Singular Higher Order Divergence-Conforming Bases of Additive Kind and Moments Method Applications to 3D Sharp-Wedge Structures

Abstract

We present new subsectional, singular divergenceconforming vector bases that incorporate the edge conditions for conducting wedges. The bases are of additive kind because obtained by incrementing the regular polynomial vector bases with other subsectional basis sets that model the singular behavior of the unknown vector field in the wedge neighborhood. Singular bases of this kind, complete to arbitrarily high order, are described in a unified and consistent manner for curved quadrilateral and triangular elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. Our singular bases guarantee normal continuity along the edges of the elements allowing for the discontinuity of tangential components, adequate modelling of the divergence, and removal of spurious solutions. These singular high-order bases provide more accurate and efficient numerical solutions of surface integral problems. Several test-case problems are considered in the paper, thereby obtaining highly accurate numerical results for the current and charge density induced on 3D sharp-wedge structures. The results are compared with other solutions when available and confirm the faster convergence of these bases on wedge problems.Index Terms-Basis functions, boundary integral equations, curvilinear geometry, electromagnetic analysis, electromagnetic diffraction, electromagnetic scattering, Galerkin method, high-order modelling, method of moments (MoM), numerical analysis, singular vector functions incorporating edge conditions, wedges.

Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformation

“…In finite element application and in boundary element application for diffraction problems [38] regular polynomial basis functions together with singular basis functions are successfully used to model the unknowns, see [35][36][37]. In particular we observe that the singular functions behave as Müntz polynomials with different irrational and/or rational exponents related to the physical parameters of the diffraction problem [38,39]: number of angular regions, aperture angles and materials.…”

“…We have extensively used this technique in computational electromagnetics for diffraction problems, see for example [35,36].…”

“…Finally, in Section 5 we present several numerical tests to validate the proposed quadrature scheme; in particular Example 6 shows the application of the technique to the practical problem of electromagnetic diffraction [35][36][37][38][39].…”

“…In particular we observe that for BEM problems the source integral contains a weakly singular kernel, which is a Hankel function of the second kind of order zero in two-dimensional problems; therefore, in the near self-region Müntz polynomial basis functions times logarithmic kernel need to be integrated [36,37].…”

“…The proposed algorithm can be applied to the more general case of three-dimensional diffraction problems [36] where the unknowns behave as finite energy Müntz polynomials, see Section 4.3.…”