Discontinious Galerkin formulations for thin bending problems

Abstract

A structural thin bending problem is essentially associated with a fourth-order partial differential equation. Within the finite element framework, the numerical solution of thin bending problems demands the use of C^1 continuous shape functions. Elements using these functions are challenging and difficult to construct. A particular discontinuous Galerkin method has been used to deal with thin bending problems. It exploits standard Lagrange finite element basis functions with displacement degrees-of-freedom only. The method relies on a lifting operation to transform jumps in the normal derivative across element boundaries to a field defined on element interiors. By introducing special integrals over element boundaries, continuity requirements associated with thin bending problems are met weakly while consistency of the Galerkin problem is preserved and the stability of the formulation is controlled. The approach is formulated for a range of bending problems. Firstly, a finite element formulation for linear Kirchhoff plates is proposed. The formulation is analysed in terms of stability and convergence. A priori error estimates are supported by a range of numerical examples, including static plate bending, plate buckling and vibration. The method has also been developed for geometrically linear and nonlinear thin shell models. For geometrically nonlinear problems, an exact linearisation of the proposed formulation is presented. Using the approach, a non-smooth shell geometry can be dealt with. The performance of the approach is demonstrated via a range of numerical benchmark tests for both geometrically linear and nonlinear thin shells. Based on the obtained results, it can be concluded that the proposed formulation is robust, accurate and relatively simple. As a thin shell theory has been addressed directly, shear locking is completely avoided. Although membrane locking exists, it can be alleviated when high-order basis functions are used. Cubic elements perform particularly well in various challenging benchmark tests. The method is general in sense that it does not rely on a particular element type or shape.