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Cut finite element methods on parametric multipatch surfaces
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2019 (English)Licentiate thesis, comprehensive summary (Other academic)
Place, publisher, year, edition, pages
Umeå: Umeå Universitet , 2019. , p. 23
Series
Research report in mathematics, ISSN 1653-0810
Keywords [en]
Cut finite element method, Nitsche method, a priori error estimation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-159748ISBN: 978-91-7855-019-7 (print)OAI: oai:DiVA.org:umu-159748DiVA, id: diva2:1320637
Presentation
2019-06-14, N420, Naturvetarhuset, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2019-06-05 Created: 2019-06-05 Last updated: 2019-08-21Bibliographically approved
List of papers
1. Cut finite element methods for elliptic problems on multipatch parametric surfaces
Open this publication in new window or tab >>Cut finite element methods for elliptic problems on multipatch parametric surfaces
2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 324, p. 366-394Article in journal (Refereed) Published
Abstract [en]

We develop a finite element method for the Laplace–Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a subdomain of the unit square which is bounded by a number of smooth trim curves. A patchwise tensor product mesh is constructed by using a structured mesh in the reference domain. Since the patches are trimmed we obtain cut elements in the vicinity of the interfaces. We discretize the Laplace–Beltrami operator using a cut finite element method that utilizes Nitsche’s method to enforce continuity at the interfaces and a consistent stabilization term to handle the cut elements. Several quantities in the method are conveniently computed in the reference domain where the mappings impose a Riemannian metric. We derive a priori estimates in the energy and L2 norm and also present several numerical examples confirming our theoretical results.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Cut finite elements, Fictitious domain, Nitsche's method, A priori error estimates, Multipatch surface, Laplace-Beltrami operator
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-138015 (URN)10.1016/j.cma.2017.06.018 (DOI)000408032700016 ()
Available from: 2017-08-02 Created: 2017-08-02 Last updated: 2019-06-05Bibliographically approved
2. A Nitsche method for elliptic problems on composite surfaces
Open this publication in new window or tab >>A Nitsche method for elliptic problems on composite surfaces
2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, p. 505-525Article in journal (Refereed) Published
Abstract [en]

We develop a finite element method for elliptic partial differential equations on so called composite surfaces that are built up out of a finite number of surfaces with boundaries that fit together nicely in the sense that the intersection between any two surfaces in the composite surface is either empty, a point, or a curve segment, called an interface curve. Note that several surfaces can intersect along the same interface curve. On the composite surface we consider a broken finite element space which consists of a continuous finite element space at each subsurface without continuity requirements across the interface curves. We derive a Nitsche type formulation in this general setting and by assuming only that a certain inverse inequality and an approximation property hold we can derive stability and error estimates in the case when the geometry is exactly represented. We discuss several different realizations, including so called cut meshes, of the method. Finally, we present numerical examples.

Place, publisher, year, edition, pages
Lausanne: Elsevier, 2017
Keywords
Nitsche method, Composite surfaces, Laplace-Beltrami operator, A priori error estimates
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-139526 (URN)10.1016/j.cma.2017.08.033 (DOI)000413322300022 ()
Funder
Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708eSSENCE - An eScience CollaborationSwedish Foundation for Strategic Research , AM13-0029
Available from: 2017-09-15 Created: 2017-09-15 Last updated: 2019-06-05Bibliographically approved
3. Graded Parametric CutFEM and CutIGA for Elliptic Boundary Value Problems in Domains with Corners
Open this publication in new window or tab >>Graded Parametric CutFEM and CutIGA for Elliptic Boundary Value Problems in Domains with Corners
2019 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 354, p. 331-350Article in journal (Refereed) Published
Abstract [en]

We develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neighborhood of a point at the boundary. Our approach is based on identification of a suitable mapping that grades the mesh towards the singularity. In particular, this mapping may be chosen without identifying the opening angle at the corner. We employ cut finite elements together with Nitsche boundary conditions and stabilization in the vicinity of the boundary. We prove that the method is stable and convergent of optimal order in the energy norm and L2 norm. This is achieved by mapping to the reference domain where we employ a structured mesh.

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
Corner singularities, a priori error estimates, Cut finite element method, Cut isogeometric analysis
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-159747 (URN)10.1016/j.cma.2019.05.024 (DOI)
Funder
Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 2013-4708Swedish Research Council, 2017-03911eSSENCE - An eScience Collaboration, -
Available from: 2019-06-05 Created: 2019-06-05 Last updated: 2019-06-12Bibliographically approved

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Jonsson, Tobias

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