Towards Infinite Tilings with Symmetric Boundaries

Abstract

Large-time coarsening and the associated scaling and statistically self-similar properties are used to construct infinite tilings. This is realized using a Cahn–Hilliard equation and special boundaries on each tile. Within a compromise between computational effort and the goal to reduce recurrences, an infinite tiling has been created and software which zooms in and out evolve forward and backward in time as well as traverse the infinite tiling horizontally and vertically. We also analyze the scaling behavior and the statistically self-similar properties and describe the numerical approach, which is based on finite elements and an energy-stable time discretization.

An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation

“…Please note that if β(t) = 0, then Equation (2) becomes the classical AC Equation (1). The nonlocal AC equation with mass conservation has been investigated numerically and analytically [13][14][15][16][17][18][19].…”

“…The phase-field model is one of the representatives of the interface capturing approach and has been widely investigated in order to interpret the interfacial dynamics [1]. The Allen-Cahn (AC) equation, which is a phenomenological model for antiphase domain coarsening in a binary mixture [2], is one of the most popular phase-field models and is given as…”