Finite element analysis and multibody dynamics are popular mathematical modeling techniques for sev- eral mechanical applications. The finite element method is a costly method from a runtime standpoint. Moreover, both multibody dynamics and finite element methods require approximating the geometry of the application either by meshing or by analytical surfaces. Therefore, IsoGeometric Analysis (IGA) has been proposed as an alternate method for parameterizing geometric surfaces. IGA enables a uni- fied mathematical representation of both geometries and solution fields using B-Splines or NURBS. NURBS are the de-facto standard in modern Computer-Aided Design (CAD) workflows, but their use for analysis in finite elements is rare. By directly using the geometric information for the analysis, IGA reduces the cost of mesh generation, therefore enabling a more efficient design process.
The IGA-based rigid body contact algorithm offers several advantages over the traditional FEM- based approach. First, it provides an accurate representation of the geometry. This improves the efficiency of the design process and reduces the need for mesh refinement. Second, by looking for the local orthogonal projection in the parameter space rather than the coordinate space, the algorithm simplifies the definition of boundary conditions. This makes the process of defining boundary conditions easier and more intuitive. Third, the algorithm approximates the contact area by mapping it from the parameter space to the physical space. This allows for the specification of the level of accuracy of the contact area, which can permit more precise solutions.
The paper compares the IGA-based rigid body contact algorithm with the existing rigid FE mesh- based contact method in Simcenter Madymo. The results show that for the same level of accuracy, the IGA-based method can largely reduce execution time. This suggests that IGA could be a valuable alternative for use in the field of contact simulation.

Topology optimization is a branch of mechanical engineering in which the topology of a structure is created and optimized to certain conditions and restrictions. In the last few decades, the demand for highly accurate and complex models of these structures has increased and it has a big effect on the computational power needed. To ease the computational load for the dynamical systems one can use model order reduction methods to reduce the size of the models.
Classic Arnoldi is a widely used method for model order reduction (MOR) with topology optimization. In this thesis, we discuss two-sided Arnoldi and IRKA to help find a suitable moment-matching MOR method for topology optimization. These two reduction methods are implemented and improved to create a high fidelity reduced model. For improvements in the accuracy, the use of orthogonalization methods is analysed and discussed as well as including rigid body modes for IRKA and a preconditioner for two-sided Arnoldi. Lastly, a participation factor is discussed and improved to help reduce the model created with two-sided Arnoldi.
In the end, we find that two-sided Arnoldi in combination with the participation factor performs better than IRKA by creating a smaller and more accurate reduced model.

The development of optical metamaterials in recent years has enabled the design of novel optical devices with exciting properties and applications ranging across many fields, including in scientific instrumentation for space missions. This in
turn has led to demand for computational methods which can produce efficient device designs. Traditional optical devices admit a closed-form solution for this inverse design problem. However, in the presence of strong multiple scattering, which is often the case when considering optical metamaterials, the inverse problem becomes ill-posed. As a result, many optimization and machine learning techniques have been applied towards discovering good solutions.
In this MSc thesis project, several of the most promising of these techniques are applied to a specific problem, the discovery of silicon metamaterial lens designs for the CoPILOT high-altitude balloon project. Ultimately, a software tool capable of producing effective and admissible designs is produced and demonstrated.
First, an overview of the CoPILOT design problem is presented. Next, relevant background material topics, including properties of metamaterials and computational methods for simulating them, are covered in some detail. After this, methods used to solve optical design problems in past literature are described and contrasted. Then, a comprehensive explanation of the method developed and used for this project, including important design considerations, is given. The best solutions found using this lens optimization method are shown and compared. Finally, fruitful areas of future work on this topic are listed.

Differentiable ray-tracing is an exciting new development in computer graphics to approach all sorts of 3D scene design problems by obtaining gradients of renders produced by ray-tracing with respect to parameters that define the scene. These gradients can then be incorporated in a gradient-descent type optimization pipeline.

One such type of design problem is caustic design with free-form lenses. This is the process of obtaining the geometry of lenses in an optical system such that the light that passes through this system forms a desired illumination distribution on a target screen. A typical application of this is distributing the light from streetlights or car headlights in a pleasing and efficient way over the street.

The non-imaging optics literature offers many methods to design free-form lenses for caustic design, but differentiable ray-tracing is largely unknown in this field. Therefore this thesis proposes a method of optimizing free-form lenses for caustic design with differentiable ray-tracing.

The optimization pipeline starts with a multi-layer perceptron neural network which outputs parameters that define one free-form lens side in the form of a B-spline surface. A specially implemented ray-tracer then produces a caustic render by tracing through this lens from either a plane wave or a (grid of) point source(s). Back-propagation and optimization takes place using automatic differentiation in PyTorch and the Adam optimizer.

The results, which are verified with LightTools, show great promise for this technique, especially for the plane wave optimizations. The point source grid optimizations proved to be more challenging, but also here the optimization was able to achieve improvement. This shows that the proposed technique also has potential in positive etendue optimizations.

In anti-cancer therapy, ntiangiogenic treatments are applied and take effect on the vascularization of tissue. To evaluate the efficacy of treatments, we adopt two methods to solve the physiological pharmacokinetic model’s parameter estimation problem, providing discrete, partial, and noisy observations of stochastic differential equations. One is to compute the exact likelihood using the Kalman filter recursion and implement numerical maximization. The other is a novel Markov Chain Monte Carlo algorithm to estimate parameters using guided proposals in a Bayesian setup, namely Backward Filtering with Forward Guiding algorithm. The identifiability of the model and parameters are established before parametric inference. We extend the BFFG algorithm to include an automatic optimal kernel finding scheme for the Metropolis-Hastings-within-Gibbs sampler. In comparison, a Conjugate Gradient algorithm is applied when employing the maximum likelihood method. Besides performing parameter estimation via different methods separately, joint estimation is performed using the Bayesian approach. After that, time delay and Arterial Input Function in the statistical model are estimated via change point detection and piecewise inference. We illustrate the goodness-of-fit of estimates and advantages of the bayesian approach towards the maximum likelihood method.

The increase in complexity of mathematical models in an attempt to approximate reality and desire to have near real-time results have emphasized the need for fast numerical simulations. Especially in areas where classic numerical methods struggle to produce valid solutions in reasonable computational time due to their
complex behaviour on multiple temporal and spatial scales, such as (cardio-vascular) fluid modelling, machine learning techniques can be of help. The aim of this study is to construct a single reduced order model, based on neural networks, for time-dependent incompressible blood flow through the aorta that can account for varying velocity inlet conditions, material parameters and geometries (computational domains). The objectives are to minimize the reduction of accuracy of simulated time series with OpenFoam, and to obtain speedup compared to the OpenFoam simulations.

In this study, aortic blood flow during one heartbeat is modelled by Navier-Stokes equations for incompressible, Newtonian fluids. Data for training and testing the neural networks was generated using the finite volume method. The network architecture consists of a convolutional encoder and decoder model for dimensionality reduction, with an additional neural network for time evolution inserted between the encoder and decoder. The network takes one time step as input and predicts five consecutive time steps ahead. Three different networks for time evolution were tested. The best performing network was tested on increasingly complex computational domains. Thereafter, the network’s ability to generalize to varying inlet velocity patterns and new variations of the computational domain was tested, for which the network for time evolution was adapted to take into account varying inlet velocity patterns. The performance of the networks was evaluated on predicting one to five time steps ahead from one input time step, and using the prediction of one time step ahead recursively to obtain the prediction of one entire heartbeat.

The best performing architecture found was an encoder, recurrent neural network with long-short term memory units for time evolution, decoder model combination. The generalizability for the tested material parameter, blood viscosity, while keeping the inlet velocity pattern fixed, appeared quite good. A relative mean absolute error (MAE) of around 10% was obtained for straight, bent and bifurcated channel geometries. When testing the network on varying inlet velocity patterns, the relative MAE in the domain increased considerably to
around 25%. In general, the relative errors were significantly higher in comparison to the absolute errors, and mainly increased by low velocity parts of the domain and time series. In addition, the full height of the velocity peak was not always captured and a spike in the domain MAE was seen for rapidly decreasing inlet velocities. Also, the network was unable generalize to new geometries.

Various attempts to improve the accuracy, including altering the loss function and augmenting the data, were not successful. However, many choices of data augmentations, normalization techniques and loss functions were not tested and could improve the performance. The generalizability to different computational
domains could be improved by adding more geometries and more variation to the geometries in the data. Nonetheless, the prediction of one heartbeat was obtained in approximately 20s, which is equivalent to an achieved speedup of around 20 compared to OpenFoam simulations. In conclusion, this research should be seen as a basis for time-dependent blood flow predictions in the aorta, from which a multitude of different paths can be explored.

The aim of this research is to provide amathematical model that describes the physics in a levee when waves are overtopping a flood embankment. Ideally, this numerical simulation can replace empirical methods based on overtopping simulations and provide more insight into the physical process of an overtopping flow on a levee. This could prove to be useful for the design andmaintenance of flood barriers. Different interpretations of the stress tensor of pore water have each lead to distinct systems of partial differential equations. For each interpretation, the resulting system has been solved, using a finite element analysis in combination with a time-stepping method, in order to assess the validity of the imposed definitions. However, only one definition lead to a mathematical framework that yielded trustworthy results. In this finalmathematical framework, both the hydrostatic water pressure and the gravitational force have been disregarded, resulting in a system only consisting of variables such as soil particles displacements, pore water velocities and a distribution function »(t ). The distribution function »(t ) represents the fraction of the exerted wave stress on the surface carried by the pore water. By definition, the fraction can vary over time, which stands in contrast to state of the art models. The applicability of the results is limited, since the problem is simplified to a one-dimensional setting in which only normal stresses are exerted. The mathematical framework could theoretically be extended to multiple dimensions. However, it remains contestable whether it sufficiently simulates the physics in a levee. Further research is needed to show whether the extension holds when shear stresses are present and whether the same distribution function »(t ) can be applied to non-axial directions. In conclusion, this research is a proof of concept and serves as a stepping stone for more research. The used code can be found at https://github.com/HaveMersie/Overtoppingfailure

To regulate pollution emission of $\text{NO}_2$ and greenhouse gasses like carbon dioxide and methane, satellite images of the TROPOspheric Monitoring Instrument (TROPOMI) are used to map the spread of air pollution. This instrument delivers near-daily 'snapshots' of the $\text{NO}_2$ concentration over the whole Earth. In this report, first, the spread of $\text{NO}_2$ pollution is modelled using a so called convection-diffusion equation. This equation is constructed from a convection term, which contributes to how much the wind transports matter, and a diffusion term which described the natural mixing of gases. Next, $\text{NO}_2$ pollution sources were reconstructed using the inverse of the time propagation of the model. Thus, known $\text{NO}_2$ concentrations are used to reconstruct temporal information. By finding out what is missing between data snapshots, sources can be reconstructed. The verification of the algorithm was first performed on artificial data, after which it was applied to experimental data.

The convection-diffusion equation that was used is a two dimensional equation, using longitude and latitude as Cartesian coordinates. The equation was solved by the method of lines, i.e. first a spatial discretisation was made using the Finite Volume Method, after which time integration methods were used to propagate the problem in time. In this report two time integration methods formed the basis for the time propagation and two source reconstruction algorithms. The algorithms take two $\text{NO}_2$ snapshots $\textbf{y}_n$ and $\textbf{y}_{n+m}$, with $m$ time steps in between. First the inverse form of Forward Euler is used, which performed well on artificial data on a small time scale. Afterwards the Backward Euler method is used to define the inverse problem of finding the source function. This algorithm performed well on artificial data with 24 hours between snapshots. The algorithms were found to be sensitive to noise and to be influenced greatly by the second $\text{NO}_2$ snapshot. After applying the algorithm to experimental data it was found that the algorithm is able to localise extended sources of emissions, but is not capable of pin-pointing individual point sources. In addition, sources with low emission rates were left undetected due to information being lost in the noise. Emission rates were not reconstructed accurately either, although it was able to find the main sources of emissions. The performance of the algorithm will take advantage from more frequent data. With this improvement, this method is a viable option for finding not only $\text{NO}_2$ sources, but can also be applied to other tropospheric gases.

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.

Eddy currents are currents which are induced in a conducting object by a varying magnetic field. These currents generate a magnetic field of their own. Modelling this phenomenon constitutes a diverse and challenging set of problems, for which many applications exist. One such application is the degaussing system of a naval ship. To guarantee safety on missions, a ship uses a degaussing system to reduce its magnetic signature. Being able to model the effect of eddy current fields is necessary to improve the accuracy of future degaussing systems.

This thesis will examine how eddy current effects can be modeled, and how such a model can be validated. An analytical solution for a sphere is derived and investigated. A boundary element method (BEM) is implemented, which is able to numerically approximate the electromagnetic fields in terms of the modified magnetic vector potential A* and the reduced magnetic scalar potential. The approximation using the BEM is compared to the analytical solution. The BEM shows promising results, being able to model the shape of the magnetic signature of a sphere accurately. The model is also applied to more realistic geometries resembling naval ships, making it a strong candidate for further development and potential implementation in a future degaussing system.

A severe incident with the cargo ship MSC Zoe, which lost hundreds of containers near the Dutch coast, shows how important it can be to detect and localise hidden objects at sea quickly. Also in defence applications localisation of hidden objects in water is important. If the objects are made of magnetic materials, they disturb the Earth magnetic field. This disturbance is called the object's magnetic signature. An aerial drone that carries magnetic field sensors could in principle be able to locate magnetic objects autonomously. Another defence-related application of such a drone would be estimating magnetic signatures of ships under water, using measurements above water - signature translation. For both applications, algorithms need to be developed that process magnetic field data, and computes estimates. In this thesis, data-driven techniques are applied to the localisation problem and the magnetic signature translation problem.

For the localisation problem, an algorithm is proposed to localise a magnetic dipole using a limited number of noisy measurements from a sensor array forming a horizontal grid. The algorithm is based on the theory of compressed sensing. In the algorithm, a number of sensors is chosen which perform each a measurement of the three magnetic field components. The sensors can be chosen randomly from the sensor array, but also optimal placement using QR pivoting is considered. Using the obtained field measurements, a sparse representation in the location domain is computed using L1-optimisation. Based on the resulting sparse representation, a classification is produced, which consists of estimates on its location and on its magnetic moment magnitude and orientation. The possibility of performing iterations is explored, where the basis and chosen sensors improve after every location estimate. Using results from simulations as well as experiments, the algorithm is shown to be effective in localising magnetic dipoles.

To solve the signature translation problem, two approaches are investigated. One algorithm was developed using the Gappy POD technique, and one using neural networks. Using only a few measurements (a gappy measurement), a full signal can be reconstructed. This method is adapted to also be able to translate from a field above a ship to a field below a ship. Tikhonov regularisation is applied in the process to obtain better results. Second, different neural networks are created, trained using data above and below a ship. Linear and non-linear networks are compared, and standard loss functions are compared to physics-guided losses. Both the Gappy POD based algorithm and linear neural networks are shown to give good magnetic signature estimates. The Gappy POD method is greatly improved by applying Tikhonov regularisation. Results show to improve when more basis modes are considered, and when more sensors are used for measurements. Linear neural networks show the same results as Gappy POD using Tikhonov regularisation, indicating that some regularisation is done while training the neural network. Non-linear neural networks show no improvement over linear ones, and a physics-guided loss function is not needed for a resulting field that obeys known laws of magnetism.

Markov chains are used to describe random processes in discrete time, which have the property of being memoryless. This report covers Markov chains on a finite space that are homogeneous in time and mainly follows the structure of ''Markov Chains and Mixing Times''. Markov chains exhibit a strong connection with electric networks. We exploit such a connection and apply the laws of physics to answer several probabilistic questions about random walks, which are certain types of Markov chains. This connection is then called the electric network approach and is built on translating the random walk into an electric network by relating the transition probability to the so-called conductance. The electric network approach provides problem simplification tools, such as the Series/Parallel Law, and powerful inequalities, such as the Nash-Williams inequality. These tools are based on physics and are often more intuitive than their probabilistic counterpart.

We simulate a two-dimensional random walk that starts at the center of a square and ''escapes'' if it reaches the perimeter of the square before returning to the center. We then compare this escape probability to an upper bound, which results from using the Nash-Williams inequality. The sharpness of the upper bound depends on the choice of edge-cutsets. We find that choosing edge-cutsets with a minimal amount of edges gives a sharper upper bound, than choosing edge-cutsets that contain all the edges of the square. The relation between the upper bound and the escape probability seems to be independent of the size of the square. Furthermore, we provide proofs that are not explicit in ''Markov Chains and Mixing Times'' and ''Reversible Markov Chains and Random Walks on Graphs'' and add to the contents of ''Markov Chains and Mixing Times'' by studying random walks from a graph theory perspective.

Lacunes of presumed vascular origin (lacunes) are small lesions in the brain and are an important indicator of cerebral small vessel disease (cSVD). To gain more insight in this disease, obtaining more information about the shape, size and location of lacunes is essential. However, manual segmentation (the voxel-wise labeling of lacunes in a scan) can be very time-consuming. In this thesis, we optimize convolutional neural networks using different loss functions to automate the process of segmenting lacunes in full brain MRI scans. A set of 111 scans was used for development and a separate test set of 111 scans was used for evaluation. As lacunes are small, they generally occupy less than 0.02% of an MRI scan. This leads to an extreme data imbalance between lacune voxels and background voxels, which complicates the optimization process. We trained networks with the binary cross-entropy (BCE) loss, the weighted binary cross-entropy (WBCE) loss and the Dice loss. Additionally, we trained networks with two proposed adaptations of the Dice loss: Dice-ReLU loss and constrained Dice-ReLU (CDR) loss. Our experiments show that all losses except the BCE loss are able to cope with the data imbalance and learn to segment lacunes. The Dice-ReLU loss performs best on detection with a sensitivity of 0.79, but produces an excessive amount of false positives (FPs) with on average 26 per image. Adding a size constraint (the constrained Dice-ReLU loss) improves the number of FPs considerably to 11.52 FPs per image with a sensitivity of 0.70. The Dice loss has just 1.93 FPs per image, but reaches a detection sensitivity of only 0.43. However, the Dice loss obtains the best segmentation performance of the true positive (TP) elements with a Dice similarity coefficient score of 0.47. Compared to the Dice loss, the WBCE loss performs slightly less on FPs per image and TP-element-wise segmentation, but slightly better on sensitivity. We developed methods that were able to learn to segment lacunes. However, the data imbalance still influences the optimization process considerably, leading to methods that either focus too much on the foreground or too much on the background. Further work on developing a loss function that can cope better with this data imbalance could greatly improve lacune segmentation performance.

In order to decrease the time that the viscous-flow solver PARNASSOS needs to simulate ship-generated waves, direct solvers are introduced in order to make the iterative solver used in PARNASSOS converge faster

With the increasing number of electric vehicles on the road, the routing problem has become more complex. As charging electric vehicles takes longer than fueling non-electric vehicles, congestion can occur at charging stations. This might lead to the shortest route not being the fastest route, due to long waiting times at the stations. By communicating the intentions of each vehicle, the vehicles can spread out over multiple stations. This thesis investigates the eect of such a routing system on the profitability of charging stations. In particular, the influence of a charging station's location on its profitability has been researched. In order to do this, a pricing model has been developed to extend the routing model. It has become clear that due to the intention-aware routing, the number of visits at stations is harder to predict, and there is not always a clear pattern to be found between the location and the number of visits. This research does, however, propose an optimisation model, which can decide where to build a new charging station, such that the number of visits will be the highest.

This essay shows a two dimensional implementation of the finite element method for the Westervelt equation. To do this the finite element method is first applied to the linear wave equation, then to non-linear diffusion and finally to the Westervelt equation. Both an element by element and a faster vectorized implementation are given for the finite element method. To verify the numerical solution two analytical solutions are used. The first is a one dimensional wave and the second a circularly symmetric wave.

We found that the two-dimensional implementation was successful in computing the Westervelt equation. The error of the solution scales with the mesh size with a power of around 1.7. It was also found that the time step used to compute the solution needs to be small enough for the implementation to converge.

In this thesis, an algorithm to model the magnetic perturbation field caused by ships is designed and implemented. A systematic description of methods used for modelling the magnetic signature of ships is given. The algorithm fits coefficients of a prolate spheroidal harmonic expansion of the scalar potential of the magnetic field using a least angle regression method (LARS) modified to implement Lasso regularisation. A Monte Carlo method with model selection based on Akaike's information criterion (AIC) is used to select optimal parameters specifying the prolate spheroidal coordinate system centred on the ship. Furthermore, a method to restrict the degree and order of the harmonic expansion is presented and an extension of the scikit-learn module in Python is given. The predictive power of the model was verified using simulated test data, which showed that the designed model is able to make adequate predictions, but improvements are needed. Different analyses on the inputs of the model showed that the model is succesful for low levels of noise, but is susceptible to overfitting for higher levels of noise. Several recommendations for further research are made.

The first part of this thesis provides a mathematical description for bipartite quantum correlations, aiming to analyze the geometry of several sets of correlations. We explain why quantum entanglement can be used to simulate shared randomness: C_loc(Γ) ⊆ C_q^d(Γ) for a sufficiently large d. The known bound for this dimension d in the literature is d ≥ dim(C_loc(Γ))+1, but we improve this by showing that the inclusion is always true for d ≥ dim(C_loc(Γ)). For the proof of this bound, we show that the set C_private(Γ) of correlations using private randomness is connected, which allows the use of an improved version of Carathéodory’s Theorem. In the second part of this thesis, we define and analyze a see-saw method to determine the state and measurement operators that reconstruct both the correlation itself as its entanglement dimension, by solving consecutive semidefinite programs. One of the strengths of the algorithm is its generality: it applies to different dimensions, question sets, and answer sets. Some numerical experiments demonstrated that the method can indeed reconstruct quantum correlations, although some highly entangled correlations failed to be reconstructed due to the computationallimitations. The numerical experiments motivated several new theorems, for example the fact that every correlation with |A|= 1 or |B|= 1 has entanglement dimension 1, which means that it can be written as a private randomness correlation. The proof of this result is based on the earlier described improvement for the dimension d.

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.

Due to an increase of precipitation reactions in water by temperature changes, cloggings canoccur in a geothermal doublet. A consequence of the cloggings is that the process of generatinggeothermal energy becomes inefficient eventually. Our goal is to determine which minerals playan important role in the clogging process in each part of the geothermal doublet. First, we havedivided the geothermal doublet into four tanks. Then we have derived a general model for ageneral chemical reaction. Afterwards, we have used this model for 5 kinds of mineralisationreactions. This led to first-order differential equations, which we have solved analytically ornumerically with the Forward Euler method. Finally, we have concluded that the mineralsbarite, calcite, dolomite and goethite play an important role in the clogging process in tank 3and tank 4.