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. ... Read more

This Bachelor thesis provides an analysis of Runge-Kutta methods using Butcher tableaus. Runge-Kutta method are numerical methods used for approximating initial value problems. A Runge-Kutta method can be classified as either an explicit or an implicit method. A special kind of implicit methods are diagonally implicit methods. The type of method can be recognised by the Butcher tableau.

Using the entries of the Butcher tableau, one can compute the amplification factor of a Runge-Kutta method. The amplification factor can then be used to compute the order of the local truncation error and the stability region. Examples of these computations are given for seven methods. Furthermore, this thesis provides an algorithm to perform time steps for each of the three types of Runge-Kutta methods. Finally, in order to analyse the global truncation error of the seven methods, the algorithm to perform time steps is used with different step sizes. ... Read more

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. ... Read more

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

In (mathematical) physics one can encounter problems with a fase change, the so called Stefan problems. Numerical methods to solve these problems often use a level-set function which indicates the distance to the fase boundary. Problems can arise in the medial axis points of the zero set. Therefore it can be usefull to calculate this medial axis, which this report focusses on. Multiple techniques to compute the medial axis are discussed. In particular test results are shown for a strategy based on the Voronoi diagram and using Fortune's Algorithm. ... Read more

An analysis of curve interpolation with splines as a means to recover the curvature of a ordered set of discrete data points that originate from a closed smooth planar curve, in the context of the levelset method. ... Read more

The level-set (LS) method uses a signed-distance function to capture the interface in two-phase flows. Geometrical properties can be easily obtained, and merging and splitting of the interface are handled automatically by the LS method. However, it is not inherently volume conserving. Several methods found in literature that aim to solve this problem are discussed in this report, including the interface-correction level-set (ICLS) method. The ICLS method uses an additional advection step with a correction-velocity field to restore global volume loss/gain. We present the volume-of-fluid-based local interface-correction level-set (VOF-LICLS) method. This is an extension to the ICLS method that aims to restore volume locally. This is achieved by coupling the ICLS method with the VoF method. In each time step we evolve both the level-set function and the volume fraction function. The VoF advection is performed using a Lagrangian-Eulerian method on a dual mesh. From the advected LS field volume fractions are constructed and compared to the advected VoF field. This allows us to use local volume fluxes for the velocity field, instead of a global volume flux. The construction of the correction-velocity is performed with the use of an analytic equation and the local volume fluxes. The novel volume correction procedure is then executed iteratively. The level-set equation is discretized in space with a finite element approach. Instabilities occur for the standard Galerkin approach for pure advection equation, so the SUPG method is used in order to obtain stable solutions. The performance of the developed method in this thesis is compared with LS and ICLS for three different test cases. We report a significant improvement for local volume conservation, and we observe that VOF-LICLS yields more accurate interface positions. ... Read more

The aim of this research is to develop an N -dimensional adaptive sampling algorithm to efficiently sample functions, meaning that with fewer samples the same accuracy is achieved compared to what homogeneously spaced samples would achieve. This algorithm is based on an existing Python package called Adaptive. The developed algorithm is applied to find and plot the Fermi surface of crystals with a higher resolution than homogeneous sampling would with the same number of points. ... Read more

The numerical errors involving the application of the level-set method to a problem can be minimized by fitting a mesh (usually a triangular mesh in two dimensions) to the zero level-set curve defined by the level-set function. These numerical errors depend on two things: the size and skewness of the triangles in the mesh. To fit the mesh, Timo Wortelboer derived a physical model using springs (Wortelboer, 2018). He also defined measures to quantify the quality of a fitted mesh. First, his model is analyzed, as well as the quality measures he defined. Then an introduction on linear isotropic elasticity is given, based on which an elastic model for meshfitting is derived. The Finite Element Method is used to solve the differential equations of the elastic model. Then an optimisation algorithm is introduced to change the Lamé parameters such that the quality of the mesh is improved even more. Lastly the spring model and the elastic model are compared, which results in the elastic model being better than the spring model. ... Read more

In the work of Den Ouden the level-set method has been used to model the growth and dissolution in a steel alloy of the precipitate cementite and the diffusive phase austenite. The movement of the interface between the two phases is controlled by the diffusion of carbon and a reaction on the interface between the two phases. In this thesis research a similar model and method is given for a three-phased model with an extra diffusive phase. This third phase is ferrite, which nucleates on the interface of austenite and cementite when the temperature is dropped below the eutectoid temperature for a high carbon steel alloy. The difference between austenite and ferrite is the matrix structure of the iron atoms. This influences how much carbon the phase can contain. The difference in structure also results in different properties of the steel with respect to strength, hardness, etc. Modeling the growth of this three-phased steel alloy could eventually lead to useful insights for high-carbon steel production.\\
Extra difficulties arise by the introduction of a second diffusive phase. These difficulties concern defining sufficient interface boundary conditions between the different phases, capturing the different phases with the level-set method, creating a qualitative good mesh capturing these phases, recovering gradients on an interface and getting correct carbon concentration value in the triple points because of the interface boundary conditions. Except for the latter problem, the results of this research show decent results. The effect of cooling on the growth of ferrite is also shown and explained. ... Read more

In applying the level-set method in the context of a finite-element method, errors can be minimized by adjusting the mesh to the shape of the level-set curve. The size of the different types of errors that occur depend on the goodness of fit to the zero levelset curve, the skewness of the triangles and the size of the triangles of the mesh. To fit the mesh basic methods where designed by Javierre, Den Ouden and Verbeek. These methods adjust the mesh only locally and add many edges in the process. The aim of this research was to find better ways in which to adjust the mesh, without having to add new edges. First quality measures were set up: the maximum skewness, average skewness, standard deviation of skewnesses, maximum size, minimum size, standard deviation of sizes, where the sizes and skewnesses are taken over all the triangles in the mesh. Then the existing ‘cut’ method was analysed in depth and an extension called the ‘flip’ method was added. Afterwards a new approach was used, using shortest path algorithms to decide which points to move, and orthogonal projection to the level-set curve to move them. A second method was designed using a physical model, modelling every non-fitted non-boundary edge as a spring, in order to improve the rest of the mesh. These last two methods combined improved the quality of the created meshes greatly, with its only downside being its more limited applicability. ... Read more

The coarse grid of numerical weather prediction and climate models requires parametrization models to resolve atmospheric processes that are smaller than the grid size. For parametrization development, these processes are simulated by a high resolution model. At the Royal Netherlands Meteorological
Institute, the Dutch Atmospheric Large-Eddy Simulation (DALES) is used. This three-dimensional high resolution model uses advection schemes that are too diffusive when steep gradients are present. In this thesis, an advection scheme based on the Discontinuous Galerkin (DG) method is implemented
for DALES.

The DG method is known to be dispersive. To remove those non-physical oscillations, the moment limiter of Krivodonova is used. Krivodonova constructed the limiter for one- and two-dimensions. In this thesis the moment limiter and limiting order are derived for three-dimensions.

DALES is a model based on the finite difference method and uses operational splitting. Therefore, the DG advection scheme needs a mapping from each cell average to all nodal values that are needed for one DG cell, and a mapping back, which we called mapping a and b respectively. Mappings a that are discussed are taking the cell average as value for all nodal points of the DG cell (cell average a), and taking the L -projection of the cell average to the continuous finite element space (L -projection). This thesis describes mappings b that calculate cell averages of nodal DG values (cell average b)
and calculate the cell averages of the tendencies of DG values (cell average of tendency). Using cell average a combined with cell average of tendency, made the DG method as diffusive as the first order upwind scheme. Substituting the cell average a method with the L -projection, the DG method became
very dispersive, meaning that there was not enough diffusion. At last, cell average b was tested with the L -projection. Its numerical results showed that the speed of the advection was slower than the theoretical velocity. Therefore, a method is suggested which does not need mappings. An option could
be a supergrid that takes multiple DALES cells as a DG cell. ... Read more