We consider the deformation problem of thin structures following linear dimension-reduced models with both displacement and rotation variables. We construct and study some Balancing Domain Decomposition Methods by Constraints (BDDC) for the discrete problems obtained with the MITC (Mixed Interpolation of Tensorial Components) finite element approximation.
In addition to the standard properties of scalability in the number of subdomains N, quasi-optimality in the ratio H/h of subdomain/element sizes, and robustness with respect to discontinuities of the material properties, our goal here is to obtain robustness also with respect to the additional small parameter t representing the plate or shell thickness. This is a challenging issue since the condition number of plates and shell problems typically diverges as O(t-2) as t tends to zero. The proposed BDDC preconditioners are based on a proper selection of primal continuity constraints, the implicit elimination of the interior degrees of freedom in each subdomain, and the solution of local (and also a global coarse) problems.
We first consider the case of a flat structure, therefore addressing the Reissner-Mindlin plate bending model. In such case we can prove that the proposed BDDC algorithm is scalable and, most important, robust with respect to the plate thickness. While this result is due to an underlying mixed formulation of the problem, both the interface plate problem and the preconditioner are positive definite. Numerical results also show that the proposed algorithm seems to be quasi-optimal and robust with respect to discontinuities of the plate material properties.
In the last part of the talk, we address shells of general geometry, namely the Naghdi model. In such case we do not have a theoretical bound and consider different choices of the primal constraints, driven by heuristic considerations and previous experience. Several numerical tests seem to indicate that the proposed BDDC preconditioners are scalable, quasi-optimal, robust with respect to discontinuities of the shell material properties, and almost robust with respect to the shell thickness.