Saad Omar

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.

O(N) iterative and O(NlogN) direct volume integral equation solvers for large-scale electrodynamic analysis

State-of-the-art volume integral equation (VIE) solvers for solving electrically large problems are iterative solvers with the complexity of each matrix-vector multiplication being O(NlogN), where N is matrix size. In this work, we reduce this complexity to strictly O(N) irrespective of electrical size. Furthermore, we develop a fast inversion based direct VIE solver of O(NlogN) complexity, which is also independent of electrical size. Numerical experiments have demonstrated the complexity, accuracy, and efficiency of the proposed new VIE solvers. Very large-scale VIE system matrices involving millions of unknowns have been directly solved in fast CPU time and modest memory consumption on a single core running at 3 GHz.

A Linear Complexity Direct Volume Integral Equation Solver for Full-Wave 3-D Circuit Extraction in Inhomogeneous Materials

An H2-matrix based linear complexity direct matrix solution is developed for the volume integral equation (VIE) based broadband full-wave extraction of general 3-D circuits. Such circuits are in general electrically small or moderate, but contain arbitrarily shaped lossy conductors immersed in inhomogeneous dielectrics with ports located anywhere in the physical layout of the circuit. In the proposed direct solver, we first develop a well-conditioned VIE formulation without sacrificing the rigor and the advantages of the prevailing formulations. This formulation facilitates a robust direct solution of good accuracy even with a rank-1 representation. We then overcome the numerical challenge of solving the resultant highly unstructured system matrix mixed with both square and rectangular dense and sparse matrices by developing a fast linear complexity direct solution. This direct solution is capable of inverting dense matrices involving over 2 million unknowns in less than 1 h on a single CPU core running at 3 GHz. Numerical simulations of large-scale 3-D circuits and comparisons with state-of-the-art linear complexity iterative VIE solvers have demonstrated the accuracy, efficiency, and linear complexity of the proposed direct VIE solver.

A New Volume Integral Formulation for Broadband 3-D Circuit Extraction in Inhomogeneous Materials With and Without External Electromagnetic Fields

A new first-principles-based volume integral equation (VIE) formulation is developed for the broadband full-wave extraction of general 3-D circuits, containing arbitrarily shaped lossy conductors with inhomogeneous dielectrics. The proposed formulation accentuates all the advantages of the VIE formulation traditionally developed for solving wave-related problems, while allowing for the extraction of multiport circuit parameters such as impedance Z-, admittance Y-, and scattering S-parameters at ports located anywhere in the physical structure of a circuit. Its first-principles-based formulation without circuit-based simplifications and approximations can also be utilized to analyze the performance of a circuit in adverse ambient conditions, such as the exposure to strong external electromagnetic fields. In addition, the magneto-quasi-static and electro-magneto-quasi-static counterparts of the proposed full-wave formulation are also given for low-frequency applications. Numerical experiments have validated the accuracy and capability of the proposed new VIE formulation.

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.

A new volume integral equation formulation for analyzing 3-D circuits in inhomogeneous dielectrics exposed to external fields

A new volume integral equation formulation is developed for the full-wave extraction of 3-D circuits, containing arbitrarily shaped lossy conductors with inhomogeneous dielectrics, in the presence of external electromagnetic fields. It exploits all the flexibilities offered by the volume integral formulation traditionally developed for solving wave-related problems, while accommodating a circuit-source based excitation in order to model both the circuit excitation and the ambient environment where the circuit is exposed to. It facilitates circuit parameter extraction at ports located anywhere in the physical structure of a circuit. An excellent agreement of numerical results with reference data validates the proposed formulation.

An analytical approach to the low-frequency breakdown of the right hand side and scattered field computation in EFIE

In addition to the breakdown caused by the loss of the contribution of the vector potential term at low frequencies due to finite machine precision, the EFIE has also suffered from the low-frequency breakdown owing to the loss of the frequency dependence of the right hand side in scattering analysis, and the same loss of the Greens function in scattered field and thereby RCS computation. We propose an analytical approach to solve the latter two breakdown problems. This approach is rigorous from electrodynamic frequencies all the way down to zero frequency. Excellent agreement with analytical results and Mie series has validated the proposed approach.

A novel volume integral formulation for wideband impedance extraction of arbitrarily-shaped 3-D lossy conductors in multiple dielectrics

A novel impedance extraction formulation based on volume integral equation has been developed for modeling arbitrarily shaped 3-D lossy conductors in inhomogeneous dielectrics. It retains all the flexibilities offered by the volume integral formulation traditionally developed for solving wave-related problems, while accommodating a circuit-source based excitation inside a field based volume integral formulation. It facilitates circuit parameter extraction at ports located anywhere in the physical structure of a circuit. This formulation can also naturally revert to a static formulation for impedance extraction at low frequencies. An excellent agreement of numerical results with reference data validates the proposed formulation.

A new volume integral formulation for fullwave extraction of 3-D circuits in inhomogeneous dielectrics exposed to external fields

A new first-principles based volume integral equation (VIE) formulation is developed for the broadband fullwave extraction of 3-D circuits, containing arbitrarily shaped lossy conductors with inhomogeneous dielectrics. It accentuates all the advantages of the VIE formulation traditionally developed for solving wave-related problems, while facilitating circuit parameter extraction such as impedance (Z)- and Scattering (S)-parameter extraction at ports located anywhere in the physical structure of a circuit. Its conformance to the wave-based VIE can also be utilized to analyze the performance of circuits exposed to external fields. An excellent agreement of numerical results with reference data validates the proposed formulation.

Minimal-rank ℋ 2 -matrix-based iterative and direct volume integral equation solvers for large-scale scattering analysis

It can be shown that the matrix structure resulting from a fast multipole method (FMM)-based algorithm is an ℋ2-matrix, but with a full-rank representation for electrically large analysis. We compare the computational complexity of a volume integral equation (VIE) solver having a minimal-rank ℋ2-representation with that of a VIE solver using an FMM-based ℋ2-representation. The former is shown to be strict O(N) in storage and matrix-vector multiplication, and O(NlogN) in inverse irrespective of electric size. Such a complexity has been demonstrated by the analysis of large-scale dielectric scattering problems involving millions of unknowns.