A novel mixed finite element method based on the volume coordinate system for stress analysis of plates

Abstract

Traditional bilinear isoparametric coordinate systems exhibit sensitivity to mesh distortion due to their fully high-order polynomials being only equivalent to first-order polynomials in Cartesian coordinate systems when confronted with mesh distortion. This paper combines the concept of 3- and 6-component volume coordinate systems (VCS) with the generalized mixed element method to develop a novel bivariate method called the non-conforming generalized mixed element method in the volume coordinate system (NGMVC). Established a bivariate field analysis mode in the VCS. Taking VCS as local coordinates significantly improves the morbidity relationship between local and Cartesian coordinate systems in conventional isoparametric elements during mesh distortion. Also avoids the calculation of the Jacobian inverse in the element strain matrix, greatly simplifies mathematical expressions, and lowers computational costs while ensuring that elements are insensitive to mesh distortion. On the other hand, in the analyzing procedure, the NGMVC describes the finite element model more objectively and rationally by considering both stress and displacement boundary conditions simultaneously. Thus, addressing the limitation of traditional displacement methods lacking consideration of stress boundary conditions. Based on these, considering the objective fact that the in-plane stress in composite laminated structures may not be continuous between layers, the non-conforming generalized partial mixed method in the volume coordinates (NGPMVC) was established by separating in-plane stress and out-of-plane stress in the mixed element formulation. The proposed method was verified through benchmark problems for laminates. The numerical results demonstrate that the method is not sensitive to mesh distortion and has a good ability to capture each stress component for different mesh densities, aspect ratios, and geometric structures.