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The second-generation Shifted Boundary Method and its numerical analysis

Publication ,  Journal Article
Atallah, NM; Canuto, C; Scovazzi, G
Published in: Computer Methods in Applied Mechanics and Engineering
December 1, 2020

Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L2-estimates without the cumbersome assumption – of earlier proofs – that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We complement these theoretical developments with numerical experiments in two and three dimensions.

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Published In

Computer Methods in Applied Mechanics and Engineering

DOI

ISSN

0045-7825

Publication Date

December 1, 2020

Volume

372

Related Subject Headings

  • Applied Mathematics
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 01 Mathematical Sciences
 

Citation

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Atallah, N. M., Canuto, C., & Scovazzi, G. (2020). The second-generation Shifted Boundary Method and its numerical analysis. Computer Methods in Applied Mechanics and Engineering, 372. https://doi.org/10.1016/j.cma.2020.113341
Atallah, N. M., C. Canuto, and G. Scovazzi. “The second-generation Shifted Boundary Method and its numerical analysis.” Computer Methods in Applied Mechanics and Engineering 372 (December 1, 2020). https://doi.org/10.1016/j.cma.2020.113341.
Atallah NM, Canuto C, Scovazzi G. The second-generation Shifted Boundary Method and its numerical analysis. Computer Methods in Applied Mechanics and Engineering. 2020 Dec 1;372.
Atallah, N. M., et al. “The second-generation Shifted Boundary Method and its numerical analysis.” Computer Methods in Applied Mechanics and Engineering, vol. 372, Dec. 2020. Scopus, doi:10.1016/j.cma.2020.113341.
Atallah NM, Canuto C, Scovazzi G. The second-generation Shifted Boundary Method and its numerical analysis. Computer Methods in Applied Mechanics and Engineering. 2020 Dec 1;372.
Journal cover image

Published In

Computer Methods in Applied Mechanics and Engineering

DOI

ISSN

0045-7825

Publication Date

December 1, 2020

Volume

372

Related Subject Headings

  • Applied Mathematics
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 01 Mathematical Sciences