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.
H. Rhein and M. Thess, October 1997