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Finite Element Methods for Thin Structures with Applications in Solid Mechanics
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)ORCID iD: 0000-0001-7838-1307
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Thin and slender structures are widely occurring both in nature and in human creations. Clever geometries of thin structures can produce strong constructions while requiring a minimal amount of material. Computer modeling and analysis of thin and slender structures have their own set of problems, stemming from assumptions made when deriving the governing equations. This thesis deals with the derivation of numerical methods suitable for approximating solutions to problems on thin geometries. It consists of an introduction and four papers.

In the first paper we introduce a thread model for use in interactive simulation. Based on a three-dimensional beam model, a corotational approach is used for interactive simulation speeds in combination with adaptive mesh resolution to maintain accuracy.

In the second paper we present a family of continuous piecewise linear finite elements for thin plate problems. Patchwise reconstruction of a discontinuous piecewise quadratic deflection field allows us touse a discontinuous Galerkin method for the plate problem. Assuming a criterion on the reconstructions is fulfilled we prove a priori error estimates in energy norm and L2-norm and provide numerical results to support our findings.

The third paper deals with the biharmonic equation on a surface embedded in R3. We extend theory and formalism, developed for the approximation of solutions to the Laplace-Beltrami problem on an implicitly defined surface, to also cover the biharmonic problem. A priori error estimates for a continuous/discontinuous Galerkin method is proven in energy norm and L2-norm, and we support the theoretical results by numerical convergence studies for problems on a sphere and on a torus.

In the fourth paper we consider finite element modeling of curved beams in R3. We let the geometry of the beam be implicitly defined by a vector distance function. Starting from the three-dimensional equations of linear elasticity, we derive a weak formulation for a linear curved beam expressed in global coordinates. Numerical results from a finite element implementation based on these equations are compared with classical results.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2013. , vi, 18 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 54
Keyword [en]
a priori error estimation, finite element method, discontinuous Galerkin, corotation, Kirchhoff-Love plate, curved beam, biharmonic equation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-79297ISBN: 978-91-7459-653-3 (print)ISBN: 978-91-7459-654-0 (print)OAI: oai:DiVA.org:umu-79297DiVA: diva2:640426
Public defence
2013-09-06, S205h, Samhällsvetarhuset, Umeå universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2013-08-16 Created: 2013-08-13 Last updated: 2013-08-16Bibliographically approved
List of papers
1. Interactive simulation of a continuum mechanics based torsional thread
Open this publication in new window or tab >>Interactive simulation of a continuum mechanics based torsional thread
2010 (English)In: Vriphys 10: 7th workshop on virtual reality interaction and physical simulation / [ed] Kenny Erleben, Jan Bender, Matthias Teschner, Copenhagen, Denmark: Eurographics Association , 2010, 49-58 p.Conference paper, Published paper (Refereed)
Abstract [en]

This paper introduces a continuum mechanics based thread model for use in real-time simulation. The model includes both rotary inertia, shear deformation and torsion. It is based on a three-dimensional beam model, using a corotational approach for interactive simulation speeds as well as adaptive mesh resolution to maintain accuracy. Desirable aspects of this model from a numerical and implementation point of view include a true constant and symmetric mass matrix, a symmetric and easily evaluated tangent stiffness matrix, and easy implementation of time-stepping algorithms. From a modeling perspective interesting features are deformation of the thread cross section and the use of arbitrary cross sections without performance penalty.

Place, publisher, year, edition, pages
Copenhagen, Denmark: Eurographics Association, 2010
Keyword
finite element method, interactive simulation, absolute nodal continuous formulation, corotation, adaptive resolution
Identifiers
urn:nbn:se:umu:diva-38195 (URN)978-3-905673-78-4 (ISBN)
Conference
VRIPHYS 10
Available from: 2010-11-29 Created: 2010-11-29 Last updated: 2013-08-14Bibliographically approved
2. Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation
Open this publication in new window or tab >>Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation
2012 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 121, no 1, 65-97 p.Article in journal (Refereed) Published
Abstract [en]

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L2 norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2012
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-50810 (URN)10.1007/s00211-011-0429-5 (DOI)000302749600003 ()
Available from: 2012-01-05 Created: 2011-12-22 Last updated: 2017-12-08Bibliographically approved
3. A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces
Open this publication in new window or tab >>A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces
2017 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 86, no 308, 2613-2649 p.Article in journal (Refereed) Published
Abstract [en]

We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in R-3. A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in L-2 norm. This can be seen as an extension of the formalism and method originally used by Dziuk ( 1988) for approximating solutions to the Laplace-Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation inverted right perpendicular(h) of an implicitly defined surface inverted right perpendicular we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on inverted right perpendicular. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-79207 (URN)10.1090/mcom/3179 (DOI)000404567600003 ()
Note

Originally published in manuscript form with title [A continuous/discontinuous Galerkin method for the biharmonic problem on surfaces]

Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2017-10-16Bibliographically approved
4. Intrinsic finite element modeling of curved beams
Open this publication in new window or tab >>Intrinsic finite element modeling of curved beams
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In the mid '90s Delfour and Zolesio [4-6] established elasticity models on surfaces described using the signed distance function, an approach they called intrinsic modeling. For problems in codimension-two, e.g. one-dimensional geometries embedded in R3, an analogous description can be done using a vector distance function. In this paper we investigate the intrinsic approach for the modeling of codimension-two problems by deriving a weak formulation for a linear curved beam expressed in three dimensions from the equilibrium equations of linear elasticity. Based on this formulation we implement a finite element model using global degrees of freedom and discuss upon the effects of curvature and locking. Comparisons with classical solutions for both straight and curved cantilever beams under a tip load are given.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-79295 (URN)
Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2013-08-14Bibliographically approved

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