit area of the chip is constantly increasing,and the operating temperature rises.Excessive chip temperature will reduce the reliability of the system and also reduce the service life of the chip.The heat dissipation problem of the chip has become one of the current difficulties,and it needs to be solved urgently as the performance of hardware equipment improves.Therefore,it is very meaningful to quickly and eficiently model and simulate the chip to guide the heat dissipation design of the chip.At present,most of the software on the market for the electromagnetic field and thermal field simulation of the chip is modeled and solved by the finite element method.But three-dimensional integrated circuit(IC)model modeling will appear multi-scale phenomenon,such as through silicon via technology(TSV)insulating layer,N channel or P channel,the spacing generally differs by many orders of magnitude relative to the size of the silicon-based substrate.The multi-scale problem will bring too much degrees of freedom of the grid when modeling and simulating the chip,resulting in a huge amount of calculation.Therefore,the simulation can only be performed on a part of the chip,and an equivalent simplified structure should be adopted at the same time.Nevertheless,in the local simulation,it is still necessary to continuously improve the calculation efficiency of the algorithm and reduce the calculation time.Domain Decomposition Method(DDM)can achieve the stability of the solution under incompatible grid conditions,and is more suitable for multi-physics coupling calculation under multi-scale conditions.This article mainly focuses on how to quickly calculate the thermal analysis of the modeling chip.The main contents of the work are:(1)The transmission conditions based on the internal penalty method are realized.This method can meet the temperature and heat flux when dealing with discontinuous grid conditions.The continuity of flux between incompatible grid areas.(2)In the sub-domain matrix formed by the nonconforming mesh,the calculation decoupling between different sub-domains is realized without forming the sparse matrix of the entire system,so the calculation time can be faster.(3)Dynamically control the time step changes according to the temperature error of the correction term and the prediction term.The temperature gradient at the initial moment should be larger compare the end time.When the temperature is close to stable,the iteration step size should be selected larger.Under the condition of satisfying the relative error,the adaptive step size can reduce the calculation step size and save the calculation time.Finally,by studying system-level simulation cases of multiple microelectronic devices,this paper verifies the numerical accuracy,computational efficiency and stability of this method under common boundary conditions such as constant temperature and convection boundary.So as to determine the reliability of the numerical results and make a detailed analysis of the operating errors.