An energetically consistent concurrent multiscale method for heterogeneous heat transfer and phase transition applications

A concurrent multiscale method is developed to model time-dependent heat transfer and phase transitions in heterogeneous media and is formulated in a way such that the energy being exchanged between scales is conserved. Ensuring this energetic consistency among scales enables the implementation of high fidelity physics-based models at critical locations within the coarse-scale to temporally and spatially resolve highly complex and localized phenomena. To achieve this, only Neumann boundary conditions are applied over the fine scale domain, ensuring a conservative formulation. The coarse-scale solution is used to reconstruct these Neumann boundary conditions on the fine scale, which are then used to evolve a separate system of governing equations. The results on the fine scale are then sent back to the coarse scale through an energy-based homogenization scheme. Transient simulations for the heat equation are implemented with the proposed method to demonstrate its accuracy in energy conservation and effectiveness, including the coupling of a phase field model at the fine scale to a coarse-scale heat equation.