Investigation of MPM inaccuracies, contact simulation and robust implementation for geotechnical problems

The material point method (MPM) is a numerical technique which has been demonstrated to be suitable for simulating numerous mechanical problems, particularly large deformation problems, while conserving mass,momentum and energy. MPM discretises material into points and solves the governing equations on a background mesh which discretises the domain space. The points are able to move through the mesh during the simulation. MPM is an improvement over other well-established numerical techniques, such as the finite element method (FEM), as it is able to simulate large deformations and therefore can simulate mechanical problems from initiation to the final outcome. It has the potential to become the preferred numerical tool to analyse many engineering problems. Nonetheless, it has been demonstrated throughout this thesis that the performance of MPM has often been far from the levels of accuracy desired in order to be considered a reliable technique for providing quantitative analyses for engineering problems. In this thesis, the implicit solution version of MPM has been taken as the starting point to investigate and solve its current main drawbacks, i.e. (i) the lack of accuracy when computing stresses (stress oscillations), and (ii) interaction between bodies, e.g. soil and structures. The stress oscillation problem is well-known in the MPM community, and is attributed mostly to material points crossing background cell boundaries, termed the cell-crossing problem. It has been shown in this thesis that cell-crossing is indeed one of the primary sources of oscillation. However, there are also other aspects contributing to the observed inaccuracies. In the literature, cell-crossing has been addressed by creating a particle domain, e.g. in the generalised interpolated material point (GIMP) method. It has been shown in this thesis that major problems also include (i) the use of linear shape function (SF) gradients to calculate (material point) strains and (ii) non-Gauss numerical quadrature to integrate material stiffness. The integration is made worse when using GIMP. In order to reduce the inaccuracies caused by integration a double mapping (DM) technique has been developed, which reduces the errors when integrating nodal stiffnesses. This is shown to also work well with GIMP (DM-G method). Additionally,DM has been combined with a Lagrangian interpolation technique, which uses a larger solution domain (through the combination of background cells to formpatches) to enhance the stresses computed at the material points (DM-C or DM-GC methods). The developed methods have been able to significantly improve the accuracy and stability of the simulated problems. This improvement will allow more robust use of more advanced constitutive models. The interaction of bodies is of benefit in large deformation simulations, although MPM can roughly simulate contact without special treatment. An MPM contact algorithm was initially proposed by other researchers for explicit time integration schemes, but no method was available for the implicit time integration scheme. An implicit contact scheme has been developed based on the original (explicit) contact formulation in order to calculate the change of nodal velocity during the Newton–Raphson iterative procedure. The results obtained with this contact methodology are shown to be as accurate as those computed using the explicit scheme, although generally with a larger time step. Additionally, it has been observed that, in most of the cases, implicit contact simulations are analysed faster than explicit simulations. However, the contact loads computed with this technique and the internal forces developed are inconsistent (i.e. not equal), reducing the energy conservation and remains an issue to be solved. An analysis of the problem is presented as a first step towards a solution. One challenge is that any method using consistent contact and internal forces is sensitive to stress oscillations, which can lead to highly unrealistic contact forces. Using the improvements developed in this thesis (i.e. DM-GC combined with the contact algorithm), soil-structure interaction problems and landslides have been successfully simulated. Incorporating the contact algorithm into the model has allowed the simulation of complex failure mechanism development during slope failure. The impact on neighbouring structures was realistic, and captured expected behaviours such as the sliding and rotation of the rigid elements. It has been demonstrated that (i) the accuracy in MPM has been improved via the combination of several (existing and novel) techniques, (ii) techniques developed for the explicit scheme (or other numerical methods) can be converted and introduced in implicit MPM, maintaining as much as possible the consistency of the formulation, and (iii) by improving diverse aspects of the formulation,more realistic simulations can be obtained. The work presented in this thesis makes several steps contributing to the improvement of MPM, which will lead towards it being used in engineering practice.