A Point Collocation Method for Geometrically Nonlinear Membranes
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Abstract
This paper describes the development of a numerical model for geometrically nonlinear membranes and evaluates its performance for membranes at static equilibrium. The scheme has several features not commonly seen in structural finite element analysis: the point collocation method, group formulation, and a staggered mesh. In the point collocation finite element method, the partial differential equations are solved at each node instead of by integrating over elements. The group formulation simplifies the handling of nonlinearities by interpolating the nonlinear products of variables, as opposed to seeking the product of independently interpolated variables. The domain is discretized with a staggered mesh of linear triangles and associated polygons. Two sequential gradient recovery operations are performed: first the gradients of the linear triangles are calculated and converted to stresses; then, polygon derivative shape functions derived in this paper are used to determine the internal forces from the stress gradients. The resulting system of nonlinear equations is solved with a Jacobian-free Newton-Krylov solver. The code is first verified using the patch test and the method of manufactured solutions. Then the results are validated using experimental data and benchmark code results in the literature for the Hencky problem (a circular membrane with a fixed perimeter and uniform inflation pressure). The observed rates of convergence for both displacement and radial strain were two. For the configurations and grids used in this investigation, the scheme was suitable for accurately predicting sub-hyperelastic deformations. Abstract © Elsevier