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Removal of spurious solutions encountered in Helmholtz scattering resonance computations in R^d
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0002-0143-5554
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs in two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can therefore be applied to graded material properties as well as piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to remove spurious solutions in electromagnetic and acoustic scattering resonances computations at low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

Keywords [en]
plasmon resonance, acoustic scattering resonances, resonance modes, nonlinear eigenvalue problems, Helmholtz problem, pseudospectrum, PML, DtN, leaky modes, resonant states, quasi-normal modes
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-159153OAI: oai:DiVA.org:umu-159153DiVA, id: diva2:1316675
Available from: 2019-05-20 Created: 2019-05-20 Last updated: 2019-06-13Bibliographically approved
In thesis
1. Reliable hp finite element computations of scattering resonances in nano optics
Open this publication in new window or tab >>Reliable hp finite element computations of scattering resonances in nano optics
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.

The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.

In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.

The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.

Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2019. p. 35
Series
Research report in mathematics, ISSN 1653-0810 ; 67
Keywords
Scattering resonances, Helmholtz problems, pseudospectrum, Lippmann-Schwinger equation, finite element methods, nonlinear eigenvalue problems, spurious solutions
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-159154 (URN)978-91-7855-076-0 (ISBN)
Public defence
2019-06-13, MA121, MIT-huset, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2019-05-23 Created: 2019-05-20 Last updated: 2019-05-21Bibliographically approved

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Araujo-Cabarcas, Juan Carlos

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CiteExportLink to record
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Citation style
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