ERRORS IN QUANTITIES OF INTEREST IN THE LAMINATED PLATE BENDING PROBLEM USING HIERARCHIC SETS OF BASIS FUNCTIONS IN GFEM

Resumo

A formulation for error estimation is developed for the bending problem of composite laminated plates based on the Mindlin-Reissner kinematic model discritized by the Generalized Finite Element Method (GFEM). The error estimation process starts with an upper bound in energy norm, which is obtained following the basic CRE (Constitutive Relation Error) framework of the Ladev`eze formulation, that is, the estimate is obtained from a statically admissible stress field computed at element level in a Neumann problem where the element boundary forces are equilibrated. The authors have previously shown that an accurate description of the in plane stresses in a laminate is essential to obtain an accurate approximation to the transverse shear stresses at the layers interfaces. Since important failure modes in laminated composite plates, like the delamination, are linked to the transverse stresses, it is essential to develop both, accurate post-processing procedures to compute improved transverse stresses, and also estimate techniques for the discretization errors. The first condition is adequately satisfied by GFEM.

Therefore, the aim of the present work is to extend the general CRE technology to develop formulations to estimation of errors in Quantity of Interest (QI) identified preferably with the stress field in the laminated plate problem. One of the steps necessary in the CRE procedure is the computation of and admissible stress field in each element, in a Neumann problem where the boundary forces have been previously equilibrated. For a GFEM basis with high order enrichment, adequate procedures have to be sought. Here we use one single higher order finite element, based on displacement FEM, to obtain an approximation to the equilibrated field. The formulation is implemented for arbitrary degree of the basis, which allows an arbitrarily close approximation to the equilibrium condition. The sharpness of the QI’s error bounds is increased with the accuracy of the primal and dual global energy norm of errors. In the present work we investigate the effectiveness of a local GFEM p-enrichment as a tool to improve the approximability of the model in capturing the local gradients which characterizes response of the dual loading. The GFEM p-enrichment is implemented in a simple and straightforward way, as opposed to some other possible forms of enrichment, e.g. local h-refinement or a sub-domain approach. Numerical tests are performed to asses the effect of the different parameters in the modeling over the errors in the quantities of interest.