A fast and accurate adaptive solution strategy for two-scale models with continuous inter-scale dependencies

This article presents a fast and accurate adaptive algorithm that numerically solves a two-scale model with continuous inter-scale dependencies. The examined sample two-scale model describes a phase transition of a binary mixture with the evolution of equiaxed dendritic microstructures. It consists of a macroscopic heat equation and a family of microscopic cell problems that model the phase transition of the mixture. Both scales are coupled: the macroscopic temperature field influences the evolution of the microstructure and the microscopic fields enter to the macroscopic heat equation via averaged coefficients. Adaptivity exploits the constitutive assumption that the evolving microstructure depends in a continuous way on the macroscopic temperature field: macroscopic nodes with similar temperature evolutions use the same microscopic data. A suitable metric compares temperature evolutions and adaptive methods select active macroscopic nodes. Microscopic cell problems are solved for active nodes only; microscopic data in inactive nodes is approximated from microscopic data of active nodes with a similar temperature evolution. The set of active nodes is updated in course of the simulation: active nodes are deactivated until all active nodes have unsimilar temperature evolutions, and inactive nodes are activated until for every inactive node there exists at least one active node with a similar temperature evolution. Numerical examples, in two and in three space dimensions, show that the adaptive solution is only slightly less accurate than the direct solution, but it is computationally much more efficient. Therefore, the adaptive algorithm enables the solution of two-scale models with continuous inter-scale dependencies on large computational macroscopic and microscopic grids within an acceptable period of time for computation. (C) 2013 Elsevier Inc. All rights reserved.