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Hansbo, P., Larson, M. G. & Larsson, K. (2018). Cut Finite Element Methods for Linear Elasticity Problems. In: Stéphane P. A. Bordas; Erik Burman; Mats G. Larson; Maxim A. Olshanskii (Ed.), Geometrically Unfitted Finite Element Methods and Applications: Proceedings of the UCL Workshop 2016 (pp. 25-63). Springer
Open this publication in new window or tab >>Cut Finite Element Methods for Linear Elasticity Problems
2018 (English)In: Geometrically Unfitted Finite Element Methods and Applications: Proceedings of the UCL Workshop 2016 / [ed] Stéphane P. A. Bordas; Erik Burman; Mats G. Larson; Maxim A. Olshanskii, Springer, 2018, p. 25-63Chapter in book (Refereed)
Abstract [en]

We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms.

Place, publisher, year, edition, pages
Springer, 2018
Series
Lecture Notes in Computational Science and Engineering, ISSN 1439-7358, E-ISSN 2197-7100 ; 121
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-145934 (URN)10.1007/978-3-319-71431-8 (DOI)978-3-319-71430-1 (ISBN)978-3-319-71431-8 (ISBN)
Available from: 2018-03-22 Created: 2018-03-22 Last updated: 2018-06-09
Burman, E., Elfverson, D., Hansbo, P., Larson, M. & Larsson, K. (2018). Shape optimization using the cut finite element method. Computer Methods in Applied Mechanics and Engineering, 328, 242-261
Open this publication in new window or tab >>Shape optimization using the cut finite element method
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2018 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 242-261Article in journal (Refereed) Published
Abstract [en]

We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity field using a transport equation. The velocity field is the largest decreasing direction of the shape derivative that satisfies a certain regularity requirement and the computation of the shape derivative is based on a volume formulation. Using the cut finite element method no re-meshing is required when updating the domain and we may also use higher order finite element approximations. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements at the boundary, which provides control of the variation of the solution in the vicinity of the boundary. We implement and illustrate the performance of the method in the two-dimensional case, considering both triangular and quadrilateral meshes as well as finite element spaces of different order.

Place, publisher, year, edition, pages
Lausanne: Elsevier, 2018
Keywords
CutFEM, Shape optimization, Level-set, Fictitious domain method, Linear elasticity
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-140281 (URN)10.1016/j.cma.2017.09.005 (DOI)000416218500011 ()
Funder
Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708eSSENCE - An eScience Collaboration
Available from: 2017-10-04 Created: 2017-10-04 Last updated: 2018-06-09Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-7352-1550

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