Abstract
The materialpoint method (MPM) is a continuumbased 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 materialpoint properties, such as displacement, velocity, and stress. In this way, MPM incorporates both Eulerian and Lagrangian descriptions. Similarly to other combined EulerianLagrangian 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 materialpoint data, and unphysical oscillations that arise when material points travel from one element to another, the socalled 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 higherorder Bspline basis functions within MPM mitigates the gridcrossing errors, thereby improving the accuracy of the method. This thesis combines the Bspline 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 cubicspline interpolation and Gauss quadrature. The numerical results show that the proposed approach further increases the accuracy of the method and leads to higherorder convergence. Moreover, the extension of BSMPM to unstructured grids using PowellSabin 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 stepbystep 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 wellknown leastsquares 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 materialpoint information to the grid. The integration of the technique into MPM, dual domain materialpoint 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 highlyaccurate cubicspline reconstruction, while preserving the physical aspects of the standard algorithm.
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 materialpoint data, and unphysical oscillations that arise when material points travel from one element to another, the socalled 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 higherorder Bspline basis functions within MPM mitigates the gridcrossing errors, thereby improving the accuracy of the method. This thesis combines the Bspline 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 cubicspline interpolation and Gauss quadrature. The numerical results show that the proposed approach further increases the accuracy of the method and leads to higherorder convergence. Moreover, the extension of BSMPM to unstructured grids using PowellSabin 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 stepbystep 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 wellknown leastsquares 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 materialpoint information to the grid. The integration of the technique into MPM, dual domain materialpoint 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 highlyaccurate cubicspline reconstruction, while preserving the physical aspects of the standard algorithm.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  18 Dec 2019 
Print ISBNs  9789463840897 
DOIs  
Publication status  Published  2019 
Keywords
 materialpoint method
 function reconstruction
 Taylor least squares
 optimal transportation meshfree method
 Bspline
 gridcrossing error
 spatial accuracy