In this thesis report, we investigate the difference in using physics-compatible elements and standard elements in the context of fluid flow through porous media. For this, we compare the H(div)-conforming Raviart-Thomas elements to Lagrange elements. Our model involves a channel flow over a porous media, resulting in a fluid and a porous subdomain. We distinguish between three approaches, as based on a research paper. The first is the one-domain approach called PE, which sets one equation on the entire domain. The other two approaches are two-domain approaches, which use different equations in the two subdomains, with interface conditions to couple them. The first of these is NSD, which couples Navier-Stokes with the Darcy equation. The second is NSF, which does the same but also incorporates the Forchheimer term, which models inertial effects.

In general, we were not able to make a completely successful model, as the PE case did not match the paper, and the results found when using Raviart-Thomas elements had some unexplained anomalies. We were also unable to carry out an accurate error analysis. We were able to see that Raviart-Thomas produced an exactly divergence-free solution, as expected from the theory, at the cost of more computing power and time needed.

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.

Convection-dominated flow problems are well-known to have non-physical oscillations near steep gradients or discontinuities in the solution when solved with standard numerical methods, such as finite elements or finite difference methods. To overcome this limitation, algebraic flux correction (AFC) can be used, which is a stabilization method. However, AFC contains time-consuming computations, therefore, alternative approaches are explored. The rapidly rising field of machine learning in the mathematical world, so called scientific machine learning, has successful applications in solving partial differential equations. In this work, the focus is on convection-dominated flow problems, in particular the steady state convection-diffusion equation in one-dimension. To solve this, two alternative approaches based on neural network-learning have been developed that are able to mimic the AFC limiter with a certain accuracy and performance. In some cases, the neural network-based limiter is outperforming the AFC limiter.

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.

An isogeometric finite element method for incompressible fluid film equations is presented. The method can be applied to numerically model the behaviour of thin cellular membranes, such as lipid bilayers. The membranes are represented by infinitely thin closed surfaces. Both the surface parametrization and analysis are based on state-of-the-art polar spline spaces. These spaces are defined such that a C1 continuous genus 0 surface can be constructed. At the discrete setting, point-wise conservation of mass is attained, using the framework of discrete exterior calculus. Therefore, the polar spline spaces are called divergence conforming. Time discretization of the highly non-linear system is done via the fixed-point iterations. It is found that for certain non-uniformly curved domains, the iterations converge and time stepping can be performed. However, for surfaces that are closely resembling a perfect sphere, the iterations are not stable for any ∆t. The solution is parameter-dependent and this indicates a possible bug in the Matlab code.

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.

The finite element method (FEM) is a numerical method that is used to approximate the solutions to partial differential equations when solutions in the classical sense do not exist or are very hard to find. The method is used to solve problems that are relevant for industries like the automotive industry, the petroleum industry, and the aviation industry. The finite element method uses a discretisation of the problem domain into sub domains and uses a variational form involving trial and test functions to arrive at a system of equations. For problems that are advection dominated (for problems with a large Péclet number), the FEM solutions start to oscillate and produce inaccurate results near boundary layers. In the past finite element schemes have been developed that use so called optimal test functions to reduce the oscillations of the solution and increase the accuracy of the method.
In this thesis an attempt was made to use Deep Operator Networks (DeepONets) to generate optimal test functions for the steady state advection-diffusion equation in 1D and 2D to improve the stability/accuracy of the finite element method. An advantage of using neural networks is that once trained they can take in problem specific parameters like the diffusion coefficient and produce optimal test functions for a wide range of problems almost instantaneously. It was found that the applicability of the DeepONets in this context varies and depends on the weak formulation for which the DeepONet generated optimal test functions are implemented, as the finite element method solution can be very sensitive to small perturbations in the optimal test functions. The approach does look promising as a DeepONet was able to improve the finite element method significantly in a 2D setting by generating optimal test functions, while using the problem specific diffusion coefficient as a variable.

In this work we investigate neural networks and subsequently physics-informed neural networks. Physicsinformed neural networks are away to solve physical models that are based on differential equations by using a neural network. The wave equation, Burgers’ equation, Euler’s equation, and the ideal magnetohydrodynamic equations are introduced and solved with physics-informed neural networks. The solutions to the first equations were captured well. The solution to the ideal magnetohydrodynamic equations contained some problems. These problems include transitions between different types of behaviour and exact values of constant sections. On the other hand, general shape and behaviour of the curve and locations of contact discontinuities were predicted well.

Physics-Informed Neural Networks (PINNs) are a new class of numerical methods for solving partial differential equations (PDEs) that have been very promising. In this paper, four different implementations will be tested and compared. These include: the original PINN functional with equal weights for the interior and boundary loss, the same functional with custom weights, and the First Order Least Squares (FOSLS) functional with equal weights and custom weights. These custom weights are chosen to be equal to the optimal weights derived by Oosterlee et al. as well as slightly bigger and smaller. These methods will be applied to the 1D stationary advection-diffusion equation where we vary the difficulty by configuring the diffusion parameter epsilon. Furthermore, for each method we have done an elaborate parameter study where we varied epsilon and the number of collocation points. We have found that the weights derived by Oosterlee et al. did not provide accurate results. Instead, equal weights usually performed best. Also, the two functionals turned out to have very similar performance.

The graduation project was conducted at the CFD department of Nuclear Research And Consultancy Group (NRG) in Petten. The modeling and simulation of Taylor bubble flow using CFD can contribute significantly to the topic of nuclear reactor safety and in particular, in the emergency cooling of nuclear reactors during a loss of coolant accident, or in the U-tubes of a stream generator during a pipe rupture. To achieve an accurate representation of the gas-liquid interface for high values of Reynolds number, a general interfacial turbulence model should be developed which adapts to local conditions automatically. Direct Numerical Simulation (DNS) of relevant large interface two-phase turbulence has the potential to contribute to this, as it can produce more refined insight while being complementary to experimental data. The current graduation project illustrates a simulation approach towards DNS of turbulent co/countercurrent Taylor bubble flow. It comprises a continuation of the novel simulation strategy indicated by (Frederix et al., 2020) in which LES of co-current turbulent Taylor bubble flow using OpenFOAM were indicated and the authors concluded that LES mesh resolution is not sufficient to capture accurately the gas-liquid interface, velocity fluctuations, and bubble disintegration rate. To counter this, in the current work, a Basilisk code is developed which due to its adaptive local grid refinement and its accurate solution of advection equation, comprises a better choice than OpenFOAM for DNS in two-phase flows (via the settings of (Hysing et al., 2009) for laminar bubble flow and (Shemer et al., 2007) for turbulent co-current flow) since it captures sharper the interface and reduces the computational cost significantly. Except for OpenFOAM, Basilisk is also successfully validated against ANSYS (Araujo et al., 2012) for the laminar Taylor bubble flow. Last but not least, the work is further extended to the simulation of laminar and turbulent counter-current Taylor bubble flows in which the effect of the choice of the pipe diameter and the initial bubble length on the bubble’s decay rate is analyzed for the experimental setting of (Mikuz et al., 2019). Overall, despite the lack of fully DNS quality, this study extends the work of (Frederix et al., 2020) and provides further insight into the performance of Basilisk in two-phase flows at a reasonable computational cost. The results and conclusions from the current study may contribute to the development of low-order turbulence models and the validation of more general two-phase modeling strategies.