Burn injuries form a very specific type of skin wounds, which can cause much suffering and inconvenience to the patients. One who has serious burn injuries, can be literally scarred for life due to the complications developed post wounding. Wound contracture is one of the symptoms that is characteristic for burn injuries and it can yield many complications. At the location of the wound, an unusual large number of fibroblasts accumulates. This is a type of skin cells that exert forces on their direct neighbourhood. Normally, skin is elastic and if forces are removed, the skin recovers to its original shape again. However, in the case of burn injuries, the forces can be so strong, that the skin, like other elastic materials, deforms permanently. Hence, contractures, excessive and pathological contractions are developed. Investigating the healing of burn injuries is necessary to obtain more insights and to know which factors can influence wound contraction and scar formation. This could be helpful in order to give a treatment that is better in tune to avoid complications asmuch as possible.

In this bachelor thesis we propose a spatial Markov Chain Cellular Automata
model for the spread of the COVID-19 virus as well as two methods for parameter
estimation. Network topologies are used to model the progression of the epidemic
by considering each individual on a grid and using stochastic principles to determine the transition between different states. The model is able to predict the
time-evolution of outbreaks under different lockdown policies. Additionally, the
impact of variation in infection probability and recovery rates on the amount of
active cases, deaths as well as the length of the epidemic is investigated. These
results can provide us with insights and predictions of the spread of the virus under different scenarios. Parameter estimation is done by using both Maximum likelihood estimation and Bayesian estimation based on simulated data. The produced estimates were relatively accurate, suggesting that these methods can be applied in order to estimate the parameters of the proposed model based on actual data.

The theory of morpho-poroelasticity is applied to tumor growth. This allows for the modeling of permanent deformation in the tissue as a result of the presence of tumor cells. The work done in this project can be divided into two parts. In the first part we review existing models for elasticity and build corresponding finite element models. The aim of the first part is to gain understanding in the behaviour of morphoelasticity and poroelasticity. In morphoelasticity the deformation tensor is decomposed into an elastic and plastic component, allowing for permanent deformations to occur in the tissue. The theory of morphoelasticity describes the interplay between a fluid in a porous tissue and the tissue's elastic properties. A common problem in poroelastic models is the occurence of non-physical oscillations in the pressure. The second part of this project contains novel work. First, a rigorous mathematical derivation is presented of the tuning parameter found in the diffusive stabilization method for poroelastic systems. Secondly, we present a finite element model for the novel combination of morpho- and poroelasticity. The derivation concerning the diffusive stabilization is also applicable to the morpho-poroelastic finite element model. We obtain a promising tumor growth model by combining a biochemcial tumor growth model with morpho-poroelasticity. This biochemical model is based on a nutrient transport equation and a tumor cell density evolution equation. Some example output of the tumor growth finite element model is shown. It can be used as a starting point for more elaborate tumor growth models.

Burn injuries occur daily and can have severe physical and mental effects both in short and long term, such as disabilities due to severe skin contraction. Even though the mortality rate has decreased over the years, the need for a higher quality of life after severe burns remains. Decreasing the probability of a severe contraction is essential for increasing the quality of life. Mathematical models have been developed to predict skin contraction over time. However, the computations are time-expensive and not suitable for applications that require many simulations, for example, when considering input uncertainty for patient-based predictions. To that end, the application of neural network surrogates is studied to accelerate the computations of a morphoelastic numerical model for the prediction of skin contraction. Two datasets are generated from a one- and two-dimensional model respectively and are used to train the neural networks. It is shown that a feedforward neural network can accurately learn the nonlinear mapping between the input parameters and the outputs of the considered morphoelastic models. The trained neural network provides fast and accurate predictions on skin contraction and strain energy. Furthermore, a first step is taken towards a hybrid model where a neural network is applied as a surrogate for computationally expensive time-stepping in the numerical model. The added value of fast neural network surrogates is demonstrated in two clinical case studies. It is shown that the surrogate can be used to perform input parameter studies by comparing an age study with the morphoelastic model with the age study using the neural network. The neural network can reproduce the age study with high accuracy in just a fraction of the time. Secondly, a concept application is designed to demonstrate patient-based predictions using Monte-Carlo simulations to cope with input parameter uncertainty. The application provides predictions for skin contraction and the strain energy, based on the age of the patient and the wound size.

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.

In this study several models for burn contraction are studied. The first ones are rudimentary, but as the thesis progresses we work towards so-called morphoelastic models. These inherently capture both elastic and plastic deformations. After establishing the deterministic part, a framework is built for the incorporation of heterogeneity.

This report has the main aim of comparing the Mixed Finite Element Method to the standard Finite Element Method. The other aim is to let the reader understand what these methods entail. The latter is done by first journeying through the theory behind the FEM. It is first explored in one dimension to keep the setting simple. Next, the Mixed FEM is explored. A new way of approximating the gradient from the found solution is constructed, using the basis of Finite Differences and the ideas from the Finite Volume Method. Following the theory is the implementation of the mentioned methods and the analysis of theresults. It yields that, when looking at the L2-norm, the classic FEM has a convergence order of 2, comparable to that of similar numerical methods. The Mixed FEM seemed to converge with an order of 4 in the same norm. Our constructed method only had a measly first order convergence, implying much greater accuracy of the Mixed FEM.

Experimental data have been extracted from wound-healing time-lapse videos. These data include the detection of the area of the wound and the detection of individual cells. Extracting the area of the wound is done by using a function in Python OpenCV called cv2.findcontours. An improved method to extract the area of the wound is introduced as well using the Sobel filter. Detecting the individual cells is done by localizing the local maxima in an image. Histogram equalization is applied to enhance the global contrast to increase the performance of cell detection. The continuum model is applied in one of the videos, where we used different parameters to model the data. A method is described to nd the optimal parameter of the continuum model by using the method of least-squares. Finally, two methods using the kernel density estimation are described which can be useful in future studies.

The present thesis discusses two modelling endeavours that serve to provide insights into both the magnetic and structural dynamics associated with a first order magnetic phase transition.
Firstly, density functional theory has been used to model the lithiation of a supercell of 2x2x2 the conventional unit cell of the ferromagnet Fe2P. Subsequently, the magnetic moments in the resulting structure were set to proxy a paramagnetic state and compute the energy difference between the ferromagnetic and paramagnetic state. It has been found that non-magnetic Li-atoms substitute for magnetic Fe-atoms at 3g-positions in the a,b-plane. Moreover, since these are the positions of Fe-atoms that carry a high magnetic moment and induce a magnetic moment on the Fe-atoms in the 3f-layer, the lithiation of the respective positions leads to a reduction of the magnetic moment per formula unit. The energy difference between the ferromagnetic state and the selected proxy of a paramagnetic state showed to decrease linearly in the fraction of lithiated 3g-positions. This deviates from experimental findings suggesting an increase of Tc of the structure upon lithiation. This difference can stem from the fact that the lithiation fractions considered here were higher than those achieved experimentally or that the selected proxy of a paramagnetic state was incorrect.
Secondly, the finite element method has been applied to approximate the displacement field of a 2D structure consisting of grains distributed over two structural phases with different associated lattice parameters. The development of shear stress discontinuities across grain boundaries has been studied in relation to the degree of porosity in a domain. The replacement of grain boundaries by voids appears, after correction for the orientation of edges at these grain boundaries, to lead to a reduction in the values for these discontinuities. More delicate development of the grid to correspond to physically realistic orientations of grain boundaries is needed to gain more robust quantitative measures for this relation. Moreover, the relation between the orientation of grain boundaries and shearing stresses at these boundaries forms an interesting venue for further research to which the present FEM-model can be readily applied.

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.

It is said that Dutch winters are becoming milder and that we experience less severe winters nowadays. In this thesis it is attempted to test meteorological data on whether there is a significant trend. This is done by using the Mann-Kendall test and Sen’s slope estimator. In the last chapter the cultural impact of the changing winters on the Eleven cities tour is also explored.

Steel is commonly coated to protect it from corrosion. One method of applying this is by using Physical Vapor Deposition, which can be done by using multiple jets. In this process jets next to each other interact. This paper's main aim is to investigate the interaction effect of two rarefied two-dimensional vapor jets in vacuum and how this is influenced by the distance between jets and inlet density. Furthermore, the analytical solution for the collisionless case for a single jet is extended to dual jets. Additionally, the objective is to maximize the processing speed for the use of coating, with a certain uniformity for the parameters researched in this paper and finding an optimal method in doing so. This study is done by using the Direct Simulation Monte Carlo (DSMC) method.
The analytical solution gave the same results as the collisionless DSMC method for both single and dual jets. Simulations with a strong interaction effect resulted in a shock. These behaved similar compared to three-dimensional jets, in the plane of the jets. The shock results in a secondary jet, which has a lower density in the middle. The interaction effect depends primarily on the inlet density. Multiple regimes are observed for different inlet density ranging from small change in properties to a shock wave, with a transitional regime inbetween. The influence of the distance between the jets is found to result in a higher density at the axis of the inlet behind the shock, for bigger distance between jets. However, for very small distances between jets compared to the inlet size the shock is weak. For the optimization, it resulted in the conclusion that the optimal coating in general is applied with the smallest distance between jets. This generally gives a better uniform coating and increased performance. However, this is not always the case when constraining the distance between the jets and sheet, as it only holds if the shock between the jets for this distance. Furthermore, an approximation is found for the optimization, which results in fewer simulations needed.

Reliable and efficient modelling of magnetic hysteresis in inhomogeneous and aniso-tropic media is an important step in developing a state-of-the-art closed-loop degaussing system for naval ships and submarines, to be developed by TNO and to be used by the Royal Netherlands Navy in an updated generation of naval vessels and submarines. Different models have been proposed to describe the nonlinear and history-dependent nature of ferromagnetic hysteresis at a material level. With a focus on three key differing aspects of models, namely linear versus nonlinear(hysteresis), isotropic versus anisotropic and homogeneous versus inhomogeneous, we attempt to discriminate between the performance of models on the basis of these criteria. More specifically, with increasing model complexity, we have combined Maxwell's equations with four different hysteresis models within the context of a prolate steel ellipsoid, whose ferromagnetic properties evolve under the influence of a uniform applied background field. Among other aspects, the hysteresis models differ in terms of physical motivation, complexity and parameter spaces. In this research, we have analysed four hysteresis models in more detail: The Induced - Permanent magnetization model, The Rayleigh} model, the Jiles-Atherton model and an Energy-Variational model, based on energy balances. The thus derived forward models have subsequently been inverted in order to estimate material hysteresis parameters. With increasing complexity also, twin experiments have been performed. This increasing complexity \textit{temporally} stems from the fact that the hysteresis models named previously, are stated in increasing order of complexity, and can all be modified in order to model anisotropic material by generalizing model parameters to tensors. Spatially, the increase in complexity is caused by the fact that in special cases, namely of uniform ellipsoid magnetization, an analytical formula relating the magnetic field, the background field and the ellipsoid magnetization exists by solving the Poisson partial differential equation on an infinite domain using direct computation with Green's functions.

In mathematical biology, a morphoelastic model, which combines elasticity and growth, has been used for, e.g., wound healing and tumour growth modelling. Here, it is used to model the growth of an embryo. In order to do so, a growth curve from literature has been fitted to a morphoelastic model which has been extended with a PDE for the normalised cell concentration. In the one dimensional situation, this fitting resulted into a curve with a shape similar to the desired shape. Despite the similar shape, it did not fulfil the total growth requirements; it fell short by a factor 100. Instead, a Gaussian curve was used to represent the cell concentration. In one dimension this gave a similar looking curve, with a relative error of 0.2316 for the first couple of weeks of the gestational age of the embryo, where the length of an embryo is measured form crown-to-rump. For the other weeks, when the length of the embryo is measured from crown-to-heel, a relative error of 0.0507 was observed. The one dimensional problem is solved using the finite element method with linear basis functions and an Euler backwards time integration method. This is implemented in MATLAB. The output is compared to results of a COMSOL Multiphysics® file created on this topic and verified to be similar. In three dimensions, the model without the cell concentration contribution is verified. For this, experiments have been done in which a force is applied in various directions separately. All results were similar to the one dimensional model with a pulling force. The model containing cell concentration is not verified in this thesis. Further, the effects of the growth term and mesh choice have been investigated. These calculations have all been done with COMSOL Multiphysics.

Chemotaxis is een verschijnsel uit de biologie dat zich op verschillende wijze
voordoet in het menselijk lichaam en bij bacteriën. Er is onderzocht hoe het
ééndimensionaal geval op te lossen is met de discontinue Galerkinmethode. Hierin
is het model eerst behoorlijk versimpeld en steeds verder uitgebreid. Zo is gekeken
naar de discontinue Galerkinmethode en de toepassing op de transportvergelijking
en kleine aanpassingen hierop én op de toepassing op de warmte
vergelijking.

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.

Over time, the width-averaged depth of estuaries changes due to a complex interaction of hydrodynamics and suspended sediment transport. In many estuaries one specic location with a suspended sediment concentration (SSC) higher than in the sea or in the upstream river, is found, which is called sediment
trapping. The location of the maximum SSC is called the estuary turbidity maximum (ETM). Understanding the dynamics is important to maintain a healthy ecosystem while making anthropogenic changes. To investigate such changes, a two-dimensional model is developed, considering the Ems estuary as a case study. The model equations consist of the width-averaged shallow water equations and a SSC equation. Assuming a morphodynamic equilibrium, these equations are solved mostly analytically by making a regular expansion of each physical variable in a relatively small parameter. Using this method, we are able to gain insight into the fundamental physical processes resulting in sediment trapping in an estuary by studying the influence of various forcings separately. One of the hydrodynamic forces is vertical mixing. This force has been assumed to be constant over time in previous studies [1]. In this thesis vertical mixing as a function that varies on the tidal timescale has been added to the model and is analysed. As a result of the salinity gradient in the estuary the mixing is stronger during flood and weaker during ebb. Using the model it is found that time variations in vertical mixing result in tidally averaged non-zero, and therefore contributing, transports. They cause a narrowing of the location where sediment is trapped. If vertical mixing is exactly maximal when the flood is maximal and minimal when ebb is maximal, the ETM shifts downstream. But if the vertical mixing is lagging the tidal stream, which is more plausible, the ETM will stay or shift upstream.

In this report, a model for modeling passenger flows on a platform with obstacles is presented. The report incorporates situations with one exit, benches as obstacles and an additional train and compares those to a basic model in which the platform has two exits and no obstacles. The present study considers the Particle Model as a basis for passenger flows on a platform. The parts of the Particle Model that are integrated are chemotaxis, random walk, repulsive forces and mechanotaxis. Therefore the model is based on both deterministic and stochastic principles. Simulation results are discussed concerning the time needed for a certain amount of passengers to leave the platform, the average number of encounters between individuals and of individuals with benches.

Dit verslag beschrijft een wiskundig model, gebaseerd op het deeltjesmodel, voor sluiting van oppervlaktewonden. Verder wordt aandacht aan verschillende soorten modellen die al bestonden en aan de biologische achtergrond van wondsluiting.