Over the last decade, an increasing amount of data has become available for data analysts to understand. Datasets containing books, images, networks, or other types of data have been studied. A recent group of methods proposes to analyze samples in datasets based on a description of their shape. This group of methods is often referred to as Topological Data Analysis (TDA). In this thesis, an extension to the most commonly used TDA method, called Persistent Homology (PH), is proposed. PH only describes topological features, while this extension additionally allows for the description of geometric properties. The new information is obtained via the persistent Laplacian, a recently proposed operator that encodes the topological information of persistent homology in its kernel and geometric information in its non-zero spectrum. The persistent Laplacian contains a lot of information and extracting the relevant parts has not yet been standardized. In this thesis, a new operator, the persistent multiplicity operator, is proposed. The new operator summarizes the information of the persistent Laplacian such that it can easily be extracted and used to extend PH. This allows the many previously studied methods based on persistent homology to additionally describe geometric properties, as opposed to only topological features. For the multiplicity operator, the trace is analyzed and the features captured by it are discussed. Besides analyzing the sum of the eigenvalues, it is argued that individual eigenvalues could contain more information. However, these are deemed hard to understand. Therefore, an adjusted multiplicity operator is proposed that contains separately interpretable eigenvalues. Finally, the operators are used to classify handwritten digits from the MNIST dataset and to make statistical tests that can detect different generation processes of artificially made cross sections of crystalline structures.

In this thesis, we study fractional differential equations with Hilfer derivative operators. Solutions are approximated using a newly developed Bernstein-splines approach and subsequently applied to the Van der Pol oscillator with fractional damping. Fractional derivatives generalize differentiation to the order α ∈ (0,∞), resulting in order α differential operators. The Hilfer-derivative of order α ∈ (0,1) and type β ∈ [0,1] is one of such operators and combines two of the most commonly used operators through parametrization: the Riemmann-Liouville-derivative (β = 0) and Caputo-derivative (β=1). The choice of β results in different kernel behavior of the operator, commonly yielding singular behavior of solutions of initial value problems (IVP's) and boundary value problems (BVP's) for β ∈ [0,1). Based on existing results for IVP's, we develop a new proof for existence and uniqueness of solutions to BVP's for Hilfer-fractional derivatives. To obtain solution approximations numerically for IVP's and BVP's, a Bernstein splines solution approach is developed and implemented, providing accurate convergence results for nonlinear IVP's and BVP's in an efficient vectorized approach. Implementation difficulties for the singular behavior of solutions for β ∈ [0,1) are successfully resolved through time-domain transformation approximation techniques. Finally, the methods are applied to numerically study the behavior of the fractionally damped Van der Pol oscillator, a nonlinear equation of interest in electrical engineering and control theory. We study the approximate limit cycle, of which behavior corresponds with existing analytical results for the Caputo-derivative (β=1). Convergence of solutions can also be obtained for Hilfer type values of β ∈ [0,1), appearing to be of little influence on the long-term limit cycle. Furthermore, when forcing is applied, we observe periodic, quasiperiodic and chaotic behavior.

This thesis investigates the flow and heat transport phenomena in a pipe and Pressurized Water Reactor (PWR) rod bundle geometries using high-fidelity Direct Numerical Simulation (DNS). These geometries are essential for nuclear reactor systems, where efficient heat transfer and stable flow patterns are vital to ensure operational safety, reliability, and performance. The study focuses on evaluating the limitations of traditional thermal-hydraulic modeling approaches and advancing the understanding of flow physics in complex geometries.
Since Computational Fluid Dynamics (CFD) simulations are computationally expensive, system codes such as RELAP5, TRACER, and SPECTRA (developed by NRG) are widely used in reactor safety and design analysis. These codes rely on conventional pipe flow correlations to approximate thermal-hydraulic behavior, which are not representative of the intricate flow patterns observed in rod bundle arrangements. Such geometries are characterized by secondary vortices, gap street vortices, and the coupling of these vortices across multiple planes, forming a complex rod bundle vortex network. This study addresses these challenges by comparing the flow and thermal characteristics of pipe and subchannel geometries to evaluate the validity and limitations of pipe-based correlations.
To identify the subchannel geometry that best represents a rod bundle arrangement, square and 2×2 subchannel configurations were selected for detailed investigation. These geometries were modeled with a pitch-to-diameter (P/D) ratio of 1.3263, typical of PWR fuel assemblies, and simulated at a Bulk Reynolds number (Reb) of 5300. The square subchannel geometry represents a simplified cross-sectional domain, while the 2×2 subchannel configuration captures inter-subchannel interactions and the enhanced coupling effects seen in rod bundles. Using the Nek5000 spectral element solver, the simulations employed advanced numerical techniques, including spatial-temporal averaging through flow-through time (FTTs) metrics and further spatial averaging over unit cells, to achieve statistically converged results. Rigorous validation of pipe configuration with reference data ensured the accuracy of the computational framework.
The results highlight significant differences in turbulence structures and heat transfer performance between the pipe, square subchannel, and 2×2 subchannel configurations. While pipe correlations provide a baseline for comparison, they fail to capture the complex flow interactions observed in rod bundles. The 2×2 subchannel geometry emerges as a more accurate representation of rod bundle dynamics due to its ability to simulate enhanced inter-subchannel mixing and vortex coupling. These findings emphasize the importance of geometry-specific modeling in accurately predicting thermal-hydraulic behavior in nuclear reactors.
This work bridges the gap between simplified system codes and detailed physics-based modeling of rod bundle flows. The insights gained from this study provide a foundation for improving thermal-hydraulic predictions and advancing the design and safety of nuclear reactor systems. Future research will extend these findings to explore higher Reynolds numbers, diverse Prandtl numbers, wall effects, and alternative rod bundle configurations, further contributing to the development of advanced engineering tools for nuclear applications.

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.

Physics Informed Neural Networks are a relatively new subject of study in the area of numerical mathematics. In this thesis, we take a look at part of the work that has been done in this area up until now, with the ultimate goal to develop a new type of PINN that improves upon the old concept. We introduce the concept of parameterized PINNs, which allow a single trained network to solve multiple partial differential equations for multiple boundary conditions and geometries by parametrizing these variables as an input for the network. Two methods are tested: one using global basis functions, and one using B-splines. The proposed methods are tested for Laplace’s equation and Poisson’s equation in multiple dimensions, most of which show that these methods are viable alternatives for the current style of collocation-based PINNs.

Mimetic formulations, also known as structure-preserving methods, are numerical schemes that preserve fundamental properties of the continuous differential operators at a discrete level. Additionally, they are well-known for satisfying constraints such as conservation of mass or momentum.

In the present work, a Mimetic Spectral Element Method based on quadrilaterals is explored. As an introduction, the framework is first implemented and tested on the classical Poisson equation, the Hartmann Flow system and several eigenvalue problems for the Laplacian operator. Solutions are attained by direct/mixed formulations and the extension to multi-element approaches is dealt with using either gathering or connectivity matrices. Different boundary conditions and various geometries are utilized.

Afterwards, the Maxwell Eigenvalue problem for the electric field E with general material properties is tackled in an attempt to generate spurious-free solutions by incorporating the condition ∇·D = 0 into the discrete system. The formulation is further scrutinized on geometries with Betti number b₁ > 0 as to verify if the proposed scheme captures the physical zero eigenvalues.

In the end, a mixed formulation for the eigenproblem is proposed in which the curl-curl operator is separated. The approximation of the electrostatic field energy is then computed with this formulation and compared to the solution obtained with a direct method allowing to create an upper and a lower bound for this variable.

High-dimensional optimization problems with expensive and non-convex cost functions pose a significant challenge, as the non-convexity limits the viability of local optimization, where the results are sensitive to initial guesses and often only represent local minima. But as the number of expensive cost function evaluations required for a full exploration of the search space grows exponentially with the increasing number of dimensions, the use of standard global optimization algorithms is also not practical. To overcome this obstacle and to lower the dimensionality of the problem, the use of an autoencoder for model order reduction is proposed. In the resulting lower dimensional space, standard global optimization methods can then be utilized, as fewer cost functions evaluations are necessary. For problems with comparatively more expensive cost functions, this optimization includes the employment of a surrogate model, which reduces the necessary number of these computationally expensive evaluations further. This proposed method is then tested firstly on a number of benchmark functions, where it shows the ability to find global optima under certain conditions. Secondly, the proposed method is used to solve a compliance minimization problem, where it shows the ability to improve upon a large number of designs generated by local optimization.

This thesis poses a new geometric formulation for compressible Euler flows. A partial decomposition of this model into Roe variables is applied; this turns mass density, momentum and kinetic energy into product quantities of the Roe variables. Lie derivative advection operators of weak forms constructed with this decomposed model naturally follow to be self-adjoint, which results in skew-symmetric discrete advection operators in any number of dimensions. Under certain conditions these conserve products of the Roe variables, leading to a discrete model formulation with advection operators that simultaneously conserve mass, momentum, kinetic energy, internal energy and total energy in compressible Euler flows. While this idea is not new the novelty of this work lies in its extension to mimetic finite element methods and its application to discontinuous compressible Euler flows.The regular geometric Euler model and its Roe variable decomposition have been discretized through mimetic isogeometric analysis. At the core of mimetic discretization methods lies the idea of retaining the De Rham sequence of differential form spaces when projecting these to finite-dimensional approximations and when constructing discrete operators. B-spline differential form spaces have been defined such that the exterior derivative maps in a topologically exact and metric-free way, while the interior product has been discretized in a weak way in order to retain its map between appropriate spaces in the De Rham sequence. Only primal grids are used; the Hodge star operator is discretized through the definition of an L2 inner product to resolve weak forms. Cartan’s homotopy formula allows for a consistent discretization of the Lie derivative through compositions of the interior product and exterior derivative.Several tests were carried out to determine the efficacy and behavior of this regular geometric Euler model, its Roe variable decomposition and the resulting discrete advection operators. Testing one-dimensional linear advection and Burgers’ equation on periodic domains shows that discretizations of the self-adjoint advection operators are consistent with conservative formulations. Sod’s shock tube is used as one-dimensional discontinuous compressible flow test. Both the regular Euler model and its Roe variable decomposition display strong oscillatory tendencies, yet solution convergence is obtained without any issues. While application of a simple moving average-filter removes the worst oscillations more sophisticated methods are necessary for obtaining solutions that are free of unphysical oscillations. The Roe variable decomposition negatively affects the accuracy of shock speed predictions. The presence of strong oscillations likely affects the numerical conservation errors of both methods. Compared to finite volume package Clawpack and the nodal Discontinuous Galerkin (DG) method of Hesthaven & Warburton both models have less numerical diffusion on coarse meshes with the regular model outperforming the Roe variable decomposition. Convergence of momentum conservation error is slow and the errors are large compared to the reference methods. For two-dimensional periodic vortices both the regular geometric Euler model and its Roe variable decomposition outperform both reference methods for stationary and moving vortices. Clawpack displays diffusive behavior, resulting in large L2 errors and high amounts of numerical diffusion. While the L2 errors of the DG method are comparable to those of the two models developed in this work for all basis function orders considered, the resulting DG discretization requires significantly more degrees of freedom to attain this. To obtain similar levels of numerical diffusion the DG method needs up to ten times as many degrees of freedom as the methods presented in the current research.

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.