Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

We present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes

Energy Stable Numerical Schemes for Ternary Cahn-Hilliard System

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. to overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, and this, in turn, yields a convex-concave decomposition of the physical energy in the ternary system. Consequently, both the unique solvability and the unconditional energy stability of the proposed numerical scheme are established at a theoretical level. in addition, an optimal rate convergence analysis in the ℓ∞(0,T;HN-1)∩ℓ2(0,T;HN1) norm is provided, with Fourier pseudo-spectral discretization in space, which is the first such result in this field. to deal with the nonlinear implicit equations at each time step, we apply an efficient preconditioned steepest descent (PSD) algorithm. a second order accurate, modified BDF scheme is also discussed. a few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme

Nonsmooth Schur-Newton methods for multicomponent Cahn-Hilliard systems

We present globally convergent nonsmooth Schur–Newton methods for the solution of discrete multicomponent Cahn–Hilliard systems with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems arising from discretization by implicit Euler methods in time and first-order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments

Numerical analysis of a coupled pair of Cahn-Hilliard equations with non-smooth free energy

A mathematical analysis has been carried out for a coupled pair of Cahn-Hilliard equations with a double well potential function with infinite walled free energy, which appears in modelling a phase separation on a thin film of binary liquid mixture coating substrate, which is wet by one component. Existence and uniqueness are proved for a weak formulation of the problem, which possesses a Lyapunov functional. Regularity results for the weak formulation are presented. Semi and fully discrete finite element approximations are proposed where existence and uniqueness of their solutions are proven. Their convergence to the solution of the continuous solutions are presented. Error bound between semi-discrete and continuous solutions, between semi-discrete and fully discrete solutions, and between fully discrete and continuous solutions are all investigated. A practical algorithm to solve the fully discrete finite element formulation at each time step is introduced and its convergence is shown. Finally, a linear stability analysis of the equations in one dimension space is presented and some numerical simulations in one and two dimension spaces are preformed

Nonsmooth Schur-Newton methods for vector-valued Cahn-Hilliard equations

We present globally convergent nonsmooth Schur-Newton methods for the solution of discrete vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems as arising from discretization by implicit Euler methods in time and first order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments

Domain growth in alloys

This thesis describes Monte-Carlo computer simulations of binary alloys, with comparisons between small angle neutron scattering (SANS) data, and numerically integrated solutions to the Cahn-Hilliard-Cook (CHC) equation. Elementary theories for droplet growth are also compared with computer simulated data. Monte-Carlo dynamical algorithms are investigated in detail, with special regard for universal dynamical times. The computer simulated systems are Fourier transformed to yield partial structure functions which are compared with SANS data for the binary Iron-Chromium system. A relation between real time and simulation time is found. Cluster statistics are measured in the simulated systems, and compared to droplet formation in the Copper-Cobalt system. Some scattering data for the complex steel PE16 is also discussed. The characterisation of domain size and its growth with time are investigated, and scaling laws fitted to real and simulated data. The simple scaling law of Lifshitz and Slyozov is found to be inadequate, and corrections such as those suggested by Huse, are necessary. Scaling behaviour is studied for the low-concentration nucleation regime and the high-concentration spinodal-decomposition regime. The need for multi-scaling is also considered. The effect of noise and fluctuations in the simulations is considered in the MonteCarlo model, a cellular-automaton (CA) model and in the Cahn-Billiard-Cook equation. The Cook noise term in the CHC equation is found to be important for correct growth scaling properties

Metal droplet entrainment by solid particles in slags : a combined phase field-experimental approach

This doctoral work investigated metal droplet entrainment by solid particles in slags with a combination of two experimental set-ups and two phase field models. The binary model with limited complexity already clarified our view of the interaction between metal droplets and nonreacting solid particles to a great extent. For example, the fact that the movement of one phase with respect to the others influenced the apparent wetting regime is very interesting for the interpretation of experimentally obtained results. Moreover, the two different types of experiments confirmed that a chemical reaction might lay at the origin of the attachment, but that it requires nucleation sites in the form of metal droplets before it takes place. However, the first phase field model assumed nonreactive solid particles. Thus, a model concerning the growth of the solid phase in a realistic quaternary oxide system was also considered. Future work needs to consider the interaction of reacting metal droplets with reacting solid particles in a realistic liquid slag

Existence results for diffuse interface models describing phase separation and damage

In this paper we analytically investigate Cahn-Hilliard and Allen-Cahn systems which are coupled with elasticity and uni-directional damage processes. We are interested in the case where the free energy contains logarithmic terms of the chemical concentration variable and quadratic terms of the gradient of the damage variable. For elastic Cahn-Hilliard and Allen-Cahn systems coupled with uni-directional damage processes, an appropriate notion of weak solutions is presented as well as an existence result based on certain regularization methods and an higher integrability result for the strain

Modeling, Characterizing and Reconstructing Mesoscale Microstructural Evolution in Particulate Processing and Solid-State Sintering

abstract: In material science, microstructure plays a key role in determining properties, which further determine utility of the material. However, effectively measuring microstructure evolution in real time remains an challenge. To date, a wide range of advanced experimental techniques have been developed and applied to characterize material microstructure and structural evolution on different length and time scales. Most of these methods can only resolve 2D structural features within a narrow range of length scale and for a single or a series of snapshots. The currently available 3D microstructure characterization techniques are usually destructive and require slicing and polishing the samples each time a picture is taken. Simulation methods, on the other hand, are cheap, sample-free and versatile without the special necessity of taking care of the physical limitations, such as extreme temperature or pressure, which are prominent issues for experimental methods. Yet the majority of simulation methods are limited to specific circumstances, for example, first principle computation can only handle several thousands of atoms, molecular dynamics can only efficiently simulate a few seconds of evolution of a system with several millions particles, and finite element method can only be used in continuous medium, etc. Such limitations make these individual methods far from satisfaction to simulate macroscopic processes that a material sample undergoes up to experimental level accuracy. Therefore, it is highly desirable to develop a framework that integrate different simulation schemes from various scales to model complicated microstructure evolution and corresponding properties. Guided by such an objective, we have made our efforts towards incorporating a collection of simulation methods, including finite element method (FEM), cellular automata (CA), kinetic Monte Carlo (kMC), stochastic reconstruction method, Discrete Element Method (DEM), etc, to generate an integrated computational material engineering platform (ICMEP), which could enable us to effectively model microstructure evolution and use the simulated microstructure to do subsequent performance analysis. In this thesis, we will introduce some cases of building coupled modeling schemes and present the preliminary results in solid-state sintering. For example, we use coupled DEM and kinetic Monte Carlo method to simulate solid state sintering, and use coupled FEM and cellular automata method to model microstrucutre evolution during selective laser sintering of titanium alloy. Current results indicate that joining models from different length and time scales is fruitful in terms of understanding and describing microstructure evolution of a macroscopic physical process from various perspectives.Dissertation/ThesisDoctoral Dissertation Materials Science and Engineering 201