Dipanjan Gope

Efficient solution of EFIE via low-rank compression of multilevel predetermined interactions

This paper describes the predetermined interaction list oct-tree (PILOT) algorithm and its application in expediting the solution of full-wave electric field integral equation (EFIE)-based scattering problems for three-dimensional arbitrarily shaped conductors. PILOT combines features of the fast multipole method (FMM) and QR decomposition-based matrix compression techniques to optimize setup times, solve times, and memory requirements. The method is kernel independent and stable for electrically small structures unlike traditional FMM. The novel features of the algorithm, namely the mixed potential compression scheme and the hierarchical multilevel predetermined matrix structure are explained in detail. A complexity estimate is presented to demonstrate the scaling in time and memory requirements. Examples exhibiting the accuracy and the time and memory performances are also presented. Finally, a quantitative study is included to address the expected but gradual degradation of QR-based compression techniques for electrically large structures.

Solving Low-Frequency EM-CKT Problems Using the PEEC Method

The partial element equivalent circuit (PEEC) formulation is an integral equation based approach for the solution of combined electromagnetic and circuit (EM-CKT) problems. In this paper, the low-frequency behavior of the PEEC matrix is investigated. Traditional EM solution methods, like the method of moments, suffer from singularity of the system matrix due to the decoupling of the charge and currents at low frequencies. Remedial techniques for this problem, like loop-star decomposition, require detection of loops and therefore present a complicated problem with nonlinear time scaling for practical geometries with holes and handles. Furthermore, for an adaptive mesh of an electrically large structure, the low-frequency problem may still occur at certain finely meshed regions. A widespread application of loop-star basis functions for the entire mesh is counterproductive to the matrix conditioning. Therefore, it is necessary to preidentify regions of low-frequency ill conditioning, which in itself represents a complex problem. In contrast, the charge and current basis functions are separated in the PEEC formulation and the system matrix is formulated accordingly. The incorporation of the resistive loss (R) for conductors and dielectric loss (G) for the surrounding medium leads to better system matrix conditioning throughout the entire frequency spectrum, and it also leads to a clean dc solution. We demonstrate that the system matrix is well behaved from a full-wave solution at high frequencies to a pure resistive circuit solution at dc, thereby enabling dc-to-daylight simulations. Finally, these techniques are applied to remedy the low-frequency conditioning of the electric field integral equation matrix

Solving low frequency EM-CKT problems using the PEEC method

The partial element equivalent circuit (PEEC) formulation is an integral equation based approach for the solution of combined electromagnetic and circuit (EM-CKT) problems. Traditional EM solvers like the electric field integral equation (EFIE) method suffer from numerical problems at low-frequencies arising from the decoupling of the charge and current basis functions. In this paper, the low frequency behavior of the PEEC matrix is investigated. Techniques leading to an excellent condition number throughout the entire frequency spectrum are discussed. Finally, these schemes are applied to remedy the low-frequency conditioning of the EFIE method.

(S)PEEC: Time- and frequency-domain surface formulation for modeling conductors and dielectrics in combined circuit electromagnetic simulations

The partial element equivalent circuit (PEEC) formulation is an integral-equation-based approach for the solution of combined circuit and electromagnetic (EM) problems. In this paper, a surface-based PEEC formulation is presented to complement the existing volume-based method. With the rise in the operating frequencies and the increasing complexity of test structures on boards, packages, and chips, a surface-based formulation is more efficient for many problems in terms of the number of unknowns generated. The composite conductor dielectric modeling is based on the PMCHWT formulation which is transformed into a PEEC representation using equivalent magnetic and electric circuits connected by mutual coupling. Both time- and frequency-domain analyses are discussed, similar to a Spice-type circuit solver. The new formulation is compared with the volume-based PEEC approach in terms of accuracy and the number of unknowns generated

Generalized Kirchoff's current and Voltage law formulation for coupled circuit-electromagnetic Simulation with surface Integral equations

In this paper, a new formulation for coupled circuit-electromagnetic (EM) simulation is presented. The formulation employs full-wave integral equations to model the EM behavior of two- or three-dimensional structures while using modified nodal analysis to model circuit interactions. A coupling scheme based on charge and current continuity and potential matching, realized as a generalization of Kirchoffs voltage and current laws, ensures that the EM and circuit interactions can be formulated as a seamless system. While rigorous port models for EM structures can be obtained using the approach discussed herein, it is shown that the coupling paradigm can reveal additional details of the EM-circuit interactions and can provide a path to analysis-based design iteration.

Oct-tree-based multilevel low-rank decomposition algorithm for rapid 3-D parasitic extraction

Fast parasitic extraction is an integral part of high-speed microelectronic simulation at the package and on-chip level. Integral equation methods and related fast solvers for the iterative solution of the resulting dense matrix systems have enabled linear time complexity and memory usage. However, these methods tend to have large disparities between setup and matrix-vector product times that affect their efficiency when applied to multiple excitation problems, i.e., problems with a large number of nets. For example, FastCap, which is based on the fast multipole method, has a significantly faster setup time than the multilevel QR decomposition-based IES/sup 3/, but relatively slow matrix-vector products. In this paper, we present a novel oct-tree-based QR compression technique for fast iterative solution. The regular cube structure of the fast multipole method and the QR compression scheme for interaction submatrices as in IES/sup 3/ are combined to achieve a predetermined compressible matrix-block structure and, consequently, superior memory, setup, and solve time efficiencies.

Block partitioned Gauss-Seidel PEEC solver accelerated by QR-based coupling matrix compression techniques

Electromagnetic (EM) integral equation solvers based on the partial element equivalent circuit (PEEC) approach have proven to be well suited for modeling combined circuit and EM problems. The solution of the full-wave electromagnetic part is transformed to the circuit domain and general well-known circuit solver techniques are applied. However owing to the mutual couplings in the PEEC formulation, the MNA matrix is not sparse as in the case of general lumped circuits. This gives rise to a time and memory bottleneck. A Gauss-Seidel relaxation (GSR) solver is presented as an appropriate alternative to SPICE sparse LU solvers, for the PEEC class of problems in the frequency domain. Circuit based block partitioning schemes similar to the ones used in waveform relaxation methods with known convergence properties are used to insure fast convergence. Furthermore, circuit coupling thinning schemes based on QR compression techniques are used to accelerate the inter block updates and also intra block solutions.

Speeding Up PEEC Partial Inductance Computations Using a QR-Based Algorithm

The partial element equivalent circuit (PEEC) approach has been used in different forms for the computation of equivalent circuit elements for quasi-static and full-wave electromagnetic models. In this paper, we focus on the topic of large scale inductance computations. For many problems as part of PEEC modeling, partial inductances need to be computed to model interactions between a large numbers of objects. These computations can be very time and memory consuming. To date, several techniques have been devised to reduce the memory and time required to compute the partial inductance entities, as well as the time required to use them in a circuit analysis compute step. Some of the existing methods use hierarchical compression while some others are based on issues like properties of the inverse of the partial inductance matrix. However, because of inherent limitations, most of these methods are less suitable for PEEC applications. In this paper, we present an approach which is based on the compression of the partial inductance matrix utilizing the QR decomposition of the far coefficients submatrices. The QR-decomposed form is represented as a compressed SPICE-compatible circuit. This yields an efficient and mathematically consistent approach for reducing the storage and time requirements

DiMES: multilevel fast direct solver based on multipole expansions for parasitic extraction of massively coupled 3D microelectronic structures

Boundary element methods are being successfully used for modeling parasitic effects in cutting-edge circuit design. The dense system matrix generated therein presents a time and memory bottleneck. Fast iterative solver techniques, developed to address the problem, suffer from convergence issues which become pronounced for large number of right hand sides as is the case for massively coupled systems. In this paper an iteration free solution scheme is presented. The dense matrix is rendered sparse by applying multilevel multipole expansions, and the resultant sparse matrix is solved by a traditional sparse matrix solver. The accuracy and time and memory requirements for the solver are compared against the regular methods. The advantage of the presented method over the corresponding iterative scheme is also demonstrated.

A three-stage preconditioner for geometries with multiple holes and handles in integral equation-based electromagnetic simulation of integrated packages

A novel three-stage approach for preconditioner including loop-tree, basis function rearrangement and incomplete LU is presented. In particular a new technique for construction of loop-tree basis functions for arbitrary-shaped geometries consisting of any number of holes and handles is described. This preconditioner enables fast iterative solution of integral modeling of electromagnetic components in integrated packages.