Multiscale analysis of membrane instability by using the Arlequin method

Abstract

This work aims at developing a multiscale model by combining the Fourier-related double scale analysis and the bridging domain method to study membrane instabilities. Towards this end, a Fourier reduced membrane model is firstly established based on the Föppl-Von Karman plate equations, where the initial unknowns are expanded into Fourier series and replaced by their Fourier coefficients. As the latter varies much more slowly than the initial functions, the computational efficiency of the reduced model is significantly improved. However, the boundary effects could not be accurately captured because the prescription of boundary conditions is always questionable in any reduced models. Thus, there is a need to develop a multiscale model, where the full shell model is used near boundaries to capture the local effects, and the Fourier reduced model is used in the rest to reduce the computational cost. These two models are then bridged by the Arlequin method, which permits to couple different mechanical models by Lagrange multipliers. Finally, the proposed multiscale model is implemented into ABAQUS as the user element (UEL) to extend its applicability for more complex membrane instabilities. Numerical results show that this multiscale model is able to simulate the membrane instabilities with high efficiency and accuracy.