Homogenization and Upscaling for Diffusion,Heat Conduction,and Wave Propagation in Heterogeneous Materials

We present a general homogenization method for diffusion,heat conduction,and wave propagation in a periodic heterogeneous material with piecewise constants.The method is relevant to the frequently encountered upscaling issues for heterogeneous materials.The dispersion relation for each problem is first expressed in the general form where the frequency ω (or wavenumber k) is expanded in terms of the wavenumber k (or frequency ω).A general homogenization model can be directly obtained with any given dispersion relation.Next step we study the unit cell of the heterogeneous material and derive the exact dispersion relation.The final homogenized equations include both leading order terms (effective properties) and high order contributions that represent the effect of the microscopic heterogeneity on the macroscopic behavior.That effect can be lumped into a single dimensionless heterogeneity parameter β,which is bounded between 1/12 ≤β≤ 0 and has a universal expression for all three problems.Numerical examples validate the proposed method and demonstrate a significant computational saving.