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R. Duy:
"Fictitious Domain Approach for Optimizing Stability Boundaries of Plates with Cutouts";
Supervisor: Y. Vetyukov; Institut für Mechanik und Mechatronik, 2021; final examination: 2021-02.

The focus of the present master´s thesis is on further developing and improving numerical methods for their application in structural mechanics. In particular, the stability problem of buckling of thin plates is considered. Depending on the geometry of the structure and ist loading conditions, analytical solutions exist only for simple problems. For more complex problems however, numerical methods like the Finite Elements Method (FEM) are applied with great success. Doing so, it is also possible to deal with complex geometries, loading, and boundary conditions.The treatment of plates with cutouts is, however, still associated with certain difficulties. There only exist semi-analytical solutions, and, more importantly, the discretization of the problem is elaborate; even more so in the case of repeated analyses which only differ in few parameters, for example the position of the cutout.In this thesis the Finite Cell Method (FCM), as an extension of traditional FEM, is applied in a way that the mesh generation burden in FEM is alleviated when the position of the hole changes. FCM is based on embedding the original problem geometry in a larger one, the so-called extended domain. The extended domain is then again split in the physical and the fictiticous domains, establishing the name "Fictitious Domain Approach" for this method. The subsequent non-geometry conforming discretization introduces discontinuous integrands. Thus, the difficulty in solving this problem now no longer lies in the geometry conforming discretization and preparation of the problem, but in the computation of the system matrices, which requires precise evaluation of the discontinuous integrands. The introduction of a non-geometry conforming discretization, which is in this case strictly Cartesian, allows the application of shape functions, which are rarely applied today due to certain limitations. In this thesis, the so-called Bogner-Fox-Schmit (BFS) shape functions are used. Finite elements based on these functions show an excellent convergence behaviour; along with the efficiency of FCM this ensures the fast and easy-to-modify solution of the buckling problem of plates.Finally, this newly developed method is applied to optimize the stability boundaries of plates with cutouts. A plate problem is introduced and the position of a hole is of interest, for which the critical buckling load is maximized. Counterintuitively, it is shown that the introduction of cutouts does not always result in a decreased critical load, but that in fact it can be increased compared to a plate without holes!

Structural mechanics; Plates; Stability; Finite elements; Numerical methods; Optimization

http://dx.doi.org/10.34726/hss.2021.70501