The Material Point Method (MPM) is a numerical method primarily used in the simulation of large deforming or multi-phase materials. An example of such a problem is a landslide or snow simulation. The MPM uses Lagrangian particles (material points) to store the interested physical quantities. These particles can move freely in the spatial domain. Each time step, these physical quantities are projected on a background grid, which is necessary to evaluate these quantities at a future time step. Once all necessary properties are projected, the acceleration on this grid can be updated using the conservation of linear momentum. At last, this updated acceleration can be projected to the particles to compute their updated velocity, position and other quantities.

The main problem addressed in this thesis arises in the projection of the particles to the background grid. The use of linear basis functions in the classical MPM causes instabilities, which can be resolved by using higher order B-splines. However, another problem arises when particles move between the background cells. Since these particles move freely through the spatial domain, it is possible for some cells to by \textit{nearly-empty}. In this thesis, the use of Extended B-splines is explored to increase the stability in these areas by deactivating unstable B-splines and \textit{extending} stable ones.

Furthermore, higher dimensional B-splines are constructed by a tensor product of one dimensional B-splines. The scalability can be a problem, as these B-splines cannot be refined locally. Therefore, Truncated Hierarchical B-splines are introduced to locally refine a geometry. The effectiveness of this technique in combination with the EB-splines is investigated.

The thesis shows that the use of EB-splines in the context of MPM increases the stability in the case of nearly-empty cells, and can also improve the quality of the solution near the boundary. Furthermore, in the neighborhood of high stress concentrations, the THB-splines have a similar accuracy as the regular B-splines, while drastically reducing the computational costs. In combination with EB-splines, THB-splines can be used to accurately refine a geometry in the presence of nearly-empty cells.

Predicting the behavior of turbulent flows and how thermochemical properties develop such as temperature, enthalpy or concentrations is an import aspect in various fields. However, numerical calculations have a high time complexity. Therefore, models were constructed to approximate these flows. This thesis focuses on using probability density functions (pdf) to predict scalar probabilities in these flows. Since the pdf does not contain spatial information, small scaled mixing has a closure problem and has to be modeled using micromixing. A micromixing model, called the Shadow Position Mixing Model (SPMM) was introduced previously to calculate this mixing. In this project, its performance is determined for the mixing in two different flows. The model shows promising results for mixing of two passive scalar, initially mainly in one of three states. Comparing the results of various mixing models to the provided DNS, the SPMM predicts the evolution of the pdf over time adequately, but still has visible differences compared to the DNS. In the second test, the model is tested using an imposed mean scalar gradient in order to check the convergence of the mixing model. The SPMM does show promising results, but can not reliably predict a correct statistically stationary state. This is due to the fact that varying number of scalars and particles lead to significant inconsistencies. Under the current conditions, the SPMM cannot be reliably applied to flows with such an additional scalar force. The SPMM uses a near-neighbor sorting algorithm which has a high algorithmic complexity. For practical purposes, a lower complexity algorithm has to be developed in order to use this model for high number of particles.