subtask: Development of DD and Multilevel preconditioners for problems of plates and shells

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Formulation of the task :

The aim is the development of domain decomposition (DD) and multilevel preconditioners for the effective numerical solution of large equation systems arising from conforming and nonconforming discretizations of plate and shell problems. We consider thin plates resp. shells in connection with small deformations; additionally the physical modeling is based on the hypotheses of Kirchhoff-Love. This leads in the case of plates to a description of the displacement field by a biharmonic differential equation, in the case of shells after different methods of simplifying (Koiter, Novozhilov, etc.) we obtain systems of elliptical differential equations of the fourth order. For the biharmonic differential equation for the conforming discretization by Bogner-Fox-Schmit (BFS) elements and by nonconforming Adini elements, there were developed asymptotically optimal DD preconditioners in [Ma1], [Ma2]. These preconditioners are based on the nonoverlapping DD method; in the interior of the subdomains the boundary potential method proposed by Bahlmann and Korneev is used as direct solver while on the couple face a BPX-like preconditioning technique is realized. In [Ma2],[T2] we describe BPX-like preconditioners for the shell equations of Novozhilov resp. Koiter which where discretized by the use of BFS and Adini elements. The iteration numbers of these preconditioners are also independent of the mesh size. These preconditioners may also be used as preconditioners in the subdomains within the framework of the DD preconditioning mentioned above. For cylindrical shells this was examined in [Ma2]. Another possible application is the use as preconditioners for the subshells in the context of junctions of shells where we have to use DD preconditioners with constrains. The last problem is another actual research topic. All algorithms described here are implemented on MIMD parallel machines. One example is the program SPC-PM EL2,5D which efficiently calculates the displacement field of shells of arbitrary forms, using the discretization by BFS and Adini elements, with BPX-like preconditioners.
See also the example of the deformation of a

cooling tower under wind load

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H. Rhein and M. Thess, October 1997