Locally Corrected Nystrom Research Articles

Fast Direct Error-Controllable Solution of Integral Equations via H,-matrix accelerated Locally Corrected Nyström Method

Computational framework for error-controlled fast direct solution of large-scale scattering problems is demonstrated. The approach is based on higher-order Locally Corrected Nystrom (LCN) discretization of the Electric and Magnetic Field Integral Equations (EFIE and MFIE) accelerated with hierarchical matrices. The latter enable $O(N\log N)$ solutions for matrices featuring quasi-constant ranks in their compressed blocks and make solutions largely insensitive to poor conditioning of the matrix equations typically resulting from LCN discretization of EFIE and MFIE.

Magnetic Field Penetration Into a Metal Enclosure Using an ELF/VLF Loop Antenna

The ability of extremely and very low frequency (ELF/VLF, 0–30 kHz) radio waves to penetrate conductive media is well established. Magnetic field penetration into a thin but highly conductive box using an ELF/VLF loop antenna transmitter is investigated. The work is relevant for electromagnetic shielding of ELF/VLF sensors, defect detection, inductive power transfer, and through container imaging. Analytical solutions are reviewed for related shielding problems, however, determining the penetration through a realistic shield geometry and finite sized near-field source requires a numerical approach. Surface integral equation (SIE) methods are well suited for shield modeling due to the low surface area to volume ratio of the shield. Method of moment techniques have been successfully applied to solving SIEs in the past, however, enforcing algorithm stability at low frequencies is known to require considerable effort. To alleviate the low-frequency concerns, a high-order locally corrected Nystrom (LCN) scheme is utilized to solve an SIE based on an augmented Muller formulation of Maxwell's equations. To validate the LCN simulations, an experiment is conducted using a loop antenna inside a 1.2 m aluminum cube of 2.7 mm thickness with an external ELF/VLF loop transmitter. Experimental results are shown to match within 3 dB of the LCN code predictions.

RWG MoM‐via‐locally corrected Nyström method in near‐field to far‐field transformation using very‐near‐field measurement

Locally corrected Nystrom (LCN) method is used to solve electric field integral equations (EFIE) of the equivalent current method in the planar very-near-field measurement of antennas and RF circuits. The exact relationship between the Rao–Wilton–Glisson (RWG) method of moments and first-order and zero-order LCN is established for both magnetic and electric currents to ensure normal current continuity between adjacent triangular patches. The proposed method is a point-based RWG discretisation of EFIE and causes a noticeable decrease in the degree of freedom. It consequently eliminates spurious charges and significantly lowers the condition number of the impedance matrix. Moreover, it is more efficient to be accelerated by fast algorithms such as multi-level fast multipole method. In what follows, the detailed explanation of the proposed method, along with the examples of current reconstruction and antenna far-field calculation, is presented.

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 today’s geometric computer-aided design systems for modeling surfaces. Combining the locally corrected Nystrom (LCN) method with NURBS requires formulating LCN on both quadrilateral and triangular Bezier surfaces, as a typical NURBS-generated Bezier 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 Bezier 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.

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 Nystrom (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 Nystrom 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.

Constrained Locally Corrected Nyström Method

A generalization of the locally corrected Nystrom (LCN) discretization method is outlined wherein sparse transformations of the LCN system matrix are obtained via singular- value decompositions of local constraint matrices. The local constraint matrices are used to impose normal continuity of the currents across boundaries shared by mesh elements. Due to the method’s simplicity and flexibility, it is straightforward to develop high-order constrained LCN (CLCN) systems for different formulations and mesh element types. Numerical examples demonstrate the memory savings provided by the CLCN method and its improved accuracy when applied to geometries with sharp edges. It is also shown that the CLCN method maintains the high-order convergence of the LCN method, and it eliminates the need to include line charges in Nystrom-based discretizations of formulations that involve the continuity equation.

Scattering from dielectric and metallic bodies using a high-order, Nystrom, multilevel fast mutipole algorithm

The multilevel fast multipole algorithm (MLFMA) is an established way to accelerate the solution of the integral equations governing scattering of electromagnetic waves. Another way is to use high-order methods, of which the locally corrected Nystrom (LCN) method is, perhaps, the best; it has been applied to metallic scatterers up to order 10. In this paper, the technique (MLFMA + LCN) is extended to handle mixed conducting/dielectric bodies. Results for coated circular and square cylinders, using orders up to 8, demonstrate the efficiency of the method