Time-domain analysis of large-scale circuits by matrix exponential method with adaptive control

We propose an explicit numerical integration method based on matrix exponential operator for transient analysis of large-scale circuits. Solving the differential equation analytically, the limiting factor of maximum time step changes largely from the stability and Taylor truncation error to the error in computing the matrix exponential operator. We utilize Krylov subspace projection to reduce the computation complexity of matrix exponential operator. We also devise a prediction-correction scheme tailored for the matrix exponential approach to dynamically adjust the step size and the order of Krylov subspace approximation. Numerical experiments show the advantages of the proposed method compared with the implicit trapezoidal method. © 1982-2012 IEEE.published_or_final_versio

An Algorithmic Framework for Efficient Large-Scale Circuit Simulation Using Exponential Integrators

We propose an efficient algorithmic framework for time domain circuit simulation using exponential integrator. This work addresses several critical issues exposed by previous matrix exponential based circuit simulation research, and makes it capable of simulating stiff nonlinear circuit system at a large scale. In this framework, the system's nonlinearity is treated with exponential Rosenbrock-Euler formulation. The matrix exponential and vector product is computed using invert Krylov subspace method. Our proposed method has several distinguished advantages over conventional formulations (e.g., the well-known backward Euler with Newton-Raphson method). The matrix factorization is performed only for the conductance/resistance matrix G, without being performed for the combinations of the capacitance/inductance matrix C and matrix G, which are used in traditional implicit formulations. Furthermore, due to the explicit nature of our formulation, we do not need to repeat LU decompositions when adjusting the length of time steps for error controls. Our algorithm is better suited to solving tightly coupled post-layout circuits in the pursuit for full-chip simulation. Our experimental results validate the advantages of our framework.Comment: 6 pages; ACM/IEEE DAC 201

MATEX: A Distributed Framework for Transient Simulation of Power Distribution Networks

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 stiff-ness 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.Comment: ACM/IEEE DAC 2014. arXiv admin note: substantial text overlap with arXiv:1505.0669

Globally stable, highly parallelizable fast transient circuit simulation via faber series

Time-domain circuit simulation based on matrix exponential has attracted renewed interested, owing to its explicit nature and global stability that enable millionth-order circuit simulation. The matrix exponential is commonly computed by Krylov subspace methods, which become inefficient when the circuit is stiff, namely, when the time constants of the circuit differ by several orders. In this paper, we utilize the truncated Faber Series for accurate evaluation of the matrix exponential even under a highly stiff system matrix arising from practical circuits. Experiments have shown that the proposed approach is globally stable, highly accurate and parallelizable, and avoids excessive memory storage demanded by Krylov subspace methods. © 2012 IEEE.published_or_final_versio

Circuit simulation via matrix exponential method for stiffness handling and parallel processing

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. © 2012 ACM.published_or_final_versio

A practical regularization technique for modified nodal analysis in large-scale time-domain circuit simulation

Fast full-chip time-domain simulation calls for advanced numerical integration techniques with capability to handle the systems with (tens of) millions of variables resulting from the modified nodal analysis (MNA). General MNA formulation, however, leads to a differential algebraic equation (DAE) system with singular coefficient matrix, for which most of explicit methods, which usually offer better scalability than implicit methods, are not readily available. In this paper, we develop a practical two-stage strategy to remove the singularity in MNA equations of large-scale circuit networks. A topological index reduction is first applied to reduce the DAE index of the MNA equation to one. The index-1 system is then fed into a systematic process to eliminate excess variables in one run, which leads to a nonsingular system. The whole regularization process is devised with emphasis on exact equivalence, low complexity, and sparsity preservation, and is thus well suited to handle extremely large circuits. © 2012 IEEE.published_or_final_versio

Métodos de diferenciación exponencial en el dominio del tiempo para el análisis de estabilidad angular

Dada la operación y expansión de los Sistemas Eléctricos de Potencia (SEP), en la actualidad se ha aumentado su vulnerabilidad a posibles fallas. Las fallas son imposibles de evitar, no obstante, es posible prever el comportamiento del sistema ante este tipo de eventos y observar su capacidad de permanecer cerca de un punto de estado estacionario después de su ocurrencia. Esto es posible mediante Estudios de Estabilidad Transitoria (EET), lo cual hace necesario describir el SEP mediante modelos dinámicos expresados por Ecuaciones Diferenciales Ordinarias (EDO). Las EDO se resuelven utilizando métodos de integración numérica. Seleccionar el algoritmo de integración numérica, su orden y su tamaño de paso a utilizar afecta de gran manera el tiempo computacional empleado al resolver las EDO. El esfuerzo computacional, la precisión de los resultados obtenidos y la estabilidad numérica son requerimientos contradictorios y en algunos casos son manejados a conveniencia del estudio realizado. Dado que disminuir el tiempo computacional sin afectar la calidad de la solución sigue siendo un tema relevante en el EET, este trabajo presenta un esquema de integración numérica implementando una familia de Métodos de Diferenciación Exponencial en el Dominio del Tiempo para el análisis de estabilidad transitoria. El esquema propuesto es validado en el SEP de prueba IEEE 39 con un algoritmo de integración numérica clásico, como es el Runge-Kutta 4. Los resultados muestran una mejora en la precisión de los resultados obtenidos con respecto a los obtenidos por el método clásico cuando son utilizados grandes pasos de integración, indicando un gran potencial para aplicaciones en SEP reales