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Cut finite element modeling of linear membranes
Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.ORCID iD: 0000-0001-7352-1550
Umeå University.
2016 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, 98-111 p.Article in journal (Refereed) Published
Abstract [en]

We construct a cut finite element method for the membrane elasticity problem on an embedded mesh using tangential differential calculus, i.e., with the equilibrium equations pointwise projected onto the tangent plane of the surface to create a pointwise planar problem in the tangential direction. Both free membranes and membranes coupled to 3D elasticity are considered. The discretization of the membrane comes from a Galerkin method using the restriction of 3D basis functions (linear or trilinear) to the surface representing the membrane. In the case of coupling to 3D elasticity, we view the membrane as giving additional stiffness contributions to the standard stiffness matrix resulting from the discretization of the three-dimensional continuum.

Place, publisher, year, edition, pages
2016. Vol. 310, 98-111 p.
Keyword [en]
Cut finite element method, Embedded membrane, Membrane shell, Tangential derivative
National Category
Applied Mechanics Computer Engineering
Identifiers
URN: urn:nbn:se:hj:diva-28420DOI: 10.1016/j.cma.2016.05.018ISI: 000384859400005Scopus ID: 2-s2.0-84979299087OAI: oai:DiVA.org:hj-28420DiVA: diva2:875536
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-04-13Bibliographically approved
In thesis
1. Finite element methods on surfaces
Open this publication in new window or tab >>Finite element methods on surfaces
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The purpose of this thesis is to improve numerical simulations of surface problems. Two novel computational concepts are analyzed and applied on two surface problems; minimal surface problems and elastic membrane problems. The concept of tangential projection implies that direct computation on the surface is made possible compared to the classical approach of mapping 2D parametric surfaces to 3D surfaces by means of differential geometry operators. The second concept presented is the cut finite element method, in which the basic idea of discretization is to embed the d- 1-dimensional surface in a d-dimensional mesh and use the basis functions of a higher dimensional mesh but integrate over the surface. The aim of this thesis is to present the basics of the two main approaches and to provide details on the implementation.

Place, publisher, year, edition, pages
Jönköping: Jönköping University, School of Engineering, 2015. 41 p.
Series
JTH Dissertation Series, 12
National Category
Mechanical Engineering Computer Engineering
Identifiers
urn:nbn:se:hj:diva-28424 (URN)978-91-87289-13-2 (ISBN)
Supervisors
Funder
Swedish Research Council, 2011-4992
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2015-12-01Bibliographically approved
2. Finite element methods for surface problems
Open this publication in new window or tab >>Finite element methods for surface problems
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The purpose of this thesis is to further develop numerical methods for solving surface problems by utilizing tangential calculus and the trace finite element method. Direct computation on the surface is possible by the use of tangential calculus, in contrast to the classical approach of mapping 2D parametric surfaces to 3D surfaces by means of differential geometry operators. Using tangential calculus, the problem formulation is only dependent on the position and normal vectors of the 3D surface. Tangential calculus thus enables a clean, simple and inexpensive formulation and implementation of finite element methods for surface problems. Meshing techniques are greatly simplified from the end-user perspective by utilizing an unfitted finite element method called the Trace Finite Element Method, in which the basic idea is to embed the surface in a higher dimensional mesh and use the shape functions of this background mesh for the discretization of the partial differential equation. This method makes it possible to model surfaces implicitly and solve surface problems without the need for expensive meshing/re-meshing techniques especially for moving surfaces or surfaces embedded in 3D solids, so called embedded interface problems. Using these two approaches, numerical methods for solving three surface problems are proposed: 1) minimal surface problems, in which the form that minimizes the mean curvature was computed by iterative update of a level-set function discretized using TraceFEM and driven by advection, for which the velocity field was given by the mean curvature flow, 2) elastic membrane problems discretized using linear and higher order TraceFEM, which makes it straightforward to embed complex geometries of membrane models into an elastic bulk for reinforcement and 3) stabilized, accurate vertex normal and mean curvature estimation with local refinement on triangulated surfaces. In this thesis the basics of the two main approaches are presented, some aspects such as stabilization and surface reconstruction are further developed, evaluated and numerically analyzed, details on implementations are provided and the current state of work is presented.

Place, publisher, year, edition, pages
Jönköping: Jönköping University, School of Engineering, 2017. 144 p.
Series
JTH Dissertation Series, 022
Keyword
trace finite element method, membrane, mean curvature, level-set method
National Category
Mechanical Engineering Computer Engineering
Identifiers
urn:nbn:se:hj:diva-35369 (URN)978-91-87289-23-1 (ISBN)
Public defence
2017-05-12, E1405, Tekniska Högskolan, Jönköping, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2011-4992
Available from: 2017-04-13 Created: 2017-04-13 Last updated: 2017-04-13Bibliographically approved

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The full text will be freely available from 2018-07-06 08:00
Available from 2018-07-06 08:00

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