A Computational Conformal Geometry Approach to Calculate the Large Deformation 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. Based on CCG, we proposed a general method for solving large deformation and nonlinear vibration of plate/shell structures with arbitrary shapes. The method can map any complex surface conformal to a rectangular region, and then FvK equations are solved in the rectangular region to study nonlinear vibration problems of any arbitrary shape plates/shells. The conform map is calculated by solving Laplace equations on a fine Delauney triangular mesh on 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 function that is equivalent to the FvK equations is provided, such that the FvK equations can be solved by multiple numerical methods. The degree-of-freedom in solving the FvK equations is usually much less than that in the finite element methods described by displacements. The effectiveness of the proposed approach is verified by several benchmark examples, and the current method is suitable for calculating the large deflections and nonlinear dynamical responses of plates/shallow shells with arbitrary shapes.