Branislav M. Notaros

Double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers

A novel double higher order Galerkin-type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for surface integral equation analysis of combined metallic and dielectric antennas and scatterers of arbitrary shapes. The technique employs generalized curvilinear quadrilaterals of arbitrary geometrical orders for the approximation of geometry (metallic and dielectric surfaces) and hierarchical divergence-conforming polynomial vector basis functions of arbitrary orders for the approximation of electric and magnetic surface currents within the elements. The geometrical orders and current-approximation orders of the elements are entirely independent from each other, and can be combined independently for the best overall performance of the method in different applications. The results obtained by the higher order technique are validated against the analytical solutions and the numerical results obtained by low-order moment-method techniques from literature. The examples show excellent accuracy, flexibility, and efficiency of the new technique at modeling of both current variation and curvature, and demonstrate advantages of large-domain models using curved quadrilaterals of high geometrical orders with basis functions of high current-approximation orders over commonly used small-domain models and low-order techniques. The reduction in the number of unknowns is by an order of magnitude when compared to low-order solutions.

Higher Order Frequency-Domain Computational Electromagnetics

A review of the higher order computational electromagnetics (CEM) for antenna, wireless, and microwave engineering applications is presented. Higher order CEM techniques use current/field basis functions of higher orders defined on large (e.g., on the order of a wavelength in each dimension) curvilinear geometrical elements, which greatly reduces the number of unknowns for a given problem. The paper reviews all major surface/volume integral- and differential-equation electromagnetic formulations within a higher order computational framework, focusing on frequency-domain solutions. With a systematic and unified review of generalized curved parametric quadrilateral, triangular, hexahedral, and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions, in both divergence- and curl-conforming arrangements, a large number of actual higher order techniques, representing various combinations of formulations, elements, bases, and solution procedures, are identified and discussed. The examples demonstrate the accuracy, efficiency, and versatility of higher order techniques, and their advantages over low-order discretizations, the most important one being a much faster (higher order) convergence of the solution. It is demonstrated that both components of the higher order modeling, namely, higher order geometrical modeling and higher order current/field modeling, are essential for accurate and efficient CEM analysis of general antenna, scattering, and microwave structures.

Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling

A novel higher order finite-element technique based on generalized curvilinear hexahedra with hierarchical curl-conforming polynomial vector basis functions is proposed for microwave modeling. The finite elements are implemented for geometrical orders from 1 to 4 and field-approximation orders from 1 to 10 in the same Galerkin-type finite-element method and applied to eigenvalue analysis of arbitrary electromagnetic cavities. Individual curved hexahedra in the model can be as large as approximately 2/spl lambda//spl times/2/spl lambda//spl times/2/spl lambda/, which is 20 times the traditional low-order modeling discretization limit of /spl lambda//10 in each dimension. The examples show excellent flexibility and efficiency of the higher order (more precisely, low-to-high order) method at modeling of both field variation and geometrical curvature, and its excellent properties in the context of p-refinement of solutions, for models with both flat and curved surfaces. The reduction in the number of unknowns is by an order of magnitude when compared to low-order solutions.

Higher order hybrid method of moments-physical optics modeling technique for radiation and scattering from large perfectly conducting surfaces

An efficient and accurate higher order, large-domain hybrid computational technique based on the method of moments (MoM) and physical optics (PO) is proposed for analysis of large antennas and scatterers composed of perfectly conducting surfaces of arbitrary shapes. The technique utilizes large generalized curvilinear quadrilaterals of arbitrary geometrical orders in both the MoM and PO regions. It employs higher order divergence-conforming hierarchical polynomial basis functions in the context of the Galerkin method in the MoM region and higher order divergence-conforming interpolatory Chebyshev-type polynomial basis functions in conjunction with a point-matching method in the PO region. The results obtained by the higher order MoM-PO are validated against the results of the full MoM analysis in three characteristic realistic examples. The truly higher order and large-domain nature of the technique in both MoM and PO regions enables a very substantial reduction in the number of unknowns and increase in accuracy and efficiency when compared to the low-order, small-domain MoM-PO solutions. The PO part of the proposed technique, on the other hand, allows for a dramatic reduction in the computation time and memory with respect to the pure MoM higher order technique, which greatly extends the practicality of the higher order MoM with a smooth transition between low- and high-frequency applications.

Higher Order Hybrid FEM-MoM Technique for Analysis of Antennas and Scatterers

A novel higher order large-domain hybrid computational electromagnetic technique based on the finite element method (FEM) and method of moments (MoM) is proposed for three-dimensional analysis of antennas and scatterers in the frequency domain. The geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders. The fields and currents on elements are modeled using curl- and divergence-conforming hierarchical polynomial vector basis functions of arbitrary approximation orders, and the Galerkin method is used for testing. The elements can be as large as about two wavelengths in each dimension. As multiple MoM objects are possible in a global exterior region, the MoM part provides much greater modeling versatility and potential for applications, especially in antenna problems, than just as a boundary-integral closure to the FEM part. The examples demonstrate excellent accuracy, convergence, efficiency, and versatility of the new FEM-MoM technique, and very effective large-domain meshes that consist of a very small number of large flat and curved FEM and MoM elements, with p-refined field and current distributions of high approximation orders. The reduction in the number of unknowns is by two orders of magnitude when compared to available data for low-order FEM-MoM modeling.

Efficient large-domain MoM solutions to electrically large practical EM problems

A numerical method is presented for the analysis and design of a wide variety of electromagnetic (EM) structures consisting of dielectric and conducting parts of arbitrary shapes. The method is based on the integral-equation formulation in frequency domain, and represents a large domain (high-order expansion) Galerkin-type version of the method of moments (MoM). The method is formulated in two versions concerning the type of the equivalence (volume and surface) utilized in the treatment of the dielectric parts of the structure. The generality, versatility, accuracy, and practicality of the method and code are demonstrated on four very diverse, electrically large, and complex EM problems. The examples are: an X-band reflector antenna modeled after a bats ear, which is about 11/spl lambda//sup 3/ large at X-band; a broad-band (0.5-4.5 GHz) nested array of crossed loaded dipoles; an EM system consisting of a dipole antenna and a human body, and a broad-band (1-5 GHz) microstrip-fed Vivaldi antenna with a high-permittivity dielectric substrate. The central processing unit times with a modest personal computer are on the order of several minutes for a single-frequency application.

Higher order large-domain FEM modeling of 3-D multiport waveguide structures with arbitrary discontinuities

A highly efficient and accurate higher order large-domain finite-element technique is presented for three-dimensional (3-D) analysis of N-port waveguide structures with arbitrary metallic and dielectric discontinuities on standard PCs. The technique implements hierarchical polynomial vector basis functions of arbitrarily high field-approximation orders on Lagrange-type curved hexahedral finite elements of arbitrary geometrical orders. Preprocessing is carried out by a semiautomatic higher order meshing procedure developed for waveguide discontinuity problems. The computational domain is truncated by coupling the 3-D finite-element method (FEM) with a two-dimensional (2-D) modal expansion technique across the waveguide ports. In cases where analytical solutions are not available, modal forms at the ports are obtained by a higher order 2-D FEM eigenvalue analysis technique. The examples demonstrate very effective higher order hexahedral meshes constructed from a very small number of large curved finite elements (large domains). When compared to the existing higher order (but small domain) finite-element solutions, the presented models require approximately 1/5 of the number of unknowns for the same (or higher) accuracy of the results.

Efficient large-domain 2-D FEM solution of arbitrary waveguides using p-refinement on generalized quadrilaterals

An efficient and accurate large-domain higher order two-dimensional (2-D) Galerkin-type technique based on the finite-element method (FEM) is proposed for analysis of arbitrary electromagnetic waveguides. The geometry of a waveguide cross section is approximated by a mesh of large Lagrangian generalized curvilinear quadrilateral patches of arbitrary geometrical orders (large domains). The fields over the elements are approximated by a set of hierarchical 2-D polynomial curl-conforming vector basis functions of arbitrarily high field-approximation orders. When compared to the conventional small-domain 2-D FEM techniques, the large-domain technique requires considerably fewer unknowns for the same (or higher) accuracy and offers a significantly faster convergence when the number of unknowns is increased. A comparative analysis of solutions using p- and h-refinements shows that the p-refinement represents a better choice for higher accuracy with lesser computation cost. In addition to increasing the field-approximation orders, the geometrical orders of elements (where needed) should also be set high for the improved accuracy of the solution without subdividing the elements. However, in general, an arbitrarily high accuracy cannot be achieved by performing the p-refinement in arbitrarily coarse meshes alone; instead, a combined hp-refinement should be utilized in order to obtain an optimal modeling performance.

OPTIMIZED ENTIRE-DOMAIN MOMENT-METHOD ANALYSIS OF 3D DIELECTRIC SCATTERERS

When compared with commonly used subdomain moment-method analysis, entire-domain analysis of 3D dielectric scatterers results in a greatly reduced number of unknowns. Unfortunately, the expressions for matrix elements tend to be quite complicated and their calculation extremely time-consuming if evaluated directly. It is shown in the paper that, in a Galerkin-type solution with large trilinear hexahedral basic volume elements and three-dimensional polynomial approximation of volume current inside them, these expressions can be manipulated analytically for optimized rapid non-redundant integration. Consequently, a method for the analysis of 3D dielectric scatterers is obtained that is efficient, rapidly converging with increasing degree of approximation for current, remarkably accurate and very moderate in computer memory requirements. The applicability of the method of moments is thereby extended to bodies of electrical sizes greatly exceeding those that can be dealt with by subdomain methods.

Continuously Inhomogeneous Higher Order Finite Elements for 3-D Electromagnetic Analysis

A novel higher order entire-domain finite element technique is presented for accurate and efficient full-wave three-dimensional (3D) analysis of electromagnetic structures with continuously inhomogeneous material regions, using large generalized curved hierarchical curl-conforming hexahedral vector finite elements that allow continuous change of medium parameters throughout their volumes. This is the first general 3D implementation and numerical demonstration of the inherent theoretical ability of the finite element method (FEM) to directly treat arbitrarily (continuously) inhomogeneous materials. The results demonstrate considerable reductions in both number of unknowns and computation time of the entire-domain FEM modeling of continuously inhomogeneous materials over piecewise homogeneous models. They indicate that, in addition to theoretical relevance and interest, large curved higher order continuous-FEM elements also have great potential for practical applications that include structures with pronounced material inhomogeneities and complexities.