Branko M. Kolundzija

Solution of large complex problems in computational electromagnetics using higher-order basis in MoM with out-of-core solvers

In a recent invited paper in the IEEE Antennas and Propagation Magazine, some of the challenging problems in computational electromagnetics were presented. One of the objectives of this note is to simply point out that challenging to one may be simple to another. This is demonstrated through an example cited in that article. The example chosen is a Vivaldi antenna array. What we discuss here also applies to the other examples presented in that article, but we have chosen the Vivaldi antenna array to help us make our point. It is shown in this short article that a higher-order basis using a surface integral equation a la a PMCHWT (Poggio-Miller-Chu-Harrington-Wu-Tsai) method-of-moments formulation may still be the best weapon that one have in todays arsenal to deal with challenging complex electromagnetic analysis problems. Here, we have used the commercially available code WIPL-D to carry out all the computations using laptop/desktop systems. The second objective of this paper is to present an out-of-core solver. The goal is to demonstrate that an out-of-core 32-bit-system-based solver can be as efficient as a 64-bit in-core solver. This is quite contrary to the popular belief that an out-of-core solver is generally much slower than an in-core solver. This can be significant, as the difference in the cost of a 32-bit system can be 1/30 of a 64-bit system of similar capabilities using current computer architectures. For the 32-bit system, we consider a Pentium 4 system, whereas for the 64-bit system, we consider an Itanium 2 system for comparison. The out-of-core solver can go beyond the 2 GB limitation for a 32-bit system and can be run on ordinary laptop/desktop; hence, we can simultaneously have a much lower hardware investment while better performance for a sophisticated and powerful electromagnetic solver. The system resources and the CPU times are also outlined.

Maximally Orthogonalized Higher Order Bases Over Generalized Wires, Quadrilaterals, and Hexahedra

This paper presents a general theory of maximally orthogonalized div- and curl-conforming higher order basis functions (HOBFs) over generalized wires, quadrilaterals, and hexahedra. In particular, all elements of such bases, necessary for fast and easy implementation, are listed up to order n=8. Numerical results, given for div-conforming bases applied in an iterative method of moments solution of integral equations, show that the condition number and the number of iterations are a) much lower than in the case of other HOBFs of polynomial type and b) practically not dependent on the applied expansion order.

Efficient iterative solution of surface integral equations based on maximally orthogonalized higher order basis functions

The goal of this paper is to propose maximally orthogonalized higher order basis functions that automatically satisfy the continuity equation at the boundary elements (BE) edges. As a result, orthogonality was imposed between all basis functions except between the two lowest order functions. Numerical results show that the proposed higher order basis functions enable convergence as fast as low order basis functions, which qualifies higher order basis functions for application with iterative solvers.

Multiminima heuristic methods for antenna optimization

Two general approaches to multiminima optimization are considered. The first approach is based on repetition of a single minima method (e.g., the Nelder-Mead simplex applied to the best solution in a set of random trials). The second approach is based on a coarse estimation of local minima using initial set of points and local optimization starting from these local minima (e.g., random search as a generator of the initial set of points and Nelder-Mead simplex as a local optimizer). A comparison of various optimization algorithms has been done on one analytical problem and two well-known examples of antenna design. It is found that: a) the multiminima method based on coarse estimation enables finding more minima with smaller number of iterations than that based on repetition, b) the best multiminima methods are comparable with the best single minima methods in a number of iterations needed for finding the global minima, and c) the multiminima method based on coarse estimation restarted with different weighting coefficients of multiobjective cost function enables efficient Pareto optimization

Efficient Analysis of Large Scatterers by Physical Optics Driven Method of Moments

A new iterative procedure is presented that enables method of moment (MoM) solution of scattered field from electrically large and complex perfectly conducting bodies using significantly reduced number of unknown coefficients. In each iteration the body is excited by a plane wave and by the currents, which are obtained as an approximate solution in the previous iteration. The physical optics (PO) and modified PO techniques are used to determine the PO and the correctional PO currents, which are expressed in terms of original MoM basis functions and grouped into macro-basis functions (MBFs). Weighting coefficients of all MBFs are determined from the condition that mean square value of residuum of original MoM matrix equation is minimized. The iterative procedure finishes when the residuum decreases below the maximum allowed value. The accuracy and efficiency of the proposed method are illustrated on two examples: cube scatterer and airplane scatterer. Since the construction of MBFs by PO and modified PO techniques ensures fast convergence to the original MoM solution, the method is named PO driven MoM.

On the locally continuous formulation of surface doublets

Exact (locally continuous) formulation of doublets and particularly rooftop basis functions based on unitary vector concept are presented. Basic properties of such a formulation are examined showing many advantages when compared with classical (approximate) formulation. In particular, in the case of rooftop basis functions based on exact formulation, the shape quality factor is defined and optimal shapes of quadrilateral patches are determined. If such quadrilaterals are used for modeling of general structures, the number of unknowns needed in the analysis is almost halved when compared with modeling by triangular doublets.

Solving electrically large EM problems by using out-of-core solver accelerated with multiple graphical processing units

We present results for frequency-domain MoM simulations of electrically large structures using out-of-core solver accelerated with multiple GPUs on a single personal computer. The structures analyzed in order to demonstrate the efficiency of proposed out-of-core solver are Cassegrain reflector antenna with up to 240 λ reflector diameter and Luneburg lens, up to 16 λ diameter, excited with a half-wavelength dipole. The acceleration of out-of-core solver is up to 10 times with one GPU compared to a standard CPU, or up to 20 times when using 3 GPUs.

On the Efficiency of Particle Swarm Optimizer when Applied to Antenna Optimization

This paper presents the results for three different antenna optimization problems that are found using the particle swarm optimizer (PSO). The outcomes found with PSO are compared to the outcomes found with other optimization algorithms to estimate the efficiency of PSO. The first problem is finding the optimal position of the feeding probe in a radiating rectangular waveguide. The second problem is finding the maximal forward gain of a Yagi antenna. The third problem is finding the optimal feeding of a broadside antenna array. The optimization problems have 2, 6, and 20 optimization variables respectively

General Entire-Domain Galerkin Method for Electromagnetic Modeling of Composite Wire and Plate Structures

Geometry modeling of wires and plates is performed by using curvilinear cylinders of variable radii and curved curvilinear rectangles and triangles. Current modeling is performed by using entire-domain approximations, which satisfy continuity equation at wire and plate ends and junctions. Unknown coefficients of these approximations are determined by solving EFIE by means of Galerkin method. In special cases this method degenerates into those of Newman and Pozar [1], Glisson and Wilton [2], or Rao et al [3]. Starting from the proposed theory various new methods can be constructed. The results obtained by a particular form of the method proposed, based on application of conical and bilinear surfaces and polynomials, show a good agreement with experimental and theoretical data. In the case of long wires satisfactory results are obtained with only 3 unknowns per wavelenght (¿), and in the case of large surfaces of simple shapes with only 10 unknowns (for both current components) per ¿2.

Matrix equilibration in method of moment solutions of surface integral equations

Basic theory of matrix equilibration is presented, relating it with other techniques for decreasing the condition number of matrix equations obtained by method of moments applied to surface integral equations. Specific variants of matrix equilibration based on diagonal and extended diagonal scaling of basis functions are proposed. Numerical examples demonstrate efficiency and robustness of proposed variants.