This thesis aims to contribute to the understanding of how waves interact with soil. It is crucial for various applications in Civil Engineering to analyze the behaviour of soil and to understand the physics behind it. This master thesis contributes to this understanding via studying the impact of the boundary conditions on the model results with the aim of being able to model interaction between waves and soil.
We assume a media that is poroelastic and fully-saturated, unless stated otherwise. We also assume that the porous media consists of incompressible soil particles and pore water particles that may either be compressible or incompressible. The main goals of this thesis are (1) to describe the response of porous media to transient hydraulic loads using numerical methods like the Finite-Element Method, and (2) to apply it to a one-dimensional case whereby a sandbed is subjected to waves. Currently, it is common to predict the changes in pore water pressures in porous media subjected to transient hydraulic loads using Biot’s model, which often assumes compressible pore water, assumes zero effective stresses on the surface of the seabed, and assumes that the wave load is completely carried by the pore water pressure only. Recently, a new model is proposed by Van Damme and Den Ouden-Van der Horst suggesting that transient hydraulic loads acting on a porous medium affect both the pore water pressures and effective stresses in soils. Note that this makes sure that the momentum balance equations are satisfied throughout the computational domain and its boundaries. The boundary conditions in this case do not satisfy Terzaghi’s effective stress principle, whereas the standard has been to impose Terzaghi’s effective stress principle when solving Biot’s equations. Terzaghi’s principle states that the sum of the effective stresses and pore water pressures must equal the hydraulic loads, whereas Biot’s model is in line with this principle.
The model of Biot and the new model of Van Damme and Den Ouden-Van der Horst describe the physics differently which can have a large impact on the results. For example, the assumption of compressibility can significantly impact the distribution of the effective stress in the soil and thus the results.
Biot’s model is more sensitive for changing the compressibility parameter than the new model. Both models give similar solutions to the water pressure. However, they give different solutions to the other variables like the volumetric strain and displacements which appear in both models. Furthermore, the new model in one dimension is in line with the momentum balance equations and satisfies the volume balance equation. On the other hand, the standard is to solve Biot’s model by imposing Terzaghi’s principle at the boundary. For the new model we found promising results for the water pressure, when validating with the data of two experiments. At the end, which model predict the best solutions for volumetric
strain, water pressure and displacements depends on what kind of problem the model is used for and the corresponding physics. The used code can be found at https://github.com/fpmklein/Compressiblevs.-incompressible-pore-water-in-fully-saturated-poroelastic-soil.

The level-set approach in the field of topology optimisation has been studied thoroughly the last two decennia, but there is a lack of challenging standard benchmark problems. Besides two standard benchmark problems, this master's thesis tackles one difficult benchmark problem in particular: the inverter. For every benchmark the minimum compliance problem is solved numerically. A level-set based algorithm has been devised to detect if a structure has a discontinuity during the optimisation process. In addition to this, a prevention method is tested, yet the results are unsatisfactory.

A one-dimensional model of respiratory deposition is developed based on an Eulerian approach. The model simulates aerosol deposition in all generations of the respiratory tract by numerically solving the aerosol general dynamics equation (GDE) for a range of aerosol diameters. The lung geometry is described by Weibel’s morphometric model, with a time varying alveolar geometry to accommodate inhalation dynamics. The model is first valuated by comparing it with numerical and experimental results. Afterwards a series of parameter studies is performed by changing breathing conditions, particle parameters and lung geometries. An increase in Tidal volume and decrease of breathing period resulted in an increase of the total deposition fraction for coarse particles and a decrease of the total deposition fraction for ultrafine particles. An increase in the particle density resulted in an increase in the total deposition fraction. A decrease in the airway diameter generally resulted in an increase of the total deposition fraction. This difference was most noticeable in the tracheobronchial region. Decreasing the airway diameter in the tracheobronchial region mostly effects coarse particles while decreasing the airway diameter in the alveolar region mostly effects ultrafine particles.

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 derivation of water quality indicators is of importance, especially in coastal areas, as most of the economic activities are located here. However, the availability of high-spatial-resolution water quality information in coastal zones is limited. Nowadays, high-resolution satellite data is becoming available and can fill in this knowledge gap. This satellite data contains spectral reflectances, so a model needs to be designed to map these reflectances to water quality indicators. In this thesis, a Gaussian process regression (GPR) method will be introduced and analyzed extensively in terms of covariance functions, hyperparameters and computational costs. Remote sensing data is collected from the Sentinel-2 mission and the in-situ data is obtained from the ODYSSEA programme. The Matérn 3/2 kernel produces the best results and these are compared with the current models that rely on machine learning techniques. GPR shows promising results in terms of estimation accuracy and chlorophyll-a maps are made for different areas and depths. Various approximation methods are tested to speed up the computation time. Singular value decomposition shows promising results for doing predictions to reduce the computation time. Moreover, GPR can handle limited availability of in-situ data well and uncertainty quantification is induced by the Bayesian framework.

In 2016 Hieber and Kashiwabara showed that the three dimensional primitive equations admit a unique, global, strong solution for all initial data in a closed subspace of the Bessel space $H^{2/p,p}(\Omega)$ provided $p\geq6/5$, being this the first result in the general $L^p$-setting. Their approach consisted in studying the properties of the hydrostatic Stokes operator $A_p$ defined on the solenoidal subspace $L^p_{\overline{\sigma}}(\Omega)$ of $L^p(\Omega)$. In 2017 Giga et. al. further proved that the hydrostatic Stokes operator $A_p$ admits a bounded $H^\infty$-calculus, obtaining maximal $L^q-L^p$ regularity estimates for the linearized primitive equations in a much simpler way. In this work we will study Giga et. al.'s and Hieber and Kashiwabara’s works particularized for the $L^2$-case as well as all the necessary literature to replicate the proofs. The goal of the thesis is to present an extended version of Giga et. al.’s proof to make it more accessible. Although the $L^p$-case is not studied for lack of time, we differentiate between the Sobolev-Slobodeckij, Bessel potential and Besov spaces to accentuate how we could extend the proofs to the $L^p$-setting.

Multiphase flows are very common in many Nuclear engineering applications. During high pressurized conditions there are possibilities of high thermal loads on the pressure vessel, leading to pipe ruptures. As part of breakdown measures, the emergency core cooling system is activated and the coolant is mixed with the fluid in the cold leg, giving rise to multiphase turbulent flow. These regimes can comprise of large scale interfaces, leading to stratified flows. These postulated accidents or events need to be identified and understood to improve nuclear reactor safety. Computational fluid dynamics can serve as an excellent tool to model these scenarios, contributes towards reactor safety. Coarse models which are widely used in industries such as RANS are known to over-predict turbulent producing unphysical gradients. Thus the turbulent mass and momentum are not yet fully understood. Using high resolution tools such as Direct Numerical Simulations (DNS), can potentially avoid these over-prediction and could model these large scale interfaces accurately. As a long term goal, the data sets generated from these simulations can be used to train such coarse models or simply support for validation.

Placing the focus on a configuration where two fluids are in a stratified scenario, this graduation thesis will show a systematic approach towards the development and modelling of air and water moving in both co-current and counter-current direction, wherein simulations are performed in RK-Basilisk. Primarily, the work starts with studying a single phase turbulent channel flow to forma basis of understanding of concepts and code. The model of (Liu et al. 2009), who use realistic properties of air-water is chosen to be implemented in RK-Basilisk. It is realized that, implementing this is in RK-Basilisk is not straightforward and thus the constraints are identified and a general mathematical framework is developed to resolve this. One of the main objectives in this thesis is to model and understand the turbulent behavior near the interface of both air and water. To do so, the physical mechanisms which govern the generation and decay of turbulence called the TKE Budgets is studied by modelling the individual terms that complete it. The budgets are modelled and validated against (Liu et al. 2009). Interesting conclusions are drawn which depict the trends of budget terms and the kinetic energy, giving a good picture of the underlying interfacial turbulent mechanisms. The same mathematical framework, along with some additional modelling lead to an extension of this study to counter-current flows, wherein another set of
conclusions are drawn.

The goal of this thesis is to create a theoreticalmodel that captures the influence of wind on the horizontal and vertical salinity distribution in an estuary, as well as the dynamics that arise within the estuary. A secondary, numerical model, that uses fewer assumptions, will be developed and analysed. This serves to determine whether the implications from the theoretical model hold only because of all the assumptions that are made, or whether the results are equally valid when some of these assumptions are relaxed. The focus of this thesis is on gaining conceptual insight in estuaries subject to wind, rather than on obtaining a high accuracy in describing a specific estuary. This thesis is focused on steady-state solution in partially orwell mixed estuaries. Moreover, the dynamics and salinity averaged over the width of the estuary are considered, such that only dynamics in the vertical and along channel direction are modelled. Estuaries are subject to periodical variations forced by tides. The variation within a tidal period is not studied in this thesis, but the dynamics and salinity averaged over a tidal period are considered. The effects of tides or variation along the width of the estuary are not neglected from this model, but are incorporated in the parametrisation of the model instead.

In this thesis the matrix product state(MPS) formalism is applied to simulate a device created for remote detection in atomic spin chains. Although simulation of the full device quickly becomes infeasible for direct algorithms, MPS allowed for highly accurate simulations while requiring only modest computational resources. It has been verified that the simulation conserves energy and maintains normalization. Furthermore, the simulation is shown to successfully perform both a coherent real-time evolution and iteration towards the ground state. Thus MPS is a very promising tool that has potential to serve as a guide for further experiment as well as a tool to understand experimental results.

Skin grafting is a common technique employed to treat patients after burn injuries. In contrast to the frequency and gravity of contractures following from skin grafts, the phenomenon itself is still poorly understood and subject of studies. The development of an accurate model of skin contractures will allow medical researchers to better understand the healing process compared to current in vitro experiments and thus enable the design of efficient and patient-specific treatments. In this work we provide a mathematical model able to capture both the mechanical and biological processes involved in skin graft healing. Two-dimensional morphoelastic equations are used to model the mechanics of contraction of skin during the healing process. A system of equations describing the cell components (myo-, fibroblasts, collagen density and signalling molecule) is solved separately to derive the forcing terms for the mechanical system. The finite element method with bilinear quadrilateral elements is used to solve the differential equations on a moving time-dependent domain. A flux corrected transport (FCT) algorithm is used to stabilize the biological equations, thus enforcing positivity of the constituents in the dermal layer and nonlinearities are treated using Picard iterations. Time stepping is performed by applying the Euler backward scheme for the mechanical system and the implicit midpoint rule for the biological system. The results are replicated using the IGA (Isogeometric Analysis) library G+Smo in C++. The results of the IGA implementation provide a solid basis for future algorithms using IGA.

With increasing attention to climate change,renewable energy generation has become a major topic for research anddevelopment. Wind and solar energy are generated on land, whereas wave, windand tidal energy generators are getting attention in the offshore domain. Anovel extension of onshore solar energy is the concept of offshore solar energyusing floating platforms. As little research has been performed on the conceptof offshore solar energy generations, main challenges in the field are relatedto consequences to the marine ecology, economics and production and structuraldesign of the platforms. In this thesis, a numerical model to assess wrinklingbehaviour of thin, floating sheets with application to the structural design ofoffshore solar platforms is developed.
Since wrinkling of thin sheets, in general, isinitiated by a structural instability (i.e. buckling), the developed modelconsists of an arc-length method that is capable to deal with bifurcationpoints and to switch to bifurcation branches. In this way, buckling andpost-buckling behaviour of thin sheets are modelled and wrinkled shapes can beassessed without imposing a priori definition of unbalancingimperfections or loads. The computational model is developed using a shelldiscretization with Isogeometric rotation-free Kirchhoff-Love elements, whichare higher-order elements with a B-spline or NURBS basis with global supportand global higher-order continuity of the solution. For the illustrativepurpose and future use, a similar Euler-Bernoulli beam model was developed andnumerical solvers for static, dynamic, modal and linear buckling analysis wereimplemented.
The model was verified using various benchmarkstudies for static, modal, (post-)buckling and dynamic analysis. In particular,the post-buckling solver was assessed by modelling the collapse of a sphericalroof and using buckling (post-)bucking of a cantilever strip. Both benchmarkshave shown excellent agreement with previous publications. Additionalverification was done on the approaching accuracy and prediction of bifurcationpoints. It was found that this accuracy showed the accurate prediction of thebifurcation point, although slightly underpredicted for finer meshes and higherorders. Additionally, the model wasapplied to three cases where wrinkling is involved. In these cases, sheets withlow bending stiffness were modelled such that their post-buckling shapes showmultiple half-waves and thus wrinkles. Based on the model of a floating sheetsubject to surface traction (e.g. wind or current), design parameters werevaried. From this case, it follows a decrease in foundation stiffness or anincrease in flexural rigidity (either by varying Young's modulus or thickness)implies the number of wrinkles to decrease and the wrinkling instability tooccur for lower loads. Thirdly, based on the wrinkling geometries of a quarterdisk, design consideration for VLFTSs for offshore solar energy generation weregiven. These are: (i) adding reinforcement to arrest wrinkles and to introducestructural hierarchy for structural reliability; (ii) consider the effect ofdifferent mooring system connections to the (reinforced) platform; and (iii)investigate the effect of holes and point loads on local wrinkling behaviour. Basedon the results of the study, it is concluded that the isogeometric thin shellformulation is suitable for different structural analyses and that in particularthat robustness and accuracy on a per-degree of freedom basis is observed inthe isogeometric post-buckling analysis. This adds post-buckling analysis tothe seamless integration of Computer Aided Design (CAD) and Analysis ofIsogeometric Analysis. Suggestions for further studies include severalimprovements of the current implementation (patch coupling, boundary conditionimplementation), utilization of nonlinear material models for modelling ofrubber-like materials, adaptive re-meshing using THB-splines to capture localwrinkling phenomena and Fluid-Structure Interaction computations with anonlinear structural and fluid description of VLTFSs in large waves.

This thesis investigates several kinds of models of describing multi-commodity dynamical flow. Three models have been developed for a spatial logistic network, on which packages are delivered by vehicles and another
model was developed to describe interactions between the customers, the stores and the deliverers, which does not explicitly take delivery time into account like before. The former three models consist of an agent-based model (ABM), and two different (differential) equation-based models (EBMs). One of the EBMs is built from the bottom up, and was created by averaging the randomness in the ABM. The other EBM was built top-down and involved optimizing a general model with respect to some parameters, such that its solution resembles the
ABM solution as closely as possible. These three models were tested and compared in the first half of the thesis.

In the second part of the thesis, the focus shifted to a generalized model (GM) approach of underlying interactions on multi-commodity dynamical
flow networks; especially the dynamics describing order placement,
scheduling and delivery. Such a network is cyclic and contains a feedback loop, where customers are less likely to order more products if they were recently delivered. This can range from someone not needing to purchase a
car if they just bought a brand new one to one not needing to order dinner when they just did so. Such networks are in general described by four kinds of elasticities in this thesis, namely the elasticities to stock, inventory level,
saturation, and co-production. These influence different parts in the network. For example, the negative elasticities to inventory level model that high inventory levels inhibit more inventory production, to prevent build-up,
and low inventory levels allow for larger inventory production, to prevent drainage. The other elasticities fulfil similar roles. This model was tested using a bifurcation analysis, a statistical ensemble method and by computing influences and sensitivities of parts and nodes in the network.

We investigate the simulation of one-phase and two-phase flow through heterogeneous porous media.The derived matrix, resulting from reservoir simulation of groundwater flow problems, can result in a large and ill-conditioned system, i.e. the matrix has a high condition number, and the modelling takes large computation time. In this thesis report, the Two-Level Preconditioned Conjugate Gradient method with deflation techniques will be considered. Recently, new Two-Level preconditioners are constructed using the AMG approach. In this research we compare this approach with the standard Two-Level preconditioners. We found that the performance of these methods can be improved by using a special starting vector and previous time-step as initial condition. From the results we see that the performance and the memory storage of the methods are similar. However, the cheapest methods per iteration resulted DEF1, DEF2, R-BNN2, and ROM.

Sandy barriers comprise 12% of coastlines around the world, and most of these barriers enclose tidal bays and lagoons. These systems accommodate human settlements vulnerable to climate change, which offer enough economic, social, and environmental utility to require further research on the impact of climate change and subsequent best management practices.

The present work aims to analyze how climate change impacts the hydrodynamics and morphodynamics of two barrier inlet systems: Katama Bay (United States of America), and the Santa Lucia Estuary (South Africa). The goal is to estimate future changes in forcing variables (e.g., sea level rise, wave climate, river discharge, tides), implement them in process-based models (coupled SWAN and Delft3D), and identify changes in the dynamics of both systems by comparing present and future state simulations.

This thesis develops a replicable and flexible methodology that can be used as a systematic tool to assess the impacts of climate change on the overall dynamics of tidal inlet systems. A novel approach (copula analysis) was used to derive the wave climate implemented in Delft3D, which was then qualitatively validated for both sites. Model results were used to compare changes to inlet stability, inlet geometry, and sediment pathways for present and future hydrodynamic conditions.

Results show that sea level rise is the primary contributor to the overall morphodynamics at both sites, whereas changes in wave direction strongly impact the rate of inlet migration. Other changes (e.g., significant wave height, wave period, and river discharge) play a secondary role in the dynamics of both systems. Comparisons with previous studies suggest that wave direction impacts each system differently. These impacts must be specifically addressed for each tidal inlet, as the results from one site should not be used to determine a general behavior for the assessment of CC impacts in tidal inlet systems.

Burn injuries can lead to serious complications that have a large influence on someone’s quality of life. In order to help patients, we need to gain insight into the wound healing process and in the development of complications that come with serious burn injuries. The final goal would be to prevent or at least reduce these complications. With a mathematical model, we simulate the time-evolution of the skin after a burn injury. The objective is to be able to quantify the impact of (patient-specific) parameters on the evolution of the skin properties. In this thesis, a cell-based model for the cell migration during wound healing is presented. Two types of cells, macrophages and fibroblasts, migrate by incentives of the strain energy density and concentration fields. Two concentration fields are implemented in the model: Platelet Derived Growth Factor (PDGF) and Transforming Growth Factor β (TGF-β). PDGF is a chemical that occurs in the wound bed and TGF-β is secreted by the macrophages. The processes such as cell division and death are modelled through stochastic processes. In this way, the data-consuming cell history does not have to be taken into account. The computational work of the corresponding simulations increases rapidly for larger numbers of cells. This holds in particular for the part that computes the strain energy density for every cell pair. This is tackled by employing the Graphics Processing Unit (GPU) for the largest bottlenecks. The CUDA framework is used to program the GPU, where certain parts of the computations can be run in parallel to make the computations more efficient. The GPU implementations are described in this report, alongside with the improved computation times. For a 2D simulation of one day, the speed-up in computation time from the CPU to the GPU implementation was a factor 58. To assess the accuracy of the computation of the concentration fields, Richardson’s Extrapolation is used to estimate the order of the error. More research is required on this part to achieve reasonable outcomes for the order. Moreover, an alternate approach for determining the TGF-β field by Green’s function was studied. The computational work for this method increased rapidly and was therefore unfit to be implemented in the model. Lastly, the influences of parameters on the model outcomes were investigated. Monte Carlo simulations are needed for the interpretation of the stochastic model. The influence of the time step and choice of normalization of the gradients were investigated. For the tested scenarios, the hypothesis that they behave similarly was not rejected. The single-precision implementation of the model did not lead to an overall speed-up. Especially, the increase in computation time of the solver for the concentration fields is unexpected. Using the GPU for efficiently modelling cell migration seems a good idea. The current model can be used as a basis for more sophisticated models in the future.

Existing multiscale solvers use a sequence of aggressive restriction, coarse-grid correction and prolongation operators to handle low-frequency modes on the coarse grid. High-frequency errors are resolved by employing a smoother on the fine grid. Deflation preconditioning improves matrix properties, i.e., damps slowly varying errors, corresponding to extreme eigenvalues, in the linear solver residuals. Vari- ous Adapted Deflated Multiscale Solvers are proposed in order to detect the low- frequency modes instead of relying on the residual map and complement today’s stet-of-the-art advanced iterative multiscale strategies.

Dit verslag geeft als eerste een overzicht van basis resultaten van de Riemann zeta functie. Vervolgens worden twee meer specifieke resultaten laten zien met betrekking tot de nulpunten van deze Riemann zeta functie.