Eintrag in der Universitätsbibliographie der TU Chemnitz
Volltext zugänglich unter
URN: urn:nbn:de:bsz:ch1-qucosa-150930
Günnel, Andreas
Stingl, Michael (Prof. Dr.) (Gutachter)
Numerical Aspects in Optimal Control of Elasticity Models with Large Deformations
Kurzfassung in englisch
This thesis addresses optimal control problems with elasticity for large deformations. A hyperelastic model with a polyconvex energydensity is employed to describe the elastic behavior of a body. The two approaches to derive the nonlinear partial differential equation, a balance of forces and an energy minimization, are compared. Besides the conventional volume and boundary loads, two novel internal loads are presented. Furthermore, curvilinear coordinates and a hierarchical plate model can be incorporated into the formulation of the elastic forward problem.
The forward problem can be solved with Newton's method, though a globalization technique should be used to avoid divergence of Newton's method. The repeated solution of the Newton system is done by a CG or MinRes method with a multigrid V-cycle as a preconditioner.
The optimal control problem consists of the displacement (as the state) and a load (as the control). Besides the standard tracking-type objective, alternative objective functionals are presented for problems where a reasonable desired state cannot be provided. Two methods are proposed to solve the optimal control problem: an all-at-once approach by a Lagrange-Newton method and a reduced formulation by a quasi-Newton method with an inverse limited-memory BFGS update.
The algorithms for the solution of the forward problem and the optimal control problem are implemented in the finite-element software FEniCS, with the geometrical multigrid extension FMG. Numerical experiments are performed to demonstrate the mesh independence of the algorithms and both optimization methods.
| Universität: | Technische Universität Chemnitz | |
| Institut: | Professur Numerik partieller Differentialgleichungen | |
| Fakultät: | Fakultät für Mathematik | |
| Dokumentart: | Dissertation | |
| Betreuer: | Herzog, Roland (Prof. Dr.) | |
| URL/URN: | http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-150930 | |
| SWD-Schlagwörter: | Optimale Kontrolle , Elastizität , Optimierung | |
| Freie Schlagwörter (Deutsch): | Optimalsteuerung , Mehrgitter , Newtonverfahren | |
| Freie Schlagwörter (Englisch): | optimal control , elasticity , multigrid , optimization , newton method | |
| DDC-Sachgruppe: | Mathematik, Numerische Analysis, Wahrscheinlichkeiten, angewandte Mathematik | |
| Tag der mündlichen Prüfung | 19.08.2014 |
