Two Finite Element Approximations of Naghdi's Shell Model in Cartesian Coordinates

Abstract

We present a penalized version of Naghdi's model and a mixed formulation of the same model, in Cartesian coordinates for linearly elastic shells with little regularity, and finite element approximations thereof. Numerical tests are given that validate and illustrate our approach.

A mixed DG method for folded Naghdi's shell in Cartesian coordinates

“…For the general shell situation, the constraint s · a 3 cannot be implemented in a standard conforming way (see [1,4]). …”

On the Obstacle Problem for a Naghdi Shell

“…In view of the discretization we observe that the tangency constraint s · a 3 = 0 which appears in the definition of V(ω) can be handled via the introduction of a Lagrange multiplier, see [7]. Let us consider the relaxed function space…”

“…in ω , (4.15) and consider the problem: It follows from the previous statements that this problem has a unique solution. Despite the further unknown ψ , it can be an efficient way to handle the constraint in V(ω), see [5, §2] or [7] for instance.…”

Error analysis for a mixed DG method for folded Naghdi's shell

“…Let us just mention the following approaches. In [4], two finite-element discretizations of Naghdi's equations are considered, a penalized version, which intends to approximate the above-mentioned tangency, the second approach consisting in handling this constraint via the introduction of a Lagrange multiplier, which involves the Laplace-Beltrami operator on the shell midsurface. The corresponding system is a second-order elliptic system of PDEs.…”