Brian Wetton

The pseudo-two-dimensional (P2D) model is a complex mathematical model that can capture the electrochemical process in Li-ion batteries. However, simulation of the model also brings a heavy computational burden. Many simplifications to the model have been introduced in the literature to reduce the complexity. We present methods for fast computation of two models: an asymptotically scaled P2D model and the full P2D model. The techniques developed in this work can be used when simplifications are not accurate enough. By rearranging the calculations, we reduce the complexity of the linear algebra problem. We also employ automatic differentiation, using the open source package JAX, for robustness, which also allows easy implementation of changes to coefficient expressions. The method alleviates the computational bottleneck in P2D models without compromising accuracy.

View record

Lithium-ion batteries are ubiquitous in modern society. The high powerand energy density of lithium-ion batteries compared to other forms of electrochemical energy storage make them very popular in a wide range of applications, most notably electric vehicles (EVs) and portable devices suchas mobile phones and laptop computers. However, despite the numerousadvantages of lithium-ion batteries over other forms of energy sources, theirperformance and durability still suffer from aging and degradation. The purpose of the work presented in this thesis is to investigate how different loadcycle properties affect the cycle life and aging processes of lithium-ion cells.To do so, two approaches are taken: physical experiments and mathematicalmodeling.In the first approach, the cycle life of commercial lithium-ion cells ofLiNiCoAlO₂ chemistry was tested using three different current rates to simulate low-, medium-, and high-power consuming applications. The batteries are discharged/charged repeatedly under the three conditions, all whiletemperature, voltage, current, and capacity are recorded. Data arising fromthe experiments are then analyzed, with the goal of quantifying batterydegradation based on capacity fade and voltage drop. The results are thenused to build two predictive models to estimate lithium-ion battery state-of-health (SoH): the decreasing battery V₀₊ model and the increasing CVcharge capacity model. Furthermore, a simple thermal model fitted fromthe battery temperature profiles is able to predict peak temperature under different working conditions, which may be the solution to temperaturesensitive applications such as cellphones.The limitation to physical experiments is that they can be costly andextremely time-consuming. On the other hand, mathematical modeling andsimulation can provide insight, such as the internal states of the battery,that is either impractical or impossible to find using physical experiments.Examples include lithium-ion intercalation and diffusion in electrodes andelectrolytes, various side-reactions, double-layer effects, and lithium concentration variations across the electrode layer. Thus, in the second approach,work focuses on implementing the pseudo-two-dimensional (P2D) model, themost widely accepted electrochemical model on lithium-ion batteries. TheP2D model comprises highly-nonlinear, tightly-coupled partial differentialequations that calculate lithium concentration, ionic flux, battery temperature and potential to significant accuracy. The unparalleled prediction abilities of the P2D model, however, are shadowed by the low computationalefficiency. Thus, much of this work focuses on reducing model complexity toshorten effective simulation time, allowing for use in applications, such as abattery management system, that have limited computational resources. Inthe end, four model reductions have been identified and successfully implemented, with each one achieving a certain standard of accuracy.

View record

It is well known that Allen-Cahn equation and Cahn-Hilliard equation are essential to study the phase separation phenomenon of a two-phase or a multiple-phase mixture. An important property of the solutions to those two equations is that the energy functional, which is defined in this thesis, decreases in time. To study these solutions, researchers developed different numerical schemes to give accurate approximations, since analytic solutions are only available in a very few simple cases. However, not all schemes satisfy the energy-decay property, which is an important standard to determine whether the scheme is stable. In recent work, Li, Qiao and Tang developed a semi-implicit scheme for the Cahn-Hilliard equation and proved the energy-decay property. In this thesis, we extend the semi-implicit scheme to the Allen-Cahn equation and fractional Cahn-Hilliard equation with a proof of the energy-decay property. Moreover, this semi-implicit scheme is practical and could be applied to more general diffusion equations while preserving the energy-decay stability.

View record

In some nonlinear reaction-diffusion equations of interest in applications, there are transition layers in solutions that separate two or more materials or phases in a medium when the reaction term is very large. Two well known equations that are of this type: The Allen-Cahn equation and the Cahn-Hillard equation. The transition layers between phases evolve over time and can move very slowly. The models have an order parameter epsilon. Fully developed transition layers have a width that scales linearly with epsilon. As epsilon goes to 0, the time scale of evolution can also change and the problem becomes numerically challenging. We consider several numerical methods to obtain solutions to these equations, in order to build a robust, efficient and accurate numerical strategy. Explicit time stepping methods have severe time step constraints, so we direct our attention to implicit schemes. Second and third order time-adaptive methods are presented using spectral discretization in space. The implicit problem is solved using the conjugate gradient method with a novel preconditioner. The behaviour of the preconditioner is investigated, and the dependence on epsilon and time step size is identified. The Allen-Cahn and Cahn-Hilliard equations have been used extensively to model phenomena in materials science. We strongly believe that our high order adaptive approach is also easily extensible to higher order models with application to pore formation in functionalized polymers and to cancerous tumor growth simulation. This is the subject of ongoing research.

View record

The removal of artificially and naturally placed oils found in the ground is of paramount concern for environmental reasons and for the extraction of crude oil for energy. While many companies are involved in this soil remediation process, the physics of the underlying problem are not very well understood. We present a mathematical model and analysis of oil extraction in a porous soil based around the injection of water. We first consider a saturation analysis based on conservation of mass between oil and water in a one-dimensional setting and we find that based on certain parameter values, the water-oil interface goes unstable producing viscous finger patterns. We then include the effects of surface tension between oil and water to determine how this affects the growth of such fingers. We conclude that in the limit of small surface tension effects, the results generalize to the original problem, but more importantly, we deduce that there is a scaling which places the formulation into a setting that is invariant with respect to the magnitude of surface tension effects. With this scaling, we notice that the effect of surface tension is to limit the growth of fingers to a maximal wave number and to prevent their formation entirely beyond a certain critical wave number. Finally with the inclusion of temperature, via heated water injection, we see the formation of a dual fingering pattern: one associated with the mass conservation analysis of the oil-water interface and one associated with the conservation of energy across an interface where thermal gradients occur. We see that the thermal gradients across the interface where temperature drops induce unstable viscous patterns with a higher wave number than would occur for an equivalent isothermal interface where there was solely a change in viscosity. The thermal gradients also promote fingering development downstream across the classical viscosity differential driven interface but at the expense of lowering the interfacial velocity. It is interesting to note that the change in saturation that occurs across the energy interface is a result of a pseudo-free boundary created by the thermal problem.

View record

This thesis is a combination of two research projects in applied mathematics, which use the applied math techniques of numerical and asymptotic analysis to study real-world problems. The first problem is in superconductivity. This section is motivated by recent experimental results at the Paul Sherrer Institute. Here, we need to determine how the surface roughness of a superconductor influences the penetration properties of an externally applied magnetic field. We apply asymptotic analysis to study the influences, and then verify the accuracy - even going well-beyond the limits of the asymptotics - by means of computational approximations. Through our analysis, we are able to offer insights into the experimental results, and we discover the influence of a few particular surface geometries. The second problem is in gas diffusion. The application for this study is in fuel cells. We compare two gas diffusion models in a particular fuel cell component, the gas diffusion layer, which allows transport of reactant gases from channels to reaction sites. These two models have very different formulations and we explore the question of how they differ qualitatively in computing concentration changes of gas species. We make use of asymptotic analysis, but also use computational methods to verify the asymptotics and to study the models more deeply. Our work leads us to a deeper understanding of the two models, both in how they differ and what similarities they share.

View record