BDDC preconditioners for Naghdi shell problems and MITC9 elements

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.