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We introduce and study a BDDC (Balancing Domain Decomposition by Constraints) method for the Naghdi shell problem discretized with MITC (Mixed Interpolation of Tensorial Components) elements. Compared with the Kirchhoff model, the Naghdi model uses both displacement and rotation as variables, and therefore is more accurate but also more complicated at the numerical level. The severe difficulties of finite element shell analysis are also reflected in the condition number of the problem, which quickly diverges as the thickness of the shell and/or the finite element mesh size tend to zero. The proposed BDDC preconditioner is based on a proper selection of primal continuity constraints, the implicit elimination of the interior degrees of freedom in each subdomain, and the iterative solution of the resulting shell Schur complement by a preconditioned conjugate gradient method. The preconditioner is built from the solutions of local shell problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse shell problem for the primal degrees of freedom. Three choices of primal constraints, hence coarse spaces, are considered, yielding three BDDC preconditioner of increasing strength and cost. Several numerical tests are performed for cylindrical, hyperbolic and elliptic shells. The results show that the proposed BDDC preconditioners are scalable in the number of subdomains, quasi-optimal in the ratio subdomain/element sizes, robust with respect to discontinuities of the shell material properties, and almost robust with respect to the shell thickness.

A Balancing Domain Decomposition Method by Constraints (BDDC) is constructed and analyzed for the Reissner-Mindlin plate bending problem discretized with Mixed Interpolation of Tensorial Components (MITC) finite elements. This BDDC algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints. After the implicit elimination of the interior degree s of freedom in each subdomain, the resulting plate Schur complement is solved by the preconditioned conjugate gradient method. The preconditioner is based on the solution of local Reissner-Mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse Reissner- Mindlin plate problem for the primal degrees of freedom. The main results of the paper are the proof and numerical verification that the proposed BDDC plate algorithm is scalable, quasi-optimal, 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. The numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties. © 2010 Society for Industrial and Applied Mathematics.

A family of plate elements introduced by Falk and Tu [25] is considered. A new stability and a-priori error analysis is given. In addition, an a-posteriori error estimate is proved. The analysis is confirmed by numerical benchmark computations. © 2008 Springer Science + Business Media B.V.

We consider the class of shell problems which are neither purely bending neither purely membrane dominated. In such cases the asymptotic energy norm behavior (which is useful not only because it represents the structure stiffness, but also for numerical comparison purposes) is not a priori known. In this work we apply a numerical procedure in order to estimate the energy behavior of a general shell problem. In order to test its reliability, the method is applied to various problems for which the theoretical energy behavior is known and the results can be compared. Among the problems tested, we have two classical engineering shell benchmarks which are neither bending neither membrane dominated, and for which an analytical evaluation has been obtained in a recent work. All the energy behavior estimates obtained with the numerical method are in perfect agreement with the theoretical values. © 2003 Elsevier Ltd. All rights reserved.