Imposition of essential boundary conditions in the material point method

Abstract

SummaryThere is increasing interest in the material point method (MPM) as a means of modelling solid mechanics problems in which very large deformations occur, e.g. in the study of landslides and metal forming; however, some aspects vital to wider use of the method have to date been ignored, in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges. In this paper, we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions, for the MPM, expanding its capabilities to model boundaries of any inclination. To the authors' knowledge, this is the first time that a method has been proposed that allows arbitrary Dirichlet boundary conditions (zero and nonzero values at any inclination) to be imposed in the MPM. The method presented in this paper is different from other MPM boundary approximation approaches, in that (1) the boundaries are independent of the background mesh, (2) artificially stiff regions of material points are avoided, and (3) the method does not rely on mirroring of the problem domain to impose symmetry. The main contribution of this work is equally applicable to standard finite elements and the MPM.

Rainfall-induced slope collapse with coupled material point method

“…Stress oscillations occur in both the effective stresses and pore pressures; although there are a number of methods to tackle this, such as GIMP ( Bardenhagen and Kober, 2004) and CMPM (Gonzalez Acosta et al, 2017), none has yet managed to eliminate them. Boundary conditions that do not align well with the mesh are difficult to apply accurately, although recent work has begun to address this (Remmerswaal, 2017;Cortis et al, 2018). Finally, it is noted that compared to analytical methods and FEM, the computational effort is very high; therefore, it is considered a useful method to simulate the full failure, but not in determining an initial failure or factor of safety.…”

Weak impositions of Dirichlet boundary conditions in solid mechanics: A critique of current approaches and extension to partially prescribed boundaries

“…The step boundary method has a similar formulation to the implicit boundary method but ends up with a system that contains a penalty parameter. It has also been applied to spline based finite element methods [53] and the material point method [54].…”

An implicit boundary finite element method with extension to frictional sliding boundary conditions and elasto-plastic analyses

“…In elasto-plastic analysis, F ext in the discretised weak form (40), which contains the body force and the traction on the Neumann boundary, is not dependent on the current deformed 400 state, such that it remains constant throughout a certain load increment, whereas both F int and R in Eq. (40) are to be linearised, that is…”

“…A special scheme for integration is developed because of this step-like weight function, leading to a final solution system that can also be categorised as a distributed penalty method. Since then, the step boundary method has also been extended to combine with B-spline bases [37], and applied to inclusion problems [38], shell-like structures [39], dynamics problems [29] and finite deformation problems in the Material Point Method [40,41]. The essential concept of the step boundary method is to use the following function, which vanishes on the Dirichlet boundary 70 and rises to unity within a step size ✏, to weight the regular finite element interpolation:…”