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Graded Parametric CutFEM and CutIGA for Elliptic Boundary Value Problems in Domains with Corners
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0001-8981-0234
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0001-7838-1307
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. Vol. 354, p. 331-350
Keywords [en]
Corner singularities, a priori error estimates, Cut finite element method, Cut isogeometric analysis
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-159747DOI: 10.1016/j.cma.2019.05.024ISI: 000474690000013Scopus ID: 2-s2.0-85066801941OAI: oai:DiVA.org:umu-159747DiVA, id: diva2:1320631
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: 2022-11-15Bibliographically approved
In thesis
1. Cut finite element methods on parametric multipatch surfaces
Open this publication in new window or tab >>Cut finite element methods on parametric multipatch surfaces
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
Cut finite element method, Nitsche method, a priori error estimation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-159748 (URN)978-91-7855-019-7 (ISBN)
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
2. Cut isogeometric methods on trimmed multipatch surfaces
Open this publication in new window or tab >>Cut isogeometric methods on trimmed multipatch surfaces
2022 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Skurna isogeometriska metoder på trimmade multipatchytor
Abstract [en]

Partial differential equations (PDE) on surfaces appear in a variety of applications, such as image processing, modeling of lubrication, fluid flows, diffusion, and transport of surfactants.  In some applications, surfaces are drawn and modeled by using CAD software, giving a very precise patchwise parametric description of the surface. This thesis deals with the development of methods for finding numerical solutions to PDE posed on such parametrically described multipatch surfaces. The thesis consists of an introduction and five papers.

In the first paper, we develop a general framework for the Laplace-Beltrami operator on a patchwise parametric surface. Each patch map induces a Riemannian metric, which we utilize to compute quantities in the simpler reference domain. We use the cut finite element method together with Nitsche’s method to enforce continuity over the interfaces between patches.

In the second paper, we extend the framework to be able to handle geometries that consist of an arrangement of surfaces, i.e., more than two per interface. By using a Kirchhoff's condition this method avoids defining any co-normal to each surface and can deal with sharp edges. This approach is shown to be equivalent to standard Nitsche interface method for flat geometries.

In the third paper, we developed a cut finite element method for elliptic problems with corner singularities. The main idea is to use an appropriate radial map that grades the finite element mesh towards the corner that counter-acts the solution's singularity.

In the fourth paper, we present a new robust isogeometric method for surfaces described by CAD patches with gaps or overlaps. The main approach here is to cover all interfaces with a three-dimensional mesh and then use a hybrid variable in a Nitsche-type formulation to transfer data over the gaps. Using this hybridized approach leads to a convenient and easy to implement method with no restriction on the number of coupled patches per interface.

In the fifth paper, we present a routine to the multipatch isogeometric framework for dealing with singular maps. To exemplify this, we consider a specific type of singular parametrization which essentially maps a square onto a triangle. One part of the boundary of the square will be transformed into a single point and the metric tensor becomes singular as we approach this boundary. In this work we propose a regularization procedure which is based on eigenvalue decomposition of the metric tensor.

Place, publisher, year, edition, pages
Umeå: Umeå University, 2022. p. 36
Series
Research report in mathematics, ISSN 1653-0810 ; 73
Keywords
cut finite element method, isogeometric analysis, Nitsche's method, a priori error estimation, parametric geometry, interface problem, surface CAD
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-201017 (URN)978-91-7855-920-6 (ISBN)978-91-7855-921-3 (ISBN)
Public defence
2022-12-08, Nat.D.360, Naturvetarhuset, 13:15 (English)
Opponent
Supervisors
Available from: 2022-11-17 Created: 2022-11-14 Last updated: 2022-11-15Bibliographically approved

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Jonsson, TobiasLarson, Mats G.Larsson, Karl

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