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This work develops a new open source API and software package called textit{SymPhas} for simulations of phase-field, phase-field crystal and reaction-diffusion models, supporting up to three dimensions and an arbitrary number of fields. textit{SymPhas} delivers two novel program capabilities: 1) User specification of models from the associated dynamical equations in an unconstrained form and 2) extensive support for integrating user-developed discrete-grid-based numerical solvers into the API. The capability to specify general phase-field models is primarily achieved by developing a novel symbolic algebra functionality that can formulate mathematical expressions at compile time, is able to apply rules of symbolic algebra such as distribution, factoring and automatic simplification, and support user-driven expression tree manipulation. A modular design based on the CC++ template meta-programming paradigm is applied to the symbolic algebra library and general API implementation to minimize application runtime and increase the accessibility of the API for third party development. textit{SymPhas} is written in C/CC++ and emphasizes high-performance capabilities via parallelization with OpenMP and the CC++ standard library. textit{SymPhas} is equipped with a forward Euler solver and a semi-implicit Fourier spectral solver. Sample implementations and simulations of several phase-field models are presented, generated using the semi-implicit Fourier spectral solver.
The study of polycrystalline materials requires theoretical and computational techniques enabling multiscale investigations. The amplitude expansion of the phase field crystal model (APFC) allows for describing crystal lattice properties on diffusive timescales by focusing on continuous fields varying on length scales larger than the atomic spacing. Thus, it allows for the simulation of large systems still retaining details of the crystal lattice. Fostered by the applications of this approach, we present here an efficient numerical framework to solve its equations. In particular, we consider a real space approach exploiting the finite element method. An optimized preconditioner is developed in order to improve the convergence of the linear solver. Moreover, a mesh adaptivity criterion based on the local rotation of the polycrystal is used. This results in an unprecedented capability of simulating large, three-dimensional systems including the dynamical description of the microstructures in polycrystalline materials together with their dislocation networks.
In this paper, we develop an efficient lattice Boltzmann (LB) model for simulating immiscible incompressible $N$-phase flows $(N geq 2)$ based on the Cahn-Hilliard phase field theory. In order to facilitate the design of LB model and reduce the calculation of the gradient term, the governing equations of the $N$-phase system are reformulated, and they satisfy the conservation of mass, momentum and the second law of thermodynamics. In the present model, $(N-1)$ LB equations are employed to capture the interface, and another LB equation is used to solve the Navier-Stokes (N-S) equations, where a new distribution function for the total force is delicately designed to reduce the calculation of the gradient term. The developed model is first validated by two classical benchmark problems, including the tests of static droplets and the spreading of a liquid lens, the simulation results show that the current LB model is accurate and efficient for simulating incompressible $N$-phase fluid flows. To further demonstrate the capability of the LB model, two numerical simulations, including dynamics of droplet collision for four fluid phases and dynamics of droplets and interfaces for five fluid phases, are performed to test the developed model. The results show that the present model can successfully handle complex interactions among $N$ ($N geq 2$) immiscible incompressible flows.
In this paper we present two unconditionally energy stable finite difference schemes for the Modified Phase Field Crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic Phase Field Crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes.
Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.
Bilayer graphene has been a subject of intense study in recent years. We extend a structural phase field crystal method to include an external potential from adjacent layer(s), which is generated by the corresponding phase field and changes over time. Moreover, multiple layers can be added into the structure. Using the thickness of the boundaries between different stacking variants of the bilayer structure as the key parameter, we quantify the strength of the adjacent layer potential by comparing with atomistic simulation results. We then test the multiple graphene structures, including bilayers, triple layers, up to 6 layers. We find that besides the initial conditions, the way of new layers added into the structure will also affect the layout of the atomic configuration. We believe tour results can help understanding the mechanism of graphene structure consists of more than one layer.