Raymond A. Wildman

A Finite Difference Delay Modeling Approach to the Discretization of the Time Domain Integral Equations of Electromagnetics

A new method for solving the time-domain integral equations of electromagnetic scattering from conductors is introduced. This method, called finite difference delay modeling, appears to be completely stable and accurate when applied to arbitrary structures. The temporal discretization used is based on finite differences. Specifically, based on a mapping from the Laplace domain to the z-transform domain, first- and second-order unconditionally stable methods are derived. Spatial convergence is achieved using the higher-order divergence-conforming vector bases of Graglia et al. Low frequency instability problems are avoided with the loop-tree decomposition approach. Numerical results will illustrate the accuracy and stability of the technique.

An accurate scheme for the solution of the time-domain Integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation

Despite the numerous advances made in increasing the computational efficiency of time-domain integral equation (TDIE)-based solvers, the stability and accuracy of TDIE solvers remain problematic. This paper introduces a new numerical method for the accurate solution of TDIEs for scattering from arbitrary perfectly conducting surfaces. The work described in this paper uses the higher order divergence-conforming basis functions of Graglia et al. for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations. Since the basis functions used for the temporal representation are noncausal, an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time. Numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization.

An accurate broad-band method of moments using higher order basis functions and tree-loop decomposition

A method is presented that combines the low-frequency accuracy of tree-loop decomposition with higher order basis functions to achieve an electric field integral equation based method of moments that is accurate over a wide band of frequencies. In general, the use of higher order divergence-conforming bases introduces more solenoidal degrees of freedom than the standard Rao-Wilton-Glisson (zeroth-order) bases. These solenoidal degrees of freedom are counted and identified, and an efficient method to construct the required higher order loops is presented. Numerical results demonstrate that the technique allows the use of a single mesh to acquire scattering results over a wide band of frequencies using arbitrary order basis functions.

Two-dimensional transverse-magnetic time-domain scattering using the Nystro/spl uml/m method and bandlimited extrapolation

A method is presented for the solution of the integral equations that describe the electromagnetic scattering from an infinitely long conducting cylinder in the time domain. The method discretizes the integral equations spatially using a high-order locally-corrected Nystro/spl uml/m method and temporally using a filtered kernel. The filtering of the kernel both controls aliasing and reduces the order of its singularity. On the other hand, filtering also gives rise to a noncausal kernel so the time marching is accomplished with a bandlimited extrapolation scheme. Numerical results demonstrate the stability and accuracy of the proposed method.

Geometry Reconstruction of Conducting Cylinders Using Genetic Programming

A genetic programming-based method for the imaging of two-dimensional conductors is presented. Geometry is encoded in this scheme using a tree-shaped chromosome to represent the Boolean combination of convex polygons into an arbitrary two-dimensional geometry. The polygons themselves are encoded as the convex hull of variable-length lists of points that reside in the terminal nodes of the tree. A set of genetic operators is defined for efficiently solving the inverse scattering problem. Specifically, the encoding scheme allows for a standard genetic programming crossover operator, and several mutation operators are designed in consideration of the encoding scheme. Several results are presented that demonstrate the method on a number of different shapes

Multi-objective optimization of interconnect geometry

The rapid increase in the number of wiring layers due to improved planarization and metallization techniques permits spatial resources to be traded for improved performance. Yield, power dissipation and propagation delay are all sensitive to the selection of the pitch and width of wires in each layer. As in many other engineering design problems, however, there exists no unique solution which simultaneously optimizes all aspects of system performance. The best that can be achieved is the identification of the optimal surface within the multi-objective performance space. A single design can be chosen from this list a posteriori using additional selection criteria which may depend, for example, on the specific details of the product application. This paper investigates the use of Pareto genetic algorithms to explore the extent of multi-objective optimal surfaces. The tradeoffs between yield, power-dissipation and cycle time for a benchmark netlist are examined as a function of in-plane geometry for a seven-layer interconnect.

Gravitational and magnetic anomaly inversion using a tree-based geometry representation

Gravitational and magnetic anomaly inversion of homogeneous 2D and 3D structures is treated using a geometric parameterization that can represent multiple, arbitrary polygons or polyhedra and a local-optimization scheme based on a hill-climbing method. This geometry representation uses a tree data structure, which defines a set of Boolean operations performed on convex polygons. A variable-length list of points, whose convex hull defines a convex polygon operand, resides in each leaf node of the tree. The overall optimization algorithm proceeds in two steps. Step one optimizes geometric transformations performed on different convex polygons. This step provides approximate size and location data. The second step optimizes the points located on all convex hulls simultaneously, giving a more accurate representation of the geometry. Though not an inherent restriction, only the geometry is optimized, not including material values such as density or susceptibility. Results based on synthetic and measured data show that the method accurately reconstructs various structures from gravity and magnetic anomaly data. In addition to purely homogeneous structures, a parabolic density distribution is inverted for 2D inversion.

Mixed-Order Testing Functions on Triangular Patches for the Locally Corrected NystrÖm Method

The use of polynomial-complete representations for the current in the locally corrected Nystrom method has been identified as a major cause of inaccuracy in the solution of the electric field integral equation. Mixed-order quadrature rules for quadrilateral patches were proposed to provide a self-consistent basis for the charge, which alleviated much of the inaccuracy. Here, this idea is extended to triangular patches. Rather than modifying the integration rule, a set of mixed-order testing functions for the current can provide a self-consistent polynomial basis for the charge. Plots of the divergence of the current on a flat plate show that the use of mixed-order testing functions provides more accurate solutions

Numerical solution of time domain integral equations using the Nystro/spl uml/m method

A wide variety of electromagnetic scattering problems are, in principle, most efficiently solved with time domain integral equations. Integral equation methods are well suited for problems involving homogenous scatterers, and time domain methods can efficiently solve broadband, nonlinear, and time-varying problems. While a Galerkin approach is usually used in their solution, recently, the locally corrected Nystro/spl uml/m method (Canino, L.F. et al., 1998) was applied to the two-dimensional time domain integral equations (Wildman, R.A. and Weile, D.S., IEEE APS Int. Symp., 2004). The Nystro/spl uml/m method has several benefits over Galerkins method. In the approach described by Wildman and Weile, a standard Nystro/spl uml/m discretization was used in space, and a low-order approximation (trapezoidal rule) was used for the temporal integrals. To obtain accurate results with low-order integration, the kernel of the integral equation was filtered, and bandlimited extrapolation was used to recover a causal representation. This paper follows a similar approach. Fortunately, in three dimensions, the filtering can be performed analytically, so the method is greatly simplified.

Accurate solution of time domain integral equations using higher order vector bases and bandlimited extrapolation

Time domain integral equations (TDIE) based solvers have been gaining a lot of attention in the past few years for the solution of electromagnetic problems. The widespread acceptance of TDIE based solvers was historically hindered by the instability and inefficiency of their original implementations. As a consequence of this, the art of TDIE based solvers is not as refined as that for more conventional methods. While higher-order spatial discretizations have been used in the past, few claims of higher order temporal convergence can be found in the literature. This paper presents a stable and accurate scheme to solve TDIEs using higher order vector bases, coupled with bandlimited temporal basis functions and bandlimited extrapolation. It will be demonstrated that the proposed scheme demonstrates enhanced convergence in space and time.