A systematic literature review of Burgers' equation with recent advances

Even if numerical simulation of the Burgers' equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers' equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective-diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers' equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers' equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers' equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers' equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations

Coupled electrochemical-thermal-mechanical stress modelling in composite silicon/graphite lithium-ion battery electrodes

Silicon is often added to graphite battery electrodes to enhance the electrode-specific capacity, but it undergoes significant volume changes during (de)lithiation, which results in mechanical stress, fracture, and performance degradation. To develop long-lasting and energy-dense batteries, it is critical to understand the non-linear stress behaviour in composite silicon-graphite electrodes. In this study, we developed a coupled electrochemical-thermal-mechanical model of a composite silicon/graphite electrode in PyBaMM (an open-source physics-based modelling platform). The model is experimentally validated against a commercially available LGM50T battery, and the effects of C-rates, depth-of-discharge (DoD), and temperature are investigated. The developed model can reproduce the voltage hysteresis from the silicon and provide insights into the stress response and crack growth/propagation in the two different phases. The stress in the silicon is relatively low at low DoD but rapidly increases at a DoD >~80%, whereas the stress in the graphite increases with decreasing temperature and DoD. At higher C-rates, peak stress in the graphite increases as expected, however, this decreases for silicon due to voltage cut-offs being hit earlier, leading to lower active material utilisation since silicon is mostly active at high DoD. Therefore, this work provides an improved understanding of stress evolution in composite silicon/graphite lithium-ion batteries