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Model Reduction of Parametrized Systems pp 183–199Cite as

Efficient Reduction of PDEs Defined on Domains with Variable Shape

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Part of the book series: MS&A ((MS&A,volume 17))

Abstract

In this work we propose a new, general and computationally cheap way to tackle parametrized PDEs defined on domains with variable shape when relying on the reduced basis method. We easily describe a domain by boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving a linear elasticity problem. The proposed procedure is built over a two-stages reduction: (1) first, we construct a reduced basis approximation for the mesh motion problem; (2) then, we generate a reduced basis approximation of the state problem, relying on finite element snapshots evaluated over a set of reduced deformed configurations. A Galerkin-POD method is employed to construct both reduced problems, although this choice is not restrictive. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing to treat geometrical deformations in a non-intrusive, efficient and purely algebraic way. In order to assess the numerical performances of the proposed technique, we address the solution of a parametrized (direct) Helmholtz scattering problem where the parameters describe both the shape of the obstacle and other relevant physical features. Thanks to its easiness and efficiency, the methodology described in this work looks promising also in view of reducing more complex problems.

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Notes

  1. 1.

    These CPU times refer to computations performed on a workstation with Intel Core i5-2400S CPU and 16 GB of RAM. The implementation of the mentioned algorithms has been done using the redbKIT library (http://redbkit.github.io/redbKIT/), developed in the MATLAB®;environment and distributed under BSD 2-clause license.

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Authors and Affiliations

  1. CMCS-MATHICSE-SB, Ecole Polytechnique Fédérale de Lausanne, Station 8, CH-1015, Lausanne, Switzerland

    Andrea Manzoni & Federico Negri

Authors
  1. Andrea Manzoni
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  2. Federico Negri
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Correspondence to Andrea Manzoni .

Editor information

Editors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Institute for Computational and Applied Mathematics, University of Münster , Münster, Germany

    Mario Ohlberger

  3. Massachusetts Institute of Technology, MIT , Cambridge, Massachusetts, USA

    Anthony Patera

  4. SISSA MathLab, International School for Adv. Studies , Trieste, Italy

    Gianluigi Rozza

  5. Institute of Numerical Mathematics, Ulm University , Ulm, Germany

    Karsten Urban

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Manzoni, A., Negri, F. (2017). Efficient Reduction of PDEs Defined on Domains with Variable Shape. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_12

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