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Computational Treatment of Interface Dynamics via Phase-Field Modeling

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Numerical Simulation in Physics and Engineering: Trends and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 24))

Abstract

This chapter describes the phase-field approach to modeling interface dynamics, with particular emphasis on the mathematical formulation and computational aspects. We describe to approaches to derive a phase-field model. The first one can be understood as a regularization approach: We start with a sharp interface model which is later replaced by a diffuse interface. The second approach starts with a free energy functional and balance laws for mass, linear momentum and energy; the final governing equations are derived using the second law of thermodynamics and the Coleman-Noll procedure. We finish by illustrating how the phase-field method can be used to solve problems on complicated geometries using cartesian grids only. Some of the opportunities opened by the phase-field approach are illustrated with numerical simulations.

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Notes

  1. 1.

    The term phase may refer to different concepts. It can be used, e.g., to define the different states of matter in a solid-liquid model, the different components in a mixture, or to determine whether there is a fracture or not in a crack propagation model.

  2. 2.

    In the case σ = 0, Eqs. (1)–(4) is referred to as the classical Stefan problem.

  3. 3.

    Although Fourier’s law is the most common theory used in heat transmission problems, there are other possibilities [28, 29, 31, 32].

  4. 4.

    Note that V n is the normal velocity of the interface, that is, V n = v LS ⋅n LS.

  5. 5.

    The co-area formula is

    $$\displaystyle \begin{aligned} \lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon} \int_\Omega q\Big( \frac{d_{\Gamma_{\mathrm{int}}}(\boldsymbol{x})}{\epsilon} \Big) \,\mathrm{d} x = \alpha_q \; \int_{\Gamma_{\mathrm{int}}} \mathrm{d} a , \end{aligned} $$
    (41)

    In this case, we use a constant value for α q, where \(\alpha _q = \int _{-\infty }^\infty q(z) \,\mathrm {d} z\). The co-area formula is valid for suitably decaying functions q(z), see [21, Lemma 2.1].

  6. 6.

    A more mathematically rigorous relation between \(\mathcal {F}^{\,\epsilon }\) and \(\mathcal {F}\) can be determined by means of the Γ-convergence theory [2, 9, 18].

  7. 7.

    The term constitutive class means that a certain function is allowed to depend on other variables. In this case, the constitutive class of Ψ allows it to depend on ϕ and ∇ϕ. Fundamental laws can be used to restrict even more the constitutive class of a function and allow only certain kinds of dependence. The term constitutive class is commonly employed in classical mechanics; see [37, 63].

  8. 8.

    Mass conservation in Eq. (54), i.e., \(\frac {\mathrm {d}}{\mathrm {d} t}(\int _{\Omega }\phi \,\mathrm {d} x)=0\), is satisfied for natural boundary conditions h ⋅n a = 0 on ∂ Ω, where n a is the unit normal vector to ∂ Ω.

  9. 9.

    The term μ h in Eq. (57) may be interpreted as an energy flux.

  10. 10.

    Note that, on multiphase systems, balance of angular momentum does not imply the symmetry of T [7].

  11. 11.

    If A is a symmetric tensor, then A : B = A : (B + B T)∕2. Consequently, it can easily be proven that if A is symmetric and B is skew-symmetric, then A : B = 0.

  12. 12.

    Usually, m and ν are assumed to be constant.

  13. 13.

    The smooth function δ Γ works as a marker of the position of the boundary of the moving domain. A suitable expression for this function could be, e.g., δ Γ = 𝜖 2|∇ϕ|2, though there are other valid expressions for δ Γ in the literature.

  14. 14.

    Note that the velocity of ρ Γ within the surface Γρ (i.e., u Γ) does not necessarily coincide with the velocity of the surface \((\boldsymbol {u}_{\Gamma _\rho })\).

  15. 15.

    The variational formulation corresponding to the reference problem is analogous.

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Authors and Affiliations

  1. School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA

    Miguel Bures, Adrian Moure & Hector Gomez

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  1. Miguel Bures
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  1. Instituto Universitario de Sistemas Inteligentes y Aplicaciones Numéricas en Ingeniería, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    David Greiner

  2. Departamento de Matemática Aplicada, Universidad de Salamanca, Salamanca, Spain

    María Isabel Asensio

  3. Instituto Universitario de Sistemas Inteligentes y Aplicaciones Numéricas en Ingeniería, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Rafael Montenegro

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Bures, M., Moure, A., Gomez, H. (2021). Computational Treatment of Interface Dynamics via Phase-Field Modeling. In: Greiner, D., Asensio, M.I., Montenegro, R. (eds) Numerical Simulation in Physics and Engineering: Trends and Applications. SEMA SIMAI Springer Series, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-030-62543-6_2

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