Abstract
In this Chapter we introduce and discuss the concept of scales and provide numerous examples. We begin with the two-scale conductivity problem which is used as a case study throughout the textbook and state the homogenization theorem for this problem in arbitrary dimension. We introduce the reader to rigorous analysis by proving the homogenization theorem for the case study problem in the one dimensional case. This analysis demonstrates that even in the simplest setting one needs to be careful because the first guess for the homogenized problem with coefficients averaged over the microscale turns out to be wrong. The significance of the asymptotic analysis in multiscale problems is demonstrated by some computational examples in dimensions one and two. This Chapter contains also an example of an inverse homogenization problem for thermo-elasticity. We describe mathematical features of this problem in a 1D setting and discuss the effect of negative effective thermal expansion observed in 2D and 3D cases, which is rather counterintuitive.
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