2026 Theses Doctoral

The Extended Finite Element Method for Static, Stability, and Stochastic Analysis of Folded-Plate Structures using Global Enrichment Functions

Zisimos, Athanasios

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, a new computationally efficient approach utilizing the eXtended Finite Element Method (XFEM) is developed within the Mindlin–Reissner thick plate theory and a flat-shell framework to perform static elastic, linearized buckling, and stochastic linearized buckling analysis of plates and folded-plate structures. As opposed to traditional XFEM, where enrichment functions are applied locally to target strong and weak discontinuities, the proposed methodology globally enriches the entire domain with functions inspired by analytical solutions of simplified plate cases.

Furthermore, to enhance accuracy in buckling problems, this work follows the more general Kirchhoff formulation by retaining both in-plane and out-of-plane terms of the nonlinear part of the Green–Lagrange strain tensor, contrasting with the commonly adopted von Kármán approach that retains only out-of-plane terms. To quantify structural performance under uncertainty, the framework is extended into a Stochastic eXtended Finite Element Method (SXFEM) scheme.

The core of the proposed method is a set of global enrichment functions incorporated into a standard coarse FE mesh. The enrichment functions related to bending behavior are based on the Navier series analytical solution for simply supported plate under uniform pressure load. For the membrane component of the flat-shell formulation, enrichment functions were developed by solving problems under various boundary conditions and exploring functions that mimic the resulting physical behavior. Although these functions are derived from specific analytical cases, they are applied globally to problems with different geometries and loading conditions where analytical solutions are unavailable.

In addition, comprehensive convergence studies are conducted to investigate the efficiency of the method for static elastic analysis and the evaluation of buckling loads in thin-walled and folded structures. The results demonstrate that the proposed XFEM approach achieves a substantial reduction in computational effort and improved convergence rates compared to standard FEM. Furthermore, the method maintains high precision even for boundary conditions that differ from those used to develop the enrichment functions, demonstrating its general applicability to complex engineering structures.

Finally, to quantify structural performance under uncertainty, the framework is extended into a Stochastic eXtended Finite Element Method (SXFEM) scheme. Geometric imperfections are modeled as two-dimensional, univariate (2D-1V) Gaussian fields synthesized via the spectral representation method. By employing the Monte Carlo Simulation (MCS) method, the variability of the buckling load is captured through a cost-effective stochastic stiffness matrix. Furthermore, sensitivity analyses identify "worst-case" buckling scenarios for stochastic parameters lacking experimental validation, proving the robustness and general applicability of the methodology for the design of complex folded plate structures.