Classical Nutrient-Phytoplankton-Zooplankton (NPZ) models typically assume a well-mixed, spatially uniform environment; a simplification that overlooks the vertical and horizontal heterogeneity fundamental to marine plankton dynamics. To determine the sensitivity of plankton dynamics to spatial structure, we extend the zero-dimensional (0D) NPZ system to a two-patch system. Linear stability analysis of this reduced system reveals that vertical diffusion and phytoplankton sinking elevate the nutrient and growth thresholds required for population persistence, thus delaying the enrichment-driven loss of stability predicted by the classical single-box model. This two-patch formulation is subsequently generalised into a flexible multi-layer and multi-column grid, where a conservative flux-balance discretisation enforces exact nitrogen conservation while enabling modular incorporation of depth-dependent diffusion, directed advection, and time-varying external forcings.

Systematic comparison against the continuous reaction-diffusion benchmark of Cowall et al. 2021 demonstrates strong numerical equivalence with the discrete solution, consistently resolving both the seasonal timing of the surface bloom and the vertical distribution of biomass beneath the retreating mixed layer. When driven by observational irradiance and stratification records from the North Atlantic, the optimised grid configuration yields a correlation of $0.714$ with satellite-derived chlorophyll-a, where the addition of a downward sinking flux is critical for correcting the model's late-season behaviour to match the observed autumnal decline. Beyond this, our simulations reveal that the horizontal propagation of idealised propagating internal-wave forcing introduces phase-dependent vertical displacements that redistribute the subsurface reservoir, generating lateral structures that persist throughout the late-season period and influence the following upward re-entrainment of biomass. From a computational standpoint, the sparse implementation of the backward differentiation formula scales near-linearly with vertical resolution, while a non-uniform mesh concentrated within the thermocline transition zone reduces the discretisation error by an order of magnitude relative to uniform grids.

Collectively, these results position the discrete grid framework as a quantitatively reliable and computationally manageable approximation for continuous PDE solvers, well-suited for incorporating additional biological complexity and for extension towards three-dimensional oceanographic simulations.
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This thesis addresses the novel double-bounded positive semidefinite Procrustes problem, which arises from the optimal model regularization problem for wafer alignment.
Despite convexity in the optimization variable A , solving the problem is challenging due to the presence of both an upper and a lower bound, both of which introduce nonlinearity. Several numerical methods have been proposed, including semidefinite programming, alternating projection, and projected gradient methods. Among these, the projected gradient method proves to be the most efficient: by decomposing the upper bound matrix and transforming the variables, the constraints simplify, allowing for straightforward projection onto the feasible region using eigenvalue decomposition.
The optimal regularization approach was tested on various customer datasets, demonstrating overlay improvements of several tens to hundreds of picometers in most cases. However, the method failed for datasets with significant deformation and high variance among wafers, due to incomplete optimization of the regularization matrix. Future research should focus on developing a more sophisticated scheme for combining data-driven optimal regularization with standard bending energy matrices, and extensively validating the method across diverse datasets for accuracy, consistency, and performance.
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The sea and the shoreline form a complex ecosystem driven by tides. So far, studies often ignore the moving boundary caused by these tides. The focus of this thesis to incorporate this boundary by using a coordinate transformation and a time-explicit numerical method. To achieve this, first the one-dimensional shallow water equations are derived from the 3D Navier Stokes equations. Then these 1D equations are non-dimensionalized and the coordinate transformation is done. This results in a system of non-linear equations. The seabed is modelled as a straight line. At the seaward side there is a periodic forced wave and at the landward side the water depth is 0. The time-explicit numerical method of Lax-Friedrichs is used. This method is stable under a more restricted Courant-Friedrichs-Lewy condition and is convergent for refined grids. For the Ameland inlet system the water depth, velocity and length of the basin results are calculated and compared to a simplified model and complemented by a Fourier analysis. The results are realistic (constant in the beginning of the basin with visible non-linearities at the landward side). An analysis is done to understand how the model behaves for dierent physical parameters, such as: the amplitude of the periodically forced wave, the undisturbed water depth, the length of the basin and the resistance. The model remains stable and the results are realistic. ... Read more

Janus particles are colloidal particles for which one half of the surface has different attributes than the other half. One property of a spherical dielectric particle with half of its surface covered by a layer of another dielectric or metal is that it has a non-uniform scattering pattern when exposed to light. However, the angle with which the light is shone on the particle has a large effect on the scattering pattern produced. Thus it is important that we are able to orient these Janus particles. The orientation can be controlled if we apply an electric field to the particle for example. The movement of colloidal particles with an electric field is widely studied and this field is called dielectrophoresis. For a Janus particle, the calculations for the force and torque become complicated. The movement and rotation of these particles have been studied, however, no analytic solution has been found. In this report, we derive a semi-analytic description of the force and torque due to an external electric field on a spherical Janus particle. For this, first the potential due to an external electric field is determined and then the force and torque are calculated with two methods: the dipole approximation and the Maxwell Stress Tensor method. In the dipole approximation, there is no force on the Janus particle. But, there is a torque on the particle in the dipole approximation. Due to this torque, the Janus particle will orient itself such that its cap points in the direction perpendicular to the applied field. For the torque calculated with the Maxwell Stress Tensor, we get a similar result as in the dipole approximation. On the other hand, according to the calculations with the stress tensor, there is a relatively small force on the particle. ... Read more