Relevant Degree Programs
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Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - April 2022)
In this dissertation, we study analytical and numerical methods on three topics in the area of partial differential equations (PDE). These topics are: the Allen-Cahn dynamics (AC) in the study of phase field models for materials science problems, the Oxygen depletion model (OD) in the study of free boundary problems, and the stationary surface quasi-geostrophic equation (SQG) in the study of fluid dynamics. We first study the behaviour in the meta-stable regime of AC and show by computation evidence and asymptotic analysis that backward Euler method satisfies energy stability with large time steps. We also give a rigorous proof for the two-dimensional radially symmetric case. In the second project, we show several mathematical formulations of OD from the literature and give a new formulation based on a gradient flow with constraint. We prove the equivalence of all formulations and study the numerical approximations of the problem that arise from the different formulations. More general (vector, higher order) implicit free boundary value problems are discussed. In the final project, we develop a new framework of ``convex integration scheme'' and construct a non-trivial solution to the stationary SQG. We thus prove the non-uniqueness of the solutions to the stationary SQG.
In this thesis we present an analysis of the Gierer-Meinhardt model with saturation (GMS) on various curve geometries in ℝ². We derive a boundary fitted coordinate framework which translates an asymptotic two-component differential equation into a single component reaction diffusion equation with singular interface conditions. We create a numerical method that generalizes the solution of such a system to arbitrary two-dimensional curves and show how it extends to other models with singularity properties that are related to the Laplace operator. This numerical method is based on integrating logarithmic singularities which we handle by the method of product integration where logarithmic singularities are handled analytically with numerically interpolated densities. In parallel with the generalized numerical method, we present some analytical solutions to the GMS model on a circular and slightly perturbed circular curve geometry. We see that for the regular circle, saturation leads to a hysteresis effect for two dynamically stable branches of equilibrium radii. For the near circle we show that there are two distinct perturbations, one resulting from the introduction of a angular dependent radius, and one caused by Fourier mode interactions which causes a vertical shift to the solution. We perform a linear stability analysis to the true circle solution and show that there are two classes of eigenvalues leading to breakup or zigzag instabilities. For the breakup instabilities we show that the saturation parameter can completely stabilize perturbations that we show are always unstable without saturation and for the zigzag instabilities we show that the eigenvalues are given by the near circle curve normal velocity. The breakup analysis is based on the reduction of an implicit non-local eigenvalue problem (NLEP) to a root finding problem. We derive conditions for which this eigenvalue problem can be made explicit and use it to analyze a stripe and ring geometry. This formulation allows us to classify certain technical properties of NLEPs such as instability bands and a Hopf bifurcation condition analytically.
Magnetized target fusion reactors are a modern idea to generate hydrogen fusion energy on earth. The design entails confining a plasma with a magnetic field and crushing it in an imploding shell of molten metal. Such a design has many unresolved questions in terms of its feasibility as a power source and ways to make it efficient. In this thesis, we will look into two of the approaches undertaken to explore these questions. Firstly, we use a coordinate transformation and implement a novel flux-limited, split-step, finite volume scheme for nonlinear coupled hyperbolic partial differential equations. With this numerical scheme, we do a parameter sensitivity analysis for the design performance. Secondly, by a careful series of asymptotic arguments, we establish a leading order asymptotic expression for the plasma compression. This expression is qualitatively consistent with the numerical work, but it also gives new insights into how the device operates. Together these approaches allow us to infer key design parameters for the success of magnetized target fusion. We will conclude with a look into the viability of magnetized target fusion and some problems for future work.
No abstract available.
Master's Student Supervision (2010 - 2021)
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.
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.
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.
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.