Double Obstacle Potential Research Articles

Framework for a chemo‐mechanical model of the breathing effect in lithium–silicon batteries

AbstractWe present a chemo‐mechanical finite element method model for the breathing effect of the silicon anode during lithiation and delithiation. The model is based on the framework for gradient extended standard dissipative solids. It couples diffusion of lithium ions, phase transformation with a moving phase boundary, and the related volume changes, which lead to mechanical stresses in the material. The driving force for diffusion is given by the gradient of the electrochemical potential. Phase transformation and phase boundary movement are described by the Allen–Cahn equation in combination with a constrained double obstacle potential. To shift the constraint from the nodes to the Gauss points, the model makes use of the micromorphic approach.

Dendrite operating state in directional solidification of AlCu binary system: numerical benchmark test with the OpenPhase software

A scientific benchmark test is carried out for a multi-phase-field model with double-obstacle potential by performing three-dimensional simulations of dendritic growth under directional solidification. The effects of key numerical parameters of the multi-phase-field model such as numerical resolution and interface width on the dendrite tip operating state are studied, optimal parameter values are set, and where the operating state becomes independent of varying these parameters is elaborated. Some uncertainties in the proper choice of effective interface mobility in the thin-interface limit are discussed and a pragmatic solution is adopted. The binary alloy Al-Cu with 4 at.% Cu is chosen as the material system because it has been used in many previous experimental and numerical studies to investigate dendrite morphology under directional solidification. The recently developed sharp phase-field model by Finel and colleagues is adapted to the double-obstacle potential function and included in the benchmark test. It is shown how the sharp phase-field model helps in achieving agreeable convergence with larger discretization, thereby reducing the computational cost significantly. The benchmarks are performed using the OpenPhase software.

Optimal Temperature Distribution for a Nonisothermal Cahn–Hilliard System with Source Term

In this note, we study the optimal control of a nonisothermal phase field system of Cahn–Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn–Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem. Crucial for the whole analysis to work is the boundedness property stating that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.

Global regularity and asymptotic stabilization for the incompressible Navier–Stokes-Cahn–Hilliard model with unmatched densities

We study an initial-boundary value problem for the incompressible Navier–Stokes–Cahn–Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\rightarrow \\infty $$\\end{document}. More precisely, the concentration function ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} is a strong solution of the Cahn–Hilliard equation for (arbitrary) positive times, whereas the velocity field u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{{u}}}$$\\end{document} becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn–Hilliard equation with divergence-free velocity belonging only to L2(0,∞;H0,σ1(Ω))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(0,\\infty ; {\ extbf{H}}^1_{0,\\sigma }(\\Omega ))$$\\end{document}, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz–Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.

Cahn–Hilliard–Brinkman model for tumor growth with possibly singular potentials

We analyze a phase field model for tumor growth consisting of a Cahn–Hilliard–Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn–Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.

A Diffuse-Interface Approach for Solid-State Dewetting with Anisotropic Surface Energies

We present a diffuse-interface model for the solid-state dewetting problem with anisotropic surface energies in {mathbb {R}}^d for din {2,3}. The introduced model consists of the anisotropic Cahn–Hilliard equation, with either a smooth or a double-obstacle potential, together with a degenerate mobility function and appropriate boundary conditions on the wall. Upon regularizing the introduced diffuse-interface model, and with the help of suitable asymptotic expansions, we recover as the sharp-interface limit the anisotropic surface diffusion flow for the interface together with an anisotropic Young’s law and a zero-flux condition at the contact line of the interface with a fixed external boundary. Furthermore, we show the existence of weak solutions for the regularized model, for both smooth and obstacle potential. Numerical results based on an appropriate finite element approximation are presented to demonstrate the excellent agreement between the proposed diffuse-interface model and its sharp-interface limit.

Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential

In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the so-called deep quench approach. Namely, the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and first-order necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential.

OPTIMAL TEMPERATURE DISTRIBUTION FOR A NONISOTHERMAL CAHN-HILLIARD SYSTEM IN TWO DIMENSIONS WITH SOURCE TERM AND DOUBLE OBSTACLE POTENTIAL

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the con­stitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails further mathematical difficulties because the mass conservation of the order parameter is no longer satisfied. In this paper, we study the case that the double-well potential driving the evolution of the phase transition is given by the nondifferentiable dou­ble obstacle potential, thereby complementing recent results obtained for the differentiable cases of regular and logarithmic potentials. Be­sides existence results, we derive first-order necessary optimality condi­tions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondifferentiable double obstacle potential is approximated by a family of potentials of logarithmic structure for which meaningful first-order necessary opti­mality conditions in terms of suitable adjoint systems and variational inequalities are available. Since the results for the logarithmic poten­tials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situ­ation, our whole analysis is restricted to the two-dimensional case.

A Cahn–Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials

A system with equation and dynamic boundary condition of Cahn–Hilliard type is considered. This system comes from a derivation performed in Liu–Wu (Arch. Ration. Mech. Anal., 233:167–247, 2019) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter.

A multi-phase field model for mesoscopic interface dynamics with large bulk driving forces

We develop a multi-phase field model for diffuse interface dynamics with large bulk chemical driving forces for phase transformation using the double-obstacle potential. We show how the classical prefactor functions for the bulk driving force significantly overestimate the velocity of multi-order-parameter junctions. We introduce a novel prefactor that properly distributes the forces for phase transformation among an arbitrary number of coexisting thermodynamic phases and order-parameters (i.e., melt patches and solid grains). The accuracy of the model is examined and we describe techniques to ensure accuracy for use with and without large bulk driving forces, including interface correction procedures that prevent profile deformation, and a recursive predictor–corrector technique for inter-parameter transfer at junctions. We explore the predictions of our models for a number of two-dimensional model configurations, including the shrinking circle, the moving tri-junction, and a shrinking circular crystal embedded in a fine-grained polycrystalline medium. The predictions for a moving tri-junction under increasingly large bulk driving forces (or length scales) are particularly notable, as the steady-state geometry of junctions deviate systematically from Young’s law for a given length scale. We provide an ansatz for the junction geometry in the case where a single-order-parameter phase (e.g., melt) consumes a multi-order-parameter phase (e.g., polycrystalline solid); an accurate solution for the opposite case remains elusive at present.

Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities

This paper treats a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the L^1-norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the so-called deep quench approximation in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls.

Multi-phase-field lattice Boltzmann model for polycrystalline equiaxed solidification with motion

The formation process of equiaxed structures during alloy solidification is a complicated multiphysics problem as it includes solid–liquid phase transformations, transport of solute and heat, liquid flow, solid motion, solid–solid interactions, and grain growth after solidification completion. In this study, a multi-phase-field (MPF) model with the double-obstacle (DO) potential is introduced to our previous model [T. Takaki et al., Comput. Mater. Sci., 147 (2018) 124–131] to express the formation process of equiaxed structures accurately and efficiently, from the growth of multiple mobile dendrites to the grain growth with grain boundary anisotropy after solidification. The accuracy of the resulting MPF-lattice Boltzmann (MPF-LB) model with the DO potential is evaluated by comparison with the results obtained using a model with the double-well potential through simulations of purely diffusive dendrite growth, the growth of an immobile dendrite with forced convection, and the growth of a mobile dendrite under shear flow. Finally, the model is applied to both large-scale polycrystalline solidification and semi-solid deformation processes.

Cahn–Hilliard equations on an evolving surface

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

Phase-field modeling of multivariant martensitic transformation at finite-strain: Computational aspects and large-scale finite-element simulations

Large-scale 3D martensitic microstructure evolution problems are studied using a finite-element discretization of a finite-strain phase-field model. The model admits an arbitrary crystallography of transformation and arbitrary elastic anisotropy of the phases, and incorporates Hencky-type elasticity, a penalty-regularized double-obstacle potential, and viscous dissipation. The finite-element discretization of the model is performed in Firedrake and relies on the PETSc solver library. The large systems of linear equations arising are efficiently solved using GMRES and a geometric multigrid preconditioner with a carefully chosen relaxation. The modeling capabilities are illustrated through a 3D simulation of the microstructure evolution in a pseudoelastic CuAlNi single crystal during nano-indentation, with all six orthorhombic martensite variants taken into account. Robustness and a good parallel scaling performance have been demonstrated, with the problem size reaching 150 million degrees of freedom.

Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms

Abstract We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.

Fast traveling waves in the phase-field theory: effective mobility approach versus kinetic energy approach

A phase-field model for small and large driving forces on solidification and melting of a pure substance or alloys is formulated. Derivations of the phase-field model are based on the effective mobility approach and on the kinetic energy approach to analyze fast phase transformation from metastable liquid to solid phase. A hodograph equation (an acceleration-velocity dependent equation of the Gibbs–Thomson type) which predicts the non-linear behavior in the velocity of the crystal–liquid interface is found at the large driving force on transformation and analyzed for different thermodynamic potentials. Traveling wave solutions of this equation are found for double-well and double-obstacle potentials. The velocity-dependent traveling waves as a function of driving force on transformation exhibit non-linearity of the solutions. Namely, in the relationship ‘velocity–driving force’ exists a maximum at a fixed undercooling which is very well known in the solidification of glass-forming metals and alloys. The predicted solidification velocity is quantitatively compared with the molecular dynamics simulation data obtained by Tang and Harrowell (2013 Nat. Mater. 12 507–11) for the solidification of congruently melting Cu–Zr binary alloy. The comparison confirms a crucial role of local non-equilibrium such as relaxation of gradient flow in the quantitative description of fast phase transformations.

Graph Merriman--Bence--Osher as a SemiDiscrete Implicit Euler Scheme for Graph Allen--Cahn Flow

In recent years there has been an emerging interest in PDE-like flows defined on finite graphs, with applications in clustering and image segmentation. In particular for image segmentation and semi-supervised learning Bertozzi and Flenner (2012) developed an algorithm based on the Allen-Cahn gradient flow of a graph Ginzburg-Landau functional, and Merkurjev, Kosti\'c and Bertozzi (2013) devised a variant algorithm based instead on graph Merriman-Bence-Osher (MBO) dynamics. This work offers rigorous justification for this use of the MBO scheme in place of Allen-Cahn flow. First, we choose the double-obstacle potential for the Ginzburg-Landau functional, and derive well-posedness and regularity results for the resulting graph Allen-Cahn flow. Next, we exhibit a "semi-discrete" time-discretisation scheme for Allen-Cahn flow of which the MBO scheme is a special case. We investigate the long-time behaviour of this scheme, and prove its convergence to the Allen-Cahn trajectory as the time-step vanishes. Finally, following a question raised by Van Gennip, Guillen, Osting and Bertozzi (2014), we exhibit results towards proving a link between double-obstacle Allen-Cahn flow and mean curvature flow on graphs. We show some promising $\Gamma$-convergence results, and translate to the graph setting two comparison principles used by Chen and Elliott (1994) to prove the analogous link in the continuum.

Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials

<p style='text-indent:20px;'>Recently, the authors derived well-posedness and regularity results for general evolutionary operator equations having the structure of a Cahn–Hilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible double-well potentials driving the phase separation process modeled by the Cahn–Hilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and first-order optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the so-called "deep quench" method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful first-order necessary optimality conditions.

Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach

In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn–Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into account a non-smooth singular nonlinearity, namely the double obstacle potential. Due to its non-smoothness behavior, the standard procedure to characterize the necessary conditions for the optimality cannot be performed. Therefore, we follow a different strategy which in the literature is known as the 'deep quench' approach in order to obtain some optimality conditions that have to be interpreted in a more general framework. We establish the existence of optimal controls and some first-order optimality conditions for the system are derived by employing suitable approximation schemes.

Comparative study of different anisotropy and potential formulations of phase-field models for dendritic solidification

Phase-field model formulations with double well and double obstacle potentials, and different anisotropy models are investigated with respect to their potential to simulate (i) tip growth on a quantitative level, (ii) well resolved side-branching. The dilute binary alloy Al-4 at%Cu is used as a model alloy. The effects of the numerical resolution (the ratio of the capillary length to the grid spacing) on the growth velocity are studied by means of convergence tests for isothermal and directional solidification in comparison to the theoretical values calculated by the Green-function method (A. Karma, W.J. Rappel, Phys. Rev. E 57 (1998) 4323). An interface stability parameter is introduced as a measure for the estimation of the maximum value of the grid spacing for effective simulations. We show that predominantly the side-branching occurs at numerical resolution lower than the limit value needed to produce correct results in accordance to the convergence analysis. The best results for dendrite growth at a relevant numerical resolution are obtained for the double well potential.