B-spline based Boundary Method for the Material Point Method

Abstract

Unlike the conventional finite element method, in which the mesh conforms to the material boundary, the material point method (MPM) does not provide a clear interpretation of the boundary. Consequently, difficulties arise when it comes to solving boundary-value problems during MPM simulations, in particular, applying traction (Neumann) and prescribed displacement (inhomogeneous Dirichlet) boundary conditions. However, little attention has been paid to this issue; no literature to date has presented an effective way to model and track boundaries in the MPM. Hence, developing new ways of boundary representation and boundary conditions application in the MPM is the focus of this research.

Formulation of the MPM is firstly presented followed by a review on current approaches to this boundary issue, where B-spline interpolation techniques and an implicit boundary method are identified as the methods to be taken forward. Essential knowledge on B-splines is then discussed. After comparing different B-spline interpolation techniques, a local cubic scheme is selected for boundary representation due to its ability to handle sharp corners and its relatively high computational stability. Next, enforcements of the boundary conditions are discussed. Tractions are applied through direct integration over the B-spline boundary and displacements are prescribed via a B-spline based implicit boundary method. Finally, this boundary method is verified through numerical examples, several of which were not possible with previous MPMs. The novelty of this thesis lies in providing a complete methodology on modelling and tracking the boundaries as well as accurately imposing both Neumann and Dirichlet boundary conditions in the MPM.