Multiple Scales: Many physical phenomena and engineering systems exhibit behavior at multiple spatial scales. Examples include composite materials with microscopic fibers, porous media with intricate pore networks, and turbulent flows with eddies of various sizes.
Separable Scales: Scale separation occurs when these multiple scales are distinct and can be separated. This means that the behavior at one scale doesn't strongly influence the behavior at another scale.
Illustrative Examples
The description provides three examples of functions with scale separation:
- Two-Scale Function:
- $2+x \cos \left(\frac{x}{\epsilon}\right)$
- This function has a slowly varying amplitude (scale 1) and a rapidly oscillating component (scale $\epsilon$).
- The parameter ϵ controls the separation between the scales; smaller $\epsilon$ means stronger separation.
- Multiple-Scale Function (Same Order):
- $2+x+\sum_{i=1}^{10} \cos \left(\frac{x r_i}{\epsilon}\right)$
- This function has multiple oscillatory components with different frequencies (due to the random numbers $r_i$), but all these components have the same scale $\epsilon$.
- Multiple-Scale Function (Different Orders):
- $2+x+\sum_{i=1}^{10} \cos \left(\frac{x r_i}{\epsilon}\right)+\sum_{i=11}^{20} \cos \left(\frac{\pi r_i}{\epsilon^i}\right)$
- This function has oscillatory components at two distinct scales, $\epsilon$ and $\epsilon^i$, demonstrating even stronger scale separation.
Visual Representation (Figure 1)
- The left plot in Figure 1 likely depicts the first example (two-scale function). You should see a smooth curve with a superimposed high-frequency oscillation.
- The right plot in Figure 1 likely depicts the second example (multiple scales of the same order). You should see a more complex pattern with multiple overlapping oscillations.
Multidimensional Examples and Figure 2
- The description mentions that similar scale separation can occur in multidimensional fields.
- Figure 2 likely shows an example of a material with high-conductivity inclusions dispersed in a matrix. This represents a case with scale separation, where the inclusions have a much smaller scale than the overall material.
https://gist.github.com/viadean/e357bc316e04a425bcbcb60335e42560
Figure 2
Figure 1