Solving the nonlinear membrane problem with the Chebyshev collocation method

Abstract In this paper, the problem of the geometrically nonlinear membrane structure is solved with the Chebyshev Collocation Method. The equations are formulated in generalized coordinates, based on a plane reference surface. Here, we do make the assumption that the undeformed configuration is a plane. We do only consider geometrical nonlinearity, while for the material, linear-elastic behavior is assumed. Boundary Conditions concerning tensorial quantities are formulated with the use of the “Practical” components of the tensors. We generate the grid using Transfinite interpolation with linear blending. For the Domain Decomposition, we use a variation of Lions’ method. The numerical procedure is verified with the Method of Manufactured Solutions. The Verification shows spectral convergence. For the Validation, we use the problem of the Plate with a Hole, the Circular Membrane Problem and the form-finding of a twin hypar membrane. One limitation of the error is the assumption of linear-elastic material behavior. The authors evaluate the method used for the Domain Decomposition to be the biggest issue concerning the efficiency of the developed methods.