Finite element approximation of a prestressed shell model

Abstract

This work deals with the finite element approximation of a prestressed shell model formulated in Cartesian coordinates system. The considered constrained variational problem is not necessarily positive. Moreover, because of the constraint, it cannot be discretized by conforming finite element methods. A penalized version of the model and its discretization are then proposed. We prove existence and uniqueness results of solutions for the continuous and discrete problems, and we derive optimal a priori error estimates. Numerical tests that validate and illustrate our approach are given.

A priori and a posteriori error analysis for a hybrid formulation of a prestressed shell model

“…Following [17] and [20] the assumption that the chart φ is isometric is crucial to obtain the considered prestressed model. Indeed, if ψ is an isometric deformation of ω obtained by applying some external forces, then the local basis of ψ is the product of an orthogonal matrix (of determinant 1) times the local basis of φ.…”

“…We assume that the shell is fixed on a part Γ 0 of positive measure of the boundary of ω. According to [17,20], the model takes the following variational form…”

“…In our previous paper [20] we have stressed on the importance and the effect of the prestressed term a p . We recall that the nonpositive character of the prestressed term may break the coercivity of the bilinear form on the space of admissible test functions space V even if the bilinear form without prestressed term is V-elliptic.…”

“…This causes a loss of coercivity of the bilinear form on the space H 1 . In order to solve this issue, a larger Hilbert space was considered in [20], where the third component r 3 is sought in the L 2 space.…”

“…• In section 2 we recall some geometrical preliminaries of surfaces and the prestressed model presented in [20].…”