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Continuous piecewise linear finite elements for the Kirchhoff-Love plate equation
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
(English)Manuscript (preprint) (Other academic)
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. 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/Basic Plate Triangle does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.

Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-38196OAI: oai:DiVA.org:umu-38196DiVA: diva2:372838
Available from: 2011-01-19 Created: 2010-11-29 Last updated: 2012-01-04Bibliographically approved
In thesis
1. Finite element methods for threads and plates with real-time applications
Open this publication in new window or tab >>Finite element methods for threads and plates with real-time applications
2010 (English)Licentiate 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 using a minimal amount of material. Computer modeling and analysis of thin and slender structures has its own set of problems stemming from assumptions made when deriving the equations modeling their behavior from the theory of continuum mechanics. In this thesis we consider two kinds of thin elastic structures; threads and plates.

Real-time simulation of threads are of interest in various types of virtual simulations such as surgery simulation for instance. In the first paper of this thesis we develop a thread model for use in interactive applications. By viewing the thread as a continuum rather than a truly one dimensional object existing in three dimensional space we derive a thread model that naturally handles both bending, torsion and inertial effects. We apply a corotational framework to simulate large deformation in real-time. On the fly adaptive resolution is used to minimize corotational artifacts.

Plates are flat elastic structures only allowing deflection in the normal direction. In the second paper in this thesis we propose a family of finite elements for approximating solutions to the Kirchhoff-Love plate equation using a continuous piecewise linear deflection field. We reconstruct a discontinuous piecewise quadratic deflection field which is applied in a discontinuous Galerkin method. Given a criterion on the reconstruction operator we prove a priori estimates in energy and L2 norms. Numerical results for the method using three possible reconstructions are presented.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2010
Series
Research report in mathematics, ISSN 1653-0810 ; 1
Keyword
finite element method, real-time simulation, absolute nodal continuous formulation, large deformation, corotation, adaptive resolution, Kirchhoff-Love plate, reconstruction, discontinuous Galerkin, a priori error estimation
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-38198 (URN)978-91-7459-080-7 (ISBN)
Presentation
2010-11-25, Naturvetarhuset, N320, Umeå universitet, Umeå, 11:12 (English)
Opponent
Supervisors
Available from: 2010-11-30 Created: 2010-11-29 Last updated: 2010-11-30Bibliographically approved

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Larsson, KarlLarson, Mats G.
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