An Explicit Formula for Computing the Sensitivity of the Effective Conductivity of Heterogeneous Composite Materials to Local Inclusion Transport Properties and Geometry

Abstract

Most natural or synthetic materials possess a multiscale structure that can have a major impact on large scale transport properties. In particular, the effective conductivity associated to thermal, electrical, or hydraulic transfers depends on its microstructure. In practice, it can be determined using some analytical formula working in limited configurations. More generally, it can be obtained numerically by solving a closure problem that maps the microscale to the macroscale. The resulting effective coefficient depends on the local properties as well on the geometry of the inclusions. The goal of this work is to give explicit formulae relating the sensitivity of large scale effective conductivity with respect to local microscale parameters directly using the solution of the mapping problem. The proposed formula is direct and does not imply solving any additional adjoint problem. It needs only evaluation of volume or surface integrals involving the mapping problem solution. The associated numerical cost is negligible once the mapping closure problem has been solved. Practical applications can be found for solving inverse problems, for optimizing a microstructure shape, fulfilling an imposed constraint like maximizing or minimizing the overall conductivity.

Properties of a Composite Material With Mixed Imperfect Contact Conditions

“…The theory and technique for such analysis of composite material properties are under development. Recently, in [16], it was derived an explicit formula allowing to compute the sensitivity of large scale conductivity of a composite material to parameters describing its microstructure (such as material microstructure properties or a set of data describing the geometrical shape of inclusions). By means of the asymptotic homogenization method, analytical formulae were obtained in [11] for the effective thermoelastic coefficients of a fiber-reinforced periodic elastic composite with hexagonal cell, where the constituents exhibit transverse isotropic properties.…”

Geological realism in hydrogeological and geophysical inverse modeling: A review

“…While there is already a set of methods available to perturb geological models while preserving the geological structures, it is still very difficult to make these perturbations efficient in the sense that they decrease rapidly the data residuals. One way to improve this could be to use explicit formulas to relate the sensitivity of the forward problem to changes in the geometry of local inclusions in heterogeneous materials as recently proposed by Noetinger (2013). We emphasize that the goal of geological realism in hydrogeophysical inverse modeling is not per se to create geologically realistic earth models, but to enable more informed conclusions and decisions under uncertainty.…”

Geological Objects and Physical Parameter Fields in the Subsurface: A Review

“…In particular, they have shown much interest in stratigraphic modeling as one single scalar field can represent a conformable stratigraphic series at once, which opens new possibilities in structural data interpolation (Calcagno et al 2008;Caumon et al 2013;Hillier et al 2014;Laurent et al 2016). Implicit surfaces also offer very nice ways to consider geometric model perturbations needed to address inverse problems in geosciences (Cardiff and Kitanidis 2009;Caumon et al 2007;Noetinger 2013;Zheglova et al 2013). A major distinction between explicit and implicit surface models is about topological control: the surface topology has to be chosen before interpolation in explicit methods, whereas it emerges from the interpolation in implicit models, see also Collon et al (2016) for more discussions.…”