Enhancing Meshfree Particle Methods

Describing quasi-brittle material failure is a challenging task in computational mechanics. While classical continuum models – usually based on softening plasticity, damage mechanics, or their combination – are basically able to represent such failure, they suffer from a lack of objectivity, leading to the loss of ellipticity in the governing differential equations. This, in turn, causes strain localization to the smallest possible entity, typically a single row of elements in finite element simulations, resulting in mesh sensitivity and non-convergent numerical behavior.
To ensure a stable and robust material model formulation, an appropriate regularization method is necessary to overcome these issues and achieve mesh-insensitive results in numerical simulations. Generalized continua, such as gradient-enhanced, Cosserat, or micromorphic continua, have proven to be effective in regularizing the previously mentioned issues in the softening regime (see Figure 1).
 

However, such localized failure behavior inherently leads to large deformations in the failure zone, causing excessive mesh distortions when using Lagrangian finite element methods (see Figure 2).

Meshfree particle methods, such as the Particle Finite Element Method (PFEM), Discrete Element Method (DEM), Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), and the Material Point Method (MPM), have proven to be suitable alternatives for solving large deformation problems.
Recently, Neuner et al. [3] successfully applied a B-spline-based implicit MPM to generalized continua for modeling large localized deformations. Unlike Lagrangian finite elements, MPM discretizes the geometry using particles that move freely over a fixed Eulerian mesh, where the partial differential equations are solved (see Figure 3).