an approach [3].
In concurrent multiscale computing, one strives to solve the problem simultaneously at several scales (in practice two‐scales) by an a priori decomposition of the computational domain. Large‐scale problems are solved, and local data (e.g., displacements) are used as boundary conditions for a more detailed part of the problem. The question of how the fine scale is coupled to the coarse scale is essential in this approach. The major difficulty in coupling occurs when different models describe fine scales and coarse scales, for example coupling FE to MD. In other words, the objective is to find a computationally inexpensive but still accurate approach to the decomposition problem. The terms ‘homogenization’ and ‘localization’ are commonly used for operations during data transfer. Again, this terminology is not well defined. Homogenization, in multiscale modelling, has two meanings. In the context of strongly coupled models, based on differential equations, it is understood as the asymptotic homogenization, the method of studying partial differential equations with rapidly oscillating coefficients. The in‐depth description of the homogenization and the averaging based on the perturbation theory, as well as their application to multiscale modelling, can be found e.g.in [12]. However, most authors use this term differently, describing it as ‘any method of transforming a heterogeneous field into a homogeneous one’. Weighted averaging is usually used; however, more advanced techniques are also present. In this book, we use the term homogenization in common understanding and asymptotic homogenization in the sense of the element of perturbation theory. Details on the multiscale classification and examples of its applications can be found in Chapter 5 of the book.
1.1.3 Prospective Applications of the Multiscale Modelling
In recent years, a gradual paradigm shift has been taking place in the selection of materials to suit particular engineering requirements, especially in high‐performance applications. The empirical approach adopted by materials scientists and engineers in choosing materials parameters from a database is being replaced by design based on the DMR concept. Features that span across a large spectrum of length scales are altered and controlled to achieve the desired properties and performance at the macroscale. Research efforts, in this aspect, include the development of engineering materials by changing the composition, morphology, and topology of their constituents at the microscopic/mesoscopic level. The objective of this book is to show multiscale methods and their applications in computational materials design. From one side, we present computational multiscale material modelling based on the bottom‐up/top‐down, one‐way coupled description of the material structure in different representative scales. On the other side, our intention is to show possibilities of a combination of multiscale methods with optimization techniques.
1.2 Optimization
Solution of optimization problems in multiscale modelling allows finding structures with better performance or strength in one scale with respect to design variables in another scale. In this case, the typical situation is to find a vector of material or geometrical parameters on the micro‐level, which minimizes an objective function dependent on state fields on a macro‐level of the structure.
A special case of optimization problems associated with the multiscale approach is the optimization of atomic clusters for the minimization of the system's potential energy. This case has an important consequence, especially in the design of new 2D nanomaterials and nanostructures.
The identification problem is formulated as the evaluation of some geometrical or material parameters of structures in one scale having measured information in another scale. The important case of identification in multiscale modelling is to find material properties, the shape of the inclusions/fibres or voids in the microstructure having measurements of state fields made on the macro‐object. The identification problem is formulated and considered as a special optimization task.
The analysis methods of multiscale models based on computational homogenization are adopted for these classes of problems. To solve optimization and identification problems, global optimization methods based on bioinspired algorithms are used.
1.3 Contents of the Book
The aspects of multiscale modelling mentioned previously have already been discussed in numerous publications, including several books [4, 12, 13, 15]. However, multiscale modelling still remains a difficult task, and its valid and reliable application is quite difficult. Moreover, the lack of an unambiguous definition leads to misunderstandings and mistakes. The book supplies some practical information concerning the development and application of multiscale material models, in particular in combination with optimization techniques.
As mentioned, the book is divided into three main sections. The first section is composed of Chapters 2 and 3, and it is focused on discussion of phenomena occurring in materials in processing and on models used to describe these phenomena. Such phenomena as recrystallization, phase transformations, cracking, fatigue, and creep are discussed very briefly. The modelling methods are divided into two groups. The first includes computational methods for continuum such as FEM, XFEM, and BEM. Discrete methods describing microscale and nanoscale phenomena include MS as well as MD, CA, and MC approaches. Computational homogenization methods, which are used in the coupled multiscale models, are also discussed in this part of the book. Basic principles of methods of optimization are also described in this part of the book, with a particular emphasis on the methods inspired by nature.
The second section of the book is composed of Chapters 4 and 5 and focuses on DMR. Case studies based on DMR are presented. Applications of methods described in Chapters 2 and 3 to modelling various processes and phenomena are shown.
The third section of the book is connected with applications of multiscale optimization methods. Such issues as optimization of atomic clusters, material parameters optimization, shape optimization, and topological optimization are discussed. The problem of identification in multiscale modelling is also presented.
Computer implementation issues for multiscale models are recapitulated in the book. Implementation of selected algorithms concerning visualization as well as scales coupling with use of commercial software are described in Chapter 7. The part will highlight the possibilities of increasing the computational efficiency, which is especially important when optimization based on a direct problem model of multiscale nature is considered.
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