The Smoothed Particle Finite Element Method (SPFEM) has gained popularity as an effective numerical method for modelling geotechnical problems involving large deformations. To promote the research and application of SPFEM in geotechnical engineering, we present ESPFEM2D, an open-source two-dimensional SPFEM solver developed using MATLAB. ESPFEM2D discretizes the problem domain into computable particle clouds and generates the finite element mesh using Delaunay triangulation and the alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha $$\end{document}-shape technique to resolve mesh distortion issues. Additionally, it incorporates a nodal integration technique based on strain smoothing, effectively eliminating defects associated with the state variable mapping after remeshing. Furthermore, the solver adopts a simple yet robust approach to prevent the rank-deficiency problem due to under-integration by using only nodes as integration points. The Drucker-Prager model is adopted to describe the soil's constitutive behavior as a demonstration. Implemented in MATLAB, this open-source solver ensures easy accessibility and readability for researchers interested in utilizing SPFEM. ESPFEM2D can be easily extended and effectively coupled with other existing codes, enabling its application to simulate a wide range of large geomechanical deformation problems. Through rigorous validation using four numerical examples, namely the oscillation of an elastic cantilever beam, non-cohesive soil collapse, cohesive soil collapse, and slope stability analysis, the accuracy, effectiveness and stability of this open-source solver have been thoroughly confirmed.

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics. (c) 2024 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/ by/4.0/).