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eXtended Hybridizable Discontinuous Galerkin (X-HDG) for Void and Bimaterial Problems

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Advances in Discretization Methods

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 12))

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Abstract

A strategy for the Hybridizable Discontinuous Galerkin (HDG) solution of problems with voids, inclusions, free surfaces, and material interfaces is proposed. It is based on an eXtended Finite Element (X-FEM) philosophy with a level-set description of interfaces where the computational mesh is not required to fit the interface (i.e. the boundary). This reduces the cost of mesh generation and, in particular, avoids continuous remeshing for evolving interfaces. Differently to previous proposals for the HDG solution with unfitting meshes, the computational mesh covers the domain in our approach, avoiding extrapolations and ensuring the robustness of the method. The local problem in elements not cut by the interface and the global problem are discretized as usual in HDG. A modified local problem is considered for elements cut by the interface. At every cut element, an auxiliary trace variable on the boundary is introduced, which is eliminated afterwards using interface conditions, keeping the original unknowns and the structure of the local problem solver. The solution is enriched with Heaviside functions in case of bimaterial problems; in case of problems with voids, inclusions, or free surfaces no such enrichment is required. Numerical experiments demonstrate how X-HDG keeps the optimal convergence, superconvergence, and accuracy of HDG with no need of adapting the computational mesh to the interface boundary.

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References

  1. Cheng, K.W., Fries, T.P.: Higher-order XFEM for curved strong and weak discontinuities. Int. J. Numer. Methods Eng. 82 (5), 564–590 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Cockburn, B.: Discontinuous Galerkin methods for computational fluid dynamics. In: Encyclopedia of Computational Mechanics. Fluids, vol. 3, chap. 4 Wiley, New York (2004)

    Google Scholar 

  3. Cockburn, B., Solano, M.: Solving convection-diffusion problems on curved domains by extensions from subdomains. J. Sci. Comput. 59 (2), 512–543 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77 (264), 1887–1916 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2), 1319–1365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.J.: Analysis of HDG methods for Stokes flow. Math. Comput. 80 (274), 723–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cockburn, B., Qiu, W., Shi, K.: Conditions for superconvergence of HDG methods for second-order elliptic problems. Math. Comput. 81 (279), 1327–1353 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Qiu, W., Solano, M.: A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity. Math. Comput. 83 (286), 665–699 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dreau, K., Chevaugeon, N., Moës, N.: Studied X-FEM enrichment to handle material interfaces with higher order finite element. Comput. Methods Appl. Mech. Eng. 199 (29–32), 1922–1936 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fries, T.P., Belytschko, T.: The extended∕generalized finite element method: An overview of the method and its applications. Int. J. Numer. Methods Eng. 84 (3), 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Giorgiani, G., Fernández-Méndez, S., Huerta, A.: Hybridizable Discontinuous Galerkin p-adaptivity for wave propagation problems. Int. J. Numer. Methods Fluids 72 (12), 1244–1262 (2013)

    Article  MathSciNet  Google Scholar 

  12. Giorgiani, G., Modesto, D., Fernández-Méndez, S., Huerta, A.: High-order continuous and discontinuous Galerkin methods for wave problems. Int. J. Numer. Methods Fluids 73 (10), 883–903 (2013)

    MathSciNet  Google Scholar 

  13. Giorgiani, G., Fernández-Méndez, S., Huerta, A.: Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier–Stokes equations. Comput. Fluids 98, 196–208 (2014)

    Article  MathSciNet  Google Scholar 

  14. Hesthaven, J., Warburton, T.: Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181 (1), 186–221 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Huerta, A., Angeloski, A., Roca, X., Peraire, J.: Efficiency of high-order elements for continuous and discontinuous Galerkin methods. Int. J. Numer. Methods Eng. 96 (9), 529–560 (2013)

    Article  MathSciNet  Google Scholar 

  16. Huynh, L., Nguyen, N., Peraire, J., Khoo, B.: A high-order hybridizable discontinuous Galerkin method for elliptic interface problems. Int. J. Numer. Methods Eng. 93, 183–200 (2012)

    Article  MathSciNet  Google Scholar 

  17. Kirby, R., Sherwin, S.J., Cockburn, B.: To CG or to HDG: A comparative study. J. Sci. Comput. 51 (1), 183–212 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Montlaur, A., Fernández-Méndez, S., Huerta, A.: Discontinuous Galerkin methods for the Stokes equations using divergence-free approximations. Int. J. Numer. Methods Fluids 57 (9), 1071–1092 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nguyen, N., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Eng. 199 (9–12), 582–597 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations. J. Comput. Phys. 230 (4), 1147–1170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peraire, J., Persson, P.O.: The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput. 30 (4), 1806–1824 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sala-Lardies, E., Fernández-Méndez, S., Huerta, A.: Optimally convergent high-order X-FEM for problems with voids and inclusions. In: ECCOMAS 2012: 6th European Congress on Computational Methods in Applied Sciences and Engineering. Programme book of abstracts, September 10–14, 2012, Vienna, Austria, pp. 1–14 (2012)

    Google Scholar 

  23. Wang, B., Khoo, B.C.: Hybridizable discontinuous Galerkin method (HDG) for Stokes interface flow. J. Comput. Phys. 247, 262–278 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the DAFOH2 project (Ministerio de Economia y Competitividad, MTM2013-46313-R), and the Erasmus Mundus Joint Doctorate SEED project (European Commission).

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Authors and Affiliations

  1. Universitat Politècnica de Catalunya, Barcelona, Spain

    Ceren Gürkan, Esther Sala-Lardies & Sonia Fernández-Méndez

  2. Technische Universität München, München, Germany

    Martin Kronbichler

Authors
  1. Ceren Gürkan
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  2. Esther Sala-Lardies
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  3. Martin Kronbichler
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  4. Sonia Fernández-Méndez
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Corresponding author

Correspondence to Ceren Gürkan .

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Editors and Affiliations

  1. Structural, Building, Geotechnical Eng., Politecnico di Torino, Torino, Italy

    Giulio Ventura

  2. Department of Engineering, University of Ferrara, Ferrara, Italy

    Elena Benvenuti

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Gürkan, C., Sala-Lardies, E., Kronbichler, M., Fernández-Méndez, S. (2016). eXtended Hybridizable Discontinuous Galerkin (X-HDG) for Void and Bimaterial Problems. In: Ventura, G., Benvenuti, E. (eds) Advances in Discretization Methods. SEMA SIMAI Springer Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-41246-7_5

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