Dan Jiao

Time-domain finite-element simulation of three-dimensional scattering and radiation problems using perfectly matched layers

An effective algorithm to construct perfectly matched layers (PMLs) for truncating time-domain finite-element meshes used in the simulation of three-dimensional (3-D) open-region electromagnetic scattering and radiation problems is presented. Both total- and scattered-field formulations are described. The proposed algorithm is based on the time-domain finite-element solution of the vector wave equation in an anisotropic and dispersive medium. The algorithm allows for the variation of the PML parameters within each element, which facilitates the efficient use of higher order vector basis functions. The stability of the resultant numerical procedure is analyzed, and it is shown that unconditionally stable schemes can be obtained. Numerical simulations of radiation and scattering problems based on both the zeroth- and higher order vector bases are presented to validate the proposed PML scheme.

A general approach for the stability analysis of the time-domain finite-element method for electromagnetic simulations

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.

Time-domain finite-element modeling of dispersive media

A general formulation is developed to model the dispersion effect in the time-domain finite element method (TDFEM). This TDFEM is based on the second-order vector wave equation, in contrast to most FDTD schemes that solve the first-order Maxwell equations. The required convolution integral is evaluated recursively without the need to store the electric fields of all past time steps. This evaluation is made to be of second order in accuracy by adopting a linear interpolation for the fields within each time step. The proposed formulation is shown to be valid for plasma, Debye, and Lorentz media with a single or multiple poles. Three-dimensional numerical examples are given to demonstrate its efficacy.

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Using an H 2 matrix as the mathematical framework, we compactly represent a dense system matrix by a reduced set of parameters, thus enabling a significant reduction in computational complexity. The error bound of the H 2-matrix-based representation of an electrodynamic problem was derived. We show that exponential convergence with respect to the number of interpolation points can be achieved irrespective of the electric size. In addition, we show that a direct application of H 2-matrix-based techniques to electrodynamic problems would result in a complexity greater than O (N), with N being the matrix size, due to the need of increasing the rank when ascending an inverted tree in order to keep a constant order of accuracy. A rank function was hence developed to maintain the same order of accuracy in a wide range of electric sizes without compromising computational complexity. With this rank function, we demonstrate that given a range of electric sizes which lead to a range of N , the dense system of O (N 2) parameters can be compactly stored in O (N) units, and the dense matrix-vector multiplication can be performed in O (N) operations. Moreover, the same order of accuracy can be kept across this range. The method is kernel independent, and hence is suitable for any integral-equation-based formulation. In addition, it is applicable to arbitrary structures. Numerical experiments from small electric sizes to 64 wavelengths have demonstrated the performance of the proposed method.

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A time-domain, finite element-boundary integral (FE-BI) method is presented for analyzing electromagnetic (EM) scattering from two-dimensional (2-D) inhomogeneous objects. The schemes finite-element component expands transverse fields in terms of a pair of orthogonal vector basis functions and is coupled to its boundary integral component in such a way that the resultant finite element mass matrix is diagonal, and more importantly, the method delivers solutions that are free of spurious modes. The boundary integrals are computed using the multilevel plane-wave time-domain algorithm to enable the simulation of large-scale scattering phenomena. Numerical results demonstrate the capabilities and accuracy of the proposed hybrid scheme.

-Matrix-Based Integral-Equation Solver of Reduced Complexity and Controlled Accuracy for Solving Electrodynamic Problems

This paper presents an efficient method-of-moments solution of the mixed-potential integral equation for a general microstrip structure in multilayer media. In this method, the general forms of the spectral-domain Greens functions for multilayer media are derived first. The spatial-domain Greens functions are then obtained by the discrete complex-image method, which obviates the time-consuming numerical evaluation of the Sommerfeld integral. The Rao-Wilton-Glisson basis functions are employed to provide necessary flexibility to model arbitrary shapes. To expedite the computation of frequency response over a broad band, a reduced-order model is presented using asymptotic waveform evaluation. Numerical results of multilayer circuits and antennas are presented to show the efficiency and accuracy of this method.

A fast time-domain finite element-boundary integral method for electromagnetic analysis

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.

Efficient electromagnetic modeling of microstrip structures in multilayer media

State-of-the-art integral-equation-based solvers rely on techniques that can perform a matrix-vector multiplication in O(N) complexity. In this work, a fast inverse of linear complexity was developed to solve a dense system of linear equations directly for the capacitance extraction of any arbitrary shaped 3D structure. The proposed direct solver has demonstrated clear advantages over state-of-the-art solvers such as FastCap and HiCap; with fast CPU time and modest memory consumption, and without sacrificing accuracy. It successfully inverts a dense matrix that involves more than one million unknowns associated with a large-scale on-chip 3D interconnect embedded in inhomogeneous materials. Moreover, we have successfully applied the proposed solver to full-wave extraction.

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

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

A direct integral-equation solver of linear complexity for large-scale 3D capacitance and impedance extraction

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