Search

It has been observed that finite element based solutions of full-wave Maxwell's equations break down at low frequencies. In this paper, we present a theoretically rigorous method to fundamentally eliminate the low-frequency breakdown problem. The key idea of this method is that the original frequency-dependent deterministic problem can be rigorously solved from a generalized eigenvalue problem that is frequency independent. In addition, we found that the zero eigenvalues of the generalized eigenvalue problem cannot be obtained as zeros because of finite machine precision. We hence correct the inexact zero eigenvalues to be exact zeros. The validity and accuracy of the proposed method have been demonstrated by the analysis of both lossless and lossy problems having on-chip circuit dimensions from dc to high frequencies. The proposed method is applicable to any frequency. Hence it constitutes a universal solution of Maxwell's equations in a full electromagnetic spectrum. The proposed method can be used to not only fundamentally eliminate the low-frequency breakdown problem, but also benchmark the accuracy of existing electromagnetic solvers at low frequencies including static solvers. Such a benchmark does not exist yet because full-wave solvers break down while static solvers involve theoretical approximations.

A fast LU factorization of linear complexity is developed to directly solve a dense system of linear equations for the capacitance extraction of any arbitrary shaped 3-D structure embedded in inhomogeneous materials. In addition, a higher-order scheme is developed to achieve any higher-order accuracy for the proposed fast solver without sacrificing its linear computational complexity. The proposed solver successfully factorizes dense matrices that involve more than one million unknowns in fast CPU run time and modest memory consumption. Comparisons with state-of-the-art integral-equation-based capacitance solvers have demonstrated its clear advantages. In addition to capacitance extraction, the proposed LU solver has been successfully applied to large-scale full-wave extraction.

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.

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.

In this paper, fast algorithms are proposed for an efficient reduction of a 3-D layered system matrix to a 2-D layered one in the framework of the frequency-domain layered finite element method. These algorithms include: 1) an effective preconditioner P that can converge the iterative solution of the volume-unknown-based matrix equation in a few iterations; 2) a fast direct computation of P -1 in linear complexity in both CPU run time and memory consumption; and 3) a fast evaluation of P -1 b in linear complexity, with b being an arbitrary vector. With these fast algorithms, the volume-unknown-based matrix equation is solved in linear complexity with a small constant in front of the number of unknowns, and hence significantly reducing the complexity of the 3-D to 2-D reduction. The algorithms are rigorous without making any approximation. They apply to any arbitrarily-shaped multilayer structure. Numerical and experimental results are shown to demonstrate the accuracy and efficiency of the proposed algorithms.

It has been observed that a full-wave finite-element-based solution breaks down at low frequencies. This hinders its application to on-chip problems in which broadband modeling from direct current to microwave frequencies is required. Although a static formulation and a full-wave formulation can be stitched together to solve this problem, it is cumbersome to implement both static and full-wave solvers and make transitions between these two when necessary. In this work, a unified finite-element solution from zero frequency to microwave frequencies is developed for full-wave modeling of large-scale three-dimensional on-chip interconnect structures. In this solution, a single full-wave formulation is used. No switching to a static formulation is needed at low frequencies. This is achieved by first identifying the reason why a full-wave eigenvalue-based solution breaks down at low frequencies, and then developing an approach to eliminate the reason. The low frequency breakdown problem was found to be attributed to the discrepant frequency dependence of the real part and the imaginary part of the eigenvalues, which leads to an ill-conditioned eigenvalue system at low frequencies. The discrepant frequency dependence of the real part and the imaginary part is further attributed to the different scaling of the transverse and longitudinal fields with respect to frequency in a transmission-line type structure. By extracting transverse and longitudinal fields separately in the framework of a full-wave formulation, we avoid the numerical difficulty of solving an ill-conditioned eigen-system at low frequencies. The validity of the proposed approach is demonstrated by numerical and experimental results.

This paper presents an efficient surface-based finite-element method for the full-wave characterization of high-density and multiterminal decoupling capacitors (2, 8, 14, and any arbitrary number of terminals). In contrast to traditional finite-element methods that involve 3-D volumetric unknowns, this method reduces the unknowns one needs to solve to those on 2-D surfaces only. In addition, the reduction from the 3-D volume-based matrix to a 2-D surface-based one is achieved efficiently by exploiting the geometrical specialty of the decap structure. The entire numerical procedure is numerically rigorous without making any approximation. Its efficiency and accuracy have been demonstrated by both measurements and numerical experiments. Based on its fast and accurate solution, different design configurations of capacitors were studied to identify the optimal configuration that can maximize the performance of a decoupling capacitor. Furthermore, the full-wave model obtained from the proposed method was employed to assess the accuracy of conventional series lumped RLC capacitor models. In addition, the full-wave model was incorporated into a high-performance microprocessor's power delivery network to investigate system performance.

Since the design of advanced microprocessors is based on simulation tools, accurate assessments of the amount of crosstalk noise are of paramount importance to avoid logic failures and less-than-optimal designs. With increasing clock frequencies, inductive effects become more important, and the validity of assumptions commonly used in simulation tools and approaches is unclear. We compared accurate experimental S-parameters with results derived from both magneto-quasi-static and full-wave simulation tools for simple crosstalk structures with various capacitive and inductive couplings, in the presence of parallel and orthogonal conductors. Our validation approach made possible the identification of the strengths and weaknesses of both tools as a function of frequency, which provides useful guidance to designers who have to balance the tradeoffs between accuracy and computation expenses for a large variety of cases