Houle Gan

Explicit Time-Domain Finite-Element Method Stabilized for an Arbitrarily Large Time Step

The root cause of the instability is quantitatively identified for the explicit time-domain finite-element method that employs a time step beyond that allowed by the stability criterion. With the identification of the root cause, an unconditionally stable explicit time-domain finite-element method is successfully created, which is stable and accurate for a time step solely determined by accuracy regardless of how large the time step is. The proposed method retains the strength of an explicit time-domain method in avoiding solving a matrix equation while eliminating its shortcoming in time step. Numerical experiments have demonstrated its superior performance in computational efficiency, as well as stability, compared with the conditionally stable explicit method and the unconditionally stable implicit method. The essential idea of the proposed method for making an explicit method stable for an arbitrarily large time step irrespective of space step is also applicable to other time domain methods.

A Time-Domain Layered Finite Element Reduction Recovery (LAFE-RR) Method for High-Frequency VLSI Design

A fast and high-capacity electromagnetic solution, time-domain layered finite element reduction recovery (LAFE-RR) method, is proposed for high-frequency modeling and simulation of large-scale on-chip circuits. This method rigorously reduces the matrix of a multilayer system to that of a single-layer system, regardless of the problem size. More importantly, the matrix reduction is achieved analytically, and hence the CPU and memory overheads are minimal. The recovery of solutions in all other layers involves only forward and backward substitution of matrices of single-layer size. The memory cost is also modest-requiring only the memory needed for the factorization of two sparse matrices of half-layer size. The superior performance applies to any arbitrarily shaped multilayer structure. Numerical and experimental results are presented to demonstrate the accuracy, efficiency, and capacity of the proposed method.

A Fast-Marching Time-Domain Layered Finite-Element Reduction-Recovery Method for High-Frequency VLSI Design

A fast-marching time-domain layered finite-element reduction-recovery (LAFE-RR) method is proposed for high-frequency modeling and simulation of large-scale integrated circuits. This method increases the time step of the LAFE-RR method by three orders of magnitude. In addition, it preserves the computational efficiency of the LAFE-RR method, i.e., the matrix reduction is achieved analytically from a three-dimensional layered system to a single-layer one regardless of the original problem size, and the sparsity of the reduced single-layer system matrix is the same as that of the original system matrix. The method applies to any arbitrarily-shaped multilayer structure. Numerical and experimental results are given to demonstrate its validity.

A fast and high-capacity electromagnetic solution for high- speed IC design

This paper proposes a fast and high-capacity electromagnetic solution, time-domain layered finite element reduction recovery (LAFE-RR) method, for high-frequency modeling and simulation of large-scale on-chip circuits. This method rigorously reduces the matrix of a multilayer system to that of a single-layer one regardless of the original problem size. More important, the matrix reduction is achieved analytically, and hence the CPU and memory overheads are minimal. The recovery of solutions in all other layers involves only forward and backward substitution of matrices of single-layer size. The memory cost is also modest-requiring only the memory needed for the factorization of a single layer sparse matrix. The improved performance applies to any arbitrarily shaped multilayer structure. Numerical and experimental results are presented to demonstrate the accuracy, efficiency, and capacity of the proposed method.

Hierarchical Finite-Element Reduction-Recovery Method for Large-Scale Transient Analysis of High-Speed Integrated Circuits

This paper proposes a hierarchical finite-element reduction-recovery method for large-scale transient analysis of high-speed integrated circuits. This method rigorously reduces the matrix of a multilayer system of O(N) to that of a single-cell system of O(1) regardless of the original problem size. More important, the matrix reduction is achieved analytically, and hence the CPU and memory overheads are minimal. In addition, the reduction preserves the sparsity of the original system matrix. As a result, the matrix factorization cost is reduced to O(1) by the proposed method. The CPU cost at each time step scales linearly with the number of unknowns. The method is applicable to any Manhattan-type integrated circuit embedded in layered dielectric media. Numerical and experimental results demonstrate the performance of the proposed method.

An unconditionally stable time-domain finite element method of significantly reduced computational complexity for large-scale simulation of IC and package problems

An unconditionally stable and computationally efficient time-domain finite-element method is developed to solve large-scale IC and package problems. In this method, an analytical expression of the time dependence is developed for the field unknowns inside conductors. And hence any time step can be used to stably solve the system of equations inside conductors. A matrix solution is involved in the analytical expression. It is efficiently obtained by the time-domain finite-element reduction-recovery method, the factorization cost of which is O(M), with M much less than the system matrix size N. The system of equations exterior to conductors is formed by a backward difference method, and hence is also unconditionally stable. The resultant system matrix is solved efficiently by an H-matrix based direct sparse solver, which is shown to outperform the state-of-the-art direct sparse solver. The system exterior to the conductors and that interior to the conductors are then solved by a staggered marching scheme, the convergence of which is theoretically proved. Applications to on-chip problems have demonstrated a time step that is three orders of magnitude larger than what is permitted by an explicit time-domain scheme, with fast CPU run time, modest memory consumption, and without sacrificing accuracy.

A Recovery Algorithm of Linear Complexity in the Time-Domain Layered Finite Element Reduction Recovery (LAFE-RR) Method for Large-Scale Electromagnetic Analysis of High-Speed ICs

Time-domain layered finite element reduction recovery (LAFE-RR) method was recently developed for large-scale electromagnetic analysis of high-speed integrated circuits (ICs). This method is capable of analytically and rigorously reducing the system matrix of a 3-D multilayer circuit to that of a single-layer one regardless of the original problem size. In addition, the reduced system matrix preserves the sparsity of the original system matrix. In this paper, an efficient algorithm is proposed to recover the volume unknowns in the time-domain LAFE-RR method. This algorithm constitutes a direct solution of the matrix formed by volume unknowns in each layer. This direct solution possesses a linear complexity in both central processing unit (CPU) time and memory consumption. The cost of matrix inversion is negligible. The cost of matrix solution scales linearly with the matrix size. Numerical and experimental results have demonstrated the accuracy and efficiency of the proposed algorithm.

A fast-marching time-domain layered finite-element reduction-recovery method for high-frequency VLSI design

In this paper, a fast-marching time-domain layered finite-element reduction-recovery (LAFE-RR) method is proposed for high-frequency modeling and simulation of large-scale integrated circuits. This method increases the time step of the LAFE-RR method by three orders of magnitude. In addition, it preserves the computational efficiency of the LAFE-RR method, i.e., the matrix reduction is achieved analytically from a three-dimensional layered system to a single-layer one regardless of the original problem size, and the sparsity of the reduced single-layer system matrix is the same as that of the original system matrix. The method applies to any arbitrarily-shaped multilayer structure. Numerical and experimental results are given to demonstrate its validity.

Hierarchical and adaptive finite-element reduction-recovery method for large-scale power and signal integrity analysis of high-speed IC and packaging structures

This paper proposes a hierarchical and adaptive finite-element reduction-recovery method for power and signal integrity analysis of high-speed IC and packaging structures. This method rigorously reduces the matrix of a multilayer system of O(N) to that of a single-cell one of O(1) regardless of the original problem size. More important, the matrix reduction is achieved analytically, and hence the CPU and memory overheads are minimal. In addition, the reduction preserves the sparsity of the original matrix. As a result, the matrix factorization cost of the proposed method is reduced to a constant. The CPU cost at each time step scales linearly with the number of unknowns to be recovered. Furthermore, an adaptive reduction-recovery scheme is developed to perform reduction and recovery in the active layers only, and hence further reducing the complexity of the proposed method. Numerical and experimental results demonstrate its performance.

A Recovery Algorithm of Linear Complexity in the Time-Domain Layered Finite Element Reduction Recovery (LAFE-RR) Method for Large Scale Electromagnetic Analysis of High-Speed ICs

Time-domain layered finite element reduction recovery (LAFE-RR) method was recently developed for large-scale electromagnetic analysis of high-speed integrated circuits (ICs). This method is capable of analytically and rigorously reducing the system matrix of a 3-D multilayer circuit to that of a single-layer one regardless of the original problem size. In addition, the reduced system matrix preserves the sparsity of the original system matrix. In this paper, an efficient algorithm is proposed to recover the volume unknowns in the time-domain LAFE-RR method. This algorithm constitutes a direct solution of the matrix formed by volume unknowns in each layer. This direct solution possesses a linear complexity in both central processing unit (CPU) time and memory consumption. The cost of matrix inversion is negligible. The cost of matrix solution scales linearly with the matrix size. Numerical and experimental results have demonstrated the accuracy and efficiency of the proposed algorithm.