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Finite element methods for surface problems
Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
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 [en]
trace finite element method, membrane, mean curvature, level-set method
National Category
Mechanical Engineering Computer Engineering
Identifiers
URN: urn:nbn:se:hj:diva-35369ISBN: 978-91-87289-23-1 (print)OAI: oai:DiVA.org:hj-35369DiVA: diva2:1088644
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
List of papers
1. Minimal surface computation using a finite element method on an embedded surface
Open this publication in new window or tab >>Minimal surface computation using a finite element method on an embedded surface
2015 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, no 7, 502-512 p.Article in journal (Refereed) Published
Abstract [en]

We suggest a finite element method for finding minimal surfaces based on computing a discrete Laplace–Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is carried out on the piecewise planar isosurface using the shape functions from the background three-dimensional mesh used to represent the distance function. A recently suggested stabilized scheme for finite element approximation of the mean curvature vector is a crucial component of the method.

Keyword
mean curvature; Laplace–Beltrami operator; level set; ghost penalty stabilization
National Category
Applied Mechanics Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-28151 (URN)10.1002/nme.4892 (DOI)000362552500003 ()2-s2.0-84928151825 (Scopus ID)
Note

Special Issue: Advances in Embedded Interface Methods

Available from: 2015-10-09 Created: 2015-10-09 Last updated: 2017-04-13Bibliographically approved
2. Cut finite element modeling of linear membranes
Open this publication in new window or tab >>Cut finite element modeling of linear membranes
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.

Keyword
Cut finite element method, Embedded membrane, Membrane shell, Tangential derivative
National Category
Applied Mechanics Computer Engineering
Identifiers
urn:nbn:se:hj:diva-28420 (URN)10.1016/j.cma.2016.05.018 (DOI)000384859400005 ()2-s2.0-84979299087 (Scopus ID)
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-04-13Bibliographically approved
3. Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement
Open this publication in new window or tab >>Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order L2-convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using L2-projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare or results to previously proposed approaches.

National Category
Mechanical Engineering Computer Engineering
Identifiers
urn:nbn:se:hj:diva-35367 (URN)
Available from: 2017-04-13 Created: 2017-04-13 Last updated: 2017-04-13Bibliographically approved
4. Numerical error estimation for a TraceFEM membrane and distance function on P1 and P2 tetrahedra
Open this publication in new window or tab >>Numerical error estimation for a TraceFEM membrane and distance function on P1 and P2 tetrahedra
(English)Manuscript (preprint) (Other academic)
National Category
Mechanical Engineering Computer Engineering
Identifiers
urn:nbn:se:hj:diva-35368 (URN)
Available from: 2017-04-13 Created: 2017-04-13 Last updated: 2017-04-13Bibliographically approved

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