Efficient XFEM/GFEM Approach for the Analysis of Thick Plates Utilizing Global Enrichments

Abstract

Plates are common flat structural elements that support loads through their flexural bending rigidity. Because plate thickness is much smaller than their surface dimensions, plate theories have emerged to simplify full three-dimensional behavior. Since analytical solutions for plate elements are not always available, especially for general loads and boundary conditions, numerical methods, such as the finite element method (FEM), are often employed to solve these problems. Due to its general polynomial approximation of the solution field, FEM often requires a large number of degrees of freedom to achieve convergence, increasing the computational burden. In this study, we propose a new approach for the analysis of plates utilizing the eXtended finite element method (XFEM) within the Mindlin-Reissner thick plate theory. As opposed to the traditional XFEM, where the enrichment functions are applied locally, mainly targeting strong and weak discontinuities in the domain, in the proposed approach, the entire domain is globally enriched with functions inspired by analytical solutions available for simple cases of plates. The objective of the current formulation is to reduce the computational burden by reducing the number of degrees of freedom in the problem compared to the standard FE configuration. The deformation of rectangular plates is studied for different boundary and loading conditions. The enrichment functions added to the FE shape functions are based on the series expansion resulting from the analytical solution of a simply supported plate under uniform pressure. However, those functions may not necessarily match the actual problem studied. To this end, higher series terms are employed to improve the accuracy and response of the plates. A convergence study is conducted to investigate the efficiency and robustness of the proposed method. The results show an improved convergence rate of the XFEM compared to the standard FEM approach.