On the order of accuracy of the generalized interpolation material point method

The Generalized Interpolation Material Point (GIMP) method is examined. Its accuracy and stability are measured precisely via manufactured solutions and several improvements to GIMP are proposed, including a new least squares method that can be easily used within the GIMP framework. This dissertation is divided into three main parts. In the first part the standard velocity projection scheme for the Material Point (MPM) and GIMP methods is examined. It is demonstrated that the fidelity of information transfer from a particle representation to the computational grid is strongly dependent on particle density and location. In addition, use of nonuniform grids, and even nonuniform particle sizes, is shown to introduce error. An enhancement to the projection operation is developed which makes use of already available velocity gradient information. In the second part the stability and accuracy of GIMP is measured directly through carefully formulated manufactured solutions over wide ranges of Courant-Friedrichs-Lewy (CFL) numbers and mesh sizes. The manufactured solutions are described in detail and the accuracy and stability of several time integration schemes are compared via numerical experiments. In the third part a new solid mechanics method based on MPM and GIMP is introduced that uses Weighted Least Squares to approximate function values and gradients at grid nodes. The method represents a compromise between Finite Element and MPM/GIMP that improves accuracy, avoids nearest neighbor searches, and allows for initialization from three-dimensional image data. The method has a few drawbacks, which are also discussed, along with ways to mediate them. Implementation of the method is discussed in detail and several example problems are solved, including one manufactured solution that allows measurement of dynamic, nonlinear, large deformation performance.