TY  - THES
T1  - C1 Continuous Methods in Computational Gradient Elasticity
A1  - Fischer,Paul
Y1  - 2011/09/07
N2  - This thesis contains the presentation and the comparison of different C1 continuous numerical methods. These are a couple of C1 continuous finite elements, the isogeometric analysis (IGA) and the natural element method (NEM). They all have in common that their shape functions are directly or indirectly related to the Bernstein-Bezier representation of polynomial splines.

The major goal of this work is the examination of the applicability of those methods to strain gradient elasticity. Therefore different problems occurring within the application of the methods are presented and demonstrated at relevant numerical benchmark tests. The errors are measured by the use of the L2, H1 and H2 Sobolev norms. These norms are computed by the use of analytical and numerical reference solutions. Within this thesis, a comprehensive comparison of the above mentioned numerical methods is presented. Furthermore, the IGA as well as some of the presented elements were applied to gradient elasticity for the first time.

For the application of the finite element method, several ideas for the improvement are proposed. It is demonstrated that a simple enhancement of the linear geometry approximation of the subparametric C1 elements improves the absolute error as well as the rate of convergence. Furthermore, for the mesh construction of the isoparametric C1 elements a new linear mesh optimization algorithm is proposed. This algorithm again improves the performance of the elements, significantly.

A complementary application is the Cahn-Hilliard equation. It is dicretized by the use of the NEM. Therefore, the C0 continuous Sibson interpolants as well as Farin's C1 interpolant are used to find the solution of the fourth order partial differential equation. For the application of the C0 continuous functions, a split of the PDE into a set of two coupled equations is used. By the use of a numerical example, it is demonstrated that the direct C1 continuous approach converges faster.

Keywords: generalized continua, gradient elasticity, Cahn-Hilliard equation, Hermite elements, natural element method (NEM), isogeometric analysis (IGA), isoparametric C1 elements, mesh generation, simulation


KW  - Isogeometrische Analysis
KW  - Gittererzeugung
KW  - Simulation
CY  - Erlangen
PB  - Universitätsbibliothek der Universität Erlangen-Nürnberg
AD  - Universitätsstraße. 4, 91054 Erlangen
L2  - http://www.opus.ub.uni-erlangen.de/opus/volltexte/2011/2769
ER  -