The time-fractional Allen–Cahn equation on geometric computational domains

Phase separation, the formation of two distinct phases from a single homogeneous mixture, has been extensively studied and observed in classical systems, typically as temperature changes. These separation phenomena depend on temperature and vary with different materials. We explore the numerical model, which shows various aspects of phase separation, such as time delay, the influence of heterogeneous materials on each domain, and the separation behavior in polyhedrons for industrial potentials.

This work investigates the time-fractional Allen–Cahn model with the Caputo fractional differential operator, which is particularly suited for capturing long waiting times in highly inhomogeneous porous media. We numerically examine the behavior of the order parameter $u$ on the surfaces of polyhedral and conical computational domains. We achieve the numerical simulation using an explicit Predictor–Corrector method in conjunction with a collocation approach, where Non-Uniform Rational B-Spline (NURBS) geometrical mapping is applied. Through this numerical method, we simulate the behavior of the order parameter $u$ on various surfaces of solids, investigating the effects of varying time-fractional derivative orders on each face of these geometries. We introduce the Schwarz alternating collocation method to handle discrepancies in boundary values when different time-fractional orders are applied to each subdomain.

Certain crystalline solids can be formed in materials science by joining different materials along their folding curves, resulting in varying material properties across boundaries. Observing phase separation in interconnected regions with varying material properties is challenging. Furthermore, phase separation that reflects the distinct characteristics of materials represented by time-fractional order differentiation in such interconnected regions with varying material properties has yet to be studied, making it a challenging problem, as far as our knowledge is concerned.

To address the aforementioned challenge, this study conducts a comprehensive series of numerical tests to investigate phase-field properties on surfaces of varying materials. Our thorough approach effectively captures how material properties are reflected across different geometries and how time-fractional behaviors vary on each surface. The numerical results illustrate critical phenomena, such as Ostwald ripening under varying time-fractional orders on the surface of geometries like cubes, tetrahedrons, prisms, and cones. Furthermore, we observe the curve shortening flow on the flattened sector of a circular conical domain, providing a complete picture of the behavior of the order parameter $u$.