The Computational Modelling of Branching Fine Structures in Constrained Crystals

We introduce a finite element method which is piecewise continuous on the microscopic scale of the spatial resolution h but discontinuous on the mesoscopic scale hδ, δε(0,1). The method is designed to capture the morphology of needle twin structures frequently found in ferric and pseudo-elastic crystals, namely, in uniaxial ferromagnets and au milieu of the Austenite-Martensite interfaces. The approach is based on a domain decomposition method that interpolates between the scale on order of the size of crystal and the microscopic scale of finite element approximation h. The scale interpolation is enabled by incorporating frequency adaptivity. The visualization and analysis of the computational results presented disclose microstructures corresponding to complex scaling laws. We document that the fine structures obtained by the presented method are not visible using classical formulation of the underlying variational problem and using conforming approximation of admissible sets. The proposed method is suitable for non-smooth relaxation and optimization when the minimizers lack the often required C1,α-regularity and when they exhibit fractal behavior.