ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics

Buhr, Andreas; Engwer, Christian; Ohlberger, Mario; Rave, Stephan

doi:10.1007/978-3-319-58786-8_9

Cited by 6 publications

(8 citation statements)

References 24 publications

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“…In this multi-query setting, the reduced basis method can perform well. Experiments have shown two dimensional time-harmonic Maxwell's to be amenable to localized model reduction [2]. However, Galerkin projection of an inf-sup stable problem is not guaranteed to be stable.…”

Section: Abstract Westfälische Wilhelms-universität Münstermentioning

confidence: 99%

Localized Reduced Basis Methods for Time Harmonic Maxwell’s Equations

Buhr

Ohlberger

Rave

2018

Preprint

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WESTFÄLISCHE WILHELMS-UNIVERSITÄT MÜNSTERLocalized model order reduction methods have attracted significant attention during the last years. They have favorable parallelization properties and promise to perform well on cloud architectures, which become more and more commonplace. We introduced ArbiLoMod [1], a localized reduced basis method targeted at the important use case of changing problem definition, wherein the changes are of local nature. This is a common situation in simulation software used by engineers optimizing a CAD model. An especially interesting application is the simulation of electromagnetic fields in printed circuit boards, which is necessary to design high frequency electronics. The simulation of the electromagnetic fields can be done by solving the time-harmonic Maxwell's equations, which results in a parameterized, inf-sup stable problem which has to be solved for many parameters. In this multi-query setting, the reduced basis method can perform well. Experiments have shown two dimensional time-harmonic Maxwell's to be amenable to localized model reduction [2]. However, Galerkin projection of an inf-sup stable problem is not guaranteed to be stable. Existing stabilization methods for the reduced basis method involve global computations and are thus not applicable in a localized setting. Replacing the Galerkin projection with the minimization of a localized a posteriori error estimator provides a stable reduction for inf-sup stable projects which retains all the advantageous properties of localized model order reduction. It allows for an offline-online decomposition and requires no global computations in the unreduced space.

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Section: Abstract Westfälische Wilhelms-universität Münstermentioning

confidence: 99%

Localized Reduced Basis Methods for Time Harmonic Maxwell’s Equations

Buhr

Ohlberger

Rave

2018

Preprint

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“…Some of the basic ideas of ArbiLo-Mod were already published by the authors [7,10]. First results for electrodynamics were published in [9].…”

Section: Existing Approachesmentioning

confidence: 99%

ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications

Buhr¹,

Engwer²,

Ohlberger³

et al. 2017

SIAM J. Sci. Comput.

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Engineers manually optimizing a structure using Finite Element based simulation software often employ an iterative approach where in each iteration they change the structure slightly and resimulate. Standard Finite Element based simulation software is usually not well suited for this workflow, as it restarts in each iteration, even for tiny changes. In settings with complex local microstructure, where a fine mesh is required to capture the geometric detail, localized model reduction can improve this workflow. To this end, we introduce ArbiLoMod, a method which allows fast recomputation after arbitrary local modifications. It employs a domain decomposition and a localized form of the Reduced Basis Method for model order reduction. It assumes that the reduced basis on many of the unchanged domains can be reused after a localized change. The reduced model is adapted when necessary, steered by a localized error indicator. The global error introduced by the model order reduction is controlled by a robust and efficient localized a posteriori error estimator, certifying the quality of the result. We demonstrate ArbiLoMod for a coercive, parameterized example with changing structure.

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“…Magnitude and scaling behavior of c pu depend on the choice of the partition of unity and the norm of the spaces. See Buhr et al (2017) for more details.…”

Section: Runtimesmentioning

confidence: 99%

“…Online enrichment is the extension of an existing reduced solution space based on the solution of the reduced model. It is used in many localized model order reduction techniques: ArbiLoMod, introduced in Buhr et al (2017), employs online enrichment. For the Localized Reduced Basis Multiscale Method (LRBMS) which was introduced in Albrecht et al (2012), online enrichment was presented in Albrecht and Ohlberger (2013) and in Ohlberger and Schindler (2015).…”

Section: Introductionmentioning

confidence: 99%

Exponential Convergence of Online Enrichment in Localized Reduced Basis Methods

Buhr

2018

IFAC-PapersOnLine

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Online enrichment is the extension of a reduced solution space based on the solution of the reduced model. Procedures for online enrichment were published for many localized model order reduction techniques. We show that residual based online enrichment on overlapping domains converges exponentially. Furthermore, we present an optimal enrichment strategy which couples the global reduced space with a local fine space. Numerical experiments on the two dimensional stationary heat equation with high contrast and channels confirm and illustrate the results.

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ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics

Cited by 6 publications

References 24 publications

Localized Reduced Basis Methods for Time Harmonic Maxwell’s Equations

Localized Reduced Basis Methods for Time Harmonic Maxwell’s Equations

ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications

Exponential Convergence of Online Enrichment in Localized Reduced Basis Methods

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