Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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
Aurialt, J.L., Boutin, C., Geindreau, C.: Homogenization of Coupled Phenomena in Heterogeneous Media. Wiley, New York (2009)
Ávila, A., Griso, G., Miara, B., Rohan, E.: Multiscale modeling of elastic waves: theoretical justification and numerical simulation of band gaps. SIAM Multiscale Model. Simul. 7, 1–21 (2008)
Baffico, L., Grandmont, C., Maday, Y., Osses, A.: Homogenization of elastic media with gaseous inclusions. SIAM Multiscale Model. Simul. 7, 432–465 (2008)
Caiazzo, A., Mura, J.: Multiscale modeling of weakly compressible elastic materials in the harmonic regime and applications to microscale structure estimation. SIAM J. Multiscale Model. Simul. 12 (2), 514–537 (2014)
Cioranescu, D., Piatnitski, A.: Homogenization in perforated domains with rapidly pulsing perforations. ESAIM Control Optim. Calc. Var. 9, 461–483 (2003)
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)
Gutiérrez, S., Mura, J.: Small amplitude homogenization applied to inverse problems. Comp. Mech. 41, 699–706 (2008)
Gutiérrez, S., Mura, J.: An adaptive procedure for inverse problems in elasticity. Comptes Rendus Mécanique 338, 402–411 (2010)
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20 (3–4), 251–265, 65Y15 (2012)
Hornung, U.: Homogenization and Porous Media. Springer, New York, (1997)
Sanchez-Palencia, E.: Non-homogeneous Media and Vibration Theory. Springer, Berlin (1980)
Sanchez-Palencia, E., Zaoui, A.: Homogenization Techniques for Composite Media. Lecture Notes in Physics, vol. 272. Springer, Berlin (1987)
Tartar, L.: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinb. 115 (3–4), 193–230 (1990)
Acknowledgements
The research of J. Mura has been supported by the Fondecyt-Initiation to Research project no. 11121606 (Conicyt/Chile).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Mura, J., Caiazzo, A. (2016). A Two-Scale Homogenization Approach for the Estimation of Porosity in Elastic Media. In: Ortegón Gallego, F., Redondo Neble, M., Rodríguez Galván, J. (eds) Trends in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-32013-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-32013-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32012-0
Online ISBN: 978-3-319-32013-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)