A Two-Scale Homogenization Approach for the Estimation of Porosity in Elastic Media
We propose a novel method for estimating the porosity of an elastic medium starting from inner displacement measurement, such as the ones that can be obtained from seismogram data for the study of soils or from magnetic resonance elastography for the diagnosis of tissue diseases. The approach is based on a two-scale homogenization, which relates geometrical characteristics of the void-elastic solid mixture at the small (mesoscopic) scale of the pore with an effective elasticity tensor at the large (macroscopic) scale of the effective material. Through semi-analytical approximations of the homogenized equations, the idea can be further extended considering slight variations in the shape of the pore. This procedure leads eventually to an inverse problem formulation that enable us to recover approximately the porosity field by means of the finite element formulation of the effective macroscale problem only. We validate the multiscale approximation and the two-scale porosity estimation method with numerical examples.
KeywordsAdjoint Problem Magnetic Resonance Elastography Cell Problem Small Porosity Numerical Interpolation
The research of J. Mura has been supported by the Fondecyt-Initiation to Research project no. 11121606 (Conicyt/Chile).
- 1.Auliac, S., Le Hyaric, A., Morice, J., Hecht, F., Ohtsuka, K., Pironneau, O.: FreeFem++, 3rd edn. Version 3.31–2 (2014). http://www.freefem.org/ff++/ftp/freefem++doc.pdf
- 7.Efendiev, Y., Hou, T.: Multiscale Finite Element method, Theory and Applications. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4. Springer, New York (2009)Google Scholar
- 10.Hecht, F.: New development in FreeFem++. J. Numer. Math. 20 (3–4), 251–265, 65Y15 (2012)Google Scholar
- 13.Sanchez-Palencia, E., Zaoui, A.: Homogenization Techniques for Composite Media. Lecture Notes in Physics, vol. 272. Springer, Berlin (1987)Google Scholar