On the stable discretization of strongly anisotropic phase field models with applications to crystal growth

We introduce unconditionally stable finite element approximations for anisotropic Allen--Cahn and Cahn--Hilliard equations. These equations frequently feature in phase field models that appear in materials science. On introducing the novel fully practical finite element approximations we prove their stability and demonstrate their applicability with some numerical results. We dedicate this article to the memory of our colleague and friend Christof Eck (1968--2011) in recognition of his fundamental contributions to phase field models.Comment: 20 pages, 8 figure

A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation

We propose a novel second order in time numerical scheme for Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is uncondition- ally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the weak coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

Numerical approximation of phase field based shape and topology optimization for fluids

We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided

Diffuse interface models of locally inextensible vesicles in a viscous fluid

We present a new diffuse interface model for the dynamics of inextensible vesicles in a viscous fluid. A new feature of this work is the implementation of the local inextensibility condition in the diffuse interface context. Local inextensibility is enforced by using a local Lagrange multiplier, which provides the necessary tension force at the interface. To solve for the local Lagrange multiplier, we introduce a new equation whose solution essentially provides a harmonic extension of the local Lagrange multiplier off the interface while maintaining the local inextensibility constraint near the interface. To make the method more robust, we develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. Asymptotic analysis is presented that shows that our new system converges to a relaxed version of the inextensible sharp interface model. This is also verified numerically. Although the model does not depend on dimension, we present numerical simulations only in 2D. To solve the 2D equations numerically, we develop an efficient algorithm combining an operator splitting approach with adaptive finite elements where the Navier-Stokes equations are implicitly coupled to the diffuse interface inextensibility equation. Numerical simulations of a single vesicle in a shear flow at different Reynolds numbers demonstrate that errors in enforcing local inextensibility may accumulate and lead to large differences in the dynamics in the tumbling regime and differences in the inclination angle of vesicles in the tank-treading regime. The local relaxation algorithm is shown to effectively prevent this accumulation by driving the system back to its equilibrium state when errors in local inextensibility arise.Comment: 25 page