An adaptive variational procedure for the conservative and positivity preserving Allen–Cahn phase-field model

Abstract

In this paper, we present an adaptive variational procedure for unstructured meshes to capture fluid–fluid interfaces in two-phase flows. The two phases are modeled by the phase-field finite element formulation, which involves the conservative Allen–Cahn equation coupled with the incompressible Navier–Stokes equations. The positivity preserving variational formulation is designed to maintain the bounded and stable solution of the Allen–Cahn equation. For the adaptivity procedure, we consider the residual-based error estimates for the underlying differential equations of the two-phase system. In particular, the adaptive refinement/coarsening is carried out by the newest vertex bisection algorithm by evaluating the residual error indicators based on the error estimates of the Allen–Cahn equation. The coarsening algorithm avoids the storage of the tree data structures for the hierarchical mesh, thus providing the ease of numerical implementation. Furthermore, the proposed nonlinear adaptive partitioned procedure aims at reducing the amount of coarsening while maintaining the convergence properties of the underlying nonlinear coupled differential equations. We investigate the adaptive phase-field finite element scheme through the spinodal decomposition in a complex curved geometry and the volume-conserved interface motion driven by the mean curvature flow for two circles in a square domain. We then assess the accuracy and efficiency of the proposed procedure by modeling the free-surface motion in a sloshing tank. In contrast to the non-adaptive Eulerian grid counterpart, we demonstrate that the mesh adaptivity remarkably reduces the degrees of freedom and the computational cost by nearly half for similar accuracy. The mass loss in the Allen–Cahn equation via adaptivity process is also reduced by nearly three times compared to the non-adaptive mesh. Finally, we apply the adaptive numerical framework to solve the application of a dam-breaking problem with topological changes.