# Casper Beentjes

A collection of small and bigger projects written during my education that have not been published.
Note that these essays have not been peer-reviewed and will inevitably contain flaws.
Code for algorithms or to generate figures for some of the essays is available in the corresponding Bitbucket directory.

### Theses

*MSc Thesis: Computing Bifurcation Diagrams with Deflation*[PDF][Bitbucket]- Contemporary numerical bifurcation analysis can yield unsatisfactory results for problems where the solution branches do not connect in the parameter domain of study. Deflation techniques can help to elude this problem and provide a scalable numerical bifurcation technique. A new theoretical investigation into convergence of Newton's method with deflation is complemented by a practical comparison with AUTO-07P for illustrative test problems.
*BSc Thesis: Symmetry-Breaking in Patterned Elastic Sheets*[PDF]- In this thesis we investigate the loading of elastic sheets, which are patterned with a square array of circular holes. An elastic buckling instability due to this loading induces a pattern transformation in the material, which breaks symmetries of the material. The pattern transformation is accompanied by a drastic change in the mechanical response, which we investigate through extensive FEM simulations, applying uni-axial as well as bi-axial loading. We find that the pre- and post-buckling response shows similarities to (thick) beam-behaviour, and therefore, we propose to model the material as single effective beams under a load. In the post-buckling phase a negative stiffness is found for certain hole sizes, which results in snap-through buckling behaviour.

### Short projects

*Pattern Formation Analysis in the Schnakenberg Model*[PDF]- Literature suggests that steady chemical patterns could serve as an explanation for pattern formation in biology. One way to generate these spatial patterns is by the Turing instability in reaction-diffusion equations. In this report we explain some of the theory of the Turing instability with a focus on a prototype model by Schnakenberg. The parameter values that allow for pattern formation are investigated in more detail. Numerical explorations using an IMEX-pseudospectral method are made to test predictions for the one-dimensional Schnakenberg model.
*Solitary Waves in Magma Dynamics*[PDF]- Movement of magma through the earth's mantle can be seen as porous media flow. To model this molten rock migration we derive a model describing a viscous fluid flow through a compacting solid matrix based on work by McKenzie. Upon simplification using, inter alia, a Boussinesq approximation we arrive at a more tractable non-linear model with rich behaviour. Solitary wave type solutions have been observed numerically for simplified models in literature and analytic expressions for a restricted class of travelling waves are derived which confirm the existence of these special waves. Strong evidence to date suggests that these special wave solutions are not solitons.
*Buckling of Constrained Hyper-Elastic Beams*[PDF]- The deformation of elastic beams in confined spaces has a wide range of applications in, inter alia, health care and the petroleum industry. Historically, the literature has mainly focussed on one-dimensional beams models. This paper, however, follows a different approach, as we focus on two dimensional hyper-elastic beams confined by a set of parallel plates. Due to non-linearity of the governing equations, a finite element approach, implemented in FEniCS, is used to find numerical solutions to the model equations.
*Dynamics of Flexible Filaments in Viscous Fluids*[PDF]- Inspired by the locomotion of organisms at the micro-scale using appendages we derive a simplified model of a flagellum-like micro-swimmer. We study a passive one-dimensional elastic filament immersed in a viscous fluid driven by an external torque on the head of the filament based on models from the literature. We derive an elastohydrodynamic model describing the fluid-filament interaction. Using weakly non-linear analysis and numerical simulations we show that this simple model indeed can describe the propulsion of a filament. The model gives a non-zero and non-trivial swimming speed which depends only on two non-dimensional model-parameters, viscous drag anisotropy and a filament property parameter.
*Computational Electrochemistry*[PDF]- We derive a mathematical model for the dynamics of redox reactions at an electrode-liquid interface when an external potential is applied to the electrode. The model couples chemical transport with electron kinetics and the set-up of the model allows for a general range of external potentials. We further investigate the resulting dynamics of five types of external potential. Numerical simulations are performed using a finite difference and a Chebyshev collocation approach yielding perfect agreement with theory whenever available.
*Quadratures on spherical surfaces*[PDF][Bitbucket]- Approximately calculating integrals over spherical surfaces in \(\mathbb{R}^3\) can be done by simple extensions of one dimensional quadrature rules. This, however, does not make use of the symmetry or structure of the integration domain and potentially better schemes can be devised by directly using the integration surface in \(\mathbb{R}^3\). We investigate several quadrature schemes for integration over a spherical surface in \(\mathbb{R}^3\), such as Lebedev quadratures and spherical designs, and numerically test their performance on a set of test functions.
*On Short Recurrences in Optimal Krylov Subspace Solvers*[PDF][Bitbucket]- To solve large sparse linear systems of equations computational fast methods are preferable above methods such as GMRES. By exploiting structure of the involved matrix one can in certain cases use short-recurrences to create an optimal Krylov subspace method. In this report we explain some of the theory regarding short-recurrence methods for solving linear systems. A central object in this theory is the Faber-Manteuffel theorem of which some extensions are given in this report as well. After the theory two actual algorithms which use these short-recurrences are discussed and tested, SUMR for shifted and scaled unitary matrices and PGMRES for nearly Hermitian matrices. Some theory is given for both methods and they are numerically tested against alternative methods.