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Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines

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Abstract

This paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an extended finite element method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

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Notes

  1. Note that the union background mesh is not attained via a set union of the separate hierarchically refined meshes. Instead, elements in this set union that fully contain smaller elements in the set union do not belong to the union background mesh.

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Acknowledgements

The first, second, fourth, and fifth authors acknowledge the support for this work from the Defense Advanced Research Projects Agency (DARPA) under the TRADES program (Agreement HR0011-17-2-0022). The first author acknowledges partial auspice of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (LLNL-JRNL-842737). The third and fifth authors acknowledge the support of Sandia National Laboratories under PO 2120843. The fourth and fifth authors acknowledge the support of the National Science foundation under Grant 2104106. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organization.

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Authors and Affiliations

  1. Computational Engineering Division, Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA, 94550, USA

    Mathias Schmidt

  2. Department of Precision and Microsystems Engineering, Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands

    Lise Noël

  3. Department of Aerospace Engineering Sciences, Aerospace Mechanics Research Center, University of Colorado Boulder, 3775 Discovery Dr, Boulder, CO, 80309-0429, USA

    Keenan Doble, John A. Evans & Kurt Maute

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Schmidt, M., Noël, L., Doble, K. et al. Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines. Comput Mech 71, 1179–1203 (2023). https://doi.org/10.1007/s00466-023-02306-x

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