Multitrace/singletrace formulations and Domain Decomposition Methods for the solution of Helmholtz transmission problems for bounded composite scatterers

Jerez-Hanckes, Carlos; Pérez‐Arancibia, Carlos; Turc, Catalin

doi:10.1016/j.jcp.2017.08.050

Cited by 11 publications

(16 citation statements)

References 32 publications

(91 reference statements)

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“…While the grating profiles considered in this paper are relatively simple, qualitatively similar results can be obtained for more geometrically complex profiles. We mention that extensions to grating profiles that involve corners is straightforward in two dimensions; graded meshes and weighted versions of RtR maps are needed to treat those cases [31]. Extensions to three-dimensional gratings with edges and corners can be done by applying existing technology [31].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Of course, for a large numbers of layers, iterative solvers such as GMRES could also provide an alternative strategy for the solution of the DDM linear system. However, large numbers of layers/subdomains produce large numbers of DDM iterations [31] if classical Robin conditions are used on subdomain interfaces.…”

Section: Ddm Nyström Discretizationmentioning

confidence: 99%

“…The integration of WGF and shifted Green functions in existing three-dimensional boundaryintegral operator discretizations presented in this contribution is relatively seamless and results in a rigorous treatment of Wood configurations in three dimensions. DDM approaches will be feasible for the solution of three-dimensional electromagnetic transmission problems in periodic layered media based on quasi-optimal transmission conditions [31,7,8] that renders them amenable to Krylov subspace iterative solvers. Quasi-optimal transmission conditions arise from a judicious choice of the complex wavenumber in the transmission operator that gives rise to a DDM whose rate of convergence is practically independent of frequency [5].…”

Section: Introductionmentioning

confidence: 99%

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Domain Decomposition for Quasi-Periodic Scattering by Layered Media via Robust Boundary-Integral Equations at All Frequencies

Pérez‐Arancibia

Shipman

Turc

et al. 2019

CICP

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We develop a non-overlapping domain decomposition method (DDM) for scalar wave scattering by periodic layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including cutoff (or Wood) frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted Green functions. Using the latter in the definition of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nyström discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.In the technologically relevant case of piecewise constant periodic layered media, simulation methods based on boundary-integral equations (BIE) and quasi-periodic Green functions are attractive candidates. Radiation conditions are enforced automatically, and discretizations of material interfaces are much smaller than volumetric discretizations and do not suffer from the pollution effect. Quasi-periodic Green functions are infinite sums of free-space Green functions with periodically distributed monopole singularities. These double sums converge, although very slowly, for all but a discrete set of "cutoff" frequencies, for a given quasi-periodicity parameters (Bloch wavevector). These are cutoff frequencies at which a Rayleigh diffraction mode transitions between propagating and evanescent and the number of propagating directions jumps. Around these frequencies, the energy is rapidly redistributed along emerging new directions and is associated with anomalous scattering behavior. These frequencies are often referred to as Wood frequencies (or Wood configurations of wavevector and frequency) because their problematic association in the literature to Wood's anomaly; see the works [30,41,45,48],[39, Ch. 1] and references therein for discussions on this phenomenon. Popular methods for accelerating the slow convergence at non-Wood frequencies include Ewald summation [25] and lattice sums [37]. At very high frequencies, asymptotic methods help to accelerate computation; see for example [34].While the underlying scattering problems are, with regard to the PDE, generically stable at Wood configurations of wavevector and frequency, the latter pose a challenge to BIE for quasiperiodic problems. In three dimensions, they become increasingly close together at high frequency, and this puts the solution of quasi-periodic problems based on the quasi-periodic Green function out of reach. For periodic layered media with large numbers of layers, such as thin films used in photovoltaic cells, the probability ...

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Ddm Nyström Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Domain Decomposition for Quasi-Periodic Scattering by Layered Media via Robust Boundary-Integral Equations at All Frequencies

Pérez‐Arancibia

Shipman

Turc

et al. 2019

CICP

Self Cite

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“…Though theoretical aspects for Maxwell scattering are now fully available for global MTFs [9,10] their implementation is extremely cumbersome. Opposingly, local versions of the MTF [1,11,12,13,14] are easily implemented and parallelized and though theoretical results for acoustic and static versions are available, in the electromagnetic case similar results remain elusive. Regarding Maxwell's equations, preconditioning is crucial as STF and MTFs incorporate electric field integral operators.…”

Section: Introductionmentioning

confidence: 99%

Local Multiple Traces Formulation for Electromagnetics: Stability and Preconditioning for Smooth Geometries

Ayala¹,

Claeys²,

Escapil-Inchauspé³

et al. 2020

Preprint

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We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple trace formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes [Adv. Comp. Math. 37 (2012), 37-91] for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators.In particular, we propose a novel second-order inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies.

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“…These lead to well-posed problems in such complex materials. The discretization of such MTF-BIEs then may be formulated by the boundary element method (BEM) [23], spectral Galerkin [25,26] or Nyström [24] methods. We chose a spectral variational scheme for the local MTF based on Chebyshev polynomials for the discretization of two-dimensional problems in order to have a high-order solution with a small number of degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

Time-Domain Multiple Traces Boundary Integral Formulation for Acoustic Wave Scattering in 2D

Jerez-Hanckes¹,

Labarca²

2020

Preprint

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We present a novel computational scheme to solve acoustic wave transmission problems over composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. Our continuous problem is cast as a multiple traces time-domain boundary integral formulation valid in two and three dimensions. Numerically, our two-dimensional non-conforming spatial discretization uses spectral elements based on second kind Chebyshev polynomials while a convolution quadrature scheme is performed in the complex frequency domain. Computational experiments reveal multistep and multistage convolution quadrature expected convergence results for a variety of complex domains.

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Multitrace/singletrace formulations and Domain Decomposition Methods for the solution of Helmholtz transmission problems for bounded composite scatterers

Cited by 11 publications

References 32 publications

Domain Decomposition for Quasi-Periodic Scattering by Layered Media via Robust Boundary-Integral Equations at All Frequencies

Domain Decomposition for Quasi-Periodic Scattering by Layered Media via Robust Boundary-Integral Equations at All Frequencies

Local Multiple Traces Formulation for Electromagnetics: Stability and Preconditioning for Smooth Geometries

Time-Domain Multiple Traces Boundary Integral Formulation for Acoustic Wave Scattering in 2D

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