Fast numerical upscaling of heat equation for fibrous materials

We are interested in numerical methods for computing the effective heat conductivities of fibrous insulation materials, such as glass or mineral wool, characterized by low solid volume fractions and high contrasts, i.e., high ratios between the thermal conductivities of the fibers and the surrounding air. We consider a fast numerical method for solving some auxiliary cell problems appearing in this upscaling procedure. The auxiliary problems are boundary value problems of the steady-state heat equation in a representative elementary volume occupied by fibers and air. We make a simplification by replacing these problems with appropriate boundary value problems in the domain occupied by the fibers only. Finally, the obtained problems are further simplified by taking advantage of the slender shape of the fibers and assuming that they form a network. A discretization on the graph defined by the fibers is presented and error estimates are provided. The resulting algorithm is discussed and the accuracy and the performance of the method are illusrated on a number of numerical experiments.