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This paper introduces an efficient approach for solving the Electric Field Integral Equation (EFIE) with highorder accuracy by explicitly enforcing the continuity of the impressed current densities across boundaries of the surface patch discretization. The integral operator involved is discretized via a Nystrom-collocation approach based on Chebyshev polynomial expansion within each patch and a closed quadrature rule is utilized such that the discretization points inside one patch coincide with those inside another patch on the shared boundary of those two patches. The continuity enforcement is achieved by constructing a mapping from those coninciding points to a vector containing unique discretization points used in the GMRES iterative solver. The proposed approach is applied to the scattering of several different geometries including a sphere, a cube, a NURBS model imported from CAD software, and a dipole structure and results are compared with the Magnetic Field Integral Equation (MFIE) and the EFIE without enforcing continuity to illustrate the effectiveness of the approach.
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Direct/Iterative Hybrid Solver for Scattering by Inhomogeneous Media
This paper presents a fast high-order method for the solution of two-dimensional problems of scattering by penetrable inhomogeneous media, with application to high-frequency configurations containing (possibly) discontinuous refractivities. The method relies on a hybrid direct/iterative combination of 1)~A differential volumetric formulation (which is based on the use of appropriate Chebyshev differentiation matrices enacting the Laplace operator) and, 2)~A second-kind boundary integral formulation (which, once again, utilizes Chebyshev discretization, but, in this case, in the boundary-integral context). The approach enjoys low dispersion and high-order accuracy for smooth refractivities, as well as second-order accuracy (while maintaining low dispersion) in the discontinuous refractivity case. The solution approach proceeds by application of Impedance-to-Impedance (ItI) maps to couple the volumetric and boundary discretizations. The volumetric linear algebra solutions are obtained by means of a multifrontal solver, and the coupling with the boundary integral formulation is achieved via an application of the iterative linear-algebra solver GMRES. In particular, the existence and uniqueness theory presented in the present paper provides an affirmative answer to an open question concerning the existence of a uniquely solvable second-kind ItI-based formulation for the overall scattering problem under consideration. Relying on a modestly-demanding scatterer-dependent precomputation stage (requiring in practice a computing cost of the order of $$O(N^{\alpha})$$ operations, with $$\alpha \approx 1.07$$, for an $$N$$-point discretization \textcolor{black}{and for the relevant Chebyshev accuracy orders $$q$$ used)}, together with fast ($O(N)$-cost) single-core runs for each incident field considered, the proposed algorithm can effectively solve scattering problems for large and complex objects possibly containing discontinuities and strong refractivity contrasts.
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- Award ID(s):
- 2109831
- PAR ID:
- 10590727
- Editor(s):
- Bruno, Oscar; Pandey, Ambuj
- Publisher / Repository:
- Siam Journal on Scientific Computing
- Date Published:
- Journal Name:
- SIAM Journal on Scientific Computing
- Edition / Version:
- 1
- Volume:
- 46
- Issue:
- 2
- ISSN:
- 1064-8275
- Page Range / eLocation ID:
- A1298 to A1326
- Subject(s) / Keyword(s):
- scattering, inhomogeneous media, direct solver, iterative solver, spectral method, integral equations
- Format(s):
- Medium: X Size: 7.31 MB Other: pdf
- Size(s):
- 7.31 MB
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/22M1521547
