10:40
MS32: Recent advances in modelling and numerics of wave phenomena (Part 2)
Chair: Vanja Nikolic
10:40
25 mins
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Parallel Controllability Methods for the Helmholtz equation
Marcus Grote, Frédéric Nataf, Jet Hoe Tang, Pierre-Henri Tournier
Abstract: The Helmholtz equation is notoriously difficult to solve with standard numerical
methods, increasingly so, in fact, at higher frequencies. Controllability methods
instead transform the problem back to the time-domain, where they seek
the time-harmonic solution of the corresponding time-dependent wave equation.
Two different approaches are considered here based either on the first (mixed)
or second-order formulation of the wave equation. Both are extended to general
boundary-value problems governed by the Helmholtz equation and lead
to robust and inherently parallel algorithms. Numerical results illustrate the
accuracy, convergence and strong scalability of controllability methods for the
solution of high frequency Helmholtz equations with up to a billion unknowns
on massively parallel architectures.
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11:05
25 mins
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Multi-Trace FEM-BEM formulation for acoustic scattering by composite objects
Marcella Bonazzoli, Xavier Claeys
Abstract: This talk is about the scattering of an acoustic wave by an object composed of piecewise homogenous parts and an arbitrarily heterogeneous part. We propose and analyze a formulation that couples, adopting a Costabel-type approach \cite{Costabel:1987}, boundary integral equations for the homogeneous subdomains with domain variational formulations for the heterogenous subdomain.
This is an extension of Costabel FEM-BEM coupling to a multi-domain configuration, with junctions points allowed, i.e. points where three or more subdomains abut. Our formulation is based on the multi-trace formalism, which was introduced in \cite{ClHi:MTF:2013,ClHi:impen:2015} for acoustic scattering by piecewise homogeneous objects; here we allow the wavenumber to vary arbitrarily in a part of the domain.
We prove that the bilinear form associated with the proposed formulation satisfies a Gårding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.
This work was supported by the French National Research Agency (ANR) in the framework of the project NonlocalDD, ANR-15-CE23-0017-01.
\begin{thebibliography}{9}
\bibitem{ClHi:MTF:2013}
X. Claeys and R. Hiptmair, Multi-Trace Boundary Integral Formulation for Acoustic Scattering by Composite Structures, \emph{Communications on Pure and Applied Mathematics}, 66(8):1163--1201, 2013.
\bibitem{ClHi:impen:2015}
X. Claeys and R. Hiptmair, Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects, \emph{Integral Equations and Operator Theory}, 81 (2015), no. 2, pp. 151--189.
\bibitem{Costabel:1987}
M. Costabel, Symmetric Methods for the Coupling of Finite Elements and Boundary Elements. In: \emph{Boundary Elements IX, Brebbia C.A., Wendland W.L., Kuhn G. (eds)}, vol 9/1 (1987).
\end{thebibliography}
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11:30
25 mins
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Computational multiscale methods for scattering in heterogeneous high-contrast media
Barbara Verfürth
Abstract: (Wave) scattering and propagation in heterogeneous high-contrast media experiences a rising interest because such materials can develop unusual optical or acoustic properties, such as a negative refractive index or wave guiding, which are important in practical applications.
From the numerical side, the solution of such problems exhibits several challenges: (i) spatially rough PDE coefficients and (ii) the high-frequency regime of wave propagation. Through the high contrast in the coefficients even moderate wave numbers can suddenly imply a high frequency regime in certain parts of the computational domain, thereby even amplifying the aforementioned numerical challenges.
In this talk, we discuss numerical multiscale methods to deal with (combinations of) rough coefficients with high contrast and the high-frequency regime. Those methods do not need to resolve all discontinuities and oscillations of the coefficients, but rather solve effective macroscopic models. Those models do not have to be derived analytically, but are automatically extracted by the method itself based on the solution of local fine-scale problems.
Periodicity and scale separation can be exploited to reduce the number of fine-scale problems \cite{OV, Ver}, but we also discuss how to deal with general rough (multiscale) coefficients \cite{PV}.
Rigorous numerical analysis allows to estimate the discretization error a priori and is confirmed by numerical examples.
Furthermore, the numerical experiments also illustrate some astonishing effects of wave scattering in heterogeneous high-contrast media, such as band gaps, flat lenses, or wave guides.
\begin{thebibliography}{100}
\bibitem{OV} M.~Ohlberger, B.~Verf\"urth. A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast. \textit{Multiscale Model. Simul.} 16(1), pp.~385--411, 2018.
\bibitem{PV} D.~Peterseim, B.~Verf\"urth. Computational high frequency scattering from high-contrast heterogeneous media. \textit{arXiv preprint} 1902.09935, 2019.
\bibitem{Ver} B.~Verf\"urth. Heterogeneous Multiscale Method for the Maxwell equations with high contrast. \textit{ESAIM M2AN} 53(1), pp.~35--61, 2019.
\end{thebibliography}
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11:55
25 mins
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Nonlinear ultrasound simulation: A priori finite element analysis and self-adaptive absorbing boundary conditions
Vanja Nikoli´
Abstract: Accurate simulation of nonlinear ultrasound offers a path to a better quality of many procedures in industry and medicine, ranging from non-destructive detection of material damages to non-invasive treatments of medical disorders. In this talk, we will discuss the spatial discretization of Westervelt’s acoustic wave equation by means of linear finite elements. The error analysis is based on the Banach fixed-point theorem combined with a priori estimates for a linear wave model with variable coefficients [2]. To reduce spurious reflections of the wave at the boundary of the computational domain, we propose a self-adaptive absorbing technique [1]. Within the method, the angle of incidence of the wave is computed based on the information provided by the wave-field gradient which is readily available in the finite element framework. The absorbing boundary conditions are then updated with the angle values in real time. Numerical experiments will illustrate the accuracy and efficiency of the proposed approach. This a joint work with Markus Muhr and Barbara Wohlmuth (TU Munich).
References
[1] M. Muhr, V. Nikolic, and B. Wohlmuth ´ , Self-adaptive absorbing boundary conditions for quasilinear acoustic wave propagation, Journal of Computational Physics, 388 (2019), pp. 279–299. [2] V. Nikolic and B. Wohlmuth ´ , A priori error estimates for the finite element approximation of Westervelt’s quasilinear acoustic wave equation, preprint, arXiv:1901.08510, (2019).
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