An Explicit Formula for Computing the Sensitivity of the Effective Conductivity of Heterogeneous Composite Materials to Local Inclusion Transport Properties and Geometry
B. Nœtinger · Multiscale Modeling and Simulation · 2013
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.