Stability of implicit material point methods for geotechnical analysis of large deformation problems

Coombs, William

Abstract

The Material Point Method (MPM, [4]) is a mesh-based continuum method where the physical body is represented by a series of Material Points (MPs), which each have a finite amount of mass, represent the material’s stress-strain behaviour, etc. These MPs are coupled to a finite element-like background grid which is used to discretise and solve the governing equations. However, unlike the Finite Element Method (FEM), between solution/time steps the background grid is reset/replaced - one way of explaining the MPM is the FEM where the quadrature points are allowed to move relative to the mesh. The use of the MPM is now widespread in the geotechnical academic community, helping to facilitate the solution of large deformation problems with history-dependent material behaviour, especially for analyses where the FEM would require extensive remeshing and projection of material history parameters. However, the MPM has yet to see robust commercialization or be widely adopted by geotechnical engineering practitioners. In the authors’ opinion this is mainly due to stability issues experienced when running MPM simulations. The most widely known issue is the so-called cell crossing instability, where the C0 continuous nature of FEM-standard polynomial basis functions results in the sudden transfer/sign change of internal force between/at nodes of the background grid. There are numerous mitigation strategies to avoid cell crossing issues, such as generalised interpolation or B-spline basis functions, which extend the continuity to at least C1 between grid cells and this issue is widely seen as solved. However, there are other stability issues that have received less attention, such as inf-sup constraints for mixed formulations [3] and the small cut issue, which is the focus of this research. The small cut issue is linked to the arbitrary nature of the interaction between the physical body, represented by the MPs, and the discretisation of the governing equations on the background grid. For large deformation analysis it is inevitable that there will be an uneven distribution of MPs within the background grid cells and highly likely that some nodes of the background grid will have very small values of mass/stiffness due to limited interaction with MPs. This small cut issue leads to loss of coercivity of the linear system being solved as the minimum eigenvalue is no longer bound and is based on the minimum dependency (i.e. basis function value) between the nodes of the background grid and the MPs - for the mass matrix the minimum is proportional to the square of the minimum basis function value. The problem is exacerbated by methods that mitigate cell crossing, such as the generalised interpolation MPM, as they are more likely to contain small interactions due to the extended stencil of the basis functions. This issue is not new and has been faced by the cut-FEM community as well as motivating explicit MPM techniques such as Modified Update Stress Last (MUSL), which reweights the nodal velocities prior to MP stress update to mitigate spuriously high values. However, these explicit techniques are not applicable to implicit MPM approaches, which have several advantages especially for coupled (soil-water) problems as they allow much larger time steps to be taken reducing the overall cost of an analysis. This research is focused on mitigating the small cut issue for implicit MPM based on the ghost stabilisation technique of Burman (2010) and extended to the MPM by Coombs (2023), which introduces a bound on the minimum eigenvalue of the linear system by constraining the gradient of the solution at the boundary of the physical body.

Citation

Coombs, W. (2024, September). Stability of implicit material point methods for geotechnical analysis of large deformation problems. Presented at ALERT Geomaterials 2024, Aussois, France

• Presentation Conference Type: Keynote
• Conference Name: ALERT Geomaterials 2024
• Start Date: Sep 30, 2024
• End Date: Oct 2, 2024
• Deposit Date: Oct 3, 2024
• Publicly Available Date: Oct 11, 2024
• Peer Reviewed: Peer Reviewed
• Public URL: https://durham-repository.worktribe.com/output/2943287