Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

Tierra, Giordano; Guillén-González, Francisco

doi:10.1007/s11831-014-9112-1

Cited by 83 publications

(70 citation statements)

References 62 publications

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“…These equations were selected because their stable time integration has garnered considerable interest in recent years [23,24,52,54], and this work generalizes some of the numerical schemes that have been put forth in the context of the Cahn-Hilliard [27,36] and phase-field crystal equations [54]. Additionally, the discretization in space is done using isogeometric analysis [13], which allows to easily generate high-order and globally continuous basis functions.…”

Section: Phase-field Modelsmentioning

confidence: 99%

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

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Section: Phase-field Modelsmentioning

confidence: 99%

An energy-stable time-integrator for phase-field models

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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“…In the case of the Cahn-Hilliard equation, various energy-stable semi-discrete (continuous in space, discrete in time) schemes of first and second order have appeared over the past years, 24,25,35,52,57,59 some of which are linear, i.e., they require only one solution of a linear-algebraic system per time step. a These schemes have been extended to NSCH systems with matched densities 31,39 and for quasiincompressible NSCH systems with a solenoidal mixture-velocity field.…”

Section: Lowengrub and Truskinovskymentioning

confidence: 99%

Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

Roudbari

Şimşek

Brummelen

et al. 2018

Math. Models Methods Appl. Sci.

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While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier-Stokes Cahn-Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman-Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with nonsolenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and break-up dynamics of droplets including pinching due to gravity.

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“…Its primary feature of interest is that it depicts the complex interactions of two (or more) constituents without the need to track their interfaces, which are diffused over boundary layers developed automatically as a feature of the solution. Numerical solutions of the CH equation have been intensively studied and many finite difference, finite volume, finite element and spectral methods have been developed to obtain numerical solutions [46][47][48]139]. Many of the finite element methods are built based on a mixed variational formulation.…”

Section: Phase-field and Mixture Theory Modelsmentioning

confidence: 99%

“…A comparative study of different mixed finite element schemes for the 1D CH equation is found in [75], where it is shown that the computational cost can vary substantially depending on properties of the spatial approximations of the model and other parameters in the scheme. A survey of some linear and nonlinear methods is likewise exhibited in [139], focusing on some desired numerical properties such as time accuracy order and energy stability. Phase-field or diffuse-interface models provide a general framework for modeling multiphase materials in which the interface between phases is handled automatically as a feature of the solution [98].…”

Section: Phase-field and Mixture Theory Modelsmentioning

confidence: 99%

Toward Predictive Multiscale Modeling of Vascular Tumor Growth

Oden

Lima

Almeida

et al. 2015

Arch Computat Methods Eng

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New directions in medical and biomedical sciences have gradually emerged over recent years that will change the way diseases are diagnosed and treated and are leading to the redirection of medicine toward patientspecific treatments. We refer to these new approaches for studying biomedical systems as predictive medicine, a new version of medical science that involves the use of advanced computer models of biomedical phenomena, high-performance computing, new experimental methods for model data calibration, modern imaging technologies, cutting-edge numerical algorithms for treating large stochastic systems, modern methods for model selection, calibration, validation, verification, and uncertainty quantification, and new approaches for drug design and delivery, all based on predictive models. The methodologies are designed to study events at multiple scales, from genetic data, to sub-cellular signaling mechanisms, to cell interactions, to tissue physics and chemistry, to organs in living human subjects. The present document surveys work on the development and implementation of predictive models of vascular tumor growth, covering aspects of what might be called modeling-and-experimentally based computational oncology. The work described is that of a multi-institutional team, centered at ICES with strong participation by members at M. D. Anderson Cancer Center and University of Texas at San Antonio. This exposition covers topics on signaling models, cell and cell-interaction models, tissue models based on multi-species mixture theories, models of angiogenesis, and beginning work of drug effects. A number of new parallel computer codes for implementing finite-element methods, multi-level Markov Chain Monte Carlo sampling methods, data classification methods, stochastic PDE solvers, statistical inverse algorithms for model calibration and validation, models of events at different spatial and temporal scales is presented. Importantly, new methods for model selection in the presence of uncertainties fundamental to predictive medical science, are described which are based on the notion of Bayesian model plausibilities. Also, as part of this general approach, new codes for determining the sensitivity of model outputs to variations in model parameters are described that provide a basis for assessing the importance of model parameters and controlling and reducing the number of relevant model parameters. Model specific data is to be accessible through careful and model-specific platforms in the Tumor Engineering Laboratory. We describe parallel computer platforms on which large-scale calculations are run as well as specific time-marching algorithms needed to treat stiff systems encountered in some phase-field mixture models. We also cover new non-invasive imaging and data classification methods that provide in vivo data for model validation. The study concludes with a brief discussion of future work and open challenges.

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Numerical Methods for Solving the Cahn–Hilliard Equation and Its Applicability to Related Energy-Based Models

Cited by 83 publications

References 62 publications

An energy-stable time-integrator for phase-field models

An energy-stable time-integrator for phase-field models

Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

Toward Predictive Multiscale Modeling of Vascular Tumor Growth

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