Subdivision-based IGA Coupled EIEQ Method for the Cahn–Hilliard Phase-Field Model of Homopolymer Blends on Complex Surfaces

In this paper, we construct an IGA-EIEQ coupling scheme to solve the phase-field model of homopolymer blends on complex subdivision surfaces, in which the total free energy contains a gradient entropy with a concentration-dependent de-Gennes type coefficient and a non-linear logarithmic Flory–Huggins type potential. Based on the EIEQ method, we develop a fully-discrete numerical scheme with the superior properties of linearity, unconditional energy stability, and second-order time accuracy. All we need to do with this fourth-order system is to solve some constant-coefficient elliptic equations by applying a new nonlocal splitting techniqueWe then provide detailed proof of the unconditional energy stability and the practical implementation process. Subdivision approaches show a robust and elegant description of the models with arbitrary topology. Subdivision basis functions serve to define the geometry of the models and represent the numerical solutions. Subdivision-based IGA approach provides us with a good candidate for solving the phase-field model on complex surfaces. We successfully demonstrate the unity of employing subdivision basis functions to describe the geometry and simulate the dynamical behaviors of the phase-field models on surfaces with arbitrary topology. This coupling strategy combining the subdivision-based IGA method and the EIEQ method could be extended to a lot of gradient flow models with complex nonlinearities on complex surfaces.