Algorithmic improvements of the material-point method and Taylor least-squares function reconstruction

The material-point method (MPM) is a continuum-based numerical tool to simulate problems that involve large deformations. Within MPM, a continuum is discretized by defining a set of Lagrangian particles, called material points, which store all relevant material properties. Themethod adopts an Eulerian background grid, where the equations of motion are solved at every time step. The solution on the background grid is used to subsequently update all material-point properties, such as displacement, velocity, and stress. In this way, MPM incorporates both Eulerian and Lagrangian descriptions. Similarly to other combined Eulerian-Lagrangian techniques, MPM attempts to avoid the numerical difficulties arising from nonlinear convective terms associated with an Eulerian problem formulation, while preventing grid distortion, typically encounteredwithin meshbased Lagrangian formulations.
Over the years,MPM has been successfully applied to many complex problems from engineering and computer graphics. Despite its impressive performance for these applications, the method still suffers from several numerical shortcomings, such as stability issues, inaccurate mapping of the material-point data, and unphysical oscillations that arise when material points travel from one element to another, the so-called grid crossing errors. This dissertation provides an overview of the existing literature that addresses these drawbacks, and introduces new mathematical techniques that improve the performance of MPM.
Previous studies have indicated that the use of higher-order B-spline basis functions within MPM mitigates the grid-crossing errors, thereby improving the accuracy of the method. This thesis combines the B-spline approach, known as BSMPM, with an alternative technique to project the information from material points to the background grid. The mapping technique is based on cubic-spline interpolation and Gauss quadrature. The numerical results show that the proposed approach further increases the accuracy of the method and leads to higher-order convergence. Moreover, the extension of BSMPM to unstructured grids using Powell-Sabin splines is discussed.
After that, this dissertation compares MPM to the optimal transportation meshfree (OTM) method. Both MPM and the OTM method have been developed to efficiently solve partial differential equations that arise from the conservation laws in continuum mechanics. However, the methods are derived in a different fashion and have been studied independently of one another. This thesis provides a direct step-by-step comparison of the MPM and OTM algorithms. Based on this comparison, the conditions, under which the two approaches can be related to each other, are derived, thereby bridging the gap between the MPM and OTM communities. In addition, the thesis introduces a novel unified approach that combines the design principles from BSMPM and the OTM method. The proposed approach is significantly cheaper and more robust than the standard OTM method and allows for the use of a consistent mass matrix without stability issues that are typically encountered in MPM computations.
Finally, this thesis introduces a novel function reconstruction technique that combines the well-known least-squares method with local Taylor basis functions, called Taylor least squares (TLS). The technique reconstructs functions from scattered data, while preserving their integral values. In conjunction with MPM or a related method, the TLS technique locally approximates quantities of interest, such as stress and density, and when used with a suitable quadrature rule, conserves the total mass and linear momentum after transferring the material-point information to the grid. The integration of the technique into MPM, dual domain material-point method (DDMPM), and BSMPM significantly improves the results of these methods. For the considered onedimensional examples, the TLS function reconstruction technique resembles the approximation properties of the highly-accurate cubic-spline reconstruction, while preserving the physical aspects of the standard algorithm.