Novel Discretization Schemes for the Numerical Simulation of Membrane Dynamics

Motivated by the demands of simulating flapping wings of Micro Air Vehicles, novel numerical methods were developed and evaluated for the dynamic simulation of membranes. For linear membranes, a mixed-form time-continuous Galerkin method was employed using trilinear space-time elements, and the entire space-time domain was discretized and solved simultaneously. For geometrically nonlinear membranes, the model incorporated two new schemes that were independently developed and evaluated. Time marching was performed using quintic Hermite polynomials uniquely determined by end-point jerk constraints. The single-step, implicit scheme was significantly more accurate than the most common Newmark schemes. For a simple harmonic oscillator, the scheme was found to be symplectic, frequency-preserving, and conditionally stable. Time step size was limited by accuracy requirements rather than stability. The spatial discretization scheme employed a staggered grid, grouping of nonlinear terms, and polygon shape functions in a strong-form point collocation formulation. Validation against existing experimental data showed the method to be accurate until hyperelastic effects dominate

A point collocation method for geometrically nonlinear membranes

AbstractThis 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

A Point Collocation Method for Geometrically Nonlinear Membranes

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.Abstract excerpt © Elsevie