C. Chinosi

Numerical analysis of some mixed finite element methods for Reissner-Mindlin plates

A class of mixed finite element methods for Reissner-Mindlin plates proposed by Arnold and Brezzi is considered. In these methods the shear energy term is split into two terms, leading to a partial selective reduced integration scheme. A parameter is involved in the splitting. In this paper an analysis of the behaviour of the approximate solution is performed in dependence of the parameter. Suggestions for a good choice of the parameter are also provided.

Robust BDDC Preconditioners for Reissner-Mindlin Plate Bending Problems and MITC Elements

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 degrees 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.

Shell elements as a coupling of plate and drill elements

Abstract In this paper a discretization of shell structures by means of flat elements is considered. The curved surface of the shell is approximated with a polyhedric surface; on each plane a finite element decomposition is introduced. Since on each element the “in plane” and bending forces cause independent deformations, a shell element is obtained by coupling in a suitable way a membrane element with a plate bending element. In this work membrane elements with drilling degrees of freedom are considered. In this way the shell elements that we shall construct will possess, in a natural way, also the degree of freedom related to the rotation along the vertical direction. This fact permits us to avoid the difficulties of programming arising from the use of classical membrane elements in which the rotational degree of freedom is absent. The numerical results performed with these new shell elements show their good convergence properties.

Remarks on partial selective reduced integration method for Reissner–Mindlin plate problem

Abstract We deal with the approximation of the Reissner–Mindlin plate problem by means of finite element techniques. We consider a non-standard mixed formulation recently proposed by Arnold and Brezzi. These methods are based on a suitable splitting, depending on a parameter, of the shear energy term into two parts, one of them being exactly integrated, while for the second one a reduced integration formula is used. In this paper we analyse the numerical behaviour of the approximate solution varying the splitting parameter and we propose a recipe for its choice.

A hierarchic family of C 1 finite elements for 4 th order elleptic problems

This paper describes a new family of high order finite elements for the approximation of plate bending problems. The elements are conforming and the C1-continuity condition is satisfied by using standard polynomials; the family is hierarchic, that is when the degree of the polynomials is increased new shape functions are added to the previous ones. Explicit expressions of the shape functions are given on general triangles. The hierarchic extension of the approximation spaces, also for a further increase of the degree of the elements, is described in details.

NONCONFORMING FINITE ELEMENTS FOR REISSNER-MINDLIN PLATES

As it is well-known (cf. [4], for instance), the numerical treatment of the Reissner-Mindlin model requires special care, in order to avoid the so-called shear locking phenomenon and the occurrence of spurious modes. The shear locking has its roots from the shear energy term, which for “small” thickness enforces the Kirchhoff constraint. It turns out that for simple low-order elements this constraint is generally too severe, thus compromising the quality of the obtained discrete solution. A general and commonly adopted strategy to overcome the problem consists in modifying, at the discrete level, the shear energy term, with the aim of reducing its influence. In most cases the modification can be interpreted as the result of (or directly arises from) a mixed approach to the problem. However, a careless choice of the shear reduction procedure may cause a loss of stability, typically enlighted by the presence of undesirable oscillating components in the discrete solution (the spurious modes). We point out that nowadays a wide choice of good elements are avalaible

Quasi-optimality of BDDC Methods for MITC Reissner-Mindlin Problems

The goal of this paper is to improve a condition number bound proven in [5] for a Balancing Domain DecompositionMethod by Constraints (BDDC) for the Reissner- Mindlin plate bending problem discretized with MITC elements. This BDDC preconditioner is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints.