Chapter 12 shows strategies to design hydrogen storage materials (example LiBH4) and Li-ion batteries (example LixMn2O4 spinel cathode) through computations. The first case shows that the dehydrogenation of LiBH4 and the role of catalysts could be understood by first-principles (FP) calculations, thermodynamic modeling, and ab initio molecular dynamics simulations. CALPHAD calculations reveal phase relations and decomposition reactions for the targeted systems. Further understanding of LiBH4 decomposition is generated by FP calculations associated with formation and migration of lattice point defects. The second case aims at understanding the performance of Li-ion batteries from a comprehensive composition-structure-property relationship. The key factors (energy density, cyclability and safety) determining the performance of the battery can be evaluated by cell voltage, capacity, electrochemical stability, extent of Jahn-Teller distortion, thermodynamic stability, and extent of oxygen gas release. All these properties are obtained by combining FP calculations with CALPHAD calculations.

The basics of atomistic simulation methods, density functional theory and molecular dynamics, are first presented in Chapter 2. Then we demonstrate how to calculate some basic materials properties (including lattice parameter, thermodynamic properties, elastic properties, and defect properties) through first-principles (FP) methods. Because of the remarkable accuracy in predicting such physical and chemical properties of materials, FP is widely used in computational materials science. Finally, we take the design of Mg–Li alloys for ultralightweight application as an example to show the important role of atomistic simulation methods in material design.

The advance of human civilization with materials development from the Stone Age to the Information Age is the starting point of Chapter 1, highlighting significant roles of computational design of materials. Important terms (model, simulation, database, and materials design) used in computational materials science are defined. The past and present development of computational design of materials is then introduced. A few milestones for alloy design, such as the Hume–Rothery rule, the Phase Computation (PHACOMP) method, and the calculation of phase diagrams (CALPHAD) approach, are highlighted. The past two-decade focus on three aspects in computational design of materials (multiscale/multilevel modeling methodologies, simulation software, and scientific database) in the core of the Materials Genome Initiative is emphasized. A general framework of materials design is demonstrated with two flowcharts: through-process simulation of Al alloys during heat treatment, and the three stages for the development of engineering materials. The two-part structure of the book – fundamentals and case studies – is explained.

Chapter 13 starts with brief summary of Chapters 1–12. Subsequently, to show that the strategy described in this book is valid for design of other materials, computational designs for other four materials (Mo2BC thin film, Cu3Sn interconnect material, slag/metal/gas LD-converter steel process, and slag recycling) were highlighted. In view of the need for establishing more quantitative relationships among four cornerstones (composition/processing-structure–properties–performance) in materials science and engineering as well as advancing product design methods, several future orientations and challenges for computational design of engineering materials are suggested. These are (1) advancement of models and approaches for more quantitative simulation in materials design, such as interfacial thermodynamics, thermodynamics under external fields, and a more quantitative phase-field model; (2) the need for scientific databases and materials informatics; (3) enhanced simulation software packages; and (4) concurrent design of materials and products (CDMP). Finally, the correlations among ICME, MGI, and CDMP are discussed.

In Chapter 4, firstly a few basic terms (object and configuration, stress, strain, and constitutive relation between stress tensor and strain tensor), three coordinate systems (shape coordinate, lattice coordinate, and laboratory coordinate), deformation gradient as well as fundamental equations in continuum mechanics are briefly recalled for the sake of understanding fundamental equations of the crystal plasticity finite element method (CPFEM). A few advantages of CPFEM (including its abilities to analyze multiparticle problems and solve crystal mechanics problems with complex boundary conditions) are highlighted. Then, representative mechanical constitutive laws of crystal plasticity including dislocation-based constitutive models and constitutive models for displacive transformation are briefly described, followed by a short introduction to the finite element method (FEM), several FEM software packages (including Adina, ABAQUS, Deform, and ANSYS) and a procedure for CPFEM simulation. Finally, a case study of plastic deformation-induced surface roughening in Al polycrystals is demonstrated to show important features of crystal plasticity finite element method in materials design.

Chapter 7 briefly introduces steels, including classification, production processes, microstructure, and properties as well as computational tools for design of steels. Two case studies for S53 and AISI H13 steels are demonstrated. For S53 steel, high strength and good corrosion resistance are needed. For that purpose, plots of thermodynamic driving forces for precipitates were established, guaranteeing the accurate precipitation of M2C strengthener in steels. In addition, a martensite model is developed, designing maximal strengthening effect and appropriate martensite start temperature to maintain an alloy with lath martensite as the matrix. The corrosion resistance was designed by analyzing thermodynamic effects to maximize Cr partitioning in spinel oxide and enhance the grain boundary cohesion. In the case of AISI H13 steel, precipitations of carbides were simulated. Then simulated microstructure was coupled with structure–property models to predict the stress–strain curve and creep properties. Subsequently, those simulated properties were coupled with FEM to predict the relaxation of internal stresses and deformation behavior at the macroscopic scale during tempering of AISI H13

Introduction

The main objective of this chapter is familiarization with a variety of numerical methods that are essential for solving advanced problems of applied physics and engineering. With the help of suitable examples, basic skills on appropriately using these methods for various applications in physics are provided.

The chapter focuses on the following special second order differential equations, which are known to have standard functional form and/or analytical solutions.

The solutions of these equations are referred to as ‘special functions’, which are significantly different from standard functions like sine/cosine, exponential and logarithmic functions. This chapter also describes the use of quadrature methods of integration for calculating improper integrals, which are either infinite in the interval of integration, or the interval of integration has an infinite bound. The quadrature methods discussed in this chapter are as follows:

The chapter has been written in a manner so as to develop the necessary skills of the reader to evaluate certain integrals that are generally not discussed in introductory physics classes because they involve advanced calculations.

Bessel Function of the First Kind

Bessel functions have several applications in physics. They arise while solving Laplace's and Helmholtz equations in spherical and cylindrical coordinates. The functions are also useful while solving problems based on electromagnetic wave propagation and Schrödinger's equation.

Introduction

Differential equations are key mathematical tools for modelling physics problems. They are frequently used in all branches of physics while expressing the variation in one quantity w.r.t. the other. There are various kinds of differential equations, such as:

This chapter introduces the necessary numerical tools for determining approximate solutions of ordinary differential equations. It focuses only on initial and boundary value problems involving first and second order ordinary linear differential equations. These equations contain functions of one independent variable, and derivatives in that variable.

There are several numerical techniques for determining the solutions of differential equations. In this chapter, some commonly used methods have been explained. Section 4.2 shows the use of Euler's method to determine the solution of a differential equation. This is followed by modified Euler's method in Section 4.3, Runge–Kutta second order method in Section 4.4 and Runge–Kutta fourth order method in Section 4.5. A graphical comparison of these four methods is presented in Section 4.6. In Section 4.7, a quick review of the finite difference method has been provided for second order boundary value problems. Some advanced application problems of physics involving the first and second order differential equations have been discussed in Section 4.8.

This chapter uses the plotting skills developed in the second chapter. The reader is encouraged to refine their understanding of the plotting techniques.

Euler's Method

This is the most basic method of numerical integration. It is a first order method for approximating solutions of differential equations. This method uses the initial value as the starting point and approximates the next point of the solution curve using a tangent line to that point. The accuracy crucially depends on the step size used to approximate the subsequent point on the solution curve.

The algorithm for writing a Scilab program based on Euler's method is explained in Section 4.2.1 (first order) and in Section 4.2.2 (second order) with the help of suitable examples.

Appendix

It has rightly been said that the mathematical theory of groups and group representations is a magnificent gift of nineteenth century mathematics to twentieth century physics. While this is particularly true within the framework of quantum mechanics, with the passage of time its relevance within classical physics has also become well understood and greatly appreciated. Today the importance of group theoretical ideas and methods for physics can hardly be overemphasised; and over the past century or so, a veritable profusion of books devoted to this theme, many of them gems of the literature, have appeared.

The present monograph is primarily based on lectures given by one of us (NM) at the Institute of Mathematical Sciences in Chennai, India, in the Fall of 2007. The lectures were prepared and presented at the invitation of Rajiah Simon, to whom both authors are indebted for his support and encouragement.

The course was titled ‘Continuous Groups for Physicists’ and consisted of about 45 extended lectures over a two month period. Its aim was to introduce the basic ideas of continuous groups and some of their applications to an audience of post graduate and doctoral students in theoretical physics. After an introduction to the basic ideas of groups and group representations (mainly in the context of finite groups and compact Lie groups), the course presented a selection of useful, interesting and quite sophisticated specific topics not often included in standard courses in physics curricula. The methods and concepts of quantum mechanics served as a backdrop for all the lectures.

The real rotation groups in two and three dimensions are followed by an account of the structures of Lie groups and Lie algebras, and then a description of the compact simple Lie groups. Their irreducible representations are described in some detail. Some of the ‘non standard’ topics that follow are: spinor representations of real orthogonal groups in both even and odd dimensions; the notion of the ‘Schwinger’ representation of a group with examples, induced representations, and systems of generalised coherent states; the properties and uses of the real symplectic groups, which are defined only in real even dimensions, and their metaplectic covering group, in a quantum mechanical setting; and the Wigner Theorem on the representation of symmetry operations in quantum mechanics.