Computational Phase‐Field Modeling

Abstract

Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

Full text

Computational phase-field modeling
Hector Gomez1, Kristoffer G. van der Zee2
1Departamento de M´etodos Matem´aticos, Universidade da Coru˜na,
Campus de A Coru˜na, 15071, A Coru˜na, Spain.
2School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD,
United Kingdom
ABSTRACT
Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The
classical description of interface problems requires the numerical solution of partial differential equations
on moving domains in which the domain motions are also unknowns. The computational treatment of
these problems requires moving meshes and is very difficult when the moving domains undergo topological
changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as
equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to
the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical
soundness of the approach. However, this is only part of the story because phase-field models do not need to
have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics.
In this context, the distinguishing feature is that constitutive models depend on the variational derivative
of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding
problems in computational mechanics, such as, the interaction of a large number of cracks in three
dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water
flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical
discretization that will excite the computational mechanics community.
key words: Phase-field modeling, Thermomechanics, Thermodynamically-consistent algorithms,
Isogeometric analysis, Multiphase flows, Fracture mechanics, Tumor growth, Cahn-Hilliard equation.
1. Introduction
1.1. Phase-field modeling in computational mechanics
There are many areas of research in computational mechanics that involve moving boundary
problems. Prime examples are fluid-structure interaction, fully three-dimensional air-water flows
and crack propagation. The mathematical formulation of these problems often involves partial-
differential equations (PDEs) which are posed on moving domains and coupled to another set of
PDEs through boundary conditions which hold on a moving and unknown interface. Although in
some cases, such as for example, fluid-structure interaction, these problems have been attacked
computationally with significant success (Bazilevs et al., 2013; Farhat, 2004), most of them remain
as outstanding problems, especially in complicated three-dimensional geometries.
The phase-field method (Emmerich., 2003; Provatas and Elder, 2010; Chen, 2002; Anderson et al.,
1998) may be understood as a mathematical theory to reformulate moving boundary problems as
PDEs which hold on a known and fixed computational domain. The usual way to achieve this
is to introduce a new field, namely, the phase field, which is defined on the entire domain and
is a marker of the location of the different phases1. The phase field is governed by an equation
1Here, the word phase should be understood in a very broad sense. It may refer, for example, to a component
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

2ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
that naturally develops internal layers which mimic the interfaces of the original moving boundary
problem. The crucial difference, however, is that the phase field is smooth and, as a consequence,
the interfaces have a small but finite width, which is usually controlled by a length scale parameter
in the model. For this reason, phase-field models are also called diffuse-interface models and, in
contrast, the corresponding moving boundary problems are referred to as sharp-interface theories.
In some cases, the easiest way to derive a phase-field model is to start from a moving boundary
problem and smooth it out as described before to obtain its phase-field analogue. This is exactly
the approach we take in Sect. 2. The process of smoothing out or diffusifying 2the original problem
can be made mathematically rigorous and in some cases the phase-field model may be shown to
converge to the moving boundary problem as a regularization parameter that controls the thickness
of the interfaces goes to zero. The enabling mathematical technique to prove the convergence of the
phase-field model to the sharp-interface equation is the theory of matched asymptotic expansions
(Caginalp, 1989; Fife, 1988).
The diffusification procedure we just sketched, however, is only a small part of the picture.
Phase-field models may also be understood in different ways and do not need to have a free-
boundary problem associated. They can be derived directly from free-energy functionals using
the classical theory of thermomechanics and Coleman-Noll-type approaches (Coleman and Noll,
1963; Truesdell and Noll, 1965). This is the point of view that we adopt in Sect. 3 and the key
feature of phase-field models in this context is that constitutive equations are allowed to depend
on the variational derivative of the free-energy itself. Using this framework, we derive the classical
Allen-Cahn and Cahn-Hilliard models, a theory of two-component immiscible fluids with surface
tension, a phase-field model of brittle fracture, a mechano-biological model of tumor growth, and
the Navier-Stokes-Korteweg equations which govern liquid-vapor phase transformations.
Regardless of the interpretation that we take, the phase-field method produces models in which
interface tracking is avoided, significantly simplifying the numerics. However, the computational
treatment of phase-field equations is far from trivial and introduces a new set of challenges for
the computational mechanics community. For example, the diffuse interfaces which travel through
a fixed computational domain manifest themselves as layers in the solution represented by large
gradients of the phase field. Of course, these features of the solution need to be resolved by the
computational mesh, which ideally should follow them dynamically. Another important aspect
is that in many cases, phase-field theories are governed by fourth- (or higher-) order PDEs.
Classical numerical methodologies, such as, for example, the finite-difference method or pseudo-
spectral collocation may be used to solve higher-order PDEs on simple geometries (Trefethen,
2000; Axelsson, 2004; Canuto and Quarteroni, 2004). When geometrical flexibility is needed, the
finite element method is usually the technique of choice, but solving higher-order PDEs in primal
form (i.e., without introducing auxiliary variables representing derivatives of the unknown) in
complicated three-dimensional geometries is still an open problem. The difficulty stems from the
fact that a standard finite element method requires Cm−1-continuous basis functions to solve a PDE
of order 2m. This requirement may be easily satisfied using the isoparametric concept for PDEs of
second order (m= 1), but it is difficult to achieve in complicated three-dimensional geometries for
higher-order PDEs (m > 1). However, a new technology called isogeometric analysis (Hughes et al.,
2005; Cottrell et al., 2009; Aaa, 201xa,x) has recently been introduced in computational mechanics.
Isogeometric analysis permits generating basis functions with tailorable global continuity on non-
trivial geometries. This, together with the increased robustness of higher-order elements, has opened
the way to more efficient algorithms for phase-field models on engineering geometries.
Another computational challenge of phase-field models is time integration (Hairer et al., 1987;
of a fluid flow composed of air and water. It may also indicate whether a solid is intact or fractured in a crack
propagation problem. The battery of phase-field theories that we show in the paper will hopefully make clear the
meaning as the reader advances in the text.
2The terms diffusify and diffusification are terms that we have introduced in this work. They have not been used
in the literature before, in this context.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 3
Hughes, 2012). Phase-field theories usually derive from non-convex potentials which locally produce
the ‘backwards’ diffusion (also referred to as ‘uphill’ diffusion) that gives rise to the natural
formation of layers in the solution. Mathematically, the equations remain well posed, because they
are regularized by higher-order operators, but usual time integration schemes may be inadequate.
Sect. 4 treats in detail time integration algorithms for phase-field models, focusing both on accuracy
and stability.
In all, we feel that the theoretical and computational developments of the last few years have
elevated phase-field modeling to a point in which it can be used as a predictive model for a
number of problems in computational mechanics which were very difficult to treat with conventional
methodologies. Among these problems, we include the interaction of a large number of cracks in
three-dimensional solids of complicated geometry (Borden et al., 2014), fully three-dimensional
air-water flows with surface tension (Ceniceros et al., 2010), liquid-vapor phase transitions and
cavitation (Liu et al., 2013), nucleate and film boiling (Liu et al., 2015), phase-change-driven
implosion of thin structures (Bueno et al., 2014), cellular migration (Shao et al., 2012) and tumor
growth (Hawkins-Daarud et al., 2012; Vilanova et al., 2013, 2014, 2012; Xu et al., 2015; Lorenzo
et al., 2015). However, even though the last few years witnessed very significant advances in the
area, there are still many challenges ahead. Among these challenges, we identify multiphysics
and coupled problems, nonlocal phase-field models, and an efficient treatment of randomness and
uncertainty in phase-field theories.
We have organized the contents of this encyclopedic chapter as follows. Following a short
subsection on notational conventions, we consider in Sect. 2 the heuristic derivation of phase-
field models via diffusification of moving-boundary problems in the context of solidification. Then
in Sect. 3, we consider phase-field models from a thermomechanical perspective, and we derive
a variety of phase-field models that are encountered in computational mechanics. In Sect. 4 we
consider various time-integration methods, and in Sect. 5 we summarize methods for the spatial
discretization of phase-field models. Finally, Sect. 6 contains numerical examples.
1.2. Notational conventions
Let f:I⊂R7→ Rbe a real-valued function defined over a real, open interval I. The function f
transforms x∈Iinto f(x). We denote the derivative of fas f0. Alternatively, we will also use the
notation df
dx. When we work on several space dimensions, we will employ the notation Ω ⊂Rndto
denote an open subset of Rnd, where ndis the number of spatial dimensions. In this paper, nd= 3
in most cases, but there are subsections in which nd= 2 or nd= 1. The closure of Ω is denoted by
Ω and its boundary by ∂Ω. The set ∂Ω is supposed to have a well-defined unit outward normal. A
spatial point x∈Ω is represented in component notation as x= (x1, x2, . . . , xnd)T. Throughout
the manuscript, the variable t∈R+∪{0}denotes time. We often use functions which depend upon
space and time, for example φ: Ω ×[0, T ]7→ R, where [0, T ] is the time interval of interest. The
partial derivatives of φare interchangeably denoted as
∂iφ=∂φ
∂xi
, ∂tφ=∂φ
∂t (1)
We also make use of standard notation based on the operator ∇. For example, ∇φdenotes the
spatial gradient of φ. We often resort to the use of material time derivatives as defined, e.g., in
(Marsden and Hughes, 1994). For example, ˙
φdenotes the material derivative of φwhich may be
interpreted as a derivative with respect to time holding a material particle fixed. For a velocity field
u, we have ˙
φ=∂tφ+u·∇φ. Throughout the manuscript, we also utilize functions which are defined
on sets of material points, rather than spatial points. Material points are also called particles and
we denote them by X. For functions defined on sets of particles, material time derivatives are
simply regular partial derivatives with respect to time. Sets of particles are denoted as Ω0or P0.
The spatial position of a set of particles at time tis usually denoted Pt. Volumetric integrals defined
on sets of spatial points are usually denoted RΩdx, while for surface integrals we use the notation
R∂Ωda. When the integrals are defined on sets of material points, we use the notation RΩ0dXand
R∂Ω0dA.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

4ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
The reader should also keep in mind that this paper describes a number of phase-field theories
using the thermomechanics framework. Many of these theories involve, for example, Cauchy stress
tensors, which are usually denoted T. The reader should not understand that Tdenotes exactly
the same quantity throughout the manuscript. Rather, Tdenotes the Cauchy stress tensor of a
particular mechanical theory ans its precise definition may vary from one section to another. This
situation is not specific to Tand happens also with other quantities such as, for example, the
Helmholtz free energy Ψ, the mass flux h, the phase field φ, the reaction term R, the dissipation D
and others. The precise definition is made clear in each subsection such that no confusion arises.
2. From classical moving-boundary problems to phase-field models via diffusification
In this section we show how classical problems involving moving boundaries and interfaces can be
regularized and written as PDEs posed on a fixed domain using the theory of phase-field modeling.
The regularization replaces sharp interfaces by diffuse interfaces that are described with the help of
the phase-field variable. As indicated before, we shall refer to this particular type of regularization
as diffusification.
We consider the classical solidification theory for pure materials, also known as the generalized
Stefan problem. The classical formulation consists of two PDEs which hold on different, but
adjacent, time-dependent domains. The PDEs are coupled through conditions on the interface,
whose location changes with time and is a priori unknown. We demonstrate heuristically how to
diffusify this moving-boundary problem and obtain a phase-field model of solidification, which is
a PDE system on a fixed domain.
2.1. Moving-boundary problem for solidification: The generalized Stefan problem
Let us consider a solid-liquid system that may undergo phase transformations. The system occupies
the spatial domain Ω, which is fixed in time. The set Ω can be decomposed as Ω = ΩS∪ΩL, where
ΩSand ΩLare the regions of Ω that host the solid and liquid phase, respectively. The sets ΩSand
ΩLchange with time, and their motions are unknowns of the problem. The solid-liquid interface,
which is located where ΩSand ΩLmeet is denoted ΓLS . Omitting the boundary conditions on ∂Ω
for simplicity, the classical sharp-interface theory is described by the following equations
∂e
∂t + div q= 0 in ΩS∪ΩL(2)
λVn=−JqK·nLS on ΓLS (3)
JθK= 0 on ΓLS (4)
ωVn+σκ =λ(θ−θm) on ΓLS (5)
The system composed by Eqs. (2)–(5) is known as the generalized Stefan problem3. Eqns. (2) and
(3) correspond to the energy balance for a body with discontinuous material properties, (4) simply
reflects that the temperature is continuous across the solid–liquid interface, and (5) is the so-called
Gibbs-Thomson condition that determines the motion of the interface. In Eq. (2), eis the internal
energy and qis the heat flux. Importantly, both eand qare discontinuous across the solid-fluid
interface ΓLS due to the phase change. The internal energy is defined as
e=Cvθ+λχS(6)
where Cvis the heat capacity, θis the temperature, λis the latent heat, and χSis the characteristic
function of ΩS, that is, a function defined on Ω that takes the value 0 in ΩLand 1 in ΩS. The heat
flux is assumed to be governed by Fourier’s law4, that is, q=−k∇θ, where the conductivity k
3The classical Stefan problem or simply Stefan problem typically refers to the particular case in which ω=σ= 0.
4Fourier’s law is the most widely utilized constitutive theory for heat transmission, but there are other interesting
possibilities (G´omez et al., 2007a,b, 2008; Gomez et al., 2010a).
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 5
equals the value kSin ΩSand the (possibly distinct) value kLin ΩL. In Eq. (3), nLS is the unit
normal to ΓLS pointing towards the solid, Vnis the normal velocity of the interface in the direction
of nLS and the operator J·Kdenotes the jump across ΓLS such that JqK=qS−qL. In Eq. (5), ω
is the coefficient of kinetic undercooling, σis the surface tension of the liquid-solid interface, θmis
the melting temperature, and κis the sum of principal curvatures of ΓLS (with the sign convention
that κis positive for liquid spherical bubbles).
2.2. The idea behind diffusification
In the above sharp-interface theory, we have three conditions at the interface ΓLS, which may be
understood as one boundary condition for ΩS, one boundary condition for ΩL, and another one
to determine the interface motion which is unknown a priori. Problem (2)–(5) is thus a moving-
boundary problem. Note that in the particular case ω=σ= 0, the Gibbs–Thomson condition
becomes θ=θmon ΓLS . In this particular case, the interface can be defined as the θmlevel set of
the temperature. However, in the general case in which ωand σare non-zero, additional information
is needed to locate the interface. In a phase-field formulation the additional information for the
interface location is addressed by introducing a new variable, which is precisely the phase field, and
which is nothing else but a marker of the phases’ location. Note that this additional variable is not
an artifact of the phase-field theory, but it is fundamentally necessary because the temperature is
not enough to determine the position of the interface.
Let us proceed to diffusify the sharp-interface theory and obtain the corresponding phase-field
model. The key idea is to introduce a smooth field, namely, φ, which is defined on the entire domain
Ω=ΩS∪ΩLand which is time-dependent, i.e., φ=φ(x, t). This field will be designed in such
a way that it takes approximately the value −1 in the liquid phase, +1 in the solid phase, and it
transitions quickly across the normal direction to the interface ΓLS. Let us use as a fundamental
hypothesis a hyperbolic tangent profile, therefore
φ(x, t) = fdt(x)
√2:= tanh dt(x)
√2(7)
where dt(x) denotes the signed distance from xto ΓLS (negative in the liquid and positive in the
solid) and is a length scale related to the thickness of the diffuse interface. Note that ΓLS can
be identified by the zero level set of φ. The subscript tin dtemphasizes that ΓLS changes with
time, but will be removed from now on for notational simplicity. By using assumption (7), we will
heuristically derive a phase-field model. Then, we will show that as →0, the ansatz (7) is indeed
recovered by the solution of the phase-field model, which gives coherence to the framework.
2.3. Derivation of a phase-field model for solidification
We proceed by suitably diffusifying the equations in (2), (3) and (5). We start with (5) and compute
the geometric terms using the phase field φ.
From basic geometry, it is known that |∇d|= 1, nLS =∇dat ΓLS , and the curvature tensor κ
satisfies κ=∇∇dat ΓLS ; see, e.g., (Deckelnick et al., 2005). From Eq. (7), one can compute the
spatial derivatives
∂iφ=1
√2f0d
√2∂id(8)
∂2
ij φ=1
22f00 d
√2∂id∂jd+1
√2f0d
√2∂2
ij d(9)
Using the hyperbolic-tangent identities f0= 1−f2, and f00 =−2ff 0, we then obtain the curvature
tensor in terms of φ:
κij =∂2
ij d=√2
1−φ2∂2
ij φ+2φ
1−φ2∂iφ∂jφ(10)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

6ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
The additive curvature, defined as κ= ∆d, can then be written as
κ=√2
(1 −φ2)∆φ+2φ
1−φ2|∇φ|2(11)
Next, using again the relation f0= 1 −f2, it may be shown that
2
2|∇φ|2=1
4(1 −φ2)2(12)
Let us introduce at this point the function
W(φ) = 1
4(1 −φ2)2(13)
referred to as a double-well potential. The key feature of the function Wis that it has two local
minima which makes possible the coexistence of solid and liquid phases. Using the expression of W,
we obtain
κ=−√2
(1 −φ2)W0(φ)
−∆φ(14)
We will now compute the normal velocity of ΓLS, which is given by the zero level set of φ. Let
rLS (t) be a position vector that points to a material particle of the interface ΓLS , and define the
function e
φ(t) = φ(rLS (t), t). The function e
φis identically zero by definition of ΓLS. Therefore, its
derivative is also identically zero. Using the chain rule, we obtain
0 = e
φ0=∇φ·drLS
dt+∂tφ(15)
If we use the notation vLS = drLS/dt, we can rewrite (15) as
∂tφ+vLS · ∇φ= 0 (16)
which is known as the level set equation (Fedkiw and Osher, 2003; Sethian, 1999). Using Eq. (8),
and the definition of nLS , we conclude that
∇φ=1
√2f0d
√2nLS (17)
Vnis the velocity normal to ΓLS, that is, Vn=vLS ·nLS , thus introducing Eq. (17) into the level
set equation, and using that f0= 1 −f2= 1 −φ2, we get
Vn=−√2
1−φ2∂tφ(18)
Substituting (14) and (18) into (5), we have now derived the phase-field equation:
ω
σ∂tφ+W0(φ)
−∆φ=−λ
√2σ(1 −φ2)(θ−θm) (19)
To obtain a regularized energy equation, we shall now diffusify (2)–(3). We consider smooth test
functions vdefined on the entire domain Ω and for which v= 0 on the external boundary ∂Ω. The
test functions vare not necessarily zero on ΓLS , but [[v]] = 0 at ΓLS . We now claim that both (2)
and (3) are equivalent to the requirement
d
dtZΩ
ev dΩ = ZΩ
q· ∇vdΩ (20)
for all test functions v. To see this, note that the left-hand side equals
d
dtZΩL∪ΩR
ev dΩ = ZΩL∪ΩR
v ∂tedΩ −ZΓLS
[[e]]v VndΓ (21)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 7
by Reynolds’ Theorem, while the right-hand side equals
ZΩL∪ΩR
q· ∇vdΩ = ZΩL∪ΩR−vdiv qdΩ + ZΓLS
v[[q]] ·nLS dΓ (22)
Collecting the interior terms and the interface terms, and noting that [[e]] = λ, reveals the
equivalence with (2) and (3).
The final ingredients that we need to diffusify (20) are interpolation functions h(φ) and k(φ),
which are required to satisfy h(±1) = 1
2±1
2,k(+1) = kSand k(−1) = kL. The simplest choices
fulfilling these conditions are
h(φ) = 1
2(1 + φ) (23)
k(φ) = 1
2(1 + φ)kS+1
2(1 −φ)kL(24)
Using these interpolations, we approximate eand qby
e=Cvθ+λh(φ) (25)
q=−k(φ)∇θ(26)
With these approximations in mind, eand qare smooth across ΓLS , and Eq. (20) is then simply
equivalent to
∂te+ div q= 0 (27)
which, upon substitution, yields the diffusified energy equation
Cv∂tθ+λh0(φ)∂tφ−div k(φ)∇θ= 0 .(28)
The system of equations (19) and (28) completes the phase-field model, which represents an
elementary phase-field solidification theory.
2.4. The diffuse-interface transition profile: Asymptotic analysis
Given the phase-field model of solidification, Eqs. (19) and (28), let us verify that it indeed gives
rise to diffuse interfaces with the hyperbolic tangent profile (7) as →0. We give a heuristic outline,
but this procedure can be formalized using the theory of matched asymptotic expansions (Caginalp,
1989; Fife, 1988).
Let us first verify separation into phases on short time-scales. Let τ=t/2denote a rescaling of
time. Then, Eq. (19) can be written as
−2∂τφ+σ
ωW0(φ)=O(−1) as →0.(29)
where we have used Landau notation on the right hand side. For small the coefficient in front of
the left-hand side is significantly larger than the right-hand side so that
∂τφ≈ −σ
ωW0(φ),(30)
which means that φquickly evolves towards the stable roots φ=±1 of −W0(φ) = −φ3+φ. In
other words, φquickly separates into phases and generates a diffuse interface between them.
Let us denote the zero level set of φby Γ
LS . It is expected that Γ
LS converges to ΓLS as
→0. We next verify the profile of φin a neighbourhood of Γ
LS . We will zoom in to an O()-
neighbourhood of Γ
LS by considering a change of the independent variables. For simplicity, we
consider a two-dimensional problem. We change from the variables (x1, x2) to (z, s), where
z(x1, x2, t;) = d(x1, x2, t;)
(31)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

8ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
is a re-scaling of the signed distance to Γ
LS (t) and s(x1, x2, t;) is the arc length of Γ
LS (t). Denoting
φ(x1, x2, t) = Φ(z, s, t), then in a neighbourhood of Γ
LS :
∂tφ=∂tΦ + −1∂zΦ∂td+∂sΦ∂ts(32)
∆φ=−2∂zz Φ + −1∂zΦ ∆d+∂ssΦ|∇s|2+∂sΦ ∆s(33)
where we have used the identity ∇s· ∇d= 0. Substitution into (19) yields:
−2W0(Φ) −∂zz Φ=O(−1),(34)
so that upon neglecting the right-hand side, one obtains, in a neighbourhood of Γ
LS ,
W0(Φ) −∂zz Φ≈0.(35)
This is a second-order ordinary differential equation subject to the boundary conditions Φ(z)→ ±1
as z→ ±∞. Its solution is the anticipated hyperbolic tangent profile:
Φ(z) = tanh z
√2= tanh d
√2(36)
which was the hypothesis in the derivation of the phase-field model in Sect. 2.2.
3. General framework of thermomechanics and energy dissipation for phase-field models
This section illustrates the rigorous thermomechanical framework of phase-field theories. Our
approach to thermomechanics follows closely the classical work of (Truesdell and Noll, 1965).
Related work in the context of phase-field models can be found in (Gurtin, 1996) and (Oden et al.,
2010). The main idea behind this approach is that theories of mechanics are fundamental balance
laws of mass, linear momentum, angular momentum, and energy, while constitutive relations are
suitably restricted by demanding that the theory be energy dissipative (or entropy productive).
One refers to models derived this way as thermomechanically or thermodynamically consistent.
In the remainder of this section, we derive using the thermomechanics framework, the following
models:
•The classical Allen-Cahn and Cahn-Hilliard equations, which may be considered as the two
canonical models of non-conserved and conserved phase dynamics,
•The Navier-Stokes-Cahn-Hilliard equations which constitute a model for two-component
immiscible flows with surface tension,
•A simple phase-field model of brittle fracture, which is the diffuse version of Griffith’s theory
of fracture,
•A phase-field model of tumor growth, which predicts the dynamics of cancerous tumors and
nutrients in avascular tissue,
•The thermal Navier-Stokes-Korteweg equations, which are a model for liquid-vapor
transformations of a single-component fluid.
Another classical problem in which phase-field modeling is widely used is solidification. A
thermomechanics framework may be used to derive solidification models (Wang et al., 1993; Penrose
and Fife, 1990), but we do not cover this here because solidification was discussed in Sect. 2 using
a different perspective. We remark that the diffusification procedure employed in Sect. 2 does not
necessarily lead to thermomechanically-consistent models.
3.1. The idea behind thermomechanically-consistent phase-field modeling
Fundamental to phase-field thermomechanics is the dependence of the Helmholtz free energy Ψ
(or entropy) on values and gradients of the phase field φ, and, additionally, the dependence of
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 9
other constitutive relations on the values and/or gradients of the variational derivative of the total
energy functional with respect to the phase.
For example, the canonical free energy, which goes back to (Cahn and Hilliard, 1958), depends
on the phase φand its gradient ∇φ, and has the form
Ψ = G(φ) + 2
2|∇φ|2(37)
for some function Gand parameter . The corresponding energy functional is known as a Ginzburg–
Landau functional:
F(φ) = ZΩ
Ψ dx=ZΩG(φ) + 2
2|∇φ|2dx(38)
The variational derivative is then
δF
δφ =G0(φ)−2∆φ(39)
which is used to define the so-called chemical potential µ=δF
δφ =G0(φ)−2∆φ. In phase-field
models, constitutive relations are allowed to depend on µand/or ∇µ. This will be worked out in
detail for the above-mentioned phase-field models in the following sub-sections.
It is possible to interpret the energy functional Fas a diffusification of the surface-energy
functional
F(Γint) = ψZΓint
da(40)
where Γint is an interior surface and ψis a surface-energy coefficient. Note that F(Γint) associates
energy to an interface which is proportional to the length of the interface. The simplest connection
is obtained by considering F(φ) and F(Γint), where φis a diffusification of the interface
Γint, such that G(φ) = 2
2|∇φ|2. For example, in case that G(φ) = W(φ), recall (13), then
φ(x) = tanh dΓint (x)
√2ε, with dΓint the distance function to Γint. Note that this particular φsatisfies
the required identity; see (12). It can then be shown that
lim
→0
1
F(φ) = F(Γint) (41)
by employing the co-area formula
lim
→0
1
ZΩ
qdΓint (x)
dx=αqZΓint
da(42)
with constant αq=R∞
−∞ q(z) dz. The co-area formula is valid for suitably decaying functions q(z),
see (Du et al., 2005, Lemma 2.1). By taking the limit in (41), the parameter ψcan be suitably
identified. To sum up, phase-field energies collapse to surface energies as the interface thickness 
approaches zero.5
3.2. Allen-Cahn and Cahn-Hilliard
The two canonical phase-field theories are the Allen-Cahn equation and the Cahn-Hilliard equation
(Emmerich., 2003; Provatas and Elder, 2010). They both emanate from the Ginzburg–Landau
energy functional in Eq. (38), with G(φ) a double-well potential. The most usual approach would
be to take G(φ) = W(φ) = 1
4(1−φ2)2, but there are other possible choices, such as the logarithmic
5A rigorous connection between the functionals Fand Fcan be made using the key mathematical theory of
Γ-convergence (Ambrosio and Tortorelli, 1990; Braides, 2002; Dal Maso, 1993).
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

10 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
double well; see Eq. (194). Essentially the Allen-Cahn and Cahn-Hilliard equations are designed
in such a way that they are gradient flows for the energy. This means that the energy is time
decreasing along solutions to the equations. The Cahn-Hilliard equation incorporates the additional
requirement that the mass of the components is conserved, while the Allen-Cahn equation allows
for non-conservative dynamics.
The thermomechanical derivation of the Allen-Cahn equation and the Cahn-Hilliard equation
starts by defining the constitutive class6for Ψ:
Ψ = b
Ψ(φ, ∇φ) (43)
Let V ⊂ Ω denote an arbitrary part of the spatial domain Ω. The postulated energy-dissipation
property may be expressed as
d
dtZVb
Ψ(φ, ∇φ)dx=W(V)− D(V) (44)
where the dissipation D(V) is ≥0 for all conceivable processes, while W(V) denotes the working,
which is associated to external forces or energy supplies coming through the boundary of V. Basic
manipulations on the left hand side of (44) lead to
d
dtZV
Ψdx=ZVh∂φb
Ψ∂tφ+∂∇φb
Ψ·∂t(∇φ)idx(45)
Let us now integrate by parts the last term of Eq. (45) after flipping the time and space derivatives.
Doing so, we obtain, d
dtZV
Ψdx=ZV
µ ∂tφdx+Z∂V
∂∇φb
Ψ·n∂tφda(46)
where µis the chemical potential defined as the variational derivative:
µ=δ
δφ ZVb
Ψ dx=∂φb
Ψ−div ∂∇φb
Ψ(47)
Now, we proceed to derive the Allen-Cahn and the Cahn-Hilliard equations.
3.2.1. Allen-Cahn equation The Allen-Cahn equation can be derived by postulating the mass
balance ∂φ
∂t =−R(48)
where Ris designed to achieve free-energy dissipation and assumed to have the dependence:
R=b
R(φ, ∇φ, µ) (49)
Note the non-standard dependence of Ron the variational derivative of the free energy. This is
one of the essential features of the phase-field method. Now, to find restrictions on R, we combine
Eqs. (46)–(48), which leads to the expression
d
dtZV
Ψdx=ZV−µRdx+Z∂V
∂∇φb
Ψ·n∂tφda(50)
Here, we identify Dand Was D(V) = RVµRdxand W(V) = R∂V∂∇φb
Ψ·n∂tφda. The nonstandard
work term is caused by internal forces ∂∇φb
Ψ conjugate to changes in φ(Gurtin, 1996). It is clear
6Constitutive class is a common nomenclature in classical mechanics; see, e.g., (Truesdell and Noll, 1965; Gurtin
et al., 2009). In this particular case, it only means that the functional Ψ is allowed to depend upon φand ∇φ.
Fundamental principles could be used to restrict further the constitutive class Ψ = b
Ψ(φ, ∇φ). For example, frame
invariance mandates that Ψ only depends on ∇φthrough |∇φ|; see, for example, (Cahn and Hilliard, 1958).
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 11
that the constitutive choice b
R(φ, ∇φ, µ) = m(φ)µwith m(φ)≥0 achieves the required property of
free-energy dissipation.
For the classical choice
b
Ψ(φ, ∇φ) = W(φ) + 2
2|∇φ|2(51)
with W(φ) a double well potential, one has
µ=W0(φ)−2∆φ(52)
and we have thus derived the Allen-Cahn equation:
∂φ
∂t =−m(φ)W0(φ)−2∆φ.
3.2.2. Cahn-Hilliard equation We start with a statement of mass conservation7, namely
∂φ
∂t + div h= 0 (53)
where the constitutive class of his given by
h=b
h(φ, ∇φ, µ, ∇µ) (54)
Using again Eq. (46), it follows that
d
dtZV
Ψdx=ZV−µdiv hdx+Z∂V
∂∇φb
Ψ·n∂tφda(55)
Integrating by parts the first term on the right-hand side, we obtain
d
dtZV
Ψdx=ZV
h· ∇µdx+Z∂V−µh+∂∇φb
Ψ∂tφ·nda(56)
where the first term on the right-hand side is identified as −D(V) and the second as W(V).
Therefore, the constitutive choice
b
h(φ, ∇φ, µ, ∇µ) = −m(φ)∇µwith m(φ)≥0 (57)
guarantees free-energy dissipation and leads to the Cahn-Hilliard equation:
∂φ
∂t = div m(φ)∇W0(φ)−2∆φ(58)
for the above choice of b
Ψ. We note that µhin Eq. (56) can be interpreted as a free-energy flux.
3.3. Navier-Stokes-Cahn-Hilliard
The Navier-Stokes-Cahn-Hilliard equations represent a model for immiscible two-component fluid
flow with surface tension. This theory has been widely studied from the physical, the mathematical
and the computational points of view (Gurtin et al., 1996; Jacqmin, 1999; Gal and Grasselli, 2010;
Kay et al., 2008; Boyer et al., 2010; Gal and Grasselli, 2011; Lowengrub and Truskinovsky, 1998;
Colli et al., 2012; Kim et al., 2004; Liu and Shen, 2003). To derive the theory, let us assume that
we have two immiscible components with volume fractions ϕ1and ϕ2, respectively. The volume
fractions verify the constraint
ϕ1+ϕ2= 1 (59)
7Note that Eq. (53) achieves mass conservation, i.e, d
dt(RΩφdx) = 0, for natural boundary conditions h·n= 0 on
∂Ω.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

12 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
The velocities of the components are denoted u1and u2, and their densities ρ1and ρ2. The density
of the mixture, namely, ρis defined as ρ=ϕ1ρ1+ϕ2ρ2. The mass-averaged velocity of the mixture
is defined as
u=1
ρ(ϕ1ρ1u1+ϕ2ρ2u2) (60)
For simplicity, both components of the mixture are supposed to be incompressible with constant
density. In addition, we will assume ρ1=ρ2. Taken together, these assumptions imply that the
mixture density ρis also a constant equal to ρ1and ρ2. Therefore, without loss of generality, we
can take
ρ=ρ1=ρ2= 1 (61)
In this scenario, mass conservation of the components implies
d
dtZPt
ϕα= 0; α= 1,2 (62)
where Ptis a set of material particles in the current configuration. Using Reynold’s theorem we
can rewrite Eq. (62) as
∂tϕα+ div(ϕαuα) = 0; α= 1,2 (63)
Summing Eqns. (63) over α, and using Eqns. (59), (60), and (61), we obtain the mixture mass
balance
div u= 0 (64)
Let us define at this point, the phase field φ=ϕ1−ϕ2, the so-called diffusion velocities wα=uα−u
for α= 1,2 and phase flux h=ϕ1w1−ϕ2w2. If we subtract Eqns. (63), we obtain the phase-field
equation
∂tφ+ div(φu) + div h= 0 (65)
If we further assume a classical mixture (Romano and Marasco, 2010, Sec. 6.2), that is, a single
momentum equation determines the mixture velocity u, then we have the following standard linear
momentum balance equation
˙
u= div T+b(66)
where Tis the Cauchy stress tensor of the mixture which we assume to be symmetric8and ˙
u
denotes the material derivative of u. The fundamental equations of the theory are (64),(65) and
(66).
Now, we will use the Coleman–Noll procedure (Coleman and Noll, 1963) to find constitutive
equations for hand Twhich ensure that the total energy is dissipated along solutions of the
theory. Under the assumption (61), the total energy (sum of free energy and kinetic energy) for an
arbitrary part Ptmay be written as9
E(φ, u) = ZPtΨ + 1
2|u|2dx, (67)
The first term on the right hand side of Eq. (67) is assumed to pertain to the constitutive class
Ψ = b
Ψ(D, φ, ∇φ),(68)
where
D=1
2(∇u+∇uT) (69)
and ∇uTdenotes the transpose of ∇u. Let us introduce the chemical potential µ, as before,
µ=δE
δφ =∂φb
Ψ−div ∂∇φb
Ψ (70)
8In general, for multiphase systems balance of angular momentum does not imply the symmetry of T(Bowen, 1976).
9Later in this section, we show why this is a reasonable energy for the physical problem we are treating.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 13
where δE/δφ denotes the variational derivative of Ewith respect to φ. We assume that the mixture
stress tensor Tand and the mass flux hbelong to the constitutive classes
T=b
T(D, φ, ∇φ, µ, ∇µ) (71)
h=b
h(D, φ, ∇φ, µ, ∇µ) (72)
We postulate now the following energy dissipation law
d
dtZPtΨ + 1
2|u|2dx=W(Pt)− D(Pt) (73)
where W(Pt) denotes the working, which is associated to external forces or energy supplies coming
through the boundary of Ptand D(Pt) denotes the dissipation. As before, we will require that
D(Pt)≥0 for all conceivable processes.
Let us compute the left-hand side of (73). Using basic results of continuum mechanics and the
constitutive class of Ψ, defined in (68), it follows that
d
dtZPt
Ψdx=ZPt˙
Ψ + Ψ div udx=ZPt∂DΨ : ˙
D+∂φΨ˙
φ+∂∇φΨ·(∇φ)·dx, (74)
where we have also made use of Eq. (64). Using the relation
(∇φ)·=∇˙
φ− ∇u∇φ(75)
in Eq. (74), and integrating by parts the term that results from ∇˙
φ, we obtain
d
dtZPt
Ψdx=ZPt∂DΨ : ˙
D+µ˙
φ−∂∇φΨ· ∇u∇φdx+Z∂Pt
˙
φ∂∇φΨ·nda(76)
Using Eq. (65) to compute ˙
φ, and integrating by parts the resulting term, we get
d
dtZPt
Ψdx=ZPt∂DΨ : ˙
D+∇µ·h−∂∇φΨ· ∇u∇φdx+Z∂Pt−µh+˙
φ∂∇φΨ·nda(77)
We can now proceed similarly with the kinetic energy. Using Reynold’s theorem, Eq. (64) and Eq.
(66), it follows
d
dtZPt
1
2|u|2dx=ZPtu·˙
u+1
2|u|2div udx=ZPt
u·(div T+b) dx(78)
If we use now integration by parts on the term u·div T, we obtain
d
dtZPt
1
2|u|2dx=ZPt
(−T:∇u+b·u) dx+Z∂Pt
u·T nda(79)
If we sum (77) and (79), and we compare it with (73), we can identify the working W(Pt) and the
dissipation D(Pt) as follows
W(Pt) = ZPt
b·udx+Z∂Pt−µh·n+˙
φ∂∇φΨ·n+u·T nda(80)
D(Pt) = ZPt−∂DΨ : ˙
D− ∇µ·h+∂∇φΨ· ∇u∇φ+T:∇udx(81)
Now, D(Pt)≥0 for all Ptimplies that the integrand has to be ≥0. There are several ways to
design constitutive laws that satisfy that requirement, but in order to keep the process as simple
as possible, we will impose that all the terms in the integrand of (81) are pointwisely positive
or zero by themselves. For example, given that the dependence of Ψ on ˙
Dhas not been allowed
in the constitutive class (68), the only way to make the term −∂DΨ : ˙
Dpointwisely positive or
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

14 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
zero is by taking ∂DΨ = 0 which implies Ψ = b
Ψ(φ, ∇φ). Considering this, and using the identity
∂∇φΨ· ∇u∇φ=∂∇φΨ⊗ ∇φ:∇u, we are left with the inequality
−∇µ·h+T:∇u+∂∇φΨ⊗ ∇φ:∇u≥0 (82)
Without loss of generality, let us consider now the following splittings,
T=Td−pIwith Td=T+pI(83)
∇u=D+Wwith W=1
2(∇u− ∇uT) (84)
where Irepresents the identity tensor and pis a scalar field that represents the mechanical pressure,
which may be interpreted as a Lagrange multiplier of the incompressibility constraint. Using Eq.
(83), it follows that
T:∇u=T: (D+W) = T:D= (Td−pI) : D=Td:D(85)
where we have used basic properties of the double-dot product10 and Eq. (64). Let us mention
now that, although ∂∇φΨ⊗ ∇φis not a symmetric tensor for arbitrary Ψ, physically relevant
free energies always make the tensor symmetric. Indeed, the symmetry of ∂∇φΨ⊗ ∇φimposes a
constraint on how the term ∇φenters the free energy density Ψ, but this constraint only rules out
terms which would be excluded anyway by using frame-invariance arguments, as shown in (Cahn
and Hilliard, 1958). Therefore, we will use the fact that ∂∇φΨ⊗ ∇φis a symmetric tensor for
physically relevant forms of Ψ, and thus,
∂∇φΨ⊗ ∇φ:∇u=∂∇φΨ⊗ ∇φ:D(86)
As a consequence, we can guarantee that the inequality (82) is satisfied by taking
h=−m∇µ(87)
Td+∂∇φΨ⊗ ∇φ= 2νD(88)
where mand νare positive functions11 that belong to the same constitutive classes as hand T,
respectively. The function νrepresents viscosity, and mis a mobility. From Eq. (88), we obtain
T=−pI+ 2νD−∂∇φΨ⊗ ∇φ(89)
Finally, considering a usual form for Ψ, that is, Ψ = γ
W(φ) + 2
2|∇φ|2, where γis the surface
tension, and Wis a double-well potential, we obtain
µ=γ1
W0(φ)−∆φ(90)
∂∇φΨ = γ∇φ(91)
which finalizes the derivation of the theory. For completeness, we rewrite here the complete model
∂φ
∂t + div(φu)−div mγ ∇1
W0(φ)−∆φ= 0 (92)
˙
u+∇p= div 2νD−γ∇φ⊗ ∇φ+b(93)
div u= 0 (94)
10In particular, we use the following property of the double dot product. If Ais a symmetric tensor, then
A:B=A: (B+BT)/2. As a consequence, if Ais symmetric and Bis skew-symmetric, A:B= 0.
11In many cases of practical interest, mand νwill be simply constants.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 15
Note that the final energy functional for a part Ptis given by Eq. (67) with Ψ =
γ
W(φ) + 2
2|∇φ|2:
E(φ, u) = ZPt1
2|u|2+γ1
W(φ) + 
2|∇φ|2dx(95)
It thus consists of two parts: a kinetic energy part and a mixing energy (or Ginzburg–Landau)
part. As explained at the end of Sect. 3.1, this energy represents a diffusification of the classical
energy of a two-phase moving boundary problem:
ZPt
1
2|u|2dx+ψZΓ∩Pt
da(96)
consisting of kinetic energy and energy associated to surface tension (Gross and Reusken, 2011,
Ch. 6), where Γ ∩ Ptis the part of the two-phase interface inside of Pt. The constant ψcan
be show to be proportional to γwith a proportionality factor depending on the double well
function W(φ) (Anderson et al., 1998).
3.4. Phase-field fracture theory
Classical approaches to treat fracture problems in computational mechanics include the virtual
crack closure technique (Krueger, 2004) and the extended finite element method (Mo¨es et al.,
1999). These methods represent cracks as discrete discontinuities. From a computational point of
view, this has been addressed by introducing discontinuity lines by way of remeshing or enriching
the displacement field by means of the partition of unity method (Babuska and Melenk, 1997).
However, this procedure has proven particularly difficult for problems in which a large number of
cracks interact dynamically in complicated three-dimensional geometries. Phase-field methods, in
contrast, permit solving the problem on a fixed mesh, which justifies their recent popularity in the
computational mechanics community (Bourdin et al., 2008; Borden et al., 2012; Miehe et al., 2010;
Abdollahi and Arias, 2011). The derivation of phase-field fracture theories can be interpreted as a
diffuse version of the classical Griffith’s theory of fracture.
To introduce the phase-field fracture theory we consider a reference configuration Ω0, which may
be associated to the initial position of the solid. Points in Ω0are called material points or particles.
The motion of the solid is defined by a time-dependent mapping ϕwhich takes Ω0into the current
configuration that we denote Ω. The displacement field is defined as d=ϕ−Id, where Id denotes
the identity mapping. To develop the crack theory we also need to introduce the quantities
F=∇ϕ,C=F F T(97)
With these definitions, we can introduce the free energy of the system, which is essentially
obtained by regularizing the sharp-energy functional associated to Griffith’s fracture theory:
E=ZΩ0Ψe+1
2ρ0|˙
d|2dX+GcZΓint
dA(98)
where Ψeis the stored elastic energy density, ρ0is the density in the reference configuration,
Gcis the fracture toughness and Γint is a two-dimensional manifold containing the cracked area.
The regularized energy functional is obtained by replacing the surface integral on Γint by a volume
integral on Ω0which suitably converges to the surface integral as explained in Sect. 3.1. Therefore,
the diffuse approximation to the energy functional may be written as
E=ZΩ0Ψe+1
2ρ0|˙
d|2dX+GcZΩ01
F(c) + 
2|∇c|2dX(99)
where the variable cis a phase field which describes whether the material is fractured or not
and which approaches the value 1 away from the crack and equals 0 inside the crack. The chief
advantage of the phase-field approach is that the integrals in (99) are posed on a fixed and known
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

16 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
domain, in contrast with the surface integral of Eq. (98). From a computational standpoint, this
avoids remeshing and tracking the interface algorithmically.
We next derive the phase-field fracture model. Consider a subset P0⊂Ω0of the reference
configuration. The total energy associated to P0is
ZP0Ψ + 1
2ρ0|˙
d|2dX(100)
where ρ0is the density in the reference configuration and
Ψ = b
Ψ(C, c, ∇c) = b
Ψe(C, c) + b
Ψc(c, ∇c) (101)
The elastic energy b
Ψeis assumed to depend on Cand c, while b
Ψcis defined as:
b
Ψc(c, ∇c) = Gc1
F(c) + 
2|∇c|2(102)
where F(c) = 1
2(c−1)2is a single-well potential. For future reference, let us define
µc=δ
δc ZP0b
Ψc(c, ∇c) dX=Gc1
F0(c)−∆c(103)
We start the theory by stating classical balance laws, namely,
˙c=−R(104)
ρ0¨
d= div P+b0(105)
where Pis the first Piola-Kirchhoff stress tensor and b0represents a body force per unit volume12.
We note that both equations are written in Lagrangian form.
Let us define at this point the following constitutive classes
P=b
P(C, c, ∇c) (106)
R=b
R(C, c, ∇c, µc) (107)
We postulate the energy dissipation inequality,
d
dtZP0Ψ + 1
2ρ0|˙
d|2dX=W(P0)− D(P0) (108)
where W(P0) is the working and D(P0) the dissipation. By requiring that the dissipation is
nonnegative for all conceivable processes we will find constitutive equations for Pand R. In fact,
d
dtZP0Ψ + 1
2ρ0|˙
d|2dX=ZP0∂Cb
Ψe:˙
C+∂cb
Ψe˙c+∂cb
Ψc˙c+∂∇cb
Ψc·(∇c)·+ρ0¨
d·˙
ddX
(109)
Since we are working in Lagrangian coordinates now, (∇c)·=∇˙c, and Eq. (109) can be rewritten
as
d
dtZP0Ψ + 1
2ρ0|˙
d|2dX=ZP0∂Cb
Ψe:˙
C+∂cb
Ψe˙c+µc˙c+˙
d·(div P+b0)dX
+Z∂P0
∂∇cb
Ψc·n˙cdA
=ZP0∂Cb
Ψe:˙
C−P:∇˙
d+b0·˙
d−(∂cb
Ψe+µc)b
RdX
+Z∂P0∂∇cb
Ψc·n˙c+P n ·˙
ddA(110)
12For a derivation of Eq. (105), the reader is referred to (Marsden and Hughes, 1994), for example.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 17
Introducing the second Piola-Kirchhoff stress tensor Sdefined as P=F S, and using the relation
P:˙
F=1
2S:˙
C, we obtain
d
dtZP0Ψ + 1
2ρ0|˙
d|2dX=ZP0∂Cb
Ψe−1
2S:˙
C−∂cb
Ψe+µcb
RdX
+Z∂P0˙c ∂∇cb
Ψc·n+P n ·˙
ddA+ZP0
b0·˙
ddX(111)
Therefore, we identify
W(P0) = Z∂P0∂∇cb
Ψc·n˙c+P n ·˙
ddA+ZP0
b0·˙
ddX(112)
D(P0) = −ZP0∂Cb
Ψe−1
2S:˙
C−∂cb
Ψe+µcb
RdX(113)
The choices
S= 2∂Cb
Ψe(114)
b
R=m∂cb
Ψe+µc(115)
where mis a positive function, guarantee that the dissipation is non-negative for all conceivable
processes. The final form of the theory is
˙c=−m∂cb
Ψe+Gc1
F0(c)−∆c(116)
ρ0¨
d= div(2F∂Cb
Ψe) + b0(117)
We note that in some applications, the term ˙cis neglected in the balance equations. Impressive
simulations using this theory and slightly modified versions (including how to address irreversibility
of the crack surface evolution) may be found in, for example, (Borden et al., 2012, 2014; Abdollahi
and Arias, 2011).
3.5. Mechano-biological mixtures: Phase-field tumor growth theory
Phase-field modeling is also important in a number of emerging fields within computational
mechanics. In this section, we consider the application of phase fields to mechano-biological
continua. In particular, we consider avascular tumor growth, which is the growth of cancerous
tumors without blood vessels. Phase-field tumor growth models have recently been proposed; see,
e.g., (Cristini et al., 2009), (Lowengrub et al., 2010) and (Oden et al., 2015). Thermomechanical
derivations of such models have been considered in (Wise et al., 2008), (Oden et al., 2010) and
(Hawkins-Daarud et al., 2012).
The starting point is the continuum theory of mixtures, as developed in (Truesdell, 1984,
Lecture 5, 6) and (Bowen, 1976). Consider an isothermal mixture with Nconstituents (or species),
α= 1, . . . , N . For tumors, such constituents include typically a tumor-cell phase, healthy-cell
phase and a nutrient phase, but may also include extracellular water, extracellular matrix, other
cell phases such as a necrotic-cell phase, and various chemicals. Each constituent can be assigned
a mass fraction (or concentration) cαwhich is the mass of constituent αper unit mass. The mass
balance for constituent αis:
∂t(ρcα) + div(ρcαuα) = γα,(118)
where ρis the density of the mixture, uαthe velocity of constituent α, and γαthe mass supply
of α(owing to reactions). We assume saturation, i.e., Pαcα= 1. Invoking the axiom of mixture
mass balance: X
α
γα= 0 ,(119)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

18 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
the global mass balance is recovered by summing (118) over all α:
∂tρ+ div(ρu) = 0 ,(120)
where u=Pαcαuαis the (mass-averaged) mixture velocity. A diffusion-form of (118) can be
obtained by employing the diffusion velocity wα=uα−u, yielding
ρ˙cα+ div(ρhα) = γα(121)
where hα=cαwαis the mass-fraction diffusion flux, and the dot ˙
( ) denotes a material derivative
with respect to the mixture velocity u.
Instead of mass fractions cα, one can work with volume fractions ϕα=cαρ/ραwhere ραis the
specific density (mass of αper unit volume of α). In this case the diffusion form is
(ραϕα)·+ραϕαdiv u+ div(ραϕαwα) = γα.(122)
As in Sec. 3.3, continuing with the assumption of a classical mixture, we additionally have the
global momentum equation
ρ˙
u= div T+b(123)
where we have taken a fluid-mixture viewpoint, Tbeing the Cauchy stress tensor and ba body
force. A homogenized equation such as u=−K∇pis also possible, with Ka tensor and pa
pressure field. Mixtures with both fluid and solid components are considered in (Oden et al.,
2010).
We now develop the constitutive theory for a particular phase-field model. Let us assume for the
sake of simplicity constant densities, all equal to 1:
ρ(x) = ρα(x) = 1 ,(124)
hence, by (120),
div u= 0 .(125)
In this case the constituent mass balance (121) simplifies to
˙cα+ div hα=γα.(126)
For volume fractions, (122) simplifies to the same equation in (126) but with cαreplaced by ϕα. In
fact, ϕα=cα. As in the Navier–Stokes–Cahn–Hilliard and phase-field fracture theory, the current
theory is based on the dissipation law:
d
dtZPtΨ + 1
2|u|2dx=W(Pt)− D(Pt) (127)
and requires D(Pt)≥0 for all conceivable processes. Following the phase-field modeling paradigm,
we choose a constitutive class with dependence on gradients:
Ψ = b
Ψ(D,{cα},{∇cα}) (128)
where Dis defined in (69) and {(·)α}is a short-hand notation for (·)1,(·)2,...,(·)N. Furthermore,
we consider the following classes for the stress, mass fluxes and reaction terms:
T=b
T(D,{cα},{∇cα},{µα},{∇µα}) (129)
hα=b
hα(D,{cα},{∇cα},{µα},{∇µα}),(130)
γα=bγα(D,{cα},{∇cα},{µα},{∇µα}) (131)
with
µα=δb
Ψ
δcα
=∂cαb
Ψ−div(∂∇cαb
Ψ) .(132)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 19
As before, by invoking the Coleman–Noll argument, suitable restrictions on the constitutive models
are obtained. In particular, the stress Tdecomposes as
T=−pI−X
α
∂∇cαb
Ψ⊗ ∇cα+b
Tvisc ,(133)
where pis the pressure field which acts as a Lagrange multiplier for the incompressibility
constraint (125), and b
Tvisc is the viscous part of the stress (typically chosen as 2νD, cf. (88)).
Furthermore, it can be shown that the dissipation D(Pt) can be identified as
D(Pt) = ZPtb
Tvisc :D−X
αµαbγα+∇µα·b
hαdx . (134)
The requirement D(Pt)≥0 displays three dissipative processes: viscous dissipation, dissipation
owing to reactions, and multi-component diffusion.
To illustrate an elementary thermodynamically-consistent tumor-growth model, we neglect the
momentum equation (123) (i.e., u=0) and consider four constituents:
c1= tumor cells ,(135)
c2= healthy cells ,(136)
c3= nutrient-rich extracellular water ,(137)
c4= nutrient-poor extracellular water ,(138)
We assume that the cell-phases occupy a fixed fraction ¯ccell ∈(0,1), i.e.,
c1(x) + c2(x) = ¯ccell ⇒c3(x) + c4(x)=1−¯ccell .(139)
Thus, it is sufficient to work with two variables, e.g., the tumor phase and nutrient concentration:
φ:= c1and σ:= c3(140)
respectively. We select a free energy
b
Ψ(φ, σ, ∇φ) = WT(φ) + 2
2|∇φ|2+1
2δσ2(141)
with WT(φ) a double well function with wells at φ= 0 and φ= ¯ccell . This free energy corresponds
to phase separation for the tumor phase φand ordinary diffusion of nutrients σ. The relevant
chemical potentials are
µ1=W0
T(φ)−2∆φand µ3=σ
δ(142)
To guarantee that D(Pt)≥0, we choose
bγ1=δ g(φ)µ3−µ1(143)
bγ3=−bγ1(144)
b
h1=−m∇µ1(145)
b
h3=−Dδ∇µ3(146)
for some growth function g(φ)≥0, mobility m > 0, diffusivity D > 0, and reversibility
parameter δ > 0 (which controls the amount of growth versus decline of the tumor phase). The
final model is then given by the constituent mass balance equations in (121). With the above
constitutive choices, and assuming that mand Dare constants, they result in:
∂tφ=m∆µ1+g(φ)(σ−δµ1),(147)
∂tσ=D∆σ−g(φ)(σ−δµ1),(148)
which is a Cahn–Hilliard/Allen–Cahn equation coupled to an reaction-diffusion equation.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

20 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
3.6. Thermally-coupled Navier-Stokes-Korteweg equations
The Navier-Stokes-Korteweg equations are a phase-field theory for single-component two-phase
flows. Here, the two phases refer to liquid and gaseous states of the same fluid and are naturally
located by the density field which identifies vapor with the low-density regions and liquid with
the high-density areas. This model is somewhat different from those discussed previously because
the phase-field plays a dual role. Indeed, the density field is used to locate the phases, but also
represents an important physical quantity used in classical compressible gas dynamics. The Navier-
Stokes-Korteweg equations allow for liquid-vapor phase transformations induced by temperature
or pressure variations. Therefore, this model opens the possibility to solve vexing problems in
computational mechanics such as for example, cavitation, film and nucleate boiling or the phase-
change-driven implosion of thin structures (Bueno et al., 2014). The Navier-Stokes-Korteweg
equations are rather new in the field, but there are interesting works of theoretical (Dunn and
Serrin, 1986) and computational nature (Gomez et al., 2010b; Liu et al., 2013, 2015; Tian et al.,
2015; Giesselmann et al., 2014; Jamet et al., 2001).
Our point of departure to derive the Navier-Stokes-Korteweg equations are classical balance laws
for mass, linear momentum, angular momentum and energy, which may be written as follows
˙ρ+ρdiv u= 0 (149)
ρ˙
u= div T+ρb(150)
T=TT(151)
ρ˙e= div(T u)−div q−div π+ρb·u+ρr (152)
Here, ρis the density, uis the velocity, Trepresents the Cauchy stress tensor, bis a body force per
unit mass, eis the total energy density, qis the heat flux, ris the radiation and πrepresents the
energy variation due to interfaces. πis the only non-standard term and will be obtained directly
from the Coleman-Noll procedure. To use the Coleman-Noll approach we will use the following
entropy-production law,
d
dtZPt
ρs dx=J(Pt) + D(Pt) (153)
where J(Pt) is an entropy flux associated to external heat sources and boundary terms and D(Pt)
is required to be positive or zero. Using Reynold’s theorem on the left hand side of Eq. (153), and
the mass conservation equation, the entropy-production law can be rewritten as
ZPt
ρ˙sdx=J(Pt) + D(Pt) (154)
Using standard thermodynamic relationships, the energy density eis computed as the sum of
the internal energy density ιand kinetic energy density, namely,
e=ι+1
2|u|2(155)
Taking the material time derivative of Eq. (155), and multiplying with ρ, we obtain
ρ˙e=ρ˙
ι+ρu˙
u(156)
Let us use now the standard definition of the Helmholtz free energy Ψ, which is,
Ψ = ι−θs (157)
Taking the time derivative of Eq. (157) and using Eq. (156) we can derive the relation
ρ˙
Ψ = ρ˙e−ρu˙
u−ρ˙
θs −ρθ ˙s(158)
Let us now use the expressions of ρ˙eand ρ˙
ufrom Eqns. (152) and (150). We substitute them in
(158), which results in
ρ˙
Ψ = T:∇u−div q−div π+ρr −ρ˙
θs −ρθ ˙s(159)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 21
We consider the following constitutive class for the Helmholtz free energy Ψ:
Ψ = b
Ψ(ρ, ∇ρ, θ) (160)
Using the chain rule, it follows that
˙
Ψ = ∂ρb
Ψ ˙ρ+∂∇ρb
Ψ(∇ρ)·+∂θb
Ψ˙
θ(161)
Substituting the value of ˙
Ψ given in Eq. (161), and applying to ρthe relation given in Eq. (75),
we obtain from (159)
ρ∂ρb
Ψ ˙ρ+ρ∂∇ρb
Ψ (∇˙ρ− ∇u∇ρ) + ρ∂θb
Ψ˙
θ=T:∇u−div q−div π+ρr −ρ˙
θs −ρθ ˙s(162)
Without loss of generality, let us split Tand ∇uas follows
T=Td+Th(163)
∇u=Ld+Lh+W(164)
where
Td=T−tr T
3Iand Th=tr T
3I(165)
and
Ld=1
2(∇u+∇uT)−1
3div uI;Lh=1
3div uI;W=1
2(∇u− ∇uT) (166)
Note that Tdand Ldare traceless and Wis skew-symmetric. This implies that
T:∇u=T: (Ld+Lh+W) = T:Ld+T:Lh
= (Td+Th) : Ld+ (Td+Th) : Lh=Td:Ld+Th:Lh
=Td:Ld+tr T
3div u=Td:Ld−˙ρ
ρ
tr T
3(167)
where in the last identity we have used the mass conservation equation (149). In what follows, we
will also use the relation
ρ∂∇ρb
Ψ∇u∇ρ=ρ∂∇ρb
Ψ⊗ ∇ρ:∇u=ρ∂∇ρb
Ψ⊗ ∇ρ: (Ld+Lh)
=ρ∂∇ρb
Ψ⊗ ∇ρ:Ld+ρ
3div utr ρ∂∇ρb
Ψ⊗ ∇ρ
=ρ∂∇ρb
Ψ⊗ ∇ρ:Ld−˙ρ
3tr ρ∂∇ρb
Ψ⊗ ∇ρ(168)
Using Eqns. (167) and (168) in expression (162), we obtain
ρ˙s=1
θ(−hρ∂ρb
Ψ + tr T
3ρ+1
3tr ρ∂∇ρb
Ψ⊗ ∇ρi˙ρ−ρ∂∇ρb
Ψ∇˙ρ−ρ∂θb
Ψ + s˙
θ
+Td+ρ∂∇ρb
Ψ⊗ ∇ρ:Ld−div q−div π+ρr)(169)
Using now the identities
ρ∂∇ρb
Ψ∇˙ρ= div ρ∂∇ρb
Ψ ˙ρ−div ρ∂∇ρb
Ψ˙ρ(170)
−1
θdiv q=−div q
θ−1
θ2∇θ·q(171)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

22 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
and doing some manipulations, we obtain
ZPt
ρ˙sdx=−Z∂Pt
q
θ·nda+ZPt
ρrdx
+ZPt
1
θ"−div π+ρ∂∇ρb
Ψ ˙ρ−ρ∂θb
Ψ + s˙
θ#dx
−ZPt
˙ρ
θ"ρ∂ρb
Ψ + tr T
3ρ−div ρ∂∇ρb
Ψ+1
3tr ρ∂∇ρb
Ψ⊗ ∇ρ#dx
+ZPt
1
θ"Td+ρ∂∇ρb
Ψ⊗ ∇ρ:Ld−1
θ∇θ·q#dx
=J(Pt) + D(Pt) (172)
where the entropy flux turns out to be the classical term
J(Pt) = −Z∂Pt
q
θ·nda+ZPt
ρrdx(173)
and D(Pt) is defined implicitly by Eqns. (172)–(173). Let us introduce at this point the following
constitutive classes,
T=b
T(ρ, ∇ρ, θ, µ) with µ=δ
δρ ZΩb
Ψdx(174)
q=b
q(∇θ) (175)
π=b
π(ρ, ∇ρ, θ, ˙ρ) (176)
s=bs(ρ, ∇ρ, θ) (177)
Now, it may be easily proven that the following constitutive choices guarantee that D(Pt) remains
nonnegative for all conceivable processes
tr T
3=−ρ2µ+ρ∇ρ·∂∇ρb
Ψ−ρ
3tr ∂∇ρb
Ψ⊗ ∇ρ(178)
s=−∂θb
Ψ (179)
Td= 2νLd−ρ∂∇ρb
Ψ⊗ ∇ρ(180)
q=−k∇θ(181)
π=−ρ∂∇ρb
Ψ ˙ρ(182)
We notice that the term −ρ2µin Eq. (178) may also be written as −ρ2∂ρb
Ψ−div(∂∇ρb
Ψ)and it is
customary to identify −ρ2∂ρb
Ψ with the thermodynamic pressure p. This concludes the derivation.
If we want to use this theory as a predictive model, we need to give a precise definition of the
Helmholtz free energy. The usual choice, associated to van der Waals fluids is
b
Ψ(ρ, ∇ρ, θ) = −aρ +Rθ log ρ
b−ρ−Cvθlog θ
θref +Cvθ+¯
λ
2ρ|∇ρ|2(183)
where Ris the specific gas constant, Cvis the specific heat capacity, θref is a reference temperature,
¯
λis associated to the surface tension of the liquid-vapor interface, and aand bare positive
constants.
4. Energy-dissipative time-integration schemes
Time discretization methods for phase-field models typically have to deal with a nonlinear term
W0(φ) originating from the nonconvex (double-well in case of a binary system) free energy W(φ).
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 23
The nonconvexity allows for ‘backwards’ diffusion, which is only mildly regularized by the higher-
order operator coming from the interfacial energy, so that naive time-discrete schemes can lack
stability. Because of this difficulty, the past decade has seen an increase in the development of
novel time-discretization methods.
To illustrate the main ideas behind the existing methods, let us consider the time discretization
of the elementary Cahn–Hilliard equation in split form13
∂tφ= ∆µin Ω ×(0, T ) (184)
µ=W0(φ)−2∆φin Ω ×(0, T ) (185)
subject to natural boundary conditions on ∂Ω
∇φ·n= 0 and ∇µ·n= 0 (186)
for all t∈(0, T ). As mentioned before, this model satisfies the free-energy dissipation
d
dtE(φ) = −ZΩ|∇µ|2dx(187)
where
E(φ) = ZΩW(φ) + 1
22|∇φ|2dx(188)
Let tn,n= 0,1,...,denote discrete time-instances at which we wish to approximate the solution.
Corresponding approximations are denoted
φn≈φ(·, tn) and µn≈µ(·, tn).(189)
For simplicity we assume tn=nτ, where τis a constant time-step size, but all schemes can be
extended to the case of nonuniform time steps. Schemes are presented without discretization in
space because we aim at understanding scheme properties which are independent of the spatial
discretization. In the consideration of time-discrete schemes (Hairer et al., 1987; Hughes, 2012),
we shall pay particular attention to
(1) the order of accuracy of the scheme,
(2) the stability of the algorithm,
(3) the solvability of the time-discrete equations.
In the context of linear problems, stability is associated to the decay of the time-discrete solution
with time, a feature also attained by the exact solution of stable linear problems. When the stability
of the time discrete solution is achieved independently of τ, the time-integration scheme is said
to be unconditionally stable. If stability holds under some constraint (e.g., on the time-step size)
then the scheme is said to be conditionally stable. Explicit time integration algorithms which are
consistent are always conditionally stable, while implicit schemes might be unconditionally stable.
For nonlinear problems the situation is much more complex than for linear equations. First,
several notions of stability may be defined for different problems. Second, designing unconditionally
stable schemes requires much more ingenuity than making the algorithm implicit. Because of the
fundamental energetic structure behind phase-field models, see e.g. Eq. (187), natural notions of
stability are those related to free-energy dissipation. In particular, a numerical scheme is said to
be unconditionally energy- (or gradient-, or nonlinearly )stable if
E(φn+1)− E(φn)≤0 for all n≥0,(190)
13For the analysis of time integration schemes, it is convenient to split the Cahn-Hilliard equation, introducing the
variable µ, which represents the chemical potential.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

24 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
independent of τ. In other words, the scheme preserves the energy-dissipative property of the
underlying model, in the sense that it dissipates energy at each time step. Analogously to the
linear case, if (190) holds under some constraint, then the scheme is said to be conditionally
energy stable.
In analyzing time-integration schemes for phase-field theories, one needs to be specific about the
smoothness of the particular chosen double well function W(φ). Two common assumptions are
(A1) W∈C0[a, b]∩C2(a, b) with −∞ ≤ a<b≤ ∞ and W00 is bounded below, i.e., Wis continuous
on the bounded interval [a, b] and at least twice differentiable on the open interval (a, b), and
there is a constant κW>0 such that
W00(φ)≥ −κW∀φ∈(a, b).(191)
(A2) W∈C2,1(R), i.e., W00 is globally Lipschitz continuous. This means that there is a
constant LW>0 such that
W00(φ)≤LW∀φ∈R.(192)
Note that if Wsatisfies (A2) then it also satisfies (A1) (with a=−∞,b=∞and κW=LW), but
not vice versa. In particular, the classical quartic potential
W(φ) = 1
4(1 −φ2)2(193)
satisfies (A1) with a=−∞ and b=∞, but it does not satisfy (A2). The logarithmic potential
W(φ) = 1
2(1 + φ) log 1 + φ
2+ (1 −φ) log 1−φ
2+θ(1 −φ2)(194)
satisfies (A1) with a=−1 and b= 1, and it also does not satisfy (A2). On the other hand, the
truncated quartic potential
W(φ) = 




(φ+ 1)2φ < −1
1
4(1 −φ2)2φ∈[−1,1]
(φ−1)2φ > 1
(195)
and the truncated logarithmic potential, which is defined e.g. in (Copetti and Elliott, 1992), satisfy
both (A1) (with a=−∞ and b=∞) and (A2).
4.1. First-order accurate schemes
We now consider first-order schemes for the Cahn–Hilliard equation. All of the schemes are implicit
in some sense, since explicit schemes for fourth-order parabolic problems are infeasible.
4.1.1. Backward Euler The backward Euler method applied to the Cahn–Hilliard equation (184)–
(185) leads to the system
φn+1 −φn
τ= ∆µn+1 (196)
µn+1 =W0(φn+1)−2∆φn+1 (197)
This method is nonlinearly implicit, because it requires the solution of a nonlinear system for the
pair (φn+1, µn+1 ). As anticipated before, implicit schemes which are unconditionally linearly stable,
such as the Backward Euler method, are generally conditionally stable for nonlinear problems. To
see this, consider the energy difference
E(φn+1)− E(φn) = ZΩW(φn+1 )−W(φn) + 1
22|∇φn+1|2−1
22|∇φn|2.(198)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 25
Given two real numbers uand vthat enclose ξ, we can use a Taylor’s series expansion to show
that
W(u)−W(v)−W0(u)(u−v) = −1
2W00(ξ)(u−v)2≤κW
2(u−v)2(199)
where the inequality holds due to assumption (A1), which is valid for all the potentials Wof
interest. One then obtains
E(φn+1)− E(φn)≤ZΩW0(φn+1 )(φn+1 −φn) + κW
2(φn+1 −φn)2+1
22|∇φn+1|2−1
22|∇φn|2dx
=ZΩµn+1(φn+1 −φn)−2∇φn+1 · ∇(φn+1 −φn) + κW
2(φn+1 −φn)2dx
+ZΩ1
22|∇φn+1|2−1
22|∇φn|2dx
=−τk∇µn+1k2+κW
2kφn+1 −φnk2−1
22k∇(φn+1 −φn)k2(200)
where k·kdenotes the L2(Ω)-norm. To derive Eq. (200) we have made use of Eqns. (196)–(197)
and the natural boundary conditions defined in (186). Next we bound kφn+1 −φnk2using (196):
ZΩ
(φn+1 −φn)2dx=−τZΩ∇µn+1 · ∇(φn+1 −φn)dx
≤τk∇µn+1kk∇(φn+1 −φn)k(by Cauchy–Schwarz’s ineq.)
≤τ1
2δZΩ|∇µn+1|2dx+τδ
2ZΩ|∇(φn+1 −φn)|2dx(by Young’s ineq.)
where the last inequality holds for for any δ > 0. Finally choosing δ=2
κW
2
τgives
E(φn+1)− E(φn)≤ −τ1−κ2
W
8
τ
2ZΩ|∇µn+1|2dx . (201)
Therefore, if
τ < 82
κ2
W
,(202)
then (190) holds. In other words, the backward Euler scheme is conditionally energy stable.
Furthermore, it can be shown, that under the same time-step constraint, the system (196)–(197)
is uniquely solvable, i.e., the nonlinear system (196)–(197) has a unique solution for (φn+1, µn+1 )
if (202) holds. The proof of this can be found in, e.g., (Elliott, 1989) and the idea behind the proof
applies to other schemes. To proof the existence of a solution, one shows that the system (196)–
(197) is the necessary condition (or Euler-Lagrange equation) corresponding to a minimization
problem for a convex functional. Next, to proof the uniqueness of a solution, one carries out steps
similar to proving the above energy stability. Note that since is generally very small, (202) implies
a severe constraint on the allowed time-step size.
4.1.2. First-order semi-implicit method A popular scheme (Provatas and Elder, 2010) is the
following first-order semi-implicit (or implicit/explicit) method:
φn+1 −φn
τ= ∆µn+1 (203)
µn+1 =W0(φn)−2∆φn+1 .(204)
Because it treats W0explicitly, it is a linear (or linearly-implicit) method requiring the solution of
a linear system at each time step. The system can be written abstractly as
Bµn+1
φn+1=φn
−W0(φn).(205)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

26 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
where Bis the differential operator defined by
B=−τ∆Id
−Id −ε2∆(206)
Here, Id denotes the identity operator. The linear system of differential equations (205) has a
unique solution, independent of τ. This follows from the coercivity estimate
ZΩµn+1
φn+1· B µn+1
φn+1dx=τk∇µn+1 k2+2k∇φn+1k2,(207)
as well as mass conservation RΩ(φn+1 −φn)dx= 0 and the condition RΩµn+1dx=RΩW0(φn)dx.
However, as may be expected, the method is only conditionally energy stable. In fact, we can only
show conditional stability if Wsatisfies (A2). Let us assume that ξis an undetermined point of
the interval (0,1). Then,
E(φn+1)− E(φn) = −τk∇µn+1 k2+1
2ZΩ
W00(φn+ξ)(φn+1 −φn)2dx−1
22k∇(φn+1 −φn)k2
≤ −τk∇µn+1k2+LW
2kφn+1 −φnk2−1
22k∇(φn+1 −φn)k2(208)
where φn+ξ≈φ(·, tn+ξτ ). The inequality (208) is identical to (200) replacing κWwith LW.
Therefore, following the same steps as in Sect. 4.1.1, leads to the similar constraint τ < 82/L2
W
for energy stability.
4.1.3. Convex-splitting method The stability issues for the backward Euler and semi-implicit
method originate from the nonconvexity of W(φ). A groundbreaking idea, which goes back to
(Elliott and Stuart, 1993) and was popularized by (Eyre, 1998), is to split Winto a convex part
and concave part, i.e.,
W(φ) = W+(φ) + W−(φ) with W00
+(φ)≥0 and W00
−(φ)≤0,(209)
The, we treat W+implicitly and W−explicitly, as
φn+1 −φn
τ= ∆µn+1 (210)
µn+1 =W0
+(φn+1) + W0
−(φn)−2∆φn+1 (211)
Using the Taylor’s formulae
W+(φn+1) = W+(φn) + W0
+(φn+1)(φn+1 −φn)−1
2W00
+(φn+ξ)(φn+1 −φn)2(212)
W−(φn+1) = W−(φn) + W0
−(φn)(φn+1 −φn) + 1
2W00
−(φn+ζ)(φn+1 −φn)2(213)
where ξ, ζ ∈(0,1), the energy difference is now
E(φn+1)− E(φn) = −τk∇µn+1 k2−1
2ZΩW00
+(φn+ξ)−W00
−(φn+ζ)(φn+1 −φn)2dx
−1
22k∇(φn+1 −φn)k2
≤ −τk∇µn+1k2−1
22k∇(φn+1 −φn)k2(214)
which shows the unconditional stability of the method.
If Wsatisfies (A1), then the above convex splitting is nonunique, but it is always possible, e.g.,
defining
W−(φ) = −κW
2φ2(215)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 27
and subsequently setting W+(φ) = W(φ)−W−(φ). Unique solvability of the system (210) follows
by equivalence with a strictly-convex minimization problem (cf. (Wise et al., 2009, Theorem 3.4)).
Hence, the convex-splitting scheme is both unconditionally stable and unconditionally solvable. If,
in addition, (A2) applies, then a splitting is possible with W+(φ) a quadratic polynomial, i.e.,
defining
W+(φ) = LW
2φ2(216)
and subsequently setting W−(φ) = W(φ)−W+(φ). In this case, the scheme becomes even linear.
See also (He et al., 2007; Shen and Yang, 2010), where splitting is seen as stabilization of the
semi-implicit scheme.
4.2. Second-order accurate schemes
The development of novel second-order time-accurate schemes with attractive stability and
solvability properties is still ongoing. The situation is much more difficult than for first-order
methods. Most investigated schemes are based on modifications of the Crank–Nicolson method.
We start by analyzing the (poor) properties of the Crank-Nicolson method and then we show how
the method can be improved.
4.2.1. Crank–Nicolson The Crank–Nicolson method is defined by:
φn+1 −φn
τ= ∆µn+1
2(217)
µn+1
2=W0(φn+1) + W0(φn)
2−2∆φn+1 +φn
2(218)
It requires the solution of a nonlinear system at each step. Using the trapezoidal quadrature rule
we have
W(v)−W(u) = Zv
u
W0(s) ds=W0(u) + W0(v)
12 (v−u)−1
2W000(ξ)(v−u)3,(219)
where ξis an unknown point that lays between uand v. Then, it can be shown that
E(φn+1)− E(φn) = −τk∇µn+1 k2−1
12 ZΩ
W000(ξ)(φn+1 −φn)3dx. (220)
The sign of the last term can not be controlled, therefore the method is generally not
unconditionally energy stable.
Furthermore, since this method corresponds to a nonconvex nonlinear system, it can be shown
(by mimicking the proof for the Backward Euler method in (Elliott, 1989)) that its solvability
suffers from a similar time-step constraint, τ≤C2/κ2
W, as the backward Euler scheme.
4.2.2. Second-order semi-implicit method The following second-order semi-implicit method
(Guill´en-Gonz´alez and Tierra, 2013) shares the same stability and solvability properties as the
Crank–Nicolson method:
φn+1 −φn
τ= ∆µn+1
2(221)
µn+1
2=W0(φn) + 1
2W00(φn)(φn+1 −φn)−2∆φn+1 +φn
2(222)
However, it is a linear method corresponding to the system
Bnµn+1/2
φn+1 =φn
−W0(φn) + 1
2W00(φn)φn+1
22∆φn.(223)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

28 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
where Bnis the differential operator defined by
Bn:= −τ∆Id
−Id 1
2W00(φn)−ε2
2∆(224)
Because of nonconvexity of the potential, W00 can have an arbitrary sign. Therefore, the method is
conditionally uniquely solvable under a time-step constraint τ < C2/κ2
W(Guill´en-Gonz´alez and
Tierra, 2013).
4.2.3. Secant method The secant method, which goes back to (Du and Nicolaides, 1991), is
designed to exactly mimic the energy dissipation. The method is defined as,
φn+1 −φn
τ= ∆µn+1
2(225)
µn+1
2=DW(φn+1, φn)−2∆φn+1 +φn
2(226)
where
DW(φ, ψ) = Z1
0
W0(φ+s(ψ−φ)) ds, (227)
or, equivalently,
DW(φn, φn+1) = (W(φn+1 )−W(φn)
φn+1−φnif φn+1 6=φn
W0(φn) if φn+1 =φn(228)
which explains the name of the method. Since
DW(φn, φn+1)(φn+1 −φn) = W(φn+1 )−W(φn) (229)
it follows that the method is unconditionally energy stable, as shown by the relation
E(φn+1)− E(φn) = −τk∇µn+1
2k2(230)
Because of nonconvexity of Whowever, the nonlinear system (225)–(226) is only conditionally
solvable (Elliott, 1989). The operator DWis usually referred to as discrete variational derivative.
The use of discrete variational derivatives is ubiquitous in the design of exact energy preserving
schemes and symplectic methods; see e.g., (Furihata and Matsuo, 2011; Simo et al., 1992; Romero,
2009).
4.2.4. Secant variants Two schemes which avoid constructing the secant, while still maintaining
stability, are the implicit Taylor method by (Kim et al., 2004), given by (225)–(226) with
DW(φn, φn+1) = W0(φn+1 )−1
2W00(φn+1 )(φn+1 −φn) + 1
3!W000(φn+1)(φn+1 −φn)2(231)
and the method by (Gomez and Hughes, 2011), given by (225)–(226) with
DW(φn, φn+1) = 1
2W0(φn) + W0(φn+1)−1
12W000(φn)(φn+1 −φn)2(232)
These two methods result in
E(φn+1)− E(φn) = −τk∇µn+1
2k2−1
24 ZΩ
W0000(φn+ξ)(φn+1 −φn)4;ξ∈(0,1) (233)
The last term is negative or zero if W0000 ≥0. The inequality W0000 ≥0 is true for all potential
functions in (193)–(195). In the case in which the potential Wdoes not satisfy the condition
W0000 ≥0, (Gomez and Hughes, 2011) propose a splitting of Wwhich achieves second-order
accuracy and unconditional stability. A different method based on a second-order perturbation
of the midpoint rule was proposed in (Liu et al., 2013). The method in (Liu et al., 2013) achieves
essentially the same properties.
As the methods discussed in Sect. 4.2.4 correspond to nonconvex nonlinear systems their
solvability is expected to be conditional.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 29
4.2.5. Second-order convex splitting In an effort to have simultaneously unconditional stability
and solvability, a second-order time-accurate convex-splitting scheme was proposed by (Hu et al.,
2009). The method is given by
φn+1 −φn
τ= ∆µn+1
2(234)
µn+1
2=DW+(φn+1, φn) + DW−(φn, φn−1)−2∆φn+1 +φn
2(235)
where
DW+(φn+1, φn) = Z1
0
W0(φn+s(φn+1 −φn)) ds(236)
DW−(φn, φn−1) = 3
2W0
−(φn)−1
2W0
−(φn−1) (237)
which corresponds to a secant treatment of the convex part and a two-step second-order backward
difference treatment of the concave part. This method is unconditionally solvable as it is equivalent
to a convex minimization problem at each time step. However, the method is not unconditionally
stable as defined in (190). A less restrictive statement referred to as weak energy stability (Hu et al.,
2009) may be proven if W−is a quadratic polynomial, which applies to many potential functions.
An alternative is the multistep scheme proposed in (Guill´en-Gonz´alez and Tierra, 2013), which
is however restricted to the quartic potential. This scheme is linear, unconditionally energy stable
for a modified energy statement, and unconditionally uniquely solvable.
4.2.6. Stabilization Most second-order linear schemes suffer from conditional stability and/or
conditional solvability. As proposed by (Wu et al., 2014) these issues can be stabilized if W
satisfies (A2). For example, the stabilized semi-implicit scheme is
φn+1 −φn
τ= ∆µn+1
2(238)
µn+1
2=W0(φn) + 1
2W00(φn)(φn+1 −φn)−2∆φn+1 +φn
2−βτ ∆(φn+1 −φn),(239)
where βis the stabilization parameter. Such terms are also referred to as artificial viscosity, and are
useful in many applications; see e.g., (Labovsky et al., 2009; Jansen et al., 2000). See also (Gomez
and Hughes, 2011) for a stabilization of the extended Crank–Nicolson method.
Using a Taylor’s formula, it can be shown that
E(φn+1)− E(φn) = −τk∇µn+1
2k2+1
2ZΩW00(ξ)−W00(φn)(φn+1 −φn)2dx
−βτ k∇(φn+1 −φn)k2(240)
Then, assuming that Wsatisfies (A2),
E(φn+1)− E(φn)≥ −τk∇µn+1
2k2+LWkφn+1 −φnk2−βτ k∇(φn+1 −φn)k2(241)
≥ −τ1−LW
2δk∇µk2−τβ−LWδ
2k∇φn+1 −φnk2.
(by Young’s ineq. with 0 < δ < LW/2)
Therefore, for β > L2
W/4, one has unconditional energy stability. We note that unconditional
solvability follows by mimicking the proof for the Backward Euler method in (Elliott, 1989).
4.3. Discussion
Based on the Cahn–Hilliard equation, we have tried to illustrate that the design of stable and
solvable time-integration schemes for phase-field models is challenging. The key idea for first-order
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

30 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
schemes has been convex-splitting, however, this idea is not sufficient for second-order schemes,
and one has to resort to particular double-well potentials, modified energy statements and/or
stabilization. There are also other popular methods that can be used, such as the generalized-α
algorithm (Jansen et al., 2000), for which little is known about their stability properties in case of
phase-field models.
For phase-field models that are coupled to additional equations (e.g., the momentum equation),
difficulties arise in maintaining energy stability and solvability. The idea of stable decoupled schemes
is also challenging, but it is attractive to achieve modularity of computer codes; see for example
(Shen and Yang, 2015) in the context of the Cahn-Hilliard-Navier-Stokes model.
5. Space discretization
The main challenges associated to the spatial discretization of phase-field theories are the
approximation of higher-order partial-differential operators, inherent to diffuse-interface models
and the presence of internal layers in the solution, which clearly calls for adaptive meshing.
Although these are long-standing problems which remain open to some extent, here we show
important developments accomplished in the last few years that have elevated phase-fields to a
point in which they can be used for practical engineering problems.
5.1. Discretization of higher-order partial-differential operators
Most phase-field theories involve higher-order spatial partial-differential operators which give rise to
diffuse interfaces in the solution. For example, the Cahn-Hilliard equation is a fourth order PDE in
space. The Navier-Stokes-Korteweg equations include third-order spatial derivatives. The fracture
theory presented in Sect. 3.4 is a second-order PDE in space, but a higher-order approximation
which involves fourth-order derivatives has been recently proposed. Data indicate that the higher-
order theory outperforms the model presented in Sect. 3.4 (Borden et al., 2014). Therefore, it seems
clear that an important requirement of a numerical algorithm for phase fields is the possibility to
discretize higher-order derivatives effectively.
The higher-order operator problem is an old one in computational mechanics. As a matter of
fact, classical theories for thin shells and plates pertaining to the Kirchhoff-Love type also involve
fourth-order operators. Numerical solutions to higher-order problems in simple geometries can be
easily obtained using classical algorithms, such as, for example, the finite difference method, or
pseudo-spectral collocation. The problem shows when one needs to solve a higher-order PDE on
a complicated three-dimensional geometry. Complex geometries typically call for the use of the
finite element method. However, solving fourth-order PDEs directly with finite elements requires
globally C1-continuous basis functions which are very difficult or even impossible to generate in
complex three-dimensional geometries.
Perhaps, the most obvious solution to the higher-order problem is to split the fourth-order
equation into a couple of second-order PDEs, but this would lead to a twofold increase in
the global number of degrees of freedom. As a consequence, different procedures have been
investigated that aim at keeping the computational cost as low as possible. Relevant examples
are the continuous/discontinuous Galerkin method (Wells et al., 2006), finite volume methods
(Cueto-Felgueroso and Peraire, 2008) or meshless methods (Rajagopal et al., 2010; Zhou and Li,
2006). Here, we will focus on a novel technology, namely isogeometric analysis, which permits simple
discretizations of higher-order operators on non-trivial geometries. The key idea is that isogeometric
analysis permits generating globally Cp−1-continuous basis functions in physical space by mapping
order-pB-Splines or NURBS (Non-uniform rational B-Splines) from a reference configuration. For
the reader not familiar with B-Splines and NURBS we recommend the references (Hughes et al.,
2005; Cottrell et al., 2009; Aaa, 201xa,x), which are intended for scholars with a background in
finite elements. In Sect. 6 we present more details on the spatial discretization of various phase-field
models.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 31
5.2. Adaptive mesh and time-step refinement
Solutions to phase-field models inherently exhibit a large range of spatial and temporal scales,
therefore adaptive mesh refinement and adaptive time-step selection are indispensable for the
efficient numerical discretization of phase-field models. Clearly, the phase-field itself is constant
everywhere throughout the domain except at interfaces, where it rapidly, but smoothly, changes
from one value to the other. This smooth transition happens on a spatial scale of order , and
therefore, in principle, its resolution requires a local mesh-size of that same order. In time, rapid
phenomena also occur, in particular when topological changes take place. For example, in case of
Cahn–Hilliard phenomena, this happens when mixed phases separate, single phases break up into
several bubbles, or when separate bubbles coalesce.
Adaptive mesh refinement crucially relies on refinement indicators, which mark elements for
refinement that are deemed important for improving the accuracy of a coarse-mesh approximation.
The indicators can be based on heuristic criteria or a posteriori error estimates. An example of a
heuristic criteria is one for which those elements are marked if the local phase-field approximation
deviates from pure phase-field values (Provatas et al., 2005), or if it has a large gradient (Rosam
et al., 2007). Typically these indicators result in reasonable refinements, but it may not work well
on very coarse meshes with very poorly resolved solutions.
Alternatively, a posteriori error estimates can be employed in a method-of-lines approach. In such
an implementation, the estimates are derived for the fully discrete system for advancing a time
step, but they estimate errors only due to the spatial part. Any a posteriori error estimator that
is suitable for the spatial operator can be employed, e.g., the Zienkiewicz–Zhu gradient-recovery
estimator (Provatas et al., 1999), or an explicit residual-based estimator (Stogner et al., 2008).
Adaptive time-step selection is important for temporal accuracy. In a method-of-lines approach
one resorts to classical techniques for adaptive time-stepping of ordinary differential equations
(S¨oderlind, 2002; Deuflhard and Bornemann, 2002). An elementary approach computes the
approximation for the next time step using a low-order and a higher-order time-stepping scheme.
Their difference can then be employed as an estimate for the temporal error and as a guide to
decrease or increase the time-step size. For phase-field models such techniques have been employed
in, e.g., (Ceniceros and Roma, 2007; Cueto-Felgueroso and Peraire, 2008; Gomez et al., 2008; Wodo
and Ganapathysubramanian, 2011).
Finally, space/time a posteriori error estimates can be employed, which can estimate errors due
to space and/or time discretization. For residual-based estimates, the difficulty is the derivation
of sharp bounds by eliminating the exponential dependence on the interface thickness; see,
e.g., (Kessler et al., 2004; Feng and Wu, 2005; Bartels, 2005) for Allen–Cahn estimates and (Feng
and Wu, 2008; Bartels and M¨uller, 2010) for Cahn–Hilliard estimates. Alternatively, estimates
based on the computation of a dual problem have recently been developed (Simsek et al., 2015) in
the spirit of duality-based estimates for nonlinear parabolic problems (Eriksson et al., 2004). It is
also possible to employ goal-oriented error estimates to control the accuracy in output quantities
of interest (van der Zee et al., 2011; Mahnken, 2013). These are based on general frameworks as
described in, e.g., (Becker and Rannacher, 2001; Stein and R¨uter, 2004).
6. Applications and numerical examples
Here we illustrate how the space discretization of fourth-order problems can be accomplished using
smooth basis functions by way of isogeometric analysis (Sect. 6.1) or resorting to classical mixed
methods (Sect. 6.2). In Sect. 6.1 we focus on the classical Cahn-Hilliard equation, while in Sect. 6.2
we illustrate the ideas with the tumor-growth model presented before. Finally, Sect. 6.3 presents
isogeometric simulations of the Navier-Stokes-Korteweg equations.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

32 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
6.1. Cahn-Hilliard equation
We start by deriving a weak form of the Cahn-Hilliard equation. For simplicity, we assume periodic
boundary conditions. Thus, given the Cahn-Hilliard equation, recall from (58),
∂φ
∂t = div m(φ)∇W0(φ)−2∆φ in Ω (242)
let us consider a smooth function w∈ V, where V=H2(Ω) is the Sobolev space of square integrable
functions with square integrable first and second derivatives in Ω. We multiply Eq. (242) with w,
integrate over the domain Ω and integrate by parts twice, which leads to the following variational
problem: Find φ∈ V such that for all w∈ V
0 = ZΩ
w∂φ
∂t dx+ZΩ∇w· ∇φW 00(φ)m(φ)dx
+ZΩ∇w· ∇φm0(φ)2∆φdx+ZΩ
∆w2m(φ)∆φdx(243)
The key point is that by way of isogeometric analysis we can construct a discrete space Vh⊂ V
using B-Splines or NURBS of order p≥2. Therefore, using the Galerkin method, the semi-discrete
problem may be stated as: Find φh∈ Vhsuch that for all wh∈ Vh
0 = ZΩ
wh∂φh
∂t dx+ZΩ∇wh· ∇φhW00(φh)m(φh)dx
+ZΩ∇wh· ∇φhm0(φh)2∆φhdx+ZΩ
∆whm(φh)2∆φhdx(244)
Eq. (244) represents a semidiscrete form of the Cahn-Hilliard equation. Time integration schemes
such as discussed in Sect. 4 can be utilized to advance in time. Our numerical examples have been
computed using the generalized-αalgorithm (Jansen et al., 2000). We omit here the implementation
details which may be found in (Gomez et al., 2008). We use adaptive time stepping because the
equation evolves on very different time scales during the simulation (Gomez et al., 2008; Gomez
and Hughes, 2011). To completely define the problem we need to choose specific forms of the
double-well potential Wand the mobility M. We take the physically-relevant logarithmic double
well potential, see (194), since the quartic polynomial is a mere qualitative approximation of the
logarithmic double well. The parameter θin (194) represents the quench temperature. In our
computations we will take θ= 3/2. Similarly, the mobility is often assumed constant, although the
so-called degenerate mobility
m(φ) = 1
4(1 −φ2) (245)
leads to better agreement with experiments and is our choice for the numerical examples. The key
feature of the degenerate mobility is that pure phases have vanishing mobility and the motion is
restricted to the (diffuse) interfaces as it happens in so-called Ostwald ripening.
Using dimensional analysis, it may be shown that the solution to the Cahn-Hilliard equation
does not depend on all its parameters, but only on the dimensionless number (Gomez et al., 2008)
C=L2
0
32(246)
where L0is a length scale of the problem. Here, we will assume that L0is a measure of the
computational domain size. For example, for a cubic domain, L0is just the length of the cube’s
edge. Without loss of generality, we will also assume L0= 1.
The usual setup for numerical simulations of the Cahn-Hilliard equation is as follows: a randomly
perturbed homogeneous solution is allowed to evolve in a sufficiently large computational domain.
Periodic boundary conditions are often used in an attempt to isolate the phase-separation physics
from boundary effects. It is possible to show that solutions of the type φ(x,·) = φ+η(x), where
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 33
Figure 1. Two-dimensional simulation of the Cahn-Hilliard equation on the computational domain
Ω = [0,1]2, with periodic boundary conditions. We started the simulation with a randomly perturbed
homogeneous solution (φ= 0.5), which spontaneously separated into two phases (blue and red colors).
As time evolves (left to right and top to bottom), the system undergoes coarsening, which manifests
itself by larger-scale morphological features. The mesh is uniform and composed of 2562C1-quadratic
elements. We used the parameter C= 104.
η(x) is a small perturbation are linearly unstable when φis in the so-called spinodal region,
that is, when W00(φ)<0 (Gomez et al., 2014). Thus, if we start the simulation with the initial
condition φ(x,0) = φ+η(x), the perturbation η(x) will grow in time, leading to spontaneous phase
separation. Phase separation manifests itself as patches where the solution is almost constant and
takes the values φ≈ ±1. The regions where φ≈ −1 and φ≈+1 are connected through a transition
layer.
To illustrate this process, we computed a two-dimensional solution using the parameters φ= 0.5
and C= 104. The top left snapshot of Fig. 1 shows the system early after phase separation. The
blue areas represent patches where φ≈ −1, while the red zones correspond to φ≈+1. After the
system is fully separated, the process of coarsening starts. During the coarsening process, small
bubbles merge together giving rise to larger-scale features (see the remaining snapshots of Fig. 1;
time increases from left to right and from top to bottom). Fig. 2 shows a similar computation in
three dimensions. The parameters are φ= 0.26 and C= 600. The time evolution may be observed
in the plot. As time evolves, the mixture coarsens until it is fully separated.
6.2. Phase-field tumor growth theory
Next, we consider a numerical illustration for the tumor-growth model derived in Sect. 3.5, subject
to the boundary conditions
∇φ·n=∇µ1·n= 0 σ=σDon ∂Ω,(247)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

34 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
Figure 2. Three-dimensional simulation of the Cahn-Hilliard equation on the computational domain
Ω = [0,1]3. We start with a randomly perturbed homogeneous solution (φ= 0.26). The physics is
analogous to that of the two-dimensional example . The mixture separates and then coarsens as time
evolves (left to right and top to bottom). The mesh is uniform and composed of 1283C1-quadratic
elements.
and initial conditions
φ(x,0) = φ0(x) and σ(x,0) = σ0∀x∈Ω,(248)
As an alternative to the spline-based discretization in the previous section, we keep the chemical
potential µ1as an unknown and consider a mixed formulation suitable for classical C0-continuous
finite element spaces Vh. After semi-discretization in space the problem is then to find φh(·, t)∈ Vh,
µh
1(·, t)∈ Vhand σh(·, t)∈ Vhsuch that
0 = ZΩ
wh∂φh
∂t dx+ZΩ
m∇wh· ∇µh
1dx−ZΩ
g(φh)(σh−δµh
1)whdx(249)
0 = −ZΩ
µh
1ηhdx+ZΩ
W0
T(φh)ηhdx+2ZΩ∇φh· ∇ηhdx(250)
0 = ZΩ
ζh∂σh
∂t dx+ZΩ
D∇σh· ∇ζhdx+ZΩ
g(φh)(σh−δµh
1)ζhdx(251)
for all (wh, ηh, ζ h)∈ Vh×Vh× Vh. Note that classical C0-continuous shape functions are possible
since ∇(·) is the highest-order derivative.
To illustrate a fully discrete method, let us employ a simple first-order time-accurate convex–
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 35
splitting scheme, extending the one for the Cahn–Hilliard equation described in Sect. 4.1.3:
ZΩ
whφh,n+1 −φh,n
τdx+ZΩ
m∇wh· ∇µh,n+1
1dx−ZΩ
g(φh,n)(σh,n+1 −δµh,n+1
1)whdx= 0
(252)
−ZΩ
µh,n+1
1ηhdx+ZΩ
W0
T+(φh,n+1)ηhdx+2ZΩ∇φh,n+1 · ∇ηhdx=−ZΩ
W0
T−(φh,n)ηhdx
(253)
ZΩ
ζhσh,n+1 −σh,n
τdx+ZΩ
D∇σh,n+1 · ∇ζhdx+ZΩ
g(φh,n)(σh,n+1 −δµh,n+1
1)ζhdx= 0 (254)
Note that we used first-order extrapolation on the nonlinearity caused by g(φh). This scheme can
be shown to be unconditionally energy stable so that large time-steps can be taken (Wu et al.,
2014).
As an example of a numerical simulation which can be performed on a laptop computer, we
consider the domain Ω = [−1,1]3, and we set the parameters m=D= 1, ¯ccell =1
2,δ= 0.01,
= 0.1, and
WT(φ) = 




8φ2(1
2−φ)2φ∈(0,1
2)
2φ2φ≤0
2(1
2−φ)2φ≥1
2
(255)
g(φ) = (256 φ2(1
2−φ)2φ∈(0,1
2)
0 otherwise (256)
For the convex spliting in the scheme, we decompose W0
T(φ) by choosing W0
T+(φ) = 4φand setting
W0
T−(φ) = W0
T(φ)−W0
T+(φ). The nutrient boundary-condition value is σD= 1 −¯ccell =1
2,
the initial nutrient distribution is σ0= 1 −¯ccell =1
2, and the initial condition for the tumor φ
corresponds to three spheres:
φ0(x) = ¯ccell
21 + tanh R1− |x−x(1) |
√2
+¯ccell
21 + tanh R2− |x−x(2) |
√2+¯ccell
21 + tanh R3− |x−x(3) |
√2(257)
with radii R1= 1/6, R2= 1/4 and R3= 1/6, and center coordinates x(1) = (−2
15 √2,−2
15 √2,0),
x(2) = ( 7
40 √2,7
40 √2,0) and x(3) = (13
30 √2,−13
30 √2,1
6). We choose Vhto correspond to linear
finite elements on a coarse uniform mesh with 403elements, and set the time-step size to a large
value: τ= 0.01. Figure 3 displays several snapshots of the simulation.
6.3. Isothermal Navier-Stokes-Korteweg equations
Here, we show how the Navier-Stokes-Korteweg equations can be discretized using globally smooth
splines. For simplicity, we consider only the isothermal version of the equations, although similar
ideas could be used to discretize the thermally-coupled theory (Liu et al., 2015). Under the
assumption of constant temperature, the Navier-Stokes-Korteweg equations reduce to
∂ρ
∂t + div(ρu) = 0 (258)
∂(ρu)
∂t + div(ρu⊗u+pI)−div σ=ρb(259)
where
σ=ν(∇u+∇Tu)−2
3div u I+¯
λρ∆ρ+1
2|∇ρ|2I−¯
λ∇ρ⊗ ∇ρ(260)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

36 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
Figure 3. Three-dimensional simulation of the phase-field tumor growth model. We start with three
small tumors. The tumors grow and merge into a single growing mass. The top row shows the level set
of φ(x) = 1
2¯ccell =1
4for t= 0, 0.01, 0.02 and 0.1. The bottom row show the computational domain
and a cut-away view of the phase-field φat time t= 0 and t= 0.1. The mesh is uniform and composed
of 403linear elements.
and
p=Rb ρθ
b−ρ−aρ2(261)
Eq. (261) is referred to as the van der Waals equation, and acts as an equation of state in the
theory. In the van der Waals equation, Ris simply the specific gas constant, while aand bare
positive parameters that may be found in scientific databases, such as for example, that of the
National Institute of Standards and Technology. The temperature θplays the role of a parameter
in the isothermal theory.
For notational simplicity, we define Tvisc as the viscous stress, that is,
Tvisc =ν(∇u+∇Tu)−2
3div u I(262)
Using the notation (262), it may be shown that
div σ= div Tvisc +¯
λρ∇(∆ρ) (263)
Substituting Eq. (263) in (259), we get the non-conservative form of the Navier-Stokes-Korteweg
equations, which may be expressed as
∂ρ
∂t + div(ρu) = 0 (264)
∂(ρu)
∂t + div(ρu⊗u+pI)−div Tvisc −¯
λρ∇(∆ρ) = ρb(265)
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 37
Eqs. (264) and (265) constitute the starting point of our weak formulation, which may be
derived by multiplying them with smooth functions qand w, integrating over the computational
domain, and then integrating by parts. Following the ideas presented in the previous subsections,
we approximate relevant functional spaces by their discrete counterparts. The discrete spaces are
subsets of H2(Ω) due to the global smoothness of B-Splines and NURBS (for p≥2) and this makes
the integrals in the weak form computable. Proceeding this way, we can recast the problem as: find
ρh∈ Vhand uh∈Vhndsuch that for all qh∈ Vhand wh∈Vhnd
0 = ZΩ
qh∂ρh
∂t dx−ZΩ∇qh·uhρhdx(266)
0 = ZΩ
wh·∂(ρhuh)
∂t dx−ZΩ∇wh:ρhuh⊗uhdx−ZΩ
div whp(ρh)dx
+ZΩ∇wh:Tvisc(vh)dx+ZΩ
wh· ∇ρh¯
λ∆ρhdx+ZΩ
div wh¯
λρh∆ρhdx
−ZΩ
whρhbhdx(267)
where we have assumed periodic boundary conditions for simplicity. We integrate in time Eqs.
(266)–(267) using the generalized-αmethod. We use this algorithm to perform two- and three-
dimensional computations.
By way of dimensional analysis, it may be shown that the solution to the unforced Navier-Stokes-
Korteweg equations depends only on the dimensionless groups
Re =L0b(ab)1/2
νand Ca =¯
λ/a1/2
L0
,(268)
where Re is the Reynolds number and Ca is the capillary number. L0= 1 represents again
the length of the computational domain. We begin by simulating the coalescence of two vapor
bubbles in the computational domain Ω = [0,1]2. The centers of the bubbles are located at points
x(1) = (0.40,0.50) and x(2) = (0.78,0.50). Their radii are R1= 0.25 and R2= 0.10, respectively.
Inside the bubbles, we set a dimensionless density ρ/b = 0.1, while outside we take ρ/b = 0.6.
These are approximate values to the so-called Maxwell states, which correspond to mechanical and
chemical equilibrium values. The density profiles are regularized using hyperbolic tangent profiles.
The initial velocity is zero. The dimensionless groups are given by Re = 512 and C a = 1/256. The
mesh is uniform and composed of 2562C1-quadratic elements.
The time evolution of the system may be observed in Fig. 4. The background color scale
represents density, while solid lines represent streamlines. The color scale superimposed to the
streamlines corresponds to the velocity magnitude. The simulation shows how mechanical forces
drive the two bubbles together, creating a velocity field. After the two bubbles are connected,
capillary forces tend to reduce the interfacial length, eventually leading to a stationary state with
a circular steady bubble (not shown).
Fig. 5 shows the three-dimensional analogue of the last example. The computational domain is
Ω = [0,1]3, and the mesh is uniform and composed of 1283C1-quadratic examples. The centers of
the bubbles are located at the points x(1) = (0.40,0.50,0.60) and x(2) = (0.75,0.50,0.50). Their
radii are R1= 0.25 and R2= 0.10, respectively. The initial conditions are the same as in the last
example, while the dimensionless groups are Re = 256 and C a = 1/128. Fig. 5 presents the time
evolution of the system. The bubbles are represented by density isosurfaces, which have a velocity
magnitude color scale superimposed. The system evolves towards a steady spherical bubble (not
shown).
REFERENCES
Aaa, A. A. (201xa). Isogeometric analysis: Analysis. In Stein, E., de Borst, R., and Hughes, T. J. R.,
editors, Encyclopedia of Computational Mechanics, volume 1: Fundamentals, chapter X, pages xx–xx.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

38 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
Figure 4. Two-dimensional vapor bubbles right before and after coalescence due to surface tension. The
background color scale represents density, while the solid lines represent streamlines of the velocity
field. The color scale superimposed to the streamlines represents velocity magnitude. The mesh is
uniform and composed of 2562C1-quadratic elements. The dimensionless groups are Re = 512, and
Ca = 1/256.
John Wiley & Sons.
Aaa, A. A. (201xb). Isogeometric analysis: Representation of geometry. In Stein, E., de Borst, R., and
Hughes, T. J. R., editors, Encyclopedia of Computational Mechanics, volume 1: Fundamentals, chapter X,
pages xx–xx. John Wiley & Sons.
Abdollahi, A. and Arias, I. (2011). Phase-field modeling of the coupled microstructure and fracture
evolution in ferroelectric single crystals. Acta Materialia, 59(12):4733–4746.
Ambrosio, L. and Tortorelli, V. M. (1990). Approximation of functional depending on jumps by elliptic
functional via Γ-convergence. Communications on Pure and Applied Mathematics, 43(8):999–1036.
Anderson, D. M., McFadden, G. B., and Wheeler, A. A. (1998). Diffuse-interface methods in fluid
mechanics. Annual Review of Fluid Mechanics, 30(1):139–165.
Axelsson, O. (2004). Finite difference methods. In Stein, E., de Borst, R., and Hughes, T. J. R., editors,
Encyclopedia of Computational Mechanics, volume 1: Fundamentals, chapter 2, pages 7–54. John Wiley
& Sons.
Babuska, I. and Melenk, J. M. (1997). The partition of unity method. International Journal for Numerical
Methods in Engineering, 40(4):727–758.
Bartels, S. (2005). A posteriori error analysis for time-dependent Ginzburg–Landau type equations. Numer.
Math., 99:557–583.
Bartels, S. and M¨uller, R. (2010). A posteriori error controlled local resolution of evolving interfaces for
generalized Cahn–Hilliard equations. Interfaces Free Bound., 12:45–73.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 39
Figure 5. Three-dimensional vapor bubbles right before and after coalescence due to surface tension.
The bubbles are represented by density isosurfaces, which are superimposed with a velocity magnitude
color scale. The mesh is uniform and composed of 1283C1-quadratic elements.
Bazilevs, Y., Takizawa, K., and Tezduyar, T. E. (2013). Computational fluid-structure interaction: methods
and applications. John Wiley & Sons.
Becker, R. and Rannacher, R. (2001). An optimal control approach to a posteriori error estimation in
finite element methods. Acta Numer., 10:1–102.
Borden, M. J., Hughes, T. J., Landis, C. M., and Verhoosel, C. V. (2014). A higher-order phase-field model
for brittle fracture: Formulation and analysis within the isogeometric analysis framework. Computer
Methods in Applied Mechanics and Engineering, 273:100–118.
Borden, M. J., Verhoosel, C. V., Scott, M. A., Hughes, T. J. R., and Landis, C. M. (2012). A phase-field
description of dynamic brittle fracture. Comput. Methods Appl. Mech. Engrg., 217–220:77–95.
Bourdin, B., Francfort, G. A., and Marigo, J.-J. (2008). The variational approach to fracture. J. Elasticity,
91:5–148.
Bowen, R. (1976). Theory of mixtures. In Eringen, A., editor, Continuum Physics, volume III: Mixtures
and EM Field Theories, pages 1–127. Academic Press.
Boyer, F., Lapuerta, C., Minjeaud, S., Piar, B., and Quintard, M. (2010). Cahn-Hilliard/Navier-Stokes
model for the simulation of three-phase flows. Transport in porous media, 82(3):463–483.
Braides, A. (2002). Γ-convergence for Beginners. Oxford University Press.
Bueno, J., Bona-Casas, C., Bazilevs, Y., and Gomez, H. (2014). Interaction of complex fluids and solids:
theory, algorithms and application to phase-change-driven implosion. Computational Mechanics, pages
1–14.
Caginalp, G. (1989). Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations.
Physical Review A, 39(11):5887.
Cahn, J. and Hilliard, J. (1958). Free energy of a non-uniform system. I. Interfacial free energy. The
Journal of Chemical Physics, 28:258–267.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

40 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
Canuto, C. and Quarteroni, A. (2004). Spectral methods. In Stein, E., de Borst, R., and Hughes, T. J. R.,
editors, Encyclopedia of Computational Mechanics, volume 1: Fundamentals, chapter 6, pages 141–155.
John Wiley & Sons.
Ceniceros, H. and Roma, A. (2007). A nonstiff, adaptive mesh refinement-based method for the Cahn–
Hilliard equation. J. Comput. Phys., 225:1849–1862.
Ceniceros, H. D., Ns, R. L., and Roma, A. M. (2010). Three-dimensional, fully adaptive simulations of
phase-field fluid models. Journal of Computational Physics, 229(17):6135 – 6155.
Chen, L.-Q. (2002). Phase-field models for microstructure evolution. Annual Review of Materials Research,
32(1):113–140.
Coleman, B. and Noll, W. (1963). The thermodynamics of elastic materials with heat conduction and
viscosity. Archive for Rational Mechanics and Analysis, 13:167–178.
Colli, P., Frigeri, S., and Grasselli, M. (2012). Global existence of weak solutions to a nonlocal Cahn-
Hilliard-Navier-Stokes system. Journal of Mathematical Analysis and Applications, 386(1):428–444.
Copetti, M. and Elliott, C. M. (1992). Numerical analysis of the Cahn–Hilliard equation with a logarithmic
free energy. Numer. Math., 63:39–65.
Cottrell, J., Hughes, T., and Bazilevs, Y. (2009). Isogeometric Analysis: Toward Integration of CAD and
FEA. Wiley.
Cristini, V., Li, X., Lowengrub, J. S., and Wise, S. M. (2009). Nonlinear simulations of solid tumor growth
using a mixture model: invasion and branching. J. Math. Biol., 58:723–763.
Cueto-Felgueroso, L. and Peraire, J. (2008). A time-adaptive finite volume method for the Cahn-Hilliard
and Kuramoto-Sivashinsky equations. Journal of Computational Physics, 227(24):9985 – 10017.
Dal Maso, G. (1993). An introduction to Γ-convergence. Springer.
Deckelnick, K., Dziuk, G., and Elliott, C. M. (2005). Computation of geometric partial differential equations
and mean curvature flow. Acta Numer., pages 139–232.
Deuflhard, P. and Bornemann, F. (2002). Scientific Computing with Ordinary Differential Equations,
volume 42 of Texts in Applied Mathematics. Springer, Berlin.
Du, Q., Liu, C., Ryham, R., and Wang, X. (2005). A phase field formulation of the Willmore problem.
Nonlinearity, 18:1249–1267.
Du, Q. and Nicolaides, R. A. (1991). Numerical analysis of a continuum model of phase transition. SIAM
J. Numer. Anal., 28(5):1310–1322.
Dunn, J. and Serrin, J. (1986). On the thermomechanics of interstitial working. In The Breadth and Depth
of Continuum Mechanics, pages 705–743. Springer Berlin Heidelberg.
Elliott, C. M. (1989). The Cahn–Hilliard model for the kinetics of phase separation. In Rodrigues, J.,
editor, Mathematical Models for Phase Change Problems: Proceedings of the European Workshop held
at ´
Obidos, Portugal, October 1–3, 1988, volume 88 of International Series of Numerical Mathematics,
pages 35–73, Basel. Birkh¨auser.
Elliott, C. M. and Stuart, A. M. (1993). The global dynamics of discrete semilinear parabolic equations.
SIAM J. Numer. Anal., 30(6):1622–1663.
Emmerich., H. (2003). The Diffuse Interface Approach in Materials Science: Thermodynamic Concepts
and Applications of Phase-Field Models. Lecture Notes in Physics. Springer, Berlin.
Eriksson, K., Johnson, C., and Logg, A. (2004). Adaptive computational methods for parabolic problems.
In Stein, E., de Borst, R., and Hughes, T. J. R., editors, Encyclopedia of Computational Mechanics,
volume 1: Fundamentals, chapter 24, pages 675–702. John Wiley & Sons.
Eyre, D. (1998). Unconditionally gradient stable time marching the Cahn–Hilliard equation. In Bullard,
J. W., Chen, L.-Q., Kalia, R. K., and Stoneham, A. M., editors, Computational and Mathematical Models
of Microstructural Evolution, volume 529 of Mater. Res. Soc. Symp. Proc., pages 39–46, Warrendale,
Pennsylvania. Materials Research Society.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 41
Farhat, C. (2004). CFD-based nonlinear computational aeroelasticity. In Stein, E., de Borst, R., and
Hughes, T. J. R., editors, Encyclopedia of Computational Mechanics, volume 3: Fluids, chapter 13,
pages 459–480. John Wiley & Sons.
Fedkiw, R. and Osher, S. (2003). Level set methods and dynamic implicit surfaces. Springer.
Feng, X. and Wu, H.-J. (2005). A posteriori error estimates and an adaptive finite element method for the
Allen–Cahn equation and the mean curvature flow. J. Sci. Comput., 24(2):121–146.
Feng, X. and Wu, H.-J. (2008). A posteriori error estimates for finite element approximations of the
Cahn–Hilliard equation and the Hele–Shaw flow. J. Comput. Math., 26(6):767–796.
Fife, P. C. (1988). Dynamics of Internal Layers and Diffusive Interfaces, volume 53 of CBMS-NSF
Reginonal Conference Series in Applied Mathematics. Society of Industrial and Applied Mathematics
(SIAM), Philadelphia, Pennsylvania.
Furihata, D. and Matsuo, T. (2011). Discrete Variational Derivative Method: A Structure-Preserving
Numerical Method for Partial Differential Equations. Chapman and Hall/CRC, Boca Raton.
Gal, C. G. and Grasselli, M. (2010). Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2d.
In Annales de l’Institut Henri Poincare (C) Non Linear Analysis, volume 27-1, pages 401–436. Elsevier.
Gal, C. G. and Grasselli, M. (2011). Instability of two-phase flows: A lower bound on the dimension
of the global attractor of the Cahn-Hilliard-Navier-Stokes system. Physica D: Nonlinear Phenomena,
240(7):629–635.
Giesselmann, J., Makridakis, C., and Pryer, T. (2014). Energy consistent discontinuous Galerkin methods
for the Navier-Stokes-Korteweg system. Mathematics of Computation, 83(289):2071–2099.
Gomez, H., Calo, V., Bazilevs, Y., and Hughes, T. (2008). Isogeometric analysis of the Cahn-Hilliard
phase-field model. Computer Methods in Applied Mechanics and Engineering, 197:4333–4352.
G´omez, H., Colominas, I., Navarrina, F., and Casteleiro, M. (2007a). A discontinuous Galerkin method
for a hyperbolic model for convection–diffusion problems in CFD. International Journal for Numerical
Methods in Engineering, 71(11):1342–1364.
G´omez, H., Colominas, I., Navarrina, F., and Casteleiro, M. (2007b). A finite element formulation for a
convection–diffusion equation based on Cattaneo’s law. Computer methods in applied mechanics and
engineering, 196(9):1757–1766.
G´omez, H., Colominas, I., Navarrina, F., and Casteleiro, M. (2008). A mathematical model and a numerical
model for hyperbolic mass transport in compressible flows. Heat and Mass Transfer, 45(2):219–226.
Gomez, H., Colominas, I., Navarrina, F., Par´ıs, J., and Casteleiro, M. (2010a). A hyperbolic theory
for advection-diffusion problems: mathematical foundations and numerical modeling. Archives of
Computational Methods in Engineering, 17(2):191–211.
Gomez, H. and Hughes, T. (2011). Provably unconditionally stable, second-order time-accurate, mixed
variational methods for phase-field models. Journal of Computational Physics, 230(13):5310–5327.
Gomez, H., Hughes, T., Nogueira, X., and Calo, V. (2010b). Isogeometric anslysis of the isothermal Navier-
Stokes-Korteweg equations. Computer Methods in Applied Mechanics and Engineering, 199:1828–1840.
Gomez, H., Reali, A., and Sangalli, G. (2014). Accurate, efficient, and (iso) geometrically flexible collocation
methods for phase-field models. Journal of Computational Physics, 262:153–171.
Gross, S. and Reusken, A. (2011). Numerical Methods for Two-phase Incompressible Flows, volume 40 of
Springer Series in Computational Mathematics. Springer, Berlin.
Guill´en-Gonz´alez, F. and Tierra, G. (2013). On linear schemes for a Cahn-Hilliard diffuse interface model.
Journal of Computational Physics, 234:140 – 171.
Gurtin, M., Fried, E., and Anand, L. (2009). The Mechanics and Thermodynamics of Continua. Cambridge
University Press.
Gurtin, M. E. (1996). Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce
balance. Phys. D, 92:178–192.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

42 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
Gurtin, M. E., Polignone, D., and Vi˜nals, J. (1996). Two-phase binary fluids and immiscible fluids described
by an order parameter. Math. Models Methods Appl. Sci., 6(6):815–832.
Hairer, E., Norsett, S., and Wanner, G. (1987). Solving Ordinary Differential Equations I (; Berlin.
Springer.
Hawkins-Daarud, A., van der Zee, K. G., and Oden, J. T. (2012). Numerical simulation of a
thermodynamically consistent four-species tumor growth model. Int. J. Numer. Meth. Biomed. Engng.,
28:3–24.
He, Y., Liu, Y., and Tang, T. (2007). On large time-stepping methods for the Cahn–Hilliard equation.
Appl. Numer. Math., 57:616–628.
Hu, Z., Wise, S. M., Wang, C., and Lowengrub, J. S. (2009). Stable and efficient finite-difference nonlinear-
multigrid schemes for the phase field crystal equation. J. Comput. Phys., 228:5323–5339.
Hughes, T., Cottrell, J., and Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS,
exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering,
194:4135–4195.
Hughes, T. J. (2012). The finite element method: linear static and dynamic finite element analysis. Courier
Corporation.
Jacqmin, D. (1999). Calculation of two-phase Navier-Stokes flows using phase-field modeling. Journal of
Computational Physics, 155(1):96–127.
Jamet, D., Lebaigue, O., Coutris, N., and Delhaye, J. (2001). The second gradient method for the
direct numerical simulation of liquid-vapor flows with phase change. Journal of Computational Physics,
169(2):624 – 651.
Jansen, K. E., Whiting, C. H., and Hulbert, G. M. (2000). A generalized-αmethod for integrating the
filtered Navier-Stokes equations with a stabilized finite element method. Comput. Methods Appl. Mech.
Engrg., 190:305–319.
Kay, D., Styles, V., and Welford, R. (2008). Finite element approximation of a Cahn-Hilliard-Navier-Stokes
system. Interfaces Free Bound, 10(1):15–43.
Kessler, D., Nochetto, R. H., and Schmidt, A. (2004). A posteriori error control for the Allen–Cahn
problem: Circumventing Gronwall’s inequality. M2AN Math. Model. Numer. Anal., 38(1):129–142.
Kim, J., Kang, K., and Lowengrub, J. (2004). Conservative multigrid methods for Cahn-Hilliard fluids.
Journal of Computational Physics, 193(2):511–543.
Krueger, R. (2004). Virtual crack closure technique: history, approach, and applications. Applied Mechanics
Reviews, 57(2):109–143.
Labovsky, A., Layton, W. J., Manica, C. C., Neda, M., and Rebholz, L. G. (2009). The stabilized
extrapolated trapezoidal finite-element method for the Navier–Stokes equations. Comput. Methods Appl.
Mech. Engrg., 198:958–974.
Liu, C. and Shen, J. (2003). A phase field model for the mixture of two incompressible fluids and its
approximation by a Fourier-spectral method. Physica D: Nonlinear Phenomena, 179(3):211–228.
Liu, J., Gomez, H., Evans, J. A., Hughes, T. J., and Landis, C. M. (2013). Functional entropy variables:
A new methodology for deriving thermodynamically consistent algorithms for complex fluids, with
particular reference to the isothermal Navier-Stokes-Korteweg equations. Journal of Computational
Physics, 248:47 – 86.
Liu, J., Landis, C. M., Gomez, H., and Hughes, T. J. (2015). Liquid-vapor phase transition:
Thermomechanical theory, entropy stable numerical formulation, and boiling simulations. Computer
Methods in Applied Mechanics and Engineering, pages –.
Lorenzo, G., Michael A., S., Kevin B., T., Thomas J.R., H., Yongjie J., Z., Lei, L., and Hector, G. (2015).
Tissue-scale, patient-specific modeling of prostate cancer growth: a new perspective. in review.
Lowengrub, J. and Truskinovsky, L. (1998). Quasi-incompressible Cahn-Hilliard fluids and topological
transitions. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and
Engineering Sciences, 454(1978):2617–2654.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

COMPUTATIONAL PHASE-FIELD MODELING 43
Lowengrub, J. S., Frieboes, H. B., Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S. M., and Cristini,
V. (2010). Nonlinear modeling of cancer: Bridging the gap between cells and tumors. Nonlinearity,
23:R1–R91.
Mahnken, R. (2013). Goal-oriented adaptive refinement for phase field modeling with finite elements.
International Journal for Numerical Methods in Engineering, 94(4):418–440.
Marsden, J. E. and Hughes, T. J. (1994). Mathematical foundations of elasticity. Courier Dover
Publications.
Miehe, C., Welschinger, F., and Hofacker, M. (2010). Thermodynamically consistent phase-field models of
fracture: Variational principles and multi-field FE implementations. International Journal for Numerical
Methods in Engineering, 83(10):1273–1311.
Mo¨es, N., Dolbow, J., and Belytschko, T. (1999). A finite element method for crack growth without
remeshing. International Journal for Numerical Methods in Engineering, 46(1):131–150.
Oden, J. T., Hawkins, A., and Prudhomme, S. (2010). General diffuse-interface theories and an approach to
predictive tumor growth modeling. Mathematical Models and Methods in Applied Sciences, 20(03):477–
517.
Oden, J. T., Lima, E. A. B. F., Almeida, R. C., Feng, Y., Rylander, M. N., Fuentes, D., Faghihi, D.,
Rahman, M. M., DeWitt, M., Gadde, M., and Zhou, J. C. (2015). Toward predictive multiscale
modeling of vascular tumor growth: Computational and experimental oncology for tumor prediction.
Arch. Comput. Methods Eng. doi: 10.1007/s11831-015-9156-x.
Penrose, O. and Fife, P. C. (1990). Thermodynamically consistent models of phase-field type for the kinetic
of phase transitions. Phys. D, 43(1):44–62.
Provatas, N. and Elder, K. (2010). Phase-Field Methods in Materials Science and Engineering. Wiley-VCH,
Weinheim.
Provatas, N., Goldenfeld, N., and Dantzig, J. (1999). Adaptive mesh refinement computation of
solidification microstructures using dynamic data structures. J. Comput. Phys., 148:265–290.
Provatas, N., Greenwood, M., Athreya, B., Goldenfeld, N., and Dantzig, J. (2005). Multiscale modeling
of solidification: Phase-field methods to adaptive mesh refinement. Internat. J. Modern Phys. B,
19(31):4525–4565.
Rajagopal, A., Fischer, P., Kuhl, E., and Steinmann, P. (2010). Natural element analysis of the Cahn-
Hilliard phase-field model. Computational Mechanics, 46(3):471–493.
Romano, A. and Marasco, A. (2010). Continuum Mechanics: Advanced Topics and Research Trends.
Modeling and Simulation in Science, Engineering and Technology. Springer, New York.
Romero, I. (2009). Thermodynamically consistent time stepping algorithms for nonlinear thermomechanical
systems. Int. J. Numer. Meth. Engng., 79(6):706–732.
Rosam, J., Jimack, P. K., and Mullis, A. (2007). A fully implicit, fully adaptive time and space discretisation
method for phase-field simulation of binary alloy solidification. J. Comput. Phys., 225(2):1271–1287.
Sethian, J. A. (1999). Level set methods and fast marching methods: evolving interfaces in computational
geometry, fluid mechanics, computer vision, and materials science, volume 3. Cambridge university
press.
Shao, D., Levine, H., and Rappel, W.-J. (2012). Coupling actin flow, adhesion, and morphology in a
computational cell motility model. Proceedings of the National Academy of Sciences, 109(18):6851–
6856.
Shen, J. and Yang, X. (2010). Energy stable schemes for Cahn-Hilliard phase-field model of two-phase
incompressible flows. Chin. Ann. Math., 31B(5):743–758.
Shen, J. and Yang, X. (2015). Decoupled, energy stable schemes for phase-field models of two-phase
incompressible flows. SIAM J. Numer. Anal., 53(1):279–296.
Simo, J. C., Tarnow, N., and Wong, K. K. (1992). Exact energy-momentum conserving algorithms and
symplectic schemes for nonlinear dynamics. Comput. Methods Appl. Mech. Engrg., 100(1):63–116.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.

44 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
Simsek, G., Wu, X., van der Zee, K. G., and van Brummelen, E. H. (2015). Duality-based two-level error
estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations. Comput.
Methods Appl. Mech. Engrg., 288:83–109.
S¨oderlind, G. (2002). Automatic control and adaptive time-stepping. Numer. Algorithms, 31(1):281–310.
Stein, E. and R¨uter, M. (2004). Finite element methods for elasticity with error-controlled discretization
and model adaptivity. In Stein, E., de Borst, R., and Hughes, T. J. R., editors, Encyclopedia of
Computational Mechanics, volume 2: Solids and Structures, chapter 2, pages 5–58. John Wiley & Sons.
Stogner, R. H., Carey, G. F., and Murray, B. T. (2008). Approximation of Cahn-Hilliard diffuse interface
models using parallel adaptive mesh refinement and coarsening with C1elements. Int. J. Numer. Meth.
Engng., 76:636–661.
Tian, L., Xu, Y., Kuerten, J., and van der Vegt, J. (2015). A local discontinuous Galerkin method for the
(non)-isothermal Navier-Stokes-Korteweg equations. Journal of Computational Physics, 295:685 – 714.
Trefethen, L. N. (2000). Spectral methods in MATLAB, volume 10. Siam.
Truesdell, C. (1984). Rational Thermodynamics. Springer, New York, 2nd edition.
Truesdell, C. and Noll, W. (1965). The Non-Linear Field Theories of Mechanics. Springer.
van der Zee, K. G., Oden, J. T., Prudhomme, S., and Hawkins-Daarud, A. (2011). Goal-oriented error
estimation for Cahn–Hilliard models of binary phase transition. Numer. Methods Partial Differential
Equations, 27(1):160–196.
Vilanova, G., Colominas, I., and Gomez, H. (2013). Capillary networks in tumor angiogenesis: From
discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis. International
journal for numerical methods in biomedical engineering, 29(10):1015–1037.
Vilanova, G., Colominas, I., and Gomez, H. (2014). Coupling of discrete random walks and continuous
modeling for three-dimensional tumor-induced angiogenesis. Computational Mechanics, 53(3):449–464.
Vilanova, G., Gomez, H., and Colominas, I. (2012). A numerical study based on the fem of a multiscale
continuum model for tumor angiogenesis. Journal of Biomechanics, 45:S466.
Wang, S.-L., Sekerka, R. F., Wheeler, A. A., Murray, B. T., Coriel, S. R., Braun, R. J., and McFadden,
G. B. (1993). Thermodynamically-consistent phase-field models for solidification. Phys. D, 69:189–200.
Wells, G. N., Kuhl, E., and Garikipati, K. (2006). A discontinuous Galerkin method for the Cahn-Hilliard
equation. Journal of Computational Physics, 218(2):860 – 877.
Wise, S. M., Lowengrub, J. S., Frieboes, H. B., and Cristini, V. (2008). Three-dimensional multispecies
nonlinear tumor growth—I Model and numerical method. J. Theoret. Biol., 253:524–543.
Wise, S. M., Wang, C., and Lowengrub, J. S. (2009). An energy-stable and convergent finite-difference
scheme for the phase field crystal equation. SIAM J. Numer. Anal., 47(3):2269–2288.
Wodo, O. and Ganapathysubramanian, B. (2011). Computationally efficient solution to the Cahn–Hilliard
equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem.
J. Comput. Phys., 230:6037–6060.
Wu, X., van Zwieten, G. J., and van der Zee, K. G. (2014). Stabilized second-order convex splitting
schemes for Cahn–Hilliard models with application to diffuse-interface tumor-growth models. Int. J.
Numer. Meth. Biomed. Engng., 30:180–203.
Xu, J., Vilanova, G., and Gomez, H. (2015). A mathematical model coupling tumor growth and
angiogenesis. in review.
Zhou, J. X. and Li, M. E. (2006). Solving phase field equations using a meshless method. Communications
in Numerical Methods in Engineering, 22(11):1109–1115.
Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´e de Borst and Thomas J.R. Hughes.
c
2004 John Wiley & Sons, Ltd.