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Interactive simulation of a continuum mechanics based torsional thread
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)
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. 49-58 p.
Keyword [en]
finite element method, interactive simulation, absolute nodal continuous formulation, corotation, adaptive resolution
Identifiers
URN: urn:nbn:se:umu:diva-38195ISBN: 978-3-905673-78-4 (print)OAI: oai:DiVA.org:umu-38195DiVA: diva2:372833
Conference
VRIPHYS 10
Available from: 2010-11-29 Created: 2010-11-29 Last updated: 2013-08-14Bibliographically 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
2. 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.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 54
Keyword
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: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)
Opponent
Supervisors
Available from: 2013-08-16 Created: 2013-08-13 Last updated: 2013-08-16Bibliographically approved

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http://www.eg.org/EG/DL/PE/vriphys/vriphys10

Authority records BETA

Larsson, KarlLarson, Mats G.

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