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

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

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

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

Bis(μ-2-methyl­quinolin-8-olato)-κ3 N,O:O;κ3 O:N,O-bis­[(acetato-κ2 O,O′)lead(II)]

Both independent PbII atoms in the title compound, [Pb2(C10H8NO)2(C2H3O2)2], are chelated by acetate and substituted quinolin-8-olate anions; the O atoms of the quinolin-8-olates also bridge to confer a five-coordinate status to each metal center. The geometry approximates a distorted Ψ-fac octa­hedron in which one of the sites is occupied by a stereochemically active lone pair

Bis(μ-5-chloro­quinolin-8-olato)-κ3 N,O:O;κ3 O:N,O-bis­[(acetato-κ2 O,O′)lead(II)]

The mol­ecule of the title compound, [Pb2(C9H5ClNO)2(C2H3O2)2], lies about a center of inversion. The PbII atom is chelated by acetate and substituted quinolin-8-olate anions; the O atoms of the quinolin-8-olates also bridge to confer a five-coordinate status to each metal center. The geometry approximates a distorted Ψ-fac octa­hedron in which one of the sites is occupied by a stereochemically active lone pair

A fast time-domain EM-TCAD coupled simulation framework via matrix exponential

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. © 2012 ACM.published_or_final_versio

Bis(μ-2-methyl­quinolin-8-olato)-κ3 N,O:O;κ3 O:N,O-bis­[(methanol-κO)(nitrato-κ2 O,O′)lead(II)]

The mol­ecule of the title compound, [Pb2(C10H8NO)2(NO3)2(CH3OH)2], lies about a centre of inversion. The Pb atom is chelated by nitrate and substituted quinolin-8-olate anions. The O atom of the quinolin-8-olate also bridges, to confer a six-coordinate status on the metal centre. When a longer Pb⋯O inter­action is considered, the geometry approximates a Ψ-cube in which one of the sites is occupied by a stereochemically active lone pair