High-order material point method

The material point method (MPM) is a meshfree mixed Lagrangian-Eulerian method which utilizes moving Lagrangian material points that store physical properties of a deforming continuum and a fixed Eulerian finite element mesh to solve the equations of motion for individual time steps. It is common practice in MPM to adopt piecewise linear basis functions to approximate the solution of the variational form. A problem arises from the discontinuity of the gradients of these basis functions at element boundaries. This leads to unphysical oscillations, for example in computed stresses, when material points cross element boundaries. Such grid crossing errors significantly affect the quality of the numerical solution and may lead to a lack of spatial convergence. As a remedy to these problems, a version of the MPM making use of quadratic B-spline basis functions is presented. Using spline interpolation allows to more accurately approximate integrals, which enables the use of a coarser mesh. This in turn results in lower computational effort. To improve spatial convergence, the use of a consistent mass matrix instead of a lumped one commonly used with the MPM is suggested to project velocities from material points to the grid more accurately. Explicitly solving the linear system is avoided by using Richardson iteration. Improvements in terms of accuracy and rate of convergence are demonstrated for 1D benchmarks involving small and large deformations. In particular a vibrating bar and a column subjected to loading are considered.