Mathematical analysis and computer simulation of dendritic growth in lithium batteries

<p dir="ltr">In this thesis, we study a system of Partial Differential Equations (PDEs) related to a model of the crystal growth inside the electrolyte of lithium batteries as presented in. Aspects of this study are of interest in the phase-field and material science communities. Our research focuses on the mathematical analysis of the aforementioned model, as well as on its numerical analysis together with simulations.</p><p dir="ltr">More precisely, in Chapter 2 we consider the Nernst–Planck–Poisson–Allen–Cahn (NPPAC) model and its derivation as presented in Chen et al. (2015). We then proceed with the “nondimensionalisation” of the proposed model, so that we can perform the mathematical analysis. We also consider the anisotropy function and the anisotropy tensor and their properties, as presented in Kobayashi (1993), Karma and Rappel (1998), Burman and Rappaz (2003), Graser, Kornhuber and Sack (2013) and Lakkis, Skouras and Styles (2023).</p><p dir="ltr">In Chapter 3 we prove existence of a solution of the proposed model by using the Rothe method. We first discretize the system of equations in time and then we prove existence and uniqueness properties of the sequence of the systems that we created. We then prove stability estimates on the norms that naturally occur from the weak formulation of the system, given that the initial data are smooth enough. Finally, we show the weak convergence of the sequence of solutions to the continuous, in time and space, solution of the initial problem.</p><p dir="ltr">In Chapter 4 the main focus is the a priori error analysis of the model. We consider a time and space discretization. For the sake of the numerical computations, we consider a semi-implicit scheme where the anisotropy tensor and the nonlinearities in the Allen– Cahn equation are given by the previous time-step, as well as the convection term in the Nernst–Planck equation. We then move on to the error analysis where we show the order of convergence in time and space. A slight difference, compared to the results of Chapter 3, is that the error estimate for the time derivative we used the Green’s operator to extract an upper bound, a technique that makes us use the dual space H1(Ω)0 instead of L2(Ω).</p><p dir="ltr">In Chapter 5 we provide figures regarding the simulation of the model at its default set of parameters, and comparison results by altering different parameters, one at a time. Chapter 6 then concludes the thesis, in it we outline future work as well as possible improvements to the model we derived here.</p>