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We present a fast time-domain multiphysics simulation framework that combines full-wave electromagnetism (EM) and carrier transport in semiconductor devices (TCAD). The proposed framework features a division of linear and nonlinear components in the EM-TCAD coupled system. The former is extracted and handled independently with high efficiency by a matrix exponential approach assisted with Krylov subspace method. The latter is treated by ordinary Newton's method yet with a much sparser Jacobian matrix that leads to substantial speedup in solving the linear system of equations. More convenient error management and adaptive control are also available through the linear and nonlinear decoupling.

We propose an advanced matrix exponential method (MEXP) to handle the transient simulation of stiff circuits and enable parallel simulation. We analyze the rapid decaying of fast transition elements in Krylov subspace approximation of matrix exponential and leverage such scaling effect to leap larger steps in the later stage of time marching. Moreover, matrix-vector multiplication and restarting scheme in our method provide better scalability and parallelizability than implicit methods. The performance of ordinary MEXP can be improved up to 4.8 times for stiff cases, and the parallel implementation leads to another 11 times speedup. Our approach is demonstrated to be a viable tool for ultra-large circuit simulations (with 1.6M ∼ 12M nodes) that are not feasible with existing implicit methods.

One well-adopted power grid simulation methodology is to factorize matrix once and perform only backward/forward substitution with a deliberately chosen step size along the simulation. Since the required simulation time is usually long for the power grid design, the costly factorization is amortized. However, such fixed step size cannot exploit larger step size for the low frequency response in the power grid to speedup the simulation. In this work, we utilize the matrix exponential method with the rational Krylov subspace approximation to enable adaptive step size in the power grid simulation. The kernel operation in our method only demands one factorization and backward/forward substitutions. Moreover, the rational Krylov subspace approximation can relax the stiffness constraint of the previous works [12] [13]. The cheap computation of adaptivity in our method could exploit the long low-frequency response in a power grid and significantly accelerate the simulation. The experimental results show that our method achieves up to 18X speedup over the trapezoidal method with fixed step size.

We proposed MATEX, a distributed framework for transient simulation of power distribution networks (PDNs). MATEX utilizes matrix exponential kernel with Krylov subspace approximations to solve differential equations of linear circuit. First, the whole simulation task is divided into subtasks based on decompositions of current sources, in order to reduce the computational overheads. Then these subtasks are distributed to different computing nodes and processed in parallel. Within each node, after the matrix factorization at the beginning of simulation, the adaptive time stepping solver is performed without extra matrix re-factorizations. MATEX overcomes the stiffness hinder of previous matrix exponential-based circuit simulator by rational Krylov subspace method, which leads to larger step sizes with smaller dimensions of Krylov subspace bases and highly accelerates the whole computation. MATEX outperforms both traditional fixed and adaptive time stepping methods, e.g., achieving around 13X over the trapezoidal framework with fixed time step for the IBM power grid benchmarks.

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