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Finite element methods for surface problems
Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling. Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
2017 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Jönköping: Jönköping University, School of Engineering , 2017. , s. 144
Serie
JTH Dissertation Series ; 022
Emneord [en]
trace finite element method, membrane, mean curvature, level-set method
HSV kategori
Identifikatorer
URN: urn:nbn:se:hj:diva-35369ISBN: 978-91-87289-23-1 (tryckt)OAI: oai:DiVA.org:hj-35369DiVA, id: diva2:1088644
Disputas
2017-05-12, E1405, Tekniska Högskolan, Jönköping, 10:00 (engelsk)
Opponent
Veileder
Forskningsfinansiär
Swedish Research Council, 2011-4992Tilgjengelig fra: 2017-04-13 Laget: 2017-04-13 Sist oppdatert: 2018-08-17bibliografisk kontrollert
Delarbeid
1. Minimal surface computation using a finite element method on an embedded surface
Åpne denne publikasjonen i ny fane eller vindu >>Minimal surface computation using a finite element method on an embedded surface
2015 (engelsk)Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, nr 7, s. 502-512Artikkel i tidsskrift (Fagfellevurdert) 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.

Emneord
mean curvature; Laplace–Beltrami operator; level set; ghost penalty stabilization
HSV kategori
Identifikatorer
urn:nbn:se:hj:diva-28151 (URN)10.1002/nme.4892 (DOI)000362552500003 ()2-s2.0-84928151825 (Scopus ID)
Merknad

Special Issue: Advances in Embedded Interface Methods

Tilgjengelig fra: 2015-10-09 Laget: 2015-10-09 Sist oppdatert: 2017-12-01bibliografisk kontrollert
2. Cut finite element modeling of linear membranes
Åpne denne publikasjonen i ny fane eller vindu >>Cut finite element modeling of linear membranes
2016 (engelsk)Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, s. 98-111Artikkel i tidsskrift (Fagfellevurdert) 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.

Emneord
Cut finite element method, Embedded membrane, Membrane shell, Tangential derivative
HSV kategori
Identifikatorer
urn:nbn:se:hj:diva-28420 (URN)10.1016/j.cma.2016.05.018 (DOI)000384859400005 ()2-s2.0-84979299087 (Scopus ID)
Tilgjengelig fra: 2015-12-01 Laget: 2015-12-01 Sist oppdatert: 2018-01-10bibliografisk kontrollert
3. Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement
Åpne denne publikasjonen i ny fane eller vindu >>Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
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.

HSV kategori
Identifikatorer
urn:nbn:se:hj:diva-35367 (URN)
Tilgjengelig fra: 2017-04-13 Laget: 2017-04-13 Sist oppdatert: 2018-01-13bibliografisk kontrollert
4. Numerical error estimation for a TraceFEM membrane and distance function on P1 and P2 tetrahedra
Åpne denne publikasjonen i ny fane eller vindu >>Numerical error estimation for a TraceFEM membrane and distance function on P1 and P2 tetrahedra
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
HSV kategori
Identifikatorer
urn:nbn:se:hj:diva-35368 (URN)
Tilgjengelig fra: 2017-04-13 Laget: 2017-04-13 Sist oppdatert: 2018-01-13bibliografisk kontrollert

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