Similarity measures

The cost functions driving the optimization of the parameters are built by accounting for the discrepancy between the numerical predictions and the measured (experimental) data. Traditionally, this discrepancy is often quantified using a sum-of-square based cost function. However, in the present case this is not possible. One problem is a potentially different number of data points on the experimental curve compared to the numerical curve. Admittedly, this circumstance could be remedied by linear interpolation - provided there are enough data points to keep the introduced error low. However, in many cases there is no unique relationship between stress and strain - the material behavior is path dependent.

Discrete Fréchet distance

The Fréchet distance, is a measure of similarity between two curves in Euclidean space and is defined as the minimum length of a path connecting two polygonal lines in a way that preserves the ordering of the vertices. It thus represents the shortest possible path that connects two polygonal lines while maintaining their relative orientation. Our implementation is based on the work of [1] and uses dynamic programming. The order of execution time is $O(mn)$ where $m$ and $n$ are the number of points in each of the two polygonal lines to be compared.

To use the discrete Fréchet distance to quantify the discrepancy between simulation and experiment choose the following setting in the class ACTglobals:

ACTglobals.setup(
  ...
  Similarity = 'frechet',
  ...
  )

References

[1] T. Devogele, L. Etienne, M. Esnault, and F. Lardy, ‘Optimized Discrete Fréchet Distance between trajectories’, in Proceedings of the 6th ACM SIGSPATIAL Workshop on Analytics for Big Geospatial Data, Redondo Beach CA USA, Nov. 2017, pp. 11–19. doi: 10.1145/3150919.3150924.