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Goal-oriented model adaptivity for viscous incompressible flows

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Abstract

In van Opstal et al. (Comput Mech 50:779–788, 2012) airbag inflation simulations were performed where the flow was approximated by Stokes flow. Inside the intricately folded initial geometry the Stokes assumption is argued to hold. This linearity assumption leads to a boundary-integral representation, the key to bypassing mesh generation and remeshing. It therefore enables very large displacements with near-contact. However, such a coarse assumption cannot hold throughout the domain, where it breaks down one needs to revert to the original model. The present work formalizes this idea. A model adaptive approach is proposed, in which the coarse model (a Stokes boundary-integral equation) is locally replaced by the original high-fidelity model (Navier–Stokes) based on a-posteriori estimates of the error in a quantity of interest. This adaptive modeling framework aims at taking away the burden and heuristics of manually partitioning the domain while providing new insight into the physics. We elucidate how challenges pertaining to model disparity can be addressed. Essentially, the solution in the interior of the coarse model domain is reconstructed as a post-processing step. We furthermore present a two-dimensional numerical experiments to show that the error estimator is reliable.

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Notes

  1. This is similar to a residual evaluation at a function with support on (part of) \({\partial \varOmega }_D\), see also [27].

  2. Except possibly locally if one chooses to define the test function \(({\varvec{v}},q)\) using a mesh.

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Acknowledgments

The meshes presented in this manuscript were generated with Gmsh [12] and implementation was based on the nutils.org framework. This research is supported by the Dutch Technology Foundation STW, which is part of the Netherlands Organisation for Scientific Research (NWO) and partly funded by the Ministry of Economic Affairs, Agriculture and Innovation (Project number 10476). Paul T. Bauman was supported by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615]. Serge Prudhomme is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He is also a participant of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    T. M. van Opstal

  2. Mechanical and Aerospace Engineering, Computational and Data-Enabled Science and Engineering, University at Buffalo, State University of New York, 318 Jarvis Hall, Buffalo, NY, 14260-440, USA

    P. T. Bauman

  3. Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal, C.P. 6079, succ. Centre-Ville, Montréal, QC, H3C 3A7, Canada

    S. Prudhomme

  4. Department of Mechanical Engineering, Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, Netherlands

    E. H. van Brummelen

Authors
  1. T. M. van Opstal
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  2. P. T. Bauman
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  3. S. Prudhomme
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  4. E. H. van Brummelen
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Correspondence to T. M. van Opstal.

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van Opstal, T.M., Bauman, P.T., Prudhomme, S. et al. Goal-oriented model adaptivity for viscous incompressible flows. Comput Mech 55, 1181–1190 (2015). https://doi.org/10.1007/s00466-015-1146-1

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  • DOI: https://doi.org/10.1007/s00466-015-1146-1

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