This thesis concerns the implementation of parallel Schwarz domain decomposition using node-level parallelism, focusing on the parallel Schwarz method in comparison with the Jacobi iterative method. The study goes into the complexities of domain decomposition methods for solving partial differential equations, which are essential in fields such as fluid dynamics, solid mechanics, quantum mechanics, and financial mathematics. The research examines the convergence and performance of these methods within a parallel computing framework. A large portion of the work involves the comparison of varying configurations of block sizes and overlaps within the use of the block Jacobi iterative method, used for the implementation of the parallel Schwarz method. Numerical experiments are conducted for a stationary heat problem on a 2-dimensional grid with 256 points in each direction. The results show optimal performance for small block sizes, attributed to the use of a dense solver for the subdomains. Larger blocks and larger overlaps show superior convergence properties, up to the limit of an overlap of half the block size. The efficiency of the parallel Schwarz method remains high for an increasing number of threads unlike the standard Jacobi iteration, showing it is better suitable to a parallel environment.

Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn wounds is crucial for developing effective treatment strategies. Traditional mathematical models, while accurate, are often computationally expensive and time-consuming, limiting their practical application. Recent advancements in machine learning, particularly in deep learning, offer promising alternatives for accelerating these predictions. This study investigates the use of a deep operator network (DeepONet), a type of neural operator, as a surrogate model for finite element simulations for predicting post-burn wound evolution. We trained DeepONets on various wound shapes, enhancing the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. The most sophisticated model achieved an R² score of 0.9960, indicating strong predictive accuracy. Additionally, the model generalised well to convex combinations of basic shapes, with an R² score of 0.9944, and provided reliable predictions over an extended period of up to one year. These findings suggest that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

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.

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.

We consider three topics motivated by the Network Exploration Toolkit (NEExT) for building unsupervised graph embeddings. NEExT vectorizes the graphs in a graph collection using the Wasserstein (optimal transport) distance between the distributions of node features of each graph. We inspect the effect of sampling only a proportion of the nodes of each graph, and show that even if the sampling fraction tends to zero (sufficiently slowly), so does the Wasserstein distance. NEExT relies on the assumption that local node features are informative to global characteristics of graphs. With this in mind, we consider triangle count: triangles are a local feature that correlate with the strength of a graph's community structure. We give asymptotic results for the number of triangles in a 2-community stochastic block model setting and the ABCD random graph model in both a scale-free and finite variance degree regime. Lastly, we show by experiment that augmenting the node features of graphs with triangle count improves a Graph Neural Network (GNN) in its ability to pick up on community structure and local clustering. For this, we use a random graph dataset with varying community structure, a random graph dataset with varying clustering coefficient, and a real-life dataset. We position random graph models as an effective tool for benchmarking the expressiveness of GNNs.

This thesis addresses the challenge of segmenting ultra-high-resolution images. Limitations of current approaches to segment these are that either detailed spatial contextual information is lost or many redundant computations are necessary. To overcome these issues, we propose a novel approach combining the U-Net architecture with domain decomposition strategies to balance incorporating spatial context and maintaining computational efficiency. Our proposed method partitions input images into non-overlapping patches, each processed independently on a separate device. A communication network facilitates the exchange of information between patches, enhancing the model's understanding of the spatial context.

Through theoretical analysis and practical experimentation on device memory usage during training, we demonstrate that our approach incurs minimal additional memory overhead for inter-device communication. Evaluation on synthetic and realistic datasets, including the Inria Aerial Image and DeepGlobe Satellite Segmentation datasets, demonstrates the effectiveness of our approach. Our model achieves competitive performance compared to the baseline U-Net model, with consistent class predictions around boundaries. Visualization of feature maps highlights the role of the communication network in transferring contextual information. Furthermore, it is shown that our approach remains scalable even when trained on limited subdomains.

In conclusion, our proposed model offers an intuitive solution for segmenting ultra-high-resolution images by effectively incorporating spatial context. Future research could explore variations of our model, such as overlapping subdomains or communication on different levels of the U-Net, to further enhance boundary consistency and information transfer.

In this thesis, a deep learning-based surrogate model for predicting sea ice dynamics is developed that is capable of predicting linear kinematic features in a high-resolution setting. Predicting sea ice dynamics at high resolutions is critical for understanding climate patterns and enabling safe navigation in Arctic regions. Traditional continuum models based on the viscous-plastic rheology are computationally intensive when employed at high resolutions to capture linear kinematic features (LKFs), which are narrow zones of deformation in sea ice.

A supervised learning approach was adopted, focusing on convolutional neural network architectures, specifically the U-Net and a classic bottleneck model. Various loss functions were explored, including traditional metrics like the MSE and novel domain-specific functions such as the strain rate error (SRE), which incorporates physical knowledge of sea ice behaviour. The models were trained on a dataset generated from numerical simulations of sea ice dynamics on a 2 km grid.

The key findings indicate that the U-Net architecture combined with the MSE+SRE loss function outperforms other models. This architecture's depth and use of skip connections allow it to capture complex, multi-scale patterns inherent in sea ice dynamics. The integration of the SRE into the loss function significantly enhances the model's ability to predict LKFs, demonstrating the benefit of incorporating domain knowledge into machine learning models.

While the surrogate model effectively predicts LKFs for short-term forecasts up to approximately 5.5 hours (10 time steps), errors accumulate over longer periods, leading to significant errors after about 11 hours (20 time steps). However, the efficiency gain of this surrogate model is striking: the computation of 10 time steps can be done within a second, compared to the 30 minutes that traditional numerical methods need. The study also underscores the critical importance of training data selection and preparation in influencing model performance and generalisation capabilities.

In conclusion, this work demonstrates that a deep learning-based surrogate model, particularly utilising the U-Net architecture and the MSE+SRE loss function, can effectively predict sea ice dynamics at high resolutions with significant computational efficiency. The model generates forecasts within seconds, offering a viable alternative to traditional numerical simulations that require hours of computation. Future work should focus on mitigating error accumulation to extend the forecasting horizon and exploring advanced learning methods to further integrate physical insights into the modelling process.

To reduce computational efforts, surrogate models have been developed for dune erosion prediction. Current surrogate models can describe the relationship between the XBeach input and output (Athanasiou, 2022) and provides a prediction of a morphological indicator based on a parameterized input (profile shape parameters and hydrodynamics). In this research, first steps are taken to set-up a surrogate model that is able to deal with spatial input and the prediction of actual profile shapes along the Holland Coast. Using a convolutional neural network structure (U-Net), several model set-ups are explored and scaled-up to a realistic storm scenario along the Holland Coast.

This paper examines whether complex high-dimensional data that describes the dynamics of a cantilever beam can be transformed into a less complex system. In particular, the transformation that is examined is the reduction of the dimension. An essential aspect of this study involves the implementation of a linear autoencoder, which is a type of machine learning model that possesses the capability to effectively reduce the dimensionality of input data while adeptly reconstructing the original dataset. The model performs well and is successful in reconstructing complex data via the less complex system. However, the model struggles if the dynamics are made more complex by adding an external force. Although the dynamics seem to be present in the results, the amplitudes differ.

Commercial finite element software like COMSOL is build to be user-friendly. For example, the user does not have to find weak formulations, or discretise the partial differential equations by hand. One of the more difficult parts of using finite element software is deciding which solver method and solver parameters to choose, as the performance of the solver, and hence the performance of the whole simulation, depends on it.
Pseudo time-stepping is a stabilisation method that is designed to lead to a stationary solution for a wider range of initial guesses compared to traditional Newton methods. During pseudo time-stepping, a CFL number is used. This CFL number is adapted each nonlinear iteration. In COMSOL the CFL number either depends on the iteration count, or the nonlinear error estimate and previous CFL number. In this case, the CFL number is a global value, but the pseudo time-step itself is local since it also depends on the mesh cell size.

In this research, a neural network is created to predict a local CFL number for pseudo time-stepping,
such that the network accelerates convergence compared to the two CFL numbers for pseudo time-stepping in COMSOL and is generalisable. The convergence is assumed to be accelerated if the
network predictions result in convergence in fewer nonlinear iterations compared to solvers that use
each one of the two CFL numbers in COMSOL. The network is assumed to be generalisable if it is able to accelerate the convergence for different laminar flow problems in COMSOL on which the network has not been trained.
A network with 2 hidden layers with each 16 neurons is trained on local data from element patches in order to make it generalizable. The element patch data points contain information of the central triangular element vertices, and of the vertices of the three adjacent elements. The local data consist of the velocities, pressure and residuals, as well as the cell Reynolds number and the element edge lengths. The loss of this network is the root mean squared error between the network output and a previously computed CFL target. This target is the optimised value for the local CFL number such
that the difference between the solution obtained from the pseudo time-step and the exact solution is
minimised.
The network predictions were able to accelerate convergence compared to to the two CFL numbers in COMSOL for several different 2D laminar flow simulations. Therefore, the conclusion is that a network trained on optimised local CFL numbers is able to generalize well and accelerate the existing pseudo time-stepping method.

The goal of this work is to evaluate the aptness of generative adversarial networks (GANs) for use as surrogate reduced order fluid models. In contrast to previously published work, the focus is placed on analyzing the specific effect of adversarial training, by comparing GAN outcomes with those from an identical generator network trained directly on ground truth (using an L1 loss). A dataset of 10000 simulated examples of stationary flow through a 2D sudden expansion geometry containing a polygonal obstacle was created, alongside two additional datasets for testing generalization. The simulation data was interpolated to a regular image grid, and the neural networks were trained to predict the velocity field based on an image encoding the geometry.
The gathered experimental data show clearly that adversarial training cannot reach the same accuracy as direct training. This was found to be true on unseen examples from the training distribution, as well as on geometries of unfamiliar type. On the other hand, GAN outcomes tend to "appear" more realistic, and exhibit a lower continuity residual. The qualitative differences were highlighted by considering bifurcation scenarios which were purposefully included in the data set. When the bifurcation parameter is at its critical value, two very different flow scenarios can occur essentially randomly. In such cases, the GAN essentially predicts just one of the possible flow outcomes, whereas the directly trained model outputs a superposition of both. This exemplifies a fundamental difference in prediction behavior.
It is shown that these results can be well understood as a direct consequence of the different cost functions used during training. Furthermore, it is demonstrated that by using a sum of both types of loss functions (adversarial & L1), advantages of both models can be combined.
In other results, the data show that the discriminator output cannot provide a reliable indication of the accuracy of a prediction, since no robust correlation between them was found. Also, it was observed that the discriminator appeared to dominate the adversarial game, and it was shown that improving this balance could lead to better results. Moreover, an investigation into predicting pressure alongside the velocity field was conducted. Results showed that adding an additional channel for pressure to the network architecture can achieve this goal, but is not necessary, as calculating the pressure field from the velocity output produces results of similar quality. In terms of resources, GAN training required a relative increase in computational time by approx. 50%. Additionally, GAN models were also found to take many more training steps to reach convergence. Similarly, for a fixed number of steps, results showed that directly trained models can benefit more from larger datasets. In conclusion, GAN models are likely not the right choice for reduced order modeling in scientific contexts due to their lower accuracy. However, they could hold potential for use in creating visual effects or as an added regularizer during training.

Perforated monopiles show promise in providing a better alternative to the commonly used jacket-like substructures used in intermediate water depths in the range of 30 to 120 m. By introducing perforations near the vicinity of the splash zone the wave loads on the monopile can be mitigated and the fatigue damage reduced, which is the main advantage jackets have over monopiles. Furthermore, perforated monopiles would be easier to install and manufacture compared to jacket structures, which has the potential to reduce the Levelized Cost of Electricity (LCoE) considerably. Perforated monopiles also have the benefit of reduced (local) scour, reduced corrosion damage and providing a habitat for marine life.

To optimize the design of perforated monopiles fast novel surrogate models are needed to determine the (wave) load reduction by the introduction of perforations on the monopile. As traditional (full) order Computational Fluid Dynamics (CFD) models are slow and expensive to evaluate. Using a surrogate model a wider range of geometries can be used in the early design optimization steps. Promising designs can then be further evaluated using full models.

This thesis develops a surrogate model for the characterization of the (hydrodynamic) loads on perforated monopiles using Convolution Neural Networks (CNN). A dataset is created of 15,561 samples, which considers geometries of varying number of perforations, porosity and ‘angle of attacks’. A Finite Element Method (FEM) CFD model is made in Gridap for a ‘creeping flow’ to determine the flow fields. And determines a load Reduction Factor (RF) by the introduction of perforations. Furthermore, using a Signed Distance Function (SDF) a representation of the geometry is made for use in the CNN model.

Using PyTorch a CNN model is used to predict the RF for a certain input. The model combines multiple convolutional layers with an optional regression head (with linear layers). Five different models are created. Three for representing the geometry by using the SDF, decomposing the SDF in its x and y distance components and the binary representation. Furthermore, two models combine the SDF or distance components with the flow fields from the CFD model.

The results show that a speedup of 386 times is achieved by using the surrogate model, showing the main benefit of using a surrogate model. This is likely to improve considerably in further research when turbulence is taken into account. Furthermore, each of the five models show a good accuracy in predicting the RF. A general trend is observed that the more information is provided in the input to the CNN model the better the accuracy tends to be. The combination of implicit representation of the geometry by using distance components with the CFD flow fields shows the best accuracy with an expected maximum relative error rate of 0.94% within the parametric space of the dataset.

Interference of light can be used to determine the concentration of a gas, called gas sensing. The absorption of the light by the gas molecules is measured based on the phase change of the light. In this report, a hollow core photonic crystal fiber is treated. The gas sample is inserted into the hollow core. Photonic crystal fibers possess the characteristic to have electromagnetic modes confined to the core with low attenuation. This means that there can be a high interference rate between the gas and light, which is desirable for gas sensing.
First, less complicated optical fibers were studied. The analytical solution of the electric and magnetic field for the TE and TM modes of the simple fiber were derived and for the TE modes of the step-index fiber. These results were compared to simulation done with COMSOL Multiphysics. Photonic crystal fiber with circular core was then simulated with COMSOL Multiphysics, but the results did not match with the literature. Therefore another photonic crystal fiber was simulated with a star-shaped core. The simulation results found modes concentrated in the core, with low attenuation.
It was attempted to get similar results with a three layer step-index fiber, by varying the imaginary refractive index of the middle layer. The attenuation of the three layer fiber was much higher than that of the photonic crystal fiber for all simulations. This indicates that the three layer step-index fiber does not support propagation modes that are confined to the core. Further research could be done by studying the effect of the radius of the middle layer and the real part of the refractive index.

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.

Computational Fluid Dynamics (CFD) is the main tool to use in industry and engineering problems including turbulent flows. Turbulence modeling relies on solving the Navier-Stokes equations. Solving these equations directly takes a lot of time and computational power. More affordable methods solve the Reynolds Averaged Navier-Stokes (RANS) equations.
The most commonly used RANS models rely on Linear Eddy Viscosity Models for the Reynolds stress closure. However, this type of modeling does not provide satisfactory accuracy in general problems, which include curvature, impingement and separation. The rapid developments of Machine Learning (ML) in fluid dynamics and the increase of available high fidelity data of turbulent flows led to the introduction of data-driven RANS turbulence modeling. ML-augmented RANS models are promising but often lack generalisability or does not meet the preference of being interpretable. In this thesis, a sparse symbolic regression method has been used to generate interpretable algebraic equations for the non linear anisotropic Reynolds stress. These equations are built from a library of candidate functions of which the best fitting functions are selected by solving a sparse regression problem.
A deeper understanding of this methodology is achieved by creating models using different input features, including a wall damping function to improve near-wall behaviour of the model. Furthermore, different sparse regression problems are investigated, including constrained regression problems for which physical knowledge is used. The discovered models are propagated in a CFD solver to obtain a corrected velocity field.
Most of the discovered models improve the prediction of the Reynolds stress and its anisotropy compared to the $k-\omega$ RANS model. A small number of the discovered models are able to improve the prediction of the streamwise velocity compared to the RANS model. By analysing the discovered models, it is recognised that a specific term in the algebraic model is the basis of an inadequate prediction of the velocity. Furthermore, it is observed that a considerable amount of the implemented models provide an unconverged solution, which is visible as instabilities in the flow variables. Suggestions are given to improve the stability of the models. The discovered models, which include more physical knowledge by applying physical constraints to the regression problem or by including near-wall treatments, give promising results.

Wave phenomena play an important role in many different applications such as MRI scans, seismology and acoustics [41, 49, 47]. At the core of such applications lies the Helmholtz equation, which represents the time-independent version of the wave equation. Simulating a Helmholtz problem numerically with accurate numerical solutions for large wave numbers is challenging. Numerical solvers for the Helmholtz problem have to balance having accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexity with respect to the grid nodes. Currently, there is no numerical Helmholtz solver that can satisfy these requirements at once.
We developed Schwarz domain decomposition preconditioners which leads to wave number independent convergence for wave numbers in 2D and 3D, while remaining to have accurate numerical solutions. The preconditioners use two-level Schwarz preconditioners, with the coarse problem being constructed using higher-order interpolation with quadratic rational Bézier curves. The developed domain decomposition preconditioners are designed to leverage parallel computing in the future in an attempt for the preconditioners to acquire the ability to solve with linear time complexity.

In this research, the preconditioner resulting in wave number independent convergence and the lowest iteration count is the two-level scaled hybrid Schwarz preconditioner with a coarse problem constructed using higher-order Bézier interpolation. This preconditioner uses a deflation method to remove unwanted eigenvalues. Removing these unwanted eigenvalues results in a clustering of the eigenvalues which is more favourable for GMRES. Currently, all the developed preconditioners suffer from high computational cost for large wave numbers, due to the coarse problem becoming large. Decreasing the coarse problem size of the preconditioners, while remaining to have wave number independent convergence, has shown to been unsuccessful. To better understand the required conditions for wave number independent convergence of the preconditioners, we investigated the relationship between the number of coarse grid nodes and the wave length, too see if there is anything generalizable about this relationship and wave number independent convergence of the solvers.
In conclusion, the balancing for a Helmholtz solver to have accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexity
is again shown to be difficult. This work provides the initial development and testing of promising wave number independent Helmholtz solvers, from which more research should follow that tackle its biggest computational problems.

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.