[For French edition see the authors, Modélisation numérique en science et génie des matériaux. Traité des Nouvelles Technologies. Série Matériaux. 10. Lausanne: Press Polytechniques et Universitaires Romandes (1998; Zbl 0914.73004).]
With the advent of technological revolution in the second half of the twentieth century, more and more accurate and refined numerical and computational methods have been discovered and employed by mathematicians, scientists and engineers in order to solve systems of complicated linear and nonlinear algebraic and differential equations. Indeed, the last three decades have produced major advances in numerical methods and their applications in the fields of materials science and engineering. Special attention has been given to more diversification and sophistication of numerical techniques, including the finite difference method (FDM), the finite element method (FDM), the Monte Carlo method, the boundary element method (BEM), stochastic methods, atomistic methods related to molecular dynamics, and homogenization methods based on average conservation laws.
This book is essentially devoted to numerical simulation and modeling in engineering and materials science. It has ten chapters and an index. The first chapter, “Continuous media”, is a summary without proofs of some of the basic equations of conservation of mass, momentum, energy, and solute. Included are the most important constitutive equations together with boundary conditions describing the exchange of mass, momentum or energy between the domain under consideration and the external world. The second chapter on the finite element method deals with numerical solutions of a simple one-dimensional problem of solute diffusion, with or without advective transport, for both stationary and nonstationary cases. The FDM is certainly the most simplest from the point of view of mathematical formalism and implementation in a programming language.
Chapter three is concerned with the fundamentals and with the basic principles of geometric discretization of space in the finite element method. The FEM is a powerful and useful method to solve differential equations of physical problems in integral forms. This integral form can be functional that results from a variational formulation, the minimum of which corresponds to the desired solution, as in elasticity or viscoelasticity, or, more generally, an integral equation. The integral form can also be obtained from an initial system of partial differential equations by a weak formulation, also called the weighted residual method. This formalism is also applied to stationary and nonstationary scalar diffusion problems, and then to a vector problem of mechanical equilibrium. However, the convergence properties of numerical solution and the associated error estimates are not included in this chapter. In chapter 4, authors consider more practical aspects of FDM and FEM methods. In particular, algorithms for mesh generations, storage of matrices in memory, basic methods for solution of linear and nonlinear systems are described in some detail. Phase transformations that are one of the fundamental aspects of materials science and technology, constitute the subject matter of chapter 5. The problems of phase transformations include the solidification of metal alloys and their solid state transformations, the processing of iron in a blast furnace or of aluminum in an electrolysis cell starting from their respective minerals, the decarburizing of steels, the production by a flux method of high melting point single crystals, the precipitation of ceramic powders starting from aqueous solutions, and corrosion or oxidation reactions on a surface. The authors successfully cover various aspects of the modeling of phase transformations, including different procedures that can be applied to the modeling of processes and to the calculation of microstructures.
The next two chapters deal with deformation of solid materials and incompressible fluid flows. Chapter 6 gives special attention to irreversible deformations of solid state metals and polymers, basic constitutive laws, and the property of some materials to undergo permanent deformation without breaking. Included are the boundary conditions characteristic of plastic deformation problem, the modeling of contact and friction, and numerical treatment of the equations to be solved. Chapter 7 is devoted to numerical analysis of mathematical models of fluid flows which arise in molten metals in steel industry, casting, the production of crystalline semiconductors, molten glass in float glass processes, polymers, plastics for the fabrication of films, and to processing of fibers, paints, and foods. Chapter 8 is concerned with inverse methods with simple examples. In fact, these deal with numerical techniques for the determination of properties or boundary conditions based on experimental results. Finally, chapter 9 introduces stochastic Monte Carlo methods, random walkers and cellular automata, and discusses some applications in materials science.
The following comments are in order. First, this book is written to meet the needs of a textbook at the undergraduate and graduate levels in materials science and engineering. It may also be useful for researchers and engineering professionals who are interested in numerical methods, numerical simulation and modeling in engineering and materials science. Second, there is a set of small number of exercises and a bibliography at the end of each chapter. However, there are no worked examples and no longer projects whose solution may involve searches in the library and deeper study. Third, the book does not contain any wrong or misleading information. Summarizing, this book is an important addition to the literature on material science.