Stress recovery for the particle-in-cell finite element method

The particle-in-cell finite element (PIC-FE) method has been widely used in geodynamic numerical modelling due to its efficiency in dealing with large deformations without the requirement of remeshing. However, material deformation within a Eulerian mesh frame will mix particles of contrasting strength properties (e.g., viscosity in Stokes problems) in a single element requiring some form of averaging to project particle properties to integration points. The numerical solutions are thus dependent on the way how the particle properties are projected to the integration points. An intra-element property discontinuity may introduce severe stress oscillations along the interfaces. In this study, we assess three preprocessing methods to smooth the viscosity contrast within one element. For simplified models with analytical solutions, the accuracy and convergence rate in L₂ norm are systematically studied with ensembles. It is found that using higher-order quadrature elements does not improve the convergence rate for either the velocity or stress solution, both close to one. Additionally, the convergence rate of the maximum stress error, which exists adjacent to the mixed-material elements, is much less than one for all cases studied here. Comparing each component of the stress tensor, we find that the stress tensor component with the highest strain rate gradient across the material interface produces the maximum stress error. Such errors can be reduced by averaging the particle properties to the Gaussian quadrature point with an inverse-distance-weighted harmonic mean.