As part of an ongoing research project between the Institute of Geotechnics (IfG), the Institute of Mechanics at TU Darmstadt (Ralf Müller), and EAFIT University (Colombia, Juan M. Rodríguez), this journal contribution investigates the influence of element formulation and state-variable mapping techniques on the accuracy and robustness of PFEM simulations within the finite element framework numgeo (developed at IfG).

The Particle Finite Element Method (PFEM) is a powerful numerical method for modelling large-deformation problems with evolving boundaries and contact interactions. However, its accuracy can be significantly affected by remeshing-induced errors, particularly since advanced geotechnical constitutive models are highly sensitive to changes in their internal state variables. Furthermore, due to the original reliance of the PFEM framework on linearly interpolated elements, volumetric locking may adversely affect the solution accuracy. This is especially critical in simulations where soils exhibit nearly incompressible behaviour as deformation paths approach the critical state. This work systematically analyses both aspects by comparing different element formulations and mapping strategies.

The study considers classical extrapolation–interpolation alongside advanced mapping techniques, including hybrid mapping, element-wise transfer, and a field-based approach using radial basis functions (RBF) to address mapping-induced errors. To mitigate volumetric locking, linearly interpolated mixed elements are compared with quadratically interpolated displacement-based elements.

The interplay between mapping techniques and element formulation is investigated using two benchmark problems: Cook’s membrane and a Hertzian contact problem. In addition, two geotechnical boundary value problems are analysed, namely the penetration of a rigid footing into Tresca soil and the back-analysis of a calibration chamber cone penetration test (CPT) in Toyoura sand. For the constitutive description of Toyoura sand, the advanced elasto-plastic Sanisand model is employed. To the best of the authors’ knowledge, this represents the first application of the Sanisand model within a PFEM framework.

The results show that mapping-induced errors strongly affect PFEM simulations with linearly interpolated elements, where the classical extrapolation–interpolation approach leads to artificial smoothing of state variables and, consequently, reduced accuracy. These deficiencies can be effectively mitigated by employing advanced mapping techniques. In contrast, quadratically interpolated elements achieve high accuracy that is largely independent of the chosen mapping strategy, even for coarse meshes and frequent remeshing.

Overall, the findings demonstrate that quadratic displacement-based elements provide a robust and efficient approach for PFEM simulations in geotechnical applications. They effectively eliminate volumetric locking, significantly reduce mapping-induced errors, and remove the need for stabilisation techniques that depend on problem-specific parameters. While advanced mapping strategies remain important for linear elements, their relevance diminishes when higher-order elements are employed, at least for the classes of problems considered in this study.

Check out the full article here.