Finite Element Methods for Thin Structures with Applications in Solid Mechanics

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. I. K. Larsson, G. Wallgren, and M.G. Larson, Interactive simulation of a continuum mechanics based torsional thread, Proceedings of VRIPHYS 10: 7th workshop on virtual reality interaction and physical simulation (2010), 49–58. 1 II. K. Larsson and M.G. Larson, Continuous piecewise linear finite elements for the Kirchhoff-Love plate equation, Numerische Mathematik, Volume 121, Number 1 (2012), 65–97. 2 III. K. Larsson and M.G. Larson, A continuous/discontinuous Galerkin method for the biharmonic problem on surfaces, Preprint IV. P. Hansbo, M.G. Larson, and K. Larsson, Intrinsic finite element modeling of curved beams, Preprint 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 to use 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 L-norm and provide numerical results to support our findings. The third paper deals with the biharmonic equation on a surface embedded in R. 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 L-norm, and we support the theoretical results by numerical convergence studies for problems on a sphere and on a torus. 1Reproduced with the kind permission of Eurographics. 2Reproduced with the kind permission of Springer.