Search

Fast algorithms are developed in this work for solving the system matrix resulting from a frequency-domain layered finite element based analysis of integrated circuits. The frequency-domain layered finite element method represents a 3-D layered system by a 2-D layered system, and further by a single-layered one. The reduced system matrix is generally denser than the original sparse matrix. In this paper, we show that 1) the dense matrix-vector multiplication can be performed in linear complexity; in addition, the reduction cost can be bypassed, 2) an effective preconditioner can be developed to converge the iterative solution of the reduced system matrix in a small number of iterations, and 3) the preconditioner can be solved in linear complexity. As a result, the reduced system matrix can be solved efficiently. The algorithms are rigorous without making any approximation. They apply to any arbitrarily-shaped multilayer structure. Numerical results demonstrated the accuracy, effectiveness, and efficiency of the proposed algorithms in analyzing on-chip circuits. [ABSTRACT FROM AUTHOR]

A time-domain orthogonal finite-element reduction-recovery method is developed to overcome the large problem sizes encountered in the simulation of large-scale integrated-circuit and package problems. In this method, a set of orthogonal prism vector basis functions is developed. Based on this set of bases, an arbitrary 3-D multilayered system such as a combined package and die is reduced to a single-layer system with negligible computational cost. More importantly, the reduced single-layer system is diagonal and, hence, can be solved readily. From the solution of the reduced system, the solution of the other unknowns is recovered in linear complexity. The method entails no theoretical approximation. It applies to any arbitrarily shaped multilayer structure involving inhomogeneous materials or any structure that can be geometrically modeled by triangular prism elements. In addition, it permits nonlinear device modeling and broadband simulation within one run. Numerical and experimental results have demonstrated its accuracy and high capacity in simulating on-chip, package, and die-package interface problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A high-capacity electromagnetic solution, layered finite element method, is proposed for high-frequency modeling of large-scale three-dimensional on-chip circuits. In this method, first, the matrix system of the original 3-D problem is reduced to that of 2-D layers. Second, the matrix system of 2-D layers is further reduced to that of a single layer. Third, an algorithm of logarithmic complexity is proposed to further speed up the analysis. In addition, an excitation and extraction technique is developed to limit the field unknowns needed for the final circuit extraction to a single layer only, as well as keep the right-hand side intact during the matrix reduction process. The entire procedure is numerically rigorous without making any theoretical approximation. The computational complexity only involves solving a single layer irrespective of the original problem size. Hence, the proposed method is equipped with a high capacity to solve large-scale IC problems. The proposed method was used to simulate a set of large-scale interconnect structures that were fabricated on a test chip using conventional Si processing techniques. Excellent agreement with the measured data has been observed from dc to 50 GHz. [ABSTRACT FROM AUTHOR]

This paper presents a general approach for the stability analysis of the time-domain finite-element method (TDFEM) for electromagnetic simulations. Derived from the discrete system analysis, the approach determines the stability by analyzing the root-locus map of a characteristic equation and evaluating the spectral radius of the finite element system matrix. The approach is applicable to the TDFEM simulation involving dispersive media and to various temporal discretization schemes such as the central difference, forward difference, backward difference, and Newmark methods. It is shown that the stability of the TDFEM is determined by the material property and by the temporal and spatial discretization schemes. The proposed approach is applied to a variety of TDFEM schemes, which include: 1) time-domain finite-element modeling of dispersive media; 2) time-domain finite element-boundary integral method; 3) higher order TDFEM; and 4) orthogonal TDFEM. Numerical results demonstrate the validity of the proposed approach for stability analysis. [ABSTRACT FROM AUTHOR]

An algorithm is presented to implement perfectly matched layers (PMLs) for the time-domain finite-element (TDFE) simulation of two-dimensional open-region electromagnetic scattering and radiation problems. The proposed algorithm is based on the TDFE solution of a special vector wave equation similar to the one in an anisotropic and dispersive medium. The impact of the PML on the stability of the resultant TDFE solution is studied for a variety of temporal discretization schemes, and it is shown that the proposed algorithm for implementing PML can support unconditionally stable TDFE schemes. Both the totaland the scattered-field formulations are described, and numerical simulations of radiation and scattering problems are presented to validate the proposed PML algorithm for the mesh truncation of the TDFE solution. [ABSTRACT FROM AUTHOR]

A novel hybrid time-domain finite element-boundary integral method for analyzing three-dimensional (3-D) electromagnetic scattering phenomena is presented. The method couples finite element and boundary integral field representations in a way that results in a sparse system matrix and solutions that are devoid of spurious modes. To accurately represent the unknown fields, the scheme employs higher-order vector basis functions defined on curvilinear tetrahedral elements. To handle problems involving electrically large objects, the multilevel plane-wave time-domain algorithm is used to accelerate the evaluation of the boundary integrals. Numerical results demonstrate the accuracy and versatility of the proposed scheme. [ABSTRACT FROM AUTHOR]