Jianfang Zhu

A Theoretically Rigorous Full-Wave Finite-Element-Based Solution of Maxwell's Equations From dc to High Frequencies

It has been observed that finite element based solutions of full-wave Maxwells 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 Maxwells 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 Unified Finite-Element Solution From Zero Frequency to Microwave Frequencies for Full-Wave Modeling of Large-Scale Three-Dimensional On-Chip Interconnect Structures

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 DC 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. Numerical and experimental results demonstrate its validity.

Solution of the Electric Field Integral Equation When It Breaks Down

With a method developed in this work, we find the solution of the original electric field integral equation (EFIE) at an arbitrary frequency where the EFIE breaks down due to low frequencies and/or dense discretizations. This solution is equally rigorous at frequencies where the EFIE does not break down and is independent of the basis functions used. We also demonstrate, both theoretically and numerically, the fact that although the problem is commonly termed low-frequency breakdown, the solution at the EFIE breakdown can be dominated by fullwave effects instead of just static or quasi-static physics. The accuracy and efficiency of the proposed method is demonstrated by numerical experiments involving inductance, capacitance, RCS extraction, and a multiscale example with a seven-orders-of-magnitude ratio in geometrical scales, at all breakdown frequencies of an EFIE. In addition to the EFIE, the proposed method is also applicable to other integral equations and numerical methods for solving Maxwells equations.

Fast Full-Wave Solution That Eliminates the Low-Frequency Breakdown Problem in a Reduced System of Order One

Full-wave solutions of Maxwells equations break down at low frequencies. Existing methods for solving this problem either are inaccurate or incur additional computational cost. In this paper, a fast full-wave finite-element-based solution is developed to eliminate the low-frequency breakdown problem in a reduced system of order one. It is applicable to general 3-D problems involving ideal conductors as well as nonideal conductors immersed in inhomogeneous, lossless, lossy, and dispersive materials. The proposed method retains the rigor of a theoretically rigorous full-wave solution recently developed for solving the low-frequency breakdown problem, while eliminating the need for an eigenvalue solution. Instead of introducing additional computational cost to fix the low-frequency breakdown problem, the proposed method significantly speeds up the low-frequency computation.

A Fast Frequency-Domain Eigenvalue-Based Approach to Full-Wave Modeling of Large-Scale Three-Dimensional On-Chip Interconnect Structures

As on-chip circuits have scaled into the deep submicron regime, electromagnetics-based analysis has increasingly become essential for high-performance integrated circuit (IC) design. Not only fast, but also high-capacity electromagnetic solutions are demanded to overcome the large problem size facing on-chip design community. In this paper, we present a novel, high-capacity, and fast approach to the full-wave modeling of 3-D on-chip interconnect structures. In this approach, the interconnect structure is decomposed into a number of seeds. In each seed, the original wave propagation problem is represented as a generalized eigenvalue problem. The resulting eigenvalue representation can comprehend both conductor and dielectric losses, arbitrary dielectric and conductor configuration in the transverse cross section, and arbitrary material. A new mode-matching technique applicable to on-chip interconnects is developed to solve large-scale 3-D problems by using 2-D-like CPU time and memory. A junction matrix acceleration technique is proposed to speed up the mode matching process. A fast frequency sweep technique is employed to obtain the response over the entire frequency band by solving at one or a few frequency points only. An extraction technique is developed to obtain S-parameters from the solution of the eigenvalue system. The entire procedure is numerically rigorous without making any theoretical approximation. Experimental and numerical results demonstrate its accuracy and efficiency.

A rigorous solution to the low-frequency breakdown in the electric field integral equation

The low-frequency breakdown problem in electric field integral equation (EFIE) is well recognized and has been extensively studied. However, existing approaches have not rigorously solved the problem yet since they rely on low-frequency approximations. In this work, we present a rigorous method to fundamentally eliminate the problem. In this method, the original frequency dependent problem is rigorously transformed to a generalized eigenvalue problem, from the solution of which the frequency dependence of the EFIE solution can be analytically derived. The rigor of the proposed method has been validated at frequencies as low as DC. As the first rigorous solution to EFIE at low frequencies, the proposed method can be used to benchmark the accuracy of existing low-frequency EFIE-based solvers, quantitatively answer critical design questions such as at which frequency full-wave effects become important, etc.

Minimal-order circuit model based fast electromagnetic simulation

State-of-the-art electromagnetics-based circuit simulators perform circuit simulation on the physical layout of an integrated circuit, the size of which can be large even for an optimal-complexity simulator to analyze it efficiently. We find a frequency- and time-independent minimal-order model of the integrated circuit from DC to high frequencies for any prescribed accuracy based on electromagnetics. The model size is only two for RC circuits and minimal such as tens for RLC and fullwave circuits, regardless of the original circuit size and the number of input/output ports. Numerical experiments demonstrate the superior performance of the proposed minimalorder model based fast electromagnetic simulation.

A rigorous solution to the low-frequency breakdown in full-wave finite-element-based analysis of general problems involving inhomogeneous lossy dielectrics and non-ideal conductors

State-of-the-art methods for solving the low-frequency breakdown problem of full-wave solvers rely on low-frequency approximations, the accuracy of which is a great concern. A rigorous method is developed in this work to fundamentally eliminate the low frequency breakdown problem for full-wave finite-element based analysis of general 3-D problems involving inhomogeneous lossy dielectrics and non-ideal conductors. In this method, the frequency dependence of the solution to Maxwells equations is explicitly derived from DC to any high frequency. The rigor of the proposed method has been validated by the analysis of realistic on-chip circuits at frequencies as low as DC. Moreover, the proposed method is applicable to both low and high frequencies, and hence constituting a universal solution to Maxwells equations in a full electromagnetic spectrum.

A theoretically rigorous solution for fundamentally eliminating the low-frequency breakdown problem in finite-element-based full-wave analysis

It has been observed that a full-wave based solution of Maxwells equations breaks down at low frequencies. Such a problem is especially severe in VLSI circuit applications because the breakdown frequency is in the range of circuit operating frequencies. To eliminate the low-frequency breakdown problem, first, one has to know the exact solution of Maxwells equations at low frequencies. However, such a benchmark solution does not exist yet because full-wave solvers break down at low frequencies whereas static solvers involve theoretical approximations. The theoretical approximation involved in static solvers is the assumption that E and H are decoupled at low frequencies. Such an assumption is not rigorous because in Maxwells equations, E and H are always coupled as long as frequency is not zero. Existing solutions to overcoming the low-frequency breakdown problem, in general, rely on static approximations, and hence are not theoretically rigorous. For example, the loop-tree and loop-star basis functions were used to achieve a natural Helmholtz decomposition of the current to overcome the low-frequency breakdown problem in integral-equation-based methods [1]. As another example, the tree-cotree splitting [2–3] was used to provide an approximate Helmholtz decomposition for edge elements in finite-element-based methods. In addition, even by using static approximations, the low-frequency breakdown problem has not been fundamentally solved.

A rigorous method for fundamentally eliminating the low-frequency breakdown in full-wave finite-element-based analysis: Combined dielectric-conductor case

It has been observed that a finite-element based solution of full-wave Maxwells equations breaks down at low frequencies. Existing approaches have not rigorously solved the problem yet since they rely on low-frequency approximations. Moreover, little work has been reported for overcoming the low-frequency breakdown for realistic circuit problems in which dielectrics and non-ideal conductors coexist. In this work, we develop a rigorous method to fundamentally eliminate the low-frequency breakdown for the analysis of general problems involving both dielectrics and conductors. Its rigor has been validated by the analysis of realistic on-chip VLSI circuits at frequencies as low as DC. Furthermore, the proposed method is applicable to any frequency, hence constituting a universal solution of Maxwells equations in a full electromagnetic spectrum. In addition, given an arbitrary integrated circuit and package structure, the proposed method can be used to quantitatively and rigorously answer critical design questions such as at which frequency full-wave effects become important and etc.