A mixed finite element model for deep excavation analysis

This study presents the development of a mixed finite element method (FEM) for analyzing deep excavation systems, with a focus on sheet pile walls. In contrast to traditional displacement-based FEM, which often suffers accuracy loss in computing bending moments and shear forces due to higher-order derivatives, this model adopts a mixed formulation that simultaneously solves for both displacement and moment as primary variables. The governing equations are discretized using the Galerkin method, and the weak form is employed to improve numerical stability and facilitate the incorporation of flexible boundary conditions and soil-structure interaction effects. The model is implemented in Python and validated against Verruijt’s (1995) analytical solution for a beam on elastic foundation, demonstrating close agreement in both displacement and moment predictions. The framework is extended to sheet pile walls, capturing deformation and internal force distributions under staged excavation scenarios. This model establishes a robust computational basis for subsequent integration with scientific artificial intelligence methods. Specifically, it marks the initial stage of a broader investigation into hybrid FEM–Physics-Informed Neural Network (PINN) frameworks for real-time simulation and inverse modeling of deep excavation behavior. As such, it offers a physically grounded, accurate, and adaptable approach for advancing data-driven geotechnical analysis in safety-critical environments.