TY  - THES
T1  - Regularizing Constraints for Mesh and Shape Optimization Problems
A1  - Scherer,Michael
Y1  - 2011/06/14
N2  - This work deals with two types of optimization problems in the context of finite element simulations:
(a) an r-adaptive mesh optimization that improves the accuracy of finite element solutions of elastostatic boundary value problems.
(b) the node-based shape optimization of elastostatic structures.
Without an adequate regularization, it is often impossible to solve these optimization problems numerically. The main goal of this work is the development of regularization strategies that make both problems solvable.
(a) r-Adaptive Mesh Optimization:
The first part of this work deals with an r-adaptive mesh
optimization that is based on the minimization of the (discretized) potential energy with respect to the positions of the element nodes. Experience shows that it is usually not possible to solve the considered mesh optimization problems numerically with an optimization algorithm for unconstrained problems, for example, a BFGS method. The reason is that the numerical
optimization process causes degenerate finite elements, which leads to a premature termination of the mesh optimization. To prevent element degeneration, a regularization technique based on distortion constraints was developed. The distortion constraints restrict the deformation of
the finite elements that results from the mesh optimization and, thus, improve the solvability of the mesh optimization problems significantly.
(b) Shape Optimization:
One reason for the unsolvability of node-based shape optimization
problems is that the shape change from the initial shape to the optimized shape is very large and not realizable with the given finite element mesh. This led to the idea to restrict the shape change. To accomplish this task, a special inequality constraint, a so called energy constraint,
was developed. The energy constraint sets an upper limit to a fictitious mechanical strain energy that serves as a measure for the shape change. The larger the energy limit is chosen the larger is the admissible shape change. Apart from the energy constraint, no further regularization techniques are applied. Node-based shape optimization problems subject to the energy constraint can be solved with gradient-based optimization algorithms for problems with inequality constraints.

KW  - Gestaltoptimierung
KW  - Finite-Elemente-Methode
KW  - Regularisierung
KW  - Kontinuumsmechanik
KW  - Adaptives Verfahren
KW  - Elastostatik
CY  - Erlangen
PB  - Universitätsbibliothek der Universität Erlangen-Nürnberg
AD  - Universitätsstraße. 4, 91054 Erlangen
L2  - http://www.opus.ub.uni-erlangen.de/opus/volltexte/2011/2615
ER  -