Gradient Preserved And Artificial Neural Network Method For Solving Heat Conduction Equations In Double Layered Structures

Layered structures have appeared in many engineering systems such as biological tissues, micro-electronic devices, thin  lms, thermal coating, metal oxide semiconductors, and DNA origami. In particular, the multi-layered metal thin  lms, gold-coated metal mirrors for example, are often used in high-powered infrared-laser systems to avoid thermal damage at the front surface of a single layer  lm caused by the high-power laser energy. With the development of new materials, functionally graded materials are becoming of more paramount importance than materials having uniform structures. For instance, in semiconductor engineering, structures can be synthesized from di erent polymers, which result in various values of conductivity. Analyzing heat transfer in layered structure is crucial for the optimization of thermal processing of such multi-layered materials. There are many numerical methods dealing with heat conduction in layered structures such as the Immersed Interface Method, the Matched Interface Method, and the Boundary Method. However, development of higher-order accurate stable nite di erence schemes using three grid points across the interface between layers for variable coe cient case is mathematically challenging. Having three grid points ensures that the nite di erence scheme leads to a tridiagonal matrix that can be solved easily using the Thomas Algorithm. But extension of such methods to higher dimensions is very tedious. Recently there have been some solution to such complex systems with the use of neural networks, that can be easily extended to higher dimensions. For the above purposes, in this dissertation, we  rst develop a gradient preserved method for solving heat conduction equations with variable coe cients in double layers. To this end, higher-order compact  nite di erence schemes based on three grid points are developed. The  rst-order spatial derivative is preserved across the interface. Unconditional stability and convergence with O(  2 + h4) are analyzed using the discrete energy method, where   and h are the time step and grid size, respectively. Numerical error and convergence rates are tested in an example. We then present an arti cial neural network (ANN) method for solving the parabolic two-step heat conduction equations in double-layered thin  lms exposed to ultrashort-pulsed lasers. Convergence of the ANN solution to the analytical solution is theoretically analyzed using the energy method. Finally, both developed methods are applied for predicting electron and lattice temperature of a solid thin  lm padding on a chromium  lm exposed to the ultrashort-pulsed lasers. Compared with the existing results, both methods provide accurate solutions that are promising

Effect of Design Parameters and Intercalation Induced Stresses in Lithium Ion Batteries

Electrochemical power sources, especially lithium ion batteries have become major players in various industrial sectors, with applications ranging from low power/energy demands to high power/energy requirements. But there are some significant issues existing for lithium ion systems which include underutilization, stress-induced material damage, capacity fade, and the potential for thermal runaway. Therefore, better design, operation and control of lithium ion batteries are essential to meet the growing demands of energy storage. Physics based modeling and simulation methods provide the best and most accurate approach for addressing such issues for lithium ion battery systems. This work tries to understand and address some of these issues, by development of physics based models and efficient simulation of such models for battery design and real time control purposes. This thesis will introduce a model-based procedure for simultaneous optimization of design parameters for porous electrodes that are commonly used in lithium ion systems. The approach simultaneously optimizes the battery design variables of electrode porosities and thickness for maximization of the energy drawn for an applied current, cut-off voltage, and total time of discharge. The results show reasonable improvement in the specific energy drawn from the lithium ion battery when the design parameters are simultaneously optimized. The second part of this dissertation will develop a 2-dimensional transient numerical model used to simulate the electrochemical lithium insertion in a silicon nanowire (Si NW) electrode. The model geometry is a cylindrical Si NW electrode anchored to a copper current collector (Cu CC) substrate. The model solves for diffusion of lithium in Si NW, stress generation in the Si NW due to chemical and elastic strain, stress generation in the Cu CC due to elastic strain, and volume expansion in the Si NW and Cu CC geometries. The evolution of stress components, i.e., radial, axial and tangential stresses in different regions in the Si NW are studied in details. Lithium-ion batteries are typically modeled using porous electrode theory coupled with various transport and reaction mechanisms with an appropriate discretization or approximation for the solid phase diffusion within the electrode particle. One of the major difficulties in simulating Li-ion battery models is the need for simulating solid-phase diffusion in the second radial dimension r within the particle. It increases the complexity of the model as well as the computation time/cost to a great extent. This is particularly true for the inclusion of pressure induced diffusion inside particles experiencing volume change. Therefore, to address such issues, part of the work will involve development of efficient methods for particle/solid phase reformulation - (1) parabolic profile approach and (2) a mixed order finite difference method. These models will be used for approximating/representing solid-phase concentration variations within the active material. Efficiency in simulation of particle level models can be of great advantage when these are coupled with macro-homogenous cell sandwich level battery models

Boundary-safe PINNs extension: Application to non-linear parabolic PDEs in counterparty credit risk

[Abstract]: The goal of this work is to develop a novel strategy for the treatment of the boundary conditions for multi-dimension nonlinear parabolic PDEs. The proposed methodology allows to get rid of the heuristic choice of the weights for the different addends that appear in the loss function related to the training process. It is based on defining the losses associated to the boundaries by means of the PDEs that arise from substituting the related conditions into the model equation itself. The approach is applied to challenging problems appearing in quantitative finance, namely, in counterparty credit risk management. Further, automatic differentiation is employed to obtain accurate approximation of the partial derivatives, the so called Greeks, that are very relevant quantities in the field.Xunta de Galicia; ED431C 2018/33Xunta de Galicia; ED431G 2019/01A.L and J.A.G.R. acknowledge the support received by the Spanish MINECO under research project number PDI2019-108584RB-I00, and by the Xunta de Galicia, Spain under grant ED431C 2018/33. All the authors thank to the support received from the CITIC research center, funded by Xunta de Galicia and the European Union (European Regional Development Fund - Galicia Program, Spain ), by grant ED431G 2019/01

Structural Design Concepts for a Multi-Megawatt Solar Electric Propulsion (SEP) Spacecraft

As a part of the Space Exploratory Initiative (SEI), NASA-Lewis is studying Solar Electric Propulsion (SEP) spacecraft to be used as a cargo transport vehicle to Mars. Two preliminary structural design concepts are offered for SEP spacecraft: a split blanket array configuration, and a ring structure. The split blanket configuration is an expansion of the photovoltaic solar array design proposed for Space Station Freedom and consists of eight independent solar blankets stretched and supported from a central mast. The ring structural concept is a circular design with the solar blanket stretched inside a ring. This concept uses a central mast with guy wires to provide additional support to the ring. The two design concepts are presented, then compared by performing stability, normal modes, and forced response analyses for varying levels of blanket and guy wire preloads. The ring structure configuration is shown to be advantageous because it is much stiffer, more stable, and deflects less under loading than the split blanket concept

Numerical Methods for Wave Turbulence: Isotropic 3-Wave Kinetic Equations

Wave turbulence theory has remained an active area of research since its inception in the early part of the last century. In the kinetic regime, the main objects of study are the wave kinetic equations. The breakthrough discovery of constant flux, time independent solutions by Zakharov in the late 1960\u27s has allowed for the theories predictions to be verified both experimentally and computationally in a wide array of physical systems. However, there remain many open questions concerning the time dependent solutions of the wave kinetic equations. In this thesis, we aim to partially address this open area of the wave turbulence theory by providing numerical methods for the time dependent solutions of the isotropic 3-wave kinetic equations. The methods we develop herein are able to confirm previous analysis for time dependent solutions, specifically the behavior of the energy cascade of these solutions

Numerical methods in phase-change problems

This paper summarizes the state of the art of the numerical solution of phase-change problems. After describing the governing equations, a review of the existing methods is presented. The emphasis is put mainly on fixed domain techniques, but a brief description of the main front-tracking methods is included. A special section is devoted to the Newton-Raphson resolution with quadratic convergence of the non-linear system of equations