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 -, admittance -, and scattering -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.

No abstract

Linear complexity iterative and log-linear complexity direct solvers are developed for the volume integral equation (VIE) based general large-scale electrodynamic analysis. The dense VIE system matrix is first represented by a new clusterbased multilevel low-rank representation. In this representation, all the admissible blocks associated with a single cluster are grouped together and represented by a single low-rank block, whose rank is minimized based on prescribed accuracy. From such an initial representation, an efficient algorithm is developed to generate a minimal-rank H 2 -matrix representation. This representation facilitates faster computation, and ensures the same minimal rank's growth rate with electrical size as evaluated from singular value decomposition. Taking into account the rank's growth with electrical size, we develop linear-complexity H 2 -matrix-based storage and matrix-vector multiplication, and thereby an O(N ) iterative VIE solver regardless of electrical size. Moreover, we develop an O(N logN ) matrix inversion, and hence a fast O(N logN ) direct VIE solver for large-scale electrodynamic analysis. Both theoretical analysis and numerical simulations of large-scale 1-, 2-and 3-D structures on a singlecore CPU, resulting in millions of unknowns, have demonstrated the low complexity and superior performance of the proposed VIE electrodynamic solvers.

An -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.

No abstract