Mohammad Shafieipour

On the Equivalence of RWG Method of Moments and the Locally Corrected Nyström Method for Solving the Electric Field Integral Equation

The first-order locally corrected Nyström (LCN) method for the electric field integral equation is modified to ensure continuity of the current between triangular elements of the mesh. Rao-Wilton-Glisson (RWG) basis functions are used to create a conversion matrix from the LCN representation of the current to the RWG method-of-moments (MoM) representation of the current in order to enforce continuity of current between adjacent triangular flat patches. Benefits of the method are two fold: first, it provides 4 × reduction in degrees of freedom and removes unacceptable error levels in first-order LCN implementations, second, the method can be viewed as a point-based discretization of the RWG MoM offering improved efficiency in its acceleration with the fast multipole algorithm.

Error Controllable Solutions of Large-Scale Problems in Electromagnetics: MLFMA-Accelerated Locally Corrected Nyström Solutions of CFIE in 3D [Open Problems in CEM]

This work provides an overview of a parallel, high-order, error-controllable framework for solving large-scale scattering problems in electromagnetics, as well as open problems pertinent to such solutions. The method is based on the higher-order locally corrected Nystrom (LCN) discretization of the combined-field integral equation (CFIE), accelerated with the error-controlled Multi-Level Fast Multipole Algorithm (MLFMA). Mechanisms for controlling the accuracy of calculations are discussed, including geometric representation, stages of the locally corrected Nystrom method, and the MLFMA. Also presented are the key attributes of parallelization for the developed numerical framework. Numerical results validate the proposed numerical scheme by demonstrating higher-order error convergence for smooth scatterers. For the problem of scattering from a sphere, the developed numerical solution is shown to have the ability to produce a solution with a maximum relative error of the order 10-9. Open-ended problems, such as treatment of general scatterers with geometric singularities, construction of well-conditioned operators, and current challenges in development of fast iterative and direct algorithms, are also discussed.

Exact Relationship Between the Locally Corrected Nyström Scheme and RWG Moment Method for the Mixed-Potential Integral Equation

The exact relationship between the Rao-Wilton-Glisson (RWG) method of moments (MoM) and the locally corrected Nyström (LCN) method for the mixed-potential (MP) electric field integral equation (EFIE) is presented as an extension to our work where we established analogous exact relationship for solving the EFIE in its vector-potential (VP) form. It is shown that in order to achieve one such relationship for the MP EFIE, the first- and zeroth-order LCN methods must be, respectively, used for the discretization of the VP and scalar-potential terms of the MP EFIE. The resulting numerical scheme is a point-based RWG MoM discretization of the MP EFIE via the Nyström method. Due to the MP formulation of the EFIE, the proposed method establishes notably higher accuracy compared to either RWG MoM or LCN discretizations of the EFIE in the VP form. The increased accuracy is attributed to the analytical cancellation of the line charge contributions in the MP formulation as opposed to numerical cancellation inherent in the VP formulation of the EFIE. The detailed study and explanations of the above cancellations is presented along with their impact on the accuracy of the respective schemes for both canonical and realistic scattering targets at different frequencies.

Solution of large multiscale EMC problems with method of moments accelerated via low-frequency MLFMA

The paper demonstrates that for large-scale electromagnetic compatibility problems not exceeding 120λ in size the low-frequency-multilevel-fast-multipole-algorithm (LF-MLFMA) based on spherical wave function expansions is advantageous to its high-frequency counterpart based on plane wave expansions. In the latter the depth of the tree is restricted by the smallest size of the leaf-level box size of 0.1λ making it inefficient at either low-frequencies or for problems with multi-scale features. The low-frequency MLFMA, however, has no limitation on the depth of the tree and allows for full-wave acceleration of Moment Method from DC to frequencies at which the models spans up to 120λ. Such broadband behaviour of the low-frequency MLFMA is made possible through construction of numerically stable translation operators for the spherical wave functions with orders reaching 180. This paper provides an overview of the algorithms allowing for stable high order translations of spherical functions. The LF-MLFMA accelerated Rao-Wilton-Glisson Moment Method utilizing one such algorithm is demonstrated in the frequency range from 1MHz to 2.5GHz for the problem of plane wave coupling to antennas onboard the F5 fighter jet at fixed discretization featuring 2 million surface elements.

Accurate transmission lines characterization via higher order moment method solution of novel single-source integral equation

A new method for high precision extraction of per-unit-length inductance and resistance in the multi-conductor transmission lines (MTLs) is presented. The approach is based on higher-order geometrical representation of the MTL cross-section followed by higher-order method of moment discretization of a novel surface single-source integral equation. Through comparison against the analytically available solutions, the method is shown to achieve 6 digits of precision in the extracted MTLs resistance (R) and inductance (L) using moderate computational resources. The proposed approach paves a way for numerically inexpensive characterization of MTLs of arbitrary cross-sections with analytic-like quality.

On New Triangle Quadrature Rules for the Locally Corrected Nyström Method Formulated on NURBS-Generated Bézier Surfaces in 3-D

Nonuniform rational b-splines (NURBS) are the most widely used technique in todays geometric computer-aided design systems for modeling surfaces. Combining the locally corrected Nyström (LCN) method with NURBS requires formulating LCN on both quadrilateral and triangular Bézier surfaces, as a typical NURBS-generated Bézier mesh includes elements of both the types. While on quadrilateral elements the product of 1-D Gaussian quadrature rules can be applied to LCN effectively, Gaussian integration rules available for triangles cannot efficiently be applied to LCN for two reasons. First, they do not possess the same number of quadrature points as the number of functions in a complete set of polynomial basis at an arbitrary order. Second, they exacerbate the condition number of the resulting matrix equation at higher orders due to an increasing density of quadrature points near the edges and corners of triangles. In this paper, we construct a new set of quadrature rules for Bézier triangles (i.e., degenerate quadrilaterals) based on the Newton-Cotes (equidistant) quadrature rules and apply these rules to the LCN solution of the electric, magnetic, and combined field integral equations. Results show that the new family of quadrature rules overcomes both the aforementioned issues and can be applied to LCN effectively for orders from 0 to 9, inclusively.

On error controlled computing of the near electromagnetic fields in the shade regions of electrically large 3D objects

Current challenges in evaluation of the near electromagnetic fields in the shadow regions of electrically large objects are discussed. It is shown that traditional low order discretization techniques commonly used for analysis of radiation and scattering problems in the presence of electrically large platforms such as Fast Multipole Method accelerated Rao-Wilton-Glisson Method of Moments are incapable of computing such fields accurately. A roadmap to the solution of this open problem using higher-order boundary element schemes is discussed as well as difficulties associated with them. The Locally Corrected Nystrom method is used for demonstration of higher-order modeling capabilities.

Large scale electromagnetic scattering problems solved using The Locally Corrected Nystrom Method and Adaptive Cross Approximation

In this work, we investigate a fast direct solution of a linear system obtained from the Locally Corrected Nystrom Method (LCN) and the Combined Field Integral Equation (CFIE). Since LCN offers a high-order (HO) solution, it allows one to solve up to several digits of precision while greatly reducing the unknown count, thereby making it preferable to typical RWG-based Method of Moments (MoM) formulations, which typically offer only 2 or 3 digits of accuracy. Furthermore, the system matrix is compressed using Adaptive Cross Approximation (ACA), allowing us to perform a direct LU factorization that is also stored in compressed form. Typical iterative solutions, such as the Multi Level Fast Multipole Algorithm (MLFMA), may suffer from conditioning issues in multiscale discretizations as well as convergence problems when the structure has cavities or other resonant features. By using the direct factorization, we can sidestep these difficulties while quickly solving multiple excitations. Also, the LCN code uses Non-Uniform Rational B-Splines (NURBs) rather than flat cells in order to accurately represent the scattering geometry and has been demonstrated to produce high-order convergence for all regions of the problem including both near- and far-fields as well as in the surface currents. The ACA implementation is multi-GPU accelerated and has been used to solve over 1 million unknowns on a single workstation. This work combines the high-order accuracy of the LCN solution with an efficient and robust fast direct solver. Finally, each stage of the solution is error-controllable enabling one to maintain the desired accuracy throughout the process.

Large-scale high-order 3D electromagnetic analysis with Locally Corrected Nystrom discretization of Combined Field Integral Equation

Todays design for Electromagnetic Compatibility (EMC) requires accurate prediction of highly isolated antennas mounted on an electrically large platforms such as ships, aircrafts, and vehicles. A modern aircraft features over 50 antennas many of which are operating simultaneously at different frequencies ranging from 30MHz to tens of GHz. Such antennas are mounted on different parts of the aircraft and produce interference with each other. The latter reduces signal to noise ratio in the channels, range of the radars, and create various other detrimental effects. Prediction of such interference which may be at the levels of -60dB and lower with the accuracy of 1% puts very stringent accuracy requirement (5 digits and higher) on the electromagnetic modeling tools used for such EMC design and verification. When the accuracy requirement of 5 digits is imposed on resolution of the electromagnetic fields throughout electrically large 3D model exceeding 500 wavelengths in size it becomes nearly impossible to reach such solutions with classical low order methods such as Rao-Wilton-Glisson (RWG) Method of Moments (MoM). In this work we present higher order EM modelling framework capable of satisfying both requirements of high accuracy and large model sizes. The approach is based on the Locally Corrected Nystrom (LCN) discretization of the Combined Field Integral Equation. The high accuracy representation of the model geometry is enabled through Non-Uniform-Rational-B-Splines (NURBS) description of the piece wise surfaces forming the model. The error-controlled LCN formulation accelerated with error-controlled Multilevel Fast Multipole Algorithm (MLFMA) formulation allows to achieve O(h^p) error behavior in antenna coupling prediction as well as the calculation of RCS and antenna radiation patterns, h being the size of the mesh elements and p the order of polynomial approximations within each element. The algorithm is in-core parallelized for distributed memory compute clusters with all of its stages establishing O(N/P) memory and CPU time scaling maintained with high efficiency for large-scale calculations, N being the number of unknowns and P the number of CPUs (I. Jeffrey, et.al., IEEE Mag. Antennas Propag, 3, 2013, pp. 294–308).

Error-controlled boundary element modeling of 3D plasmonic nano-structures via higher-order Locally Corrected Nystrom method

Traditional low order numerical method of computational electromagnetics produce notably higher error when applied to analysis of plasmonic nano-structures compared to the structures with conventional values of permittivity and permeability. The high error levels are typically observed at the junctions of the low-order elements on the surface of such structures. It is caused by the artificial geometrical discontinuities resulted from flat panelled approximation of the physically smooth surface. Increase of the low-order discretization density typically does not reduce such error effectively. In this work we describe higher-order boundary element modelling approach which eliminates such errors and provides error-controlled approximation of the fields in arbitrary smooth 3D structures down to machine precision if necessary. The approach is based on higher-order Locally Corrected Nystrom discretization of the traditional surface Electric Field and Magnetic Field integral equations formulated for multi-region penetrable objects.