Multiscale Modelling and Optimisation of Materials and Structures. Tadeusz Burczynski
different time regimes. Usually, the temporal scale is unified across the different length scales to provide physically relevant results, which sometimes makes the computational model expensive.
Figure 1.1 Multiscale concept diagram which illustrates ‘macro’, ‘meso’, ‘micro’, and ‘nano’ scales.
The lack of a definition for the term multiscale in the numerical modelling community is another problem. The broad understanding of multiscale modelling includes all problems with important features in multiple scales (temporal and spatial) [15]. Stricter definitions are also used. In the narrow sense of the term, modelling is multiscale when the models at different scales are simultaneously considered [13]. There is a fundamental difference between these two definitions. Important features at more than one scale are present in almost all problems of material science. A microstructural base of work hardening is one of the most common examples. Hardening is a process at a clearly different scale than plastic deformation modelled with a continuity assumption. Hence we can describe the simulation of plastic deformation and work hardening as the multiscale model. One can also define the multiscale model as a system of submodels at two or more scales but not necessarily simultaneously computed. Some examples are discussed in the book. Two submodels must work simultaneously when both are dependent on results from the second one. Such models are two‐way coupled, contrary to one‐way coupling where one of the models is independent.
The range of possible applications of multiscale modelling is extensive. It includes such different topics as social issues, economy, ecosystems, weather forecast, and physical and chemical processes occurring in gases, fluids, and solids. Material models are an essential branch of multiscale modelling, combining physical and chemical processes, mainly in solids. Such a wide variety of topics makes revising all multiscale modelling applications impossible. Even in material science, several families of applications can be distinguished. The main areas are modelling of polymers, composites, solidification, thin layers, or polycrystalline materials. Each of these areas has its own characteristic features, distinguishing them from the others. While the bases are common for all areas, possible solutions to particular problems are different.
In this book, applications of multiscale modelling to processes occurring in polycrystalline materials, mainly metals and their alloys, are reviewed and addressed. The most distinguishing feature of metals and their alloys is their unrepeatable microstructure. However, the atomistic structure of crystals is regular, while properties of these materials depend mostly on imperfections of atomic structures (dislocations, solid solutions of atoms, grain boundaries, etc.). The mesoscale picture of the material structures includes disordered (usually) distribution of grain shapes, sizes and orientations, as well as strong anisotropy of single crystals and grain boundaries. It is a significant difference in comparison with composites and polymers (with its repeatable microstructure). Then, the difference between the modelling of crystallization and thin layers deposition lies within lack or a very small amount of fluids. It must be remembered that modelling of fluids flow is governed by different types of equations. Moreover, numerical techniques for microscale models and coupling of scales are different in the presence of fluids. Due to this diversity in materials and different techniques used for their description, the book focuses mostly on mentioned metals and their alloys to provide in‐depth information on the practical application of dedicated multiscale modelling and optimization techniques.
1.1.2 Review of Problems Connected with Multiscale Modelling Techniques
As mentioned, in the field of metallic materials, several classifications of multiscale models exist. They depend on such issues as the character of coupling between the scales or methodology of data transformation between the scales. Coupling between the scales can be either strong or weak. Strong coupling combines a description of both (or more) scales into one equations system. In weak coupling, only some data are transferred between the subsequent scales. In the fine scale, the boundary or initial conditions can be developed based on coarse scale data (top‐down approach/downscaling), while data from the fine scale can be used as material properties or a material state in the coarse scale (bottom‐up approach/upscaling). Weakly coupled models have a relatively flexible structure and can be classified into top‐down and bottom‐up approaches.
The strength of coupling has some significant consequences. The strongly coupled models are usually faster and have a better mathematical and theoretical background. Usually, such models are solved with a single numerical method. However, phenomena in all scales must be described with consistent mathematical formulations (usually partial differential equations), which is rather hard or even not possible for some phenomena characterized, e.g.by stochastic behaviour. Moreover, the strongly coupled submodels cannot be separated, and their parts cannot be replaced with other submodels, which make their adaptation difficult. The weakly coupled models are more flexible, from both mathematical and numerical points of view, which makes their development and adaptation much easier. The coupling strength is linked with a methodology of data transfer between the scales.
Various classifications of the multiscale modelling methods were proposed having in mind all the aforementioned features of these techniques. Since this book is dedicated to simulations of materials processing methods, the multiscale models are classified into two groups: upscaling and concurrent approaches. In the upscaling class of methods, constitutive models at higher scales are constructed from observations and models at lower, more elementary scales [2]. By a sophisticated interaction between experimental observations at different scales and numerical solutions of constitutive models at increasingly larger scales, physically based models and their parameters can be derived at the macroscale. In this approach, it is natural that the microscale problem has to be solved at several locations in the macro‐model. For example, in the application of the concept of computational homogenization for each integration point of finite elements in the macroscale, the representative volume element (RVE) is considered in the microscale (Figure 1.2).
Figure 1.2 Illustration of computational homogenization concept.
Figure 1.3(a) Internal variable method as a precursor of upscaling and (b) the idea of the CAFE model.
Thus, classification of the multiscale problems with respect to the upscaling goals and the upscaling costs is needed, which is one of the book's objectives. The idea of upscaling has, in an intuitive manner, been used in engineering for decades. A commonly used approach based on a calculation of the flow stress in the FE model by a solution of the differential equation for the evolution of dislocation populations (Figure 1.3a) can be considered the precursor of upscaling.
Significant progress was made when more advanced discrete methods were used in the microscale; see Figure 1.3b, where the CAFE method (cellular automata finite element) is shown as an example. Now we witness further development of multiscale methods in computational science, which couple fine and coarse scales more systematically. A variety of models and methods is used in both microscale/nanoscale and macroscale. The models are generally different for the two scales, but solutions using the same model in the coarse and fine scales are