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Finite element methods on surfaces
Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
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: urn:nbn:se:hj:diva-28424ISBN: 978-91-87289-13-2 (print)OAI: oai:DiVA.org:hj-28424DiVA: diva2:875741
Supervisors
Funder
Swedish Research Council, 2011-4992
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2015-12-01Bibliographically 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: 2017-12-01Bibliographically 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: 2017-12-01Bibliographically approved
3. Multi-objective optimization of material model parameters of an adhesive layer by using SPEA2
Open this publication in new window or tab >>Multi-objective optimization of material model parameters of an adhesive layer by using SPEA2
2015 (English)In: Advances in structural and multidisciplinary optimization: Proceedings of the 11th World Congress of Structural and Multidisciplinary Optimization (WCSMO-11) / [ed] Qing Li, Grant P Steven, Zhongpu (Leo) Zhang, 2015, 249-254 p.Conference paper, Published paper (Refereed)
Abstract [en]

The usage of multi material structures in industry, especially in the automotive industry are increasing. To overcome the difficulties in joining these structures, adhesives have several benefits over traditional joining methods. Therefore, accurate simulations of the entire process of fracture including the adhesive layer is crucial. In this paper, material parameters of a previously developed meso mechanical finite element (FE) model of a thin adhesive layer are optimized using the Strength Pareto Evolutionary Algorithm (SPEA2). Objective functions are defined as the error between experimental data and simulation data. The experimental data is provided by previously performed experiments where an adhesive layer was loaded in monotonically increasing peel and shear. Two objective functions are dependent on 9 model parameters (decision variables) in total and are evaluated by running two FEsimulations, one is loading the adhesive layer in peel and the other in shear. The original study converted the two objective functions into one function that resulted in one optimal solution. In this study, however, a Pareto frontis obtained by employing the SPEA2 algorithm. Thus, more insight into the material model, objective functions, optimal solutions and decision space is acquired using the Pareto front. We compare the results and show good agreement with the experimental data.

Keyword
Multi-objective optimization, parameter identification, micro mechanical model, adhesive, CZM
National Category
Computer Engineering Mechanical Engineering
Identifiers
urn:nbn:se:hj:diva-28422 (URN)978-0-646-94394-7 (ISBN)
Conference
11th World Congress of Structural and Multidisciplinary Optimization (WCSMO-11)
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2016-11-10Bibliographically approved

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