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 extit{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 extit{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.

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.

Computational fluid dynamics is nowadays one of the pillars of modern aircraft design, just as impor­tant as experimental wind tunnel testing. Very ambitious goals in regards to performance, efficiency and sustainability are being asked of the aviation industry, the kind that warrant a virtual exploration of the edges of the flight envelope. High­ order has come to be regarded as a necessary ingredient to achieve breakthrough advances in this direction. So much so, in fact, that the major aircraft manufac­turers, governmental aerospace research agencies and top universities worldwide have been acting in coordination to increase the technology readiness level of these approaches, for the last ten years. In this work, I study in significant detail the one aspect that makes high-­order methods ideal candi­dates for enabling the use of more advanced turbulence models in industry: the cost­-efficiency of their discretization—they offer minimal amounts of dispersion and dissipation errors, for a given number of degrees of freedom. I consider three research objects: discontinuous Galerkin spectral method (DGSEM), flux reconstruction (FR) and a novel B-­spline-based discontinuous Galerkin formulation stabilized via algebraic flux correc­tion (DGIGA­-AFC), and characterize their order of accuracy, linear and nonlinear stability characteristics, as well as dis­persion and dissipation errors as a function of wavenumber. Afterwards, I experimentally investigate their relative suitability towards scale­-resolving simulation of compressible and turbulent flows, by solv­ing a number of simple test cases of increasing difficulty (linear advection, inviscid Burgers and Euler equations; all in 1D) using a purpose­-made MATLAB implementation. The proposed isogeometric method (DGIGA) has been found to be at least as viable as the other two, a priori, for the resolution of high­-speed turbulent flows. Moreover, I have found that low dispersion and dissipation need not always be associated to high order, but to a high number of degrees of freedom per patch instead; these two coincide in the more conventional schemes, yet not necessarily in DGIGA. As a nonlinear stabilization mechanism, however, the proposed combination of DGIGA with AFC has turned out to be inferior to existing limiters.

Computational Fluid Dynamics (CFD) offers numerous benefits, notably the ability to study flows that are challenging or costly to investigate using experiments. A central challenge in CFD lies in simulating fluid flow around complex geometries. Additionally, the governing equations follow conservation laws. This thesis aims to establish the foundation for constructing Immersed divergence-conforming finite element spaces that address both challenges. To tackle the first issue, an immersed method is considered. Instead of generating a mesh that conforms to the object where the flow occurs, the approach involves placing the object within a predefined mesh. However, this introduces new challenges, the most significant of which is the ill-conditioning of the associated linear system. The condition number depends on the location of the immersed object and can lead to an extremely large condition number. The second challenge is resolved by discretizing the problem in a way that preserves essential topological and homological structures at a discrete level, utilizing the principles of finite element exterior calculus. Within this thesis, a structure-preserving subcomplex is developed for the de Rham complex in 1D, and its accuracy is validated through numerical experimentation. Optimal convergence is achieved, the discrete inf-sup test is satisfied, and the resulting linear system's conditioning remains unaffected by the placement of the immersed object. However, the constructed structure-preserving subcomplex for the de Rham complex in 2D, known as the immersed divergence-conforming finite element spaces, does not exhibit convergence. For future research, I recommend focusing on simpler vertical and skewed cuts before exploring more intricate immersed shapes. These simpler cuts involve straightforward choices for edge basis functions/1-form basis functions that are relevant. Additionally, if the outcomes remain unsatisfactory, considering a global approach instead of a local approach could be worthwhile.

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 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.