An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry

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Abstract

Generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries. The paper at hand introduces a B-spline finite element method, which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved. Its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement, which adds several levels of local basis functions in the close vicinity of interfaces, but unfitted to their exact location, so that a simple regular grid of knot span elements can be maintained. Numerical experiments show that an hp-refinement strategy, which simultaneously increases the polynomial degree of the B-spline basis and the levels of refinement around interfaces, achieves exponential rates of convergence despite the presence of discontinuities. It is also demonstrated that the hierarchical B-spline FEM can be used to transfer the recently introduced Finite Cell concept to geometrically nonlinear problems. Its computational performance, imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one, two and three dimensions, and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite.

Original languageEnglish (US)
Pages (from-to)3358-3380
Number of pages23
JournalComputer Methods in Applied Mechanics and Engineering
Volume200
Issue number47-48
DOIs
StatePublished - Nov 1 2011

Bibliographical note

Funding Information:
The first author gratefully acknowledges financial support from the Centre of Advanced Computing (MAC) and the International Graduate School of Science and Engineering (IGSSE) at the Technische Universität München. The authors thank J. Frisch and R.-P. Mundani for their valuable support during processing of the large voxel data sets, and A. Düster, S. Kollmannsberger, M. Ruess and Z. Yosibash for helpful comments.

Keywords

  • Complex geometries
  • Finite Cell Method
  • Hierarchical refinement
  • High-order B-spline finite elements
  • Interface problems
  • Regular grid

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