Understanding multiphase flows is critical in nuclear engineering, particularly for processes such as coolant dynamics in nuclear reactors and safety scenario analyses involving different fluid phases. Numerical simulations are a valuable tool for studying these phenomena, especially when experimental approaches are impractical due to cost or safety concerns. While direct numerical simulations (DNS) offer detailed insights, their computational expense makes them impractical for turbulent flows, necessitating the use of turbulence models for efficiency. This thesis introduces a novel machine learning framework designed to improve Reynolds-averaged Navier-Stokes models in turbulent stratified gas-liquid flows while employing the Boussinesq approximation. The framework encompasses two methods for turbulent viscosity field inversion and introduces correction terms in the turbulence model equations to ensure an accurate prediction of the turbulent viscosity field. Through sparse symbolic regression, the framework consistently discovers models that improve the accuracy of the baseline RANS model, even in untrained flow scenarios, though further testing is needed for varied flow regimes. Key findings include the superior performance of sparse symbolic regression models over neural network (NN) models in improving the baseline RANS model accuracy. Notably, LASSO and elastic net techniques yielded the most successful models, significantly reducing baseline errors. However, these models did not surpass the Egorov damping approach in terms of accuracy, indicating the need for further refinement. The developed models were numerically stable and robust, which is important for practical use. However, a main limitation is that the models' accuracy during training did not always correlate with the results when coupled with the RANS equations. Moreover, data from more varied flow conditions is needed to properly assess the generalizability of the models. Overall, this research highlights the potential of data-driven turbulence modelling to enhance two-phase flow simulations, marking a significant step forward while also identifying areas for future improvement and exploration.

The Navier-Stokes equations govern the flow of viscous fluids such as air or water. Since no general solution is known, computer simulations are used to obtain approximate solutions. As computers are unable to handle continuous representations of fields, finite-dimensional projection of fields is required, resulting in a loss of information. A difficulty in constructing these discrete solutions, is to obtain discrete solutions that have the same conservation properties as continuous solutions. Other challenges include the efficient handling of the nonlinear term in the equations. In this thesis we aim to construct a numerical method whose solutions satisfy the conservation of mass, kinetic energy and helicity exactly. To achieve this, a structure preserving (mimetic) spectral element method is employed. Such methods aim to represent properties of the continuous problem exactly in the discrete problem. These methods have their theoretic roots in differential geometry, where there is a clear connection between fields and the geometric objects they are associated with. From this association we note that the fields in the Navier-Stokes equations show a dual nature, where each field can be associated to two types of geometric objects. In [1] this dual nature is used to construct a mass, kinetic energy and helicity conserving method that handles the nonlinear term efficiently by using a leapfrog scheme for problems on periodic domains. In this thesis this work is extended to handle Dirichlet boundary conditions. When considering periodic boundary conditions, the constructed numerical method exhibits the same conservation properties of energy and helicity as the Navier-Stokes equations. When considering Dirichlet boundary conditions, some additional contributions to the dissipation rates of energy and helicity are noted in the cases of inflow into the domain and nonzero normal vorticity at the boundary respectively. These additional contributions cannot be meaningfully quantified within the employed approach, and are a result of strongly enforced boundary conditions on velocity. Furthermore, the numerical method constructs a pointwise divergence free velocity field at every other time step. The mimetic spectral element method allows for high order spatial approximation of fields. Optimal spatial convergence orders are observed for all fields. Temporal integration was done via a combination of the trapezoidal and midpoint rule and the expected second order temporal convergence is observed.

In this work residual error estimates are constructed using Neural Networks for Finite Element Method. These can be used to do adaptive mesh refinement. Two neural networks are developed the Multilayer Perceptron and the Transformer model. The error estimates are made for 1d poisson equations but the idea will generalise to higher dimensions as well.

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.

Existence of commuting THB-spline projectors is of importance in the field of numerical mathematics. These projectors are required to show that numerical solutions to the abstract Hodge Laplace problem are stable and consistent. We have introduced a local THB-spline projector based on Bezier projection, that commutes in the one-dimensional case. Additionally, to show that the projector is well posed in the univariate and multivariate case, we have derived so called projection elements, that are grouped mesh elements, on which the THB-splines are linearly independent.

Artificial Intelligence in the form of neural networks is becoming wide spread. This report focuses on a specific form of neural networks, Simplicial Neural Networks. After presenting their advantages and how they were implemented in Python by using the code of [1], they are tested in 2 experiments to explore their applicability in solving physical problems. The first experiment aimed to test the feasibility of the approach as well as to compare them to traditional neural networks. The second experiment aimed at testing the use of SNNs in predicting pressures through a Stokes Flow when the flow is known. Although the experiment was carried out incorrectly the network still produced accurate results in the context of the experiment, and SNNs could present an alternative to Finite Element Solvers.