A Computational Conformal Geometry Approach to Calculate the Large Deformations of Plates/shells with Arbitrary Shapes

Abstract

High accuracy numerical methods to solve the nonlinear Föppl-von Kármán (FvK) equations usually work well only in simple domains such as rectangular regions. Computational conformal geometry (CCG) provides a systematic method to transform complicated surfaces into simple domains, preserving the orthogonal frames, such that the corresponding FvK equations can be solved by more effective numerical methods. The conform map is calculated by solving a pair of Laplace equations on a fine Delauney triangular mesh of the surface, which is numerically robust, and the map is harmonic and subsequently C∞ smooth, such that all the evaluations and spatial derivatives required by high accuracy methods at the regular nodes can be accurately and efficiently calculated. A variational functional corresponding to the FvK equations is derived for shells, which enable the problem to be solved by the finite element methods and compared with the commercial software Abaqus; fewer degrees of freedom are required in solving the transverse displacements and Airy functions of the FvK equations. The effectiveness of the proposed approach is verified by several benchmark examples, and the current method is suitable to calculate the large deflections and nonlinear dynamical responses of plates/shallow shells with arbitrary shapes.