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978-3-8439-1681-3, Reihe Mathematik

Joachim Krenciszek
Proper Orthogonal Decomposition for Contact and Free Boundary Problems

107 Seiten, Dissertation Technische Universität Kaiserslautern (2014), Softcover, A5

Zusammenfassung / Abstract

In this thesis an approach to model reduction of contact and free boundary problems using proper orthogonal decomposition (POD) is presented. Contact problems are examined on the basis of the obstacle problem and the classical and dynamic Signorini problem. In structural mechanics frequently used model reduction techniques, such as Craig-Bampton or dynamic condensation, are not applicable due to the nonlinearity inherited by the contact condition. The projection based method of proper orthogonal decomposition is a suitable choice for nonlinear problems of high order.

Both an extensive numerical analysis as well as an a priori error estimate are presented. Additionally, lossy and lossless methods to reduce the complexity of the contact condition are presented.

The reduction of free boundary problems using POD is discussed using the example of a single-phase 1D Stefan problem and a two-phase 2D Stefan problem. A detailed numerical analysis of three different approaches to the Stefan problem is performed for the 1D problem. The two-phase 2D problem is treated using a enthalpy formulation on a fixed grid. Here, an efficient update of the finite element matrices in the reduced space is introduced.

Finally, the application of POD to second order dynamical systems, as arising in a lot of mechanical problems, and their equivalent first order representation are compared. The numerical difficulties when dealing with the first order system are disclosed and the advantages of the second order system are motivated.