A discontinuous Galerkin formulation of non‐linear Kirchhoff–Love shells

Abstract

SUMMARYDiscontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknownfield derivatives and have particular appeal in problems involving high-order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197:2901-2929 to develop a formulation of linear Kirchhoff-Love shells considering only the membrane and bending responses. In this proposed one-field method-the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non-linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couplestress from the displacement field at the mid-surface of the shell, the non-linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple-stress at the edges of the shells, the extension to non-linear deformations is straightforward.

A one field full discontinuous Galerkin method for Kirchhoff–Love shells applied to fracture mechanics

“…The continuum mechanics of thin structures is well established and can be found in several references [34,35,37,38]. For this reason, this section presents only the important results and the notations required to develop the full DG theory.…”

“…(51) is deduced from (7), with the Kirchhoff-Love assumption. Finally as it is well known that, for elliptic problems, such a formulation is unstable, the method is stabilized by introducing quadratic terms as it is suggested in [30,34,35]. Such an introduction of interior penalty term is usual for the DG method applied to solid mechanics (see [40][41][42][43][44][45] among others).…”

“…Toward this end, a new one-field full discontinuous Galerkin (full-DG) discretization of the shell equations is obtained from the extension of the work of several authors [31][32][33][34][35][36] who have developed a C 0 /DG formulation for thin bodies. In these works a DG method is used to weakly enforce the C 1 continuity required by high-order formulations of beams, plates and shells, which leads to displacement-field only methods (nodes have no degree of freedom of rotation).…”

A one-field discontinuous Galerkin formulation of non-linear Kirchhoff-Love shells

“…Although jump and mean operators are meaningful on the interior boundary ∂ I A 0 , jump definition can be extended on ∂ T A 0 as a way of enforcing weakly the boundary conditions, see [10]. From these definitions, the boundary term dependent on δt is rewritten…”

“…If, in the context of solid mechanics, DG can be developed for problems involving discontinuities in the unknown field, see [2][3][4][5] for non-linear solid mechanics, but it has also been exploited in the case of C 0 displacement unknown fields, which suffer from discontinuities in their derivative. This method has been exploited for applications to beams and plates [6][7][8], and more recently for linear and non-linear Kirchhoff-Love shells [9,10]. In this resulting one-field formulation, the jump discontinuities are related to the derivatives of the continuous unknown field.…”