Change search
ReferencesLink to record
Permanent link

Direct link
A continuous/discontinuous Galerkin method for the biharmonic problem on surfaces
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0001-7838-1307
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in R3. A priori error estimates, taking both the approximation of the surface andthe approximation of surface differential operators into account, are proven in a discrete energy norm and in L2-norm. This can be seen as an extension of the formalism and method originally used in [Dzi88] for approximatingsolutions to the Laplace-Beltrami problem. Using a polyhedral approximation Σh of an implicitly defined surface Σ embedded in R3 we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on Σ. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.

National Category
Computational Mathematics
URN: urn:nbn:se:umu:diva-79207OAI: diva2:640280
Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2013-08-14Bibliographically approved
In thesis
1. Finite Element Methods for Thin Structures with Applications in Solid Mechanics
Open this publication in new window or tab >>Finite Element Methods for Thin Structures with Applications in Solid Mechanics
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.
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 54
a priori error estimation, finite element method, discontinuous Galerkin, corotation, Kirchhoff-Love plate, curved beam, biharmonic equation
National Category
Computational Mathematics
Research subject
urn:nbn:se:umu:diva-79297 (URN)978-91-7459-653-3 (ISBN)978-91-7459-654-0 (ISBN)
Public defence
2013-09-06, S205h, Samhällsvetarhuset, Umeå universitet, Umeå, 10:15 (English)
Available from: 2013-08-16 Created: 2013-08-13 Last updated: 2013-08-16Bibliographically approved

Open Access in DiVA

No full text

Other links


Search in DiVA

By author/editor
Larsson, KarlLarson, Mats G
By organisation
Department of Mathematics and Mathematical Statistics
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 61 hits
ReferencesLink to record
Permanent link

Direct link