C. Lovadina

Energy norm a posteriori error estimates for mixed finite element methods

This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields.

A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes

Summary.In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poisson’s equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

An isogeometric method for the Reissner–Mindlin plate bending problem

We present a new isogeometric method for the discretization of the Reissner–Mindlin plate bending problem. The proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections Wh and a space of smooth discrete rotations Θh such that the Kirchhoff constraint is exactly satisfied at the limit. Therefore we obtain a formulation which is natural from the theoretical/mechanical viewpoint and locking-free by construction. We prove that the method is uniformly stable and satisfies optimal convergence estimates. Finally, the theoretical results are fully supported by numerical tests.

Analysis of kinematic linked interpolation methods for Reissner-Mindlin plate problems

The approximation to the solution of Reissner-Mindlin plate problem is considered in the framework of finite element techniques. A general strategy, involving a linking operator between rotations and vertical displacements, is analyzed. An abstract convergence result is provided. Examples of elements falling into this framework are presented and shown to be stable and locking-free. Numerical tests detail the performances of the elements. EMAIL:: lovadina@ing.unitn.it

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.

A New Class of Mixed Finite Element Methods for Reissner--Mindlin Plates

A new class of finite elements for the Reissner--Mindlin plate problem is presented. The family is based on a modified mixed formulation recently introduced by Arnold and Brezzi [Boundary Value Problems for Partial Differential Equations and Applications, J. L. Lions and C. Baiocchi, eds., Masson, Paris, 1993]. A result of stability and convergence uniformly in the thickness is provided.

On the enhanced strain technique for elasticity problems

The enhanced strain technique is considered in the context of incompressible elasticity problems. A new triangular element is proposed and proved to be convergent and stable when applied to linear analysis. Moreover, a model problem arising from large deformations is presented. Such a model is used to develop some theoretical considerations about the use of enhanced strain methods in nonlinear elasticity.

A Low-order Nonconforming Finite Element for Reissner--Mindlin Plates

We propose a locking-free element for plate bending problems, based on the use of nonconforming piecewise linear functions for both rotations and deflections. We prove optimal error estimates with respect to both the meshsize and the analytical solution regularity.

A SHELL CLASSIFICATION BY INTERPOLATION

The shell problem and its asymptotic are investigated. A connection between the asymptotic behavior and real interpolation theory is established. Thus, a detailed study of the cases when neither the bending energy nor the membrane energy dominate is provided. An application to a cylindrical shell is also detailed. Although only the Koiter shells have been considered, the same procedure can be used for other models, such as Naghdis one, for example.

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.