Numerical upscaling of electrical conductivity: A problem specific approach to generate coarse-scale models

Abstract

In this work we propose a new approach for the upscaling problem of electrical conductivity in the context of electromagnetic methods. We pose the upscaling problem as a parameter estimation problem, which allow us to develop a goaloriented, quantitative framework that combines widely used simulation tools such as Mimetic Finite Volume, inversion and optimization techniques. We thus create a flexible methodology that allows the users to estimate, in an affordable manner, coarse-scale conductivity models that approximate, in some sense, the fine-scale ones. Our framework is based on the observation that for any conductivity model a number of different criteria can be considered for the homogenization problem. In particular, different physical fields and fluxes can be considered. Our framework allows the choice of the criteria that is the most appropriate for the goal of the simulation. Results are illustrated with a couple of simulations that demonstrate the capabilities of our method as well as the challenges that this different perspective offers.