Chapter 4 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. An extensive section discusses the conditioning of linear-algebra problems: borrowing ideas and examples from matrix perturbation theory, this teaches students what to look out for. Roughly half of the remaining chapter is dedicated to the solution of linear systems of equations, employing methods of varying sophistication: Gaussian elimination, LU decomposiion, pivoting, and the Jacobi iterative method. The second half addresses the eigenvalue problem, again with a variety of methods, including the power method, eigenvalue shifting, and the QR method. Crucially, this includes explicit mathematical derivations of these approaches. A brief introduction to the singular value decomposition is also given, including both an existence proof and a programming implementation. The chapter is rounded out by an extensive physics project, which studies the eigenvalue problem of interacting spins, and a problem set. The physics project patiently builds up matrix quantum mechanics, allowing students to tackle problems of increasing difficulty.

After a discussion of best programming practices and a brief summary of basic features of the Python programming language, chapter 1 discusses several modern idioms. These include the use of list comprehensions, dictionaries, the for-else idiom, as well as other ways to iterate Pythonically. Throughout, the focus is on programming in a way which feels natural, i.e., working with the language (as opposed to working against the language). The chapter also includes basic information on how to make figures using Matplotlib, as well as advice on how to effectively use the NumPy library, with an emphasis on slicing, vectorization, and broadcasting. The chapter is rounded out by a physics project, which studies the visualization of electric fields, and a problem set.

Chapter 5 focuses on the CALPHAD approach and its thermodynamic basis with the crucial concept of “phase." The origins, development, and principles of the CALPHAD method are briefly explained and current software is compiled (Thermo-Calc, Pandat, FactSage, and more). Thermodynamic modeling of Gibbs energy is introduced, from simple pure substances to complex solution phases. Examples of how to establish a thermodynamic database are given, and key issues on the consistency, coherency, quality assurance, and safety of the database are emphasized. The most important application examples in the computational design of alloys and their processing are separated in two levels. In the first level, solely thermodynamic CALPHAD databases are required. It is shown which type of calculations have proved most useful to guide design. In the second level, applications using extended CALPHAD-type databases with kinetic and thermophysical material parameters are outlined for casting, solidification, and heat treatment processes. The use of advanced CALPHAD-type software packages is demonstrated. Finally, a case study on design of Al alloys with improved hot cracking resistance is presented with these tools.

Chapter 10 starts with category and production processes of cemented carbides. Subsequently, case studies for three cemented carbides are demonstrated. In the case of ultrafine cemented carbide, thermodynamic calculations were utilized to select composition and sintering temperature to avoid segregation of the (Ta,W)C phase. Optimal mechanical properties were obtained via adding VC and Cr3C2 inhibitors and the selected sintering temperature and composition. For WC–Co–Ni–Al cemented carbides, calculated phase diagrams and interfacial energy were employed to optimize the composition of Co–Ni–Al binder phase and sintering temperature. The morphology of WC was controlled through phase-field simulation and microstructure characterization. The best trade-off between transverse rupture strength and Rockwell hardness is obtained accordingly. For gradient cemented carbides, thermodynamic and diffusion calculations were performed to select composition and sintering schedule to provide microstructure parameters. A microstructure-based model was then developed to predict the hardness distribution. This simulation-driven materials design leads to development of these products within three years.

In Chapter 3, we mainly focus on the fundamentals of typical mesoscale simulation methods, which can provide a bridge between atomistic structures and macroscopic properties of materials. Among many mesoscale simulation methods, the phase-field and cellular automaton methods are extremely popular and powerful for simulating microstructure evolution. Consequently, we first give a detailed introduction on the fundamentals of the two methods, briefly describing some other mesoscale simulation methods, such as level set and front tracking. After that, application examples using individual mesoscale simulation methods and integrations of the phase-field method with other simulation methods such as atomistic simulation, crystal plasticity, CALPHAD, and machine learning are described in detail. Finally, a case study for design of high-energy-density polymer nanocomposites using the phase-field method is very briefly presented.

Chapter 8 focuses on the design of important Al- and Mg-based light alloys. Selected examples show how CALPHAD simulation tools can be used to understand and predict the effect of alloying elements and processing conditions on alloy properties and how to use that in the design of alloys. For Al alloys, two case study examples using the extended CALPHAD-type databases are demonstrated. For cast alloy A356 (Al–Si,Mg), the solidification simulation involving dedicated microsegregation modeling is presented. For the wrought alloy 7xxx (Al–Zn,Mg/Cu), elaborate heat treatment simulation with precipitation kinetics is the design tool. For Mg alloy structural components, simulations of solidification path and T6 heat treatment of AZ series (Mg–AlZn) and the development of Mg–Al–Sn-based (AT) cast alloys involving also microsegregation simulation are demonstrated. Finally, the design of biomedical Mg alloy implants utilizing the CALPHAD method and the state-of-the-art bioresorbable Mg alloy stent to cure coronary artery disease is presented.

In Chapter 11, first an introduction to cutting tools is presented, followed by case studies for two hard coatings. For the TiAlN PVD coating case, we describe how to adjust the formation of metastable phase, select the deposition temperature, and manipulate microstructure to obtain desired mechanical properties through first-principles calculations and thermodynamic calculations. The deposition of the TiAlN/TiN and TiAlN/ZrN multilayer guided by first-principles calculations is also briefly mentioned. For the TiCN CVD coating, we demonstrate that computed CVD phase diagrams can accurately describe phases and their compositions under the given temperature, total pressure, and pressures of various gases. Subsequently, computational fluid dynamics (CFD) is used to provide temperature field, velocity, and distributions of various gases inside the CVD reactor. From that information, calculations-designed experiments were conducted and TiCN coatings were deposited highly efficiently. These simulation-driven designs for the hard coatings have found industrial applications in just two years, much quicker compared to the costly experimental approach.

Chapter 9 focuses on superalloys operating at high temperature where high strength as well as creep and corrosion resistance are demanded. We take Ni-based single-crystal superalloys and Ni–Fe-based superalloys for advanced ultrasupercritical (A-USC) power plants as examples to demonstrate how alloy design is accomplished in these multicomponent alloy systems. The first case study introduces the design procedure of Ni-based single-crystal superalloy by using a multicriterion constrained multistart optimization algorithm. In the second case study, the design procedure of an Ni–Fe-based superalloy with the artificial neural network (ANN) model combined with a genetic algorithm (GA) based on an experimental dataset is presented.

Chapter 6 starts with a definition of thermophysical properties, followed by detailed descriptions of important terms and equations in diffusion, including Fick’s laws on diffusion; four types of diffusion coefficients (self-diffusion, impurity diffusion, intrinsic diffusion, and interdiffusion); atomic mechanisms of diffusion; diffusion equations in binary, ternary, and multicomponent phases; as well as phases with narrow homogeneity range. Short-circuit diffusion is also briefly mentioned. Subsequently, several computational methods, including first-principles calculations, MD simulation, semi-empirical approaches, and DICTRA software, are presented to calculate or estimate diffusivity and atomic mobilities from which various diffusivities can be computed. Modeling of selected important thermophysical properties, including interfacial energy, viscosity, volume, and thermal conductivity, is briefly introduced. A procedure to establish thermophysical databases is described from a materials design point of view. A case study for simulating age hardening in AA6005 Al alloys is demonstrated mainly using thermophysical properties as input to show their importance for materials design.