Approximation of Cahn-Hilliard diffuse interface models using …roystgnr/libmeshpdfs/roystgnr/chpap… · c 2008 John Wiley & Sons, Ltd. KEY WORDS: Cahn-Hilliard, diffuse interface,

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng2008;1:1–23 Prepared usingnmeauth.cls [Version: 2002/09/18 v2.02]

Approximation of Cahn-Hilliard diffuse interface models using paralleladaptive mesh refinement and coarsening withC

1 elements

Roy H. Stogner1,∗, Graham F. Carey1, Bruce T. Murray2

1Institute for Computational Engineering and Sciences, University of Texas, Austin, Texas 787122Department of Mechanical Engineering, SUNY at Binghamton,Binghamton, New York 13902

SUMMARY

A variational formulation andC1 finite element scheme with adaptive mesh refinement and coarsening is developedfor phase-separation processes described by the Cahn-Hilliard diffuse interface model of transport in a mixture oralloy. The adaptive scheme is guided by a Laplacian jump indicator based on the corresponding term arising fromthe weak formulation of the fourth-order nonlinear problem, and is implemented in a parallel solution framework. Itis then applied to resolve complex evolving interfacial solution behavior for2D and 3D simulations of the classicspinodal decomposition problem from a random initial mixture and to other phase-transformation applications of interest.Simulation results and adaptive performance are discussed. The scheme permits efficient, robust multiscale resolution andinterface characterization.

Copyright c© 2008 John Wiley & Sons, Ltd.

KEY WORDS: Cahn-Hilliard, diffuse interface, multiscale, adaptive, finite elements

1. Introduction

The basic Cahn-Hilliard treatment of interfacial free energy [1] involves a diffuse-interface region ofsmall, finite thickness separating two distinct phases. There are a wide variety of applications of Cahn-Hilliard-based models in materials science [2]. Some recent interesting examples of practical applicationsare microscale annealing and nanoscale void self-assembly[3, 4]. These examples demonstrate that Cahn-Hilliard-based models can be applied on a very broad range ofspatial scales. By adding additional variablesfor concentrations of additional components [5] and/or material phases [6], systems of Cahn-Hilliard-likeequations can be used to describe the evolution of more complicated mixtures with known free-energyfunctions. Even nominally immiscible fluid mixtures can be modeled by coupling the Cahn-Hilliard andNavier-Stokes equations [7, 8]. For physical problems withsharp material interfaces, explicit front-trackingmethods [9] are useful. However in Cahn-Hilliard problems such as spinodal decomposition, nonlocalphysical processes such as nuclei “melting” and interface topology changes may be difficult to describevia a sharp-interface model.

The diffuse-interface model arising from Cahn-Hilliard theory may be described mathematically by ascalar, nonlinear partial differential equation (PDE) having fourth-order spatial partial derivatives as described

∗Correspondence to: Roy Stogner, The University of Texas at Austin, Institute for Computational Engineering and Sciences, Austin, TX78712Email: [email protected]

Copyright c© 2008 John Wiley & Sons, Ltd.

2 R. H. STOGNER, G. F. CAREY AND B. T. MURRAY

below in Section 2. There are a number of numerical studies ofthe Cahn-Hilliard equation in the literature,primarily for simple domains using finite difference discretization on structured meshes or spectral methods.Solutions based on the finite element method are less common.Early efforts were limited to one-dimensionalstudies [10]. For examples of recent finite element work on this topic see [11, 12]. Also note that most ofthe work in this area has focused on numerical simulations with fine structured fixed grids. These methodsare limited in their ability to treat practical applications involving more complex domains or unstructuredevolving meshes capable of capturing the behavior of movingtortuous diffuse interfaces that exhibit largelocal gradients in the solution structure. Galerkin finite element schemes and similar integral formulations,that can support local adaptive mesh refinement and coarsening (AMR/C) to resolve multiscale features, arebetter suited to such diffuse-interface models.

Following the presentation of the Cahn Hilliard model and diffuse-interface properties, an approximateGalerkin formulation for the resulting fourth-order equation is developed using conformingC1 elementsin Section 3. For the 2D formulation, an AMR/C scheme is implemented with hanging-node constraintsenforced onC1 macroelement triangles and tensor product bicubic Hermiterectangles. In the 3D simulationsHermite tricubic bases are used, again with hanging-node constraints enforced for conformity usingunbalanced octrees. The software frameworklibMesh [13] has been extended to accomodate this problemclass and the related AMR/C data structures. Some details ofthe solution algorithms, parallel AMR/Cimplementation, and data structures are provided in Section 5.

As a fundamental test case, consider the classical spinodaldecomposition problem [14], where an initialhomogeneous two-component mixture phase separates below aspecific transition temperature. In this casethe initial solution exhibits numerous tortuous diffuse interfaces with a complex topology that coalesces toform gradually less complex configurations, simplifying toward an eventual steady state. Simulation resultsin 2 and 3 dimensions are computed using both uniform high resolution meshes and adaptive meshes forcomparison purposes. The implications for AMR applied withprogressive coarsening are also discussed.Some of the main phenomenological and numerical observations arising from the study are reviewed in theConcluding Remarks.

2. Cahn-Hilliard model

2.1. Model Derivation

We consider an isothermal binary mixture. The concentration c of one of the two components of the systemis used to characterize the two phases of the binary system, with fixed values ofc representing the bulkphases. The model for time evolution of concentration comesfrom thermodynamic arguments and massconservation. In the conservation equation, the flux is based on the local free-energy density to ensure amonotonic decrease in the global free energy of the system. More specifically, the Cahn-Hilliard fourth-ordermodel may be derived by introducing the stationary free energy functional to define a chemical potential interms of concentration, then using this potential in the associated transport equation.

The free energy functionalF is the integral over the volumeΩ of the free-energy densityf , which isassumed to be a function ofc and its spatial derivatives. Following [1],f is approximated by the leadingterms in a truncated Taylor expansion as

f(c,∇c) = f(c) + ~L · ∇c+∇c · [K] · ∇c (1)

wheref(c) is the free-energy density of a homogeneous system at concentration c. Symmetry argumentsrequire~L = 0 (otherwise the value off would be different for gradients in opposite directions). For anisotropic system, the matrix[K] will be diagonal with equal entriesǫ

2

2 , where parameterǫ is referred to asthegradient coefficient. Note that a wide variety of symbols and names have been used for these quantities in

Copyright c© 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng2008;1:1–23Prepared usingnmeauth.cls

APPROX. OF CAHN-HILLIARD MODELS BY PARALLEL AMR/C WITHC1 ELEMENTS 3

the literature;f(c) is commonly referred to as the “configurational”, “homogeneous”, or “bulk” free-energydensity, andǫ

2

2 |∇c|2 may be called the “gradient”, “surface”, or “interfacial” free-energy density.Using this truncated Taylor series form for the free-energydensity, the corresponding approximation to the

total free energy for the system is

F =

∫

Ω

(

ǫ2

2|∇c|2 + f(c)

)

dV. (2)

Variational calculus may be used to determine the effect of achange in the concentration on the local free-energy density. Taking a variation ofF , the Frechet derivative ofF with respect to concentration defines thechemical potentialµ,

µ =δFδc

=∂f

∂c−∇ · ∂f

∂∇c= f ′(c)− ǫ2∆c (3)

This potential represents the driving force for local reduction of c. The fourth-order model for diffusion in abinary system then follows from applying conservation of mass with the constitutive relation for flux specifiedin terms of the chemical potential.

Conservation of mass implies that the time rate of change of concentration due to diffusion is related to thedivergence of the flux~J by

dc

dt= −∇ · ~J. (4)

Since transport is from regions where chemical potentialµ is high to regions where it is low, the constitutiveequation for the flux sets it proportional to the negative of the chemical potential gradient,

~J = −M(c)∇µ, (5)

whereM(c) is the mobility. In general tensor-valued mobility functions are consistent with the rest of theCahn-Hilliard equation, but in the literature scalar-valued functions and constants are used. Substituting (5)into (4) and setting convective velocity as zero yields

dc

dt=

∂c

∂t= ∇ · (M(c)∇µ) . (6)

where the chemical potentialµ is given in (3) in terms of second order partial derivatives of theconcentration. Substituting this relationship for the chemical potential into (6), for a given homogeneousfree-energy densityf(c) the resulting fourth-order Cahn-Hilliard equation has thegeneral form:

∂c

∂t= ∇ ·

[

M(c)∇(

−ǫ2∆c+ f ′(c))]

. (7)

The competing gradient and homogeneous free-energy density terms determine the structure and equilibriumthickness of the diffuse-interface regions between the bulk phases as discussed in section 2.3.

2.2. Homogeneous Free Energy

The homogeneous free-energy density depends on the chemistry of a particular physical problem. The initialwork by Cahn and Hilliard [1], as well as much of the subsequent material science literature, uses a regularsolution model that is referred to here as the “chemical” free-energy density function

f(c) = fc(c) ≡ NkT (c ln (c) + (1− c) ln (1− c)) +Nωc(1− c) (8)

for small molecule mixtures, whereN is the molecular density,k is Boltzmann’s constant,T is temperature,and ω is a scalar parameter related to fluid miscibility. Below a critical temperature (2kT < ω) thisfree-energy density is a double well which will lead to phaseseparation. In this formulation the phase is



distinguished by the concentrationc approaching either of it’s limit values (0 or 1). This function is alsocalled the “Flory-Huggins” free-energy density, due to itsoriginal derivation by Huggins [15] and Flory [16]in studies of polymer solutions.

For the chemical free energy functionfc, the derivativef ′c is

f ′c = NkT (ln (c)− ln (1− c)) +Nω(1− 2c) (9)

and becomes singular at the concentration limitsc = 0 andc = 1. This has the effect of preventingc(x, t)from taking non-physical values beyond those limits.

In the literature, a polynomial approximation off is often used, which avoids the restricted range and thesingularities of the chemical free energy expression (8). This “mathematical” free-energy density function isa quartic double-well, which for a concentration range between0 and1 is

f(c) = fm(c) ≡ c2 (1− c)2 (10)

and has derivativef ′m = 2c− 6c2 + 4c3 (11)

Another common form forf in the mathematics literature is(

1− c2)2

, in which the phasec is defined asthe normalized difference between the two concentrations in a binary mixture, and thus is expected to vary inthe range from−1 to 1. In either case,fm is just an approximation, and depending on initial conditions andon other terms in the Cahn-Hilliard equationc can take on values outside of the physically meaningful range.

2.3. Interface Property Specification

When diffuse-interface theories are used to represent real physical interfaces, the interface thicknessestypically are on the order of nanometers. The classic example of Cahn-Hilliard theory is modeling thephenomena of spinodal decomposition in a binary system. In this application the true thermodynamicfree-energy density functionf is modeled and the theory results in the appropriate interfacial energy andequilibrium interface thickness. In the last ten years, diffuse-interface or phase-field models have beendeveloped to model a wide range of phenomena. In many of theseapplications, an artificially large interfacethickness is employed to regularize the mathematical problem and make it numerically more tractable. Thisapproach is reasonable for problems where the scale of the solution structures of interest is not influencedby the regularization. Where the behavior of fine scale interface structures is of interest, regularization maynot be an option and multiresolution strategies such as AMR/C may be necessary to reliably approximate thebehavior in the real diffuse-interface layer.

The interface is characterized by a very small but finite thicknessδ, and the interpretation ofδ will dependon the specific phase-change problem addressed. In physicalterms the phase boundary is also characterizedby a surface energy or energy per unit areaσ. For diffuse-interface theories, the interface thicknessand thesurface energy can be related to the gradient coefficient (ǫ), and to the heightW of the barrier between wellsin the free-energy densityf .

By analyzing an equilibrium solution for a simple geometry,estimates for the interface thickness andsurface energy in terms of the model parameters can be obtained [1] as, respectively,

δ ∼ ǫ√W

(12)

andσ ∼ ǫ

√W. (13)

These terms influence the fine-scale and macro-scale behavior of the system.



3. Variational Formulation

The highest-order derivatives in the governing fourth-order equation correspond to the linear biharmonicoperator. Taking a weighted integral residual projection of equation (7) with test functionsφ, and integratingthe second order terms by parts once and the fourth-order term by parts twice, yields

(

∂c

∂t, φ

)

Ω

= F (c, φ)

≡ − (M(c)∇f ′(c),∇φ)Ω − ǫ2(

∆c,∇ ·M(c)T∇φ)

Ω

+((

M(c)∇(

f ′(c)− ǫ2∆c))

· ~n, φ)

∂Ω+ ǫ2

(

∆c,M(c)T∇φ · ~n)

∂Ω(14)

where the usual inner product notation is used for the respective interior and boundary integrals. In generaleven the fourth-order terms in the equation can be nonlinear, and the domain of the functional depends on themobility functionM(c) and the free-energy density functionf(c).

To complete the weak statement, an initial condition and twosets of boundary conditions are applied forthe fourth-order problem. Natural boundary conditions forthe fourth-order problem involve second and thirdderivatives and can be specified via substitution in the boundary integrals above in the same way as in platebending and viscous flow formulations[17]. Essential boundary conditions forc and∂~nc are specified directlyas constraints on the trial functions or via penalties in an augmented integral formulation. For instance, assumeessential boundary conditions are to be enforced on open subsetsΓD1,ΓD2 ⊂ ∂Ω − ΓP , with the tracesc = h1 onΓD1 and∂~nc = h2 onΓD2. Along the remainder of the boundary, assumec is required to satisfythe natural boundary conditions

(

M(c)∇(

f ′(c)− ǫ2∆c))

·~n = g1 onΓN1 ≡ ∂Ω−ΓP −ΓD1 and∆c = g2onΓN2 ≡ ∂Ω− ΓP − ΓD2.

Substituteg1 andg2 for the corresponding boundary terms in (14). The resultingfunctional is

FN (c, φ) = − (M(c)∇ (f ′(c)) ,∇φ)Ω − ǫ2(

∆c,∇ ·M(c)T∇φ)

Ω

+(g1, φ)ΓN1+ ǫ2

(

g2,M(c)T∇φ · ~n)

ΓN2

(15)

To simplify the implementation of this problem in the numerical studies presented later, the essentialboundary conditions are replaced with penalty boundary integral terms. Choosing some positiveεp << 1,add penalty terms to the weighted residual functional:

FB(c, φ) = FN (c, φ) +1

εp(c− h1, φ)ΓD1

+1

εp(∂~nc− h2, ∂~nφ)ΓD2

(16)

Comparing these terms with the boundary integrals in (14), for scalarM(c) it can be seen that the penaltymethod is equivalent to weakly enforcing the boundary conditions

∂~nc = h2 + εpM(c)ǫ2∆c (17)

onΓD2 andc = h1 + εpM(c)∂~n

(

f ′(c)− ǫ2∆c)

(18)

onΓD1.As εp → 0, solutions obtained with this boundary condition approachsolutions obtained with the exact

boundary condition.In the weak form of the problem, then, findc(t, ~x) such thatc(0, ~x) = c0(~x), and for all admissible phi,

(

∂c

∂t, φ

)

= FB(c, φ) (19)



For cases where one is interested in bulk properties in a large region, an artificially restricted domain withperiodic boundary conditions may be used. Along an edge in 2Dor face in 3D of such a periodic boundaryΓP , the traces of admissiblec and∂~nc are restricted to be equal to the traces on the opposite periodic side. Inthe weak formulation of such a problem, both trial and test functions are simply restricted to the subspace offunctions with suitably periodic boundary data. Ifp : ΓP → ΓP is the function translating any point on theperiodic boundaries to the boundary on the opposite side, this restrics a function spaceH to

HP ≡

v ∈ H : (v(x)− v(p(~x)))|ΓP= 0, (∂~nv(~x) + ∂~nv(p(~x)))|ΓP

= 0

(20)

In numerical simulations, these periodic constraints are added to the system in the same way as the hanging-node constraints required forh adaptivity.

3.1. Time Discretization

In the following results, the transient PDE system is integrated with respect to time using a generalized Euleror a trapezoidal method. In the experiments using generalized Euler, the new approximate solution at timetn+1 ≡ tn + δtn is defined recursively from the initial vectorc0 for n=1,2,... as the solution of the stationaryPDE

(

cn+1 − cn

δt, φ

)

Ω

= FB(cθ, φ) (21)

for givenθ ∈ [0, 1] with the implied approximationcθ ≡ θcn+1 +(1− θ)cn and the nonlinear weighted timederivativeF with boundary conditions as defined in (16). Of course,θ = 1 corresponds to implicit (BackwardEuler) integration, andθ = 1/2 corresponds to the second-order accurate Crank-Nicolson method. The timestep sizeδtn can be varied adaptively as described in Section 4.4. For other techniques in time truncationerror estimation and time step control see [18, 19].

When taking relatively large time steps with the weak Cahn-Hilliard equation using the chemical freeenergy density, the Crank-Nicolson method may be unsuitable. If at some point~x ∈ Ω the concentrationc(~x) < 0 or c(~x) > 1, thenF (c) can become undefined. With the Crank-Nicolson method, it is possible forF (cθ) to be well-defined (and for the time step totn+1 to complete successfully) while the correspondingF (cn+1) is undefined. At the next time step, using the obvious initialiterate ofcn+2

0 ≡ cn+1, a nonlinearsolver will then fail to calculate the solutioncn+2. In our computational experiments such failures have notbeen encountered while using uniform time step lengths, butthey often occur in the process of integratingwith adaptiveδtn selection.

This problem is not encountered when using Backward Euler, but the lower convergence rate may beunacceptable. The trapezoidal rule provides a reliable second-order accurate time discretization.

(

cn+1 − cn

δt, φ

)

Ω

=1

2FB(c

n, φ) +1

2FB(c

n+1, φ) (22)

Because this method interpolates the functionalF rather than the solutionc, it is slightly more expensiveto compute. However, becauseF (cn+1) is successfully evaluated in the solution of time stepn, it will alwaystake a well-defined value in the initial iteration at time step n+ 1.

3.2. Spatial Discretization

In the past, most approximation of the Cahn-Hilliard equation using finite element methods has focused onmixed methods, introducing a separate scalar variable for∆c or for f ′

0(c)− ǫ2∆c, then solving the resultingsystem of equations on aC0 finite element space. See e.g. [20, 21]. By usingC1 continuous finite elementspaces, one can directly determine conforming approximatesolutions to the weak form of the fourth-orderCahn-Hilliard equation without introducing the additional variable and degrees of freedom required by amixed system, while also circumventing the stability issues associated with mixed formulations.



The spatial discretization of the problem follows on posingthe weak statement on an appropriate finiteelement subspace defined by simply restrictingch to the finite dimensional function spaceHh of C1

piecewise-polynomial basis functions on a finite element mesh. Applying this process to equation (21) leavesa fully discretized set of nonlinear algebraic equations ateach time step: findch ∈ Hh such that for alladmissible test basis functionsφi ∈ Hh,

(

∂c

∂t, φi

)

= FB(c, φi) (23)

C1 finite elements have been used for some time on conforming meshes, in the 2D context of plate bendingand streamfunction formulations. The spaces of tensor products of C1 Hermite cubic polynomials (theBogner-Fox-Schmidt rectangle and its 3D analogue) are not designed for general quadrilateral or hexahedralmeshes, but achieve optimal accuracy on smooth mappings of rectilinear grids [22]. Because many interestingCahn-Hilliard problems can be posed on rectilinear domains, several of the following numerical experimentsuse Hermite elements. On arbitrary unstructured meshes,C1 spaces of complete piecewise polynomials mayrequire undesirably high polynomial degree to enable optimal accuracy and locally supported bases [23, 24].To construct low-order, locally-supportedC1 functions on general mesh geometries, each element in a meshis treated as a piecewise polynomial macroelement. Sections 4.1 and 5.2 will discuss solutions to the newchallenges encountered when creating adaptively refined meshes of any of these element types.

Dividing a macroelement into subelements adds additional “raw” degrees of freedom. Some degreesof freedom are constrained away to enforce internalC1 continuity between subelements; others are usedto enable inter-elementC1 continuity along the interfaces with adjoining macroelements. Convenientmacroelement classes in 2D include the quadratic 12-split Powell-Sabin-Heindl [25] triangle and the cubic 3-split Hsieh-Clough-Tocher [26] triangle. In this paper thenumerical results on triangles use the HCT element,which for smooth functions achieves higher approximation accuracy than PSH elements using the same setof degrees of freedom.

Each HCT macrotriangle is composed of three subtriangles separated by lines connecting the macrotrianglecentroid to each vertex. Because this geometry is preservedunder affine transformations, finite elementcalculations can be performed on HCT spaces using the traditional transformation to a “master element”,after an additional calculation to construct global finite element shape function from linear combinations oflocal master element shape functions, as described in [27].In the global HCT element space, three degrees offreedom exist on each macroelement vertex, corresponding to the function value and two function gradientcomponents at the vertex location. An additional degree of freedom corresponding to the midpoint normalderivative exists on each macroelement edge.

Because a partially developed Cahn-Hilliard solution can be characterized as a collection of single-phaseregions separated by an evolving network of thin interfaces, it is natural to consider the use of adaptive meshrefinement and coarsening to obtain more numerically efficient solutions. A wide class of macroelementsincluding the HCT triangles can be successfully used to provide optimally accurateC1-conforming functionson adapted meshes [27]. The same constraint construction techniques can be used on non-macroelements aswell, including the Hermite rectangles and rectangular prisms mentioned above.

4. Adaptive Mesh Refinement/Coarsening

At each time step, the time discretization gives an ellipticboundary value problem, with the solution at theprevious time step as a forcing function. As the character ofthe time-evolving solution changes, so will thecharacter of the optimal meshes on which to discretize that solution.

In the case of the Cahn-Hilliard problem, the typical evolving solution can be described as a set ofnearly homogenous single-phase material regions separated by narrow diffuse interfaces which exhibit large



gradients in concentration. Accurate approximation of thesingle-phase-region interiors requires relatively fewelements when compared to the requirements for accurate approximation of the interface regions. Optimally,the approximate solution at each time step should be found ona mesh which has been adapted to representboth the new and old solutions at that time step efficiently and accurately. Using AMR/C to simulate atransient problem like the Cahn-Hilliard equation demandsfour key developments:

1. The ability to construct conforming function spaces on non-conforming adaptively refined meshes, asexplained in Section 4.1.

2. An error indicator which can reliably guide mesh refinement, as in Section 4.23. An algorithm to use the information provided by those error estimates to generate the new mesh at each

time step, as described in Section 4.34. A projection operator to accurately and efficiently transfer the solution from each time step onto the

new mesh for the next time step, such as in Section 5.2.

4.1. Hanging-node Refinement

At each time step, a new adapted mesh is generated via local operations on the existing mesh. Beginningwith a coarse mesh to describe the domain geometry, a series of related adapted meshes can be generated viaindependent refinement decisions on each element. When a “parent” triangle, quad, or hex inRd is refined,it is replaced by2d “child” elements of the same type. The parent elements are retained in the softwaredata structure, a “tree” whose branches are refined parent elements and whose leaves are the “active” childelements. The function spacesHh are then defined on the mesh of child elements.

Meshes created in this fashion are non-conforming: if two elements are neighbors, but one element isthe product of more levels of refinement, then the side of the coarser element will be shared by the sidesof multiple refined neighbors. The refined-element nodes which do not correspond to coarse-element nodesare known as “hanging nodes”, and to maintain the desired continuity level of the finite element functionspace, all refined-element degree-of-freedom values on these shared sides are constrained in terms of thedegree-of-freedom values on the coarse element.

To apply these constraints onC1 element spaces, small local projection matrices on each nonconforminginternal side are calculated and inverted. On a sideγ between refined elementKF and its more coarseneighborKC , constraints are needed to ensure continuity for the function c and its firstr = 1 derivatives,∀ ~x ∈ γ, ∀ |α| ≤ r,

DαcF (~x) = DαcC(~x) (24)

Expanding the refined-element solutioncF and the coarse-element solutioncC in the respective bases,

Dα∑

i

cFi φFi (~x) = Dα

∑

j

cCj φCj (~x) (25)

Equality is enforced via projections onγ:∑

|α|≤r

(DαcF , DαφFk )γ =

∑

|α|≤r

(DαcC , DαφFk )γ (26)

Expanding and using linearity of derivatives:∑

i

cFi∑

|α|≤r

(DαφFi , D

αφFk )γ =

∑

j

cCj∑

|α|≤r

(DαφCj , D

αφFk )γ (27)

This linear system has the formAkicFi = Bkjc

Cj . Solving gives the constraint matrixA−1B, which is used

to pre- and post-multiply element stiffness matrices as in [28]. To solve, a Cholesky decomposition ofA isperformed on each nonconforming interface after each mesh adaptation; any non-trivial constraint equationcoefficients are then stored for repeated application in residual and Jacobian assemblies during nonlinearsolves.



4.2. Error Indicator

A variety of error indicators of different degrees of complexity and computational cost can be devised foradaptive simulations. These include interpolation-basedindicators, residual indicators, flux jump indicators,patch recovery indicators and adjoint-based dual indicators [29, 30, 31, 32, 33, 34]. The simple flux jumpindicator based on the Laplace operator has been used successfully for a variety of second order problems, sothe natural extension here is to use an analogous indicator based on the biharmonic operator.

Accordingly, consider the biharmonic problem: find the solutionu to∆2u = f in domainΩ, with essentialboundary conditionsu = h1 on ΓD1 ⊂ ∂Ω and∂~nu = h2 on ΓD2 ⊂ ∂Ω, and with natural boundaryconditions∂~n∆u = g1 onΓN1 ≡ ∂Ω− ΓD1 and∆u = g2 onΓN2 ≡ ∂Ω− ΓD2.

The weak formulation of the biharmonic problem follows immediately by integrating a weighted residualby parts, analogous to the process in Section 3. Given a finiteelement spaceHh ⊂ H2(Ω), the Galerkinsolutionuh ∈ Hh is obtained by enforcing the weak formulation for all test functions inHh

B . Then for thisproblem the jump-based indicator can be computed on elementsides as follows:

Define the Laplacian jump on its natural boundaryΓN2 as

[[∆uh]]ΓN2≡ 2(g2 −∆uh) (28)

Define the Laplacian jump as zero onΓD2, and on internal sides shared by subelementsS, S′ as

[[∆uh(~s)]]∂S ≡ lim~x→~s,~x∈S′

∆uh(~x)− lim~x→~s,~x∈S

∆uh(~x) (29)

Define the Laplacian flux jump on its natural boundary as

[[∂~n∆uh]]ΓN1≡ 2(g1 − ∂~n∆uh) (30)

Define the Laplacian flux jump as zero onΓD1, and on internal sides as

[[∂~n∆uh(~s)]]∂S ≡ lim~x→~s,~x∈S′

∂ ~nS∆uh(~x)− lim

~x→~s,~x∈S∂ ~nS

∆uh(~x) (31)

As derived in [27], under suitable conditions on the regularfamily of conforming finite element spacesHh, the boundaryΩ, and the boundary conditions, there exists aC dependent only onΩ for which the errore ≡ u− uh is bounded by the sum over all subelementsS ⊂ Ω

||e||H2(Ω) ≤ C∑

S

[

∣

∣

∣

∣f −∆2uh

∣

∣

∣

∣

Sh2S +

1

2||[[∂~n∆uh]]||∂S h

3/2S +

1

2||[[∆uh]]||∂S h

1/2S

]

(32)

This error estimate leads to effective local error indicators for the biharmonic problem even after muchsimplification. The simplified Laplacian jump error indicator evaluates only theh1/2

S term and only on theboundaries∂K of elementsK rather than subelementsS. An indicator that includes the contributions ofother terms in the residual would be a straightforward extension, and adjoint based dual indicators wouldallow goal-oriented adaptivity for functional quantitiesof interest. However, for the present problem class asdemonstrated in the numerical results below, the solution is benign except in the diffuse interface regions andthe simplified jump indicator is appropriate and efficient:

ηK =√

hK ||[[∆uh]]||∂K (33)

In the following results, this error indicator is used with the approximate Cahn-Hilliard solution at eachtime step to guide adaptive refinement.



4.3. Refinement/Coarsening Strategy

At each adaptive step, some elements may be refined and other elements coarsened to adapt the mesh tothe evolving solution. Every active element in the mesh is a potential candidate for immediate refinement,which replaces that element with2d “child” elements. Coarsening introduces additional complications. First,not all child elements are candidates for immediate coarsening. This is because coarsening replaces a childelement and all its “sibling” elements with their common parent, and we do not want to reverse more thanone level of refinement at each adaptive step. Hence, only elements without refined siblings are eligible forcoarsening in a single AMR/C step. Second, it is necessary tohave not just an indication of how much error isin the current solution, but how much new error would be introduced by coarsening. This can be obtained ina general fashion from any element-wise error indicator. The indicator gives a scalar error valueηK on eachactive finite elementK in the mesh. If a set of sibling elements are all active, calculate an error indicator ontheir common parent elementKp as:

ηKp= cη2

n

√

∑

K⊂Kp

η2K (34)

where n is the expected asymptotic order of convergence of the finiteelement family in the normapproximated by the error indicator, andcη > 1 is a constant used to make coarsening behavior moreconservative. This use ofcη is equivalent to using a lower error tolerance for coarsening than for refinement.The asymptotic convergence raten is used to makeηKp

a reasonable predictor of the error indicator onKp during subsequent adaptive steps, andcη makes it less likely that coarsening will be based on anunderpredictedKp and thus followed by immediate refinement on subsequent adaptive steps.

In the numerical simulations to follow, using the Laplacianjump indicator on cubic elements, the expectedconvergence raten is 2. Refinement of macroelements requires the macroelement subdivision of sides tobe compatible with the refinement subdivision. For macroelement families such as the HCT triangles wherethis is the case, refinement first deletes the subelements of the “parent” macroelement, and each new child isdivided into subelements in the canonical way.

To optimize the accuracy of the transient simulation, theseAMR/C tools roughly equidistribute the errorindicator values between mesh cells, refining to improve accuracy where the error indicator values rise andcoarsening where the error indicator values fall. There aremany possible strategies for AMR/C. In theadaptive results in Section 6, we choose to maintain a bounded degree-of-freedom count (and thus an upperbound on simulation memory usage and cost) by refining to a desired element countnd(t) as follows:

At time tn, if the number of active elementsna in the mesh exceedsnd(tn), then elements are flagged forcoarsening to eliminate the excess, with those elements having the lowest error indicatorsηKp

chosen first.If insteadnd(tn) > na, then enough active elements are flagged for refinement to eliminate the deficit, withthose elements having the highest error indicatorsηK chosen first.

The remaining sets of unflagged refinable active elements andunflagged coarsenable parent elements arethen sorted by error indicator values. Beginning with the parent element having lowestηKp

and the activeelement having highestηK , elements are flagged in pairs, one active element for refinement and one parentelement for coarsening, for as long as pairs can be found satisfying ηKp

< ηK . If K is refined while thechildren ofKp are coarsened, then the number of elements in the mesh will beunchanged, the number of“raw” degrees of freedom in the mesh will be unchanged, and even after hanging-node constraints are applied,the number of degrees of freedom in the mesh will be approximately unchanged. IfηKp

< ηK , then this tradeshould provide a reduction in the total error on the mesh at approximately the same computational cost.

4.4. Adaptive Time Discretization

In physical problems of interest such as the spinodal decomposition problem discussed in Section 6.2,adaptive time stepping is also necessary to efficiently resolve widely varying time scales. The rapid changes in



material concentration induced by the early stages of phaseseparation and by the coalescence of labyrinthineinterfaces require small time steps to simulate accurately. Carrying out an entire simulation with such a small,fixed δt would be highly inefficient, especially for later times whenthe phase structure evolves more slowlyand smoothly.

To choose efficient and accurate time step sizes, again the goal is to equidistribute locally computablediscretization error estimates. For each time step, the simulation is advanced from timetn to tn+1 ≡ tn+δtnin two different ways, by taking two time steps of sizeδtn/2 to calculatecn+1 from cn, and by taking a singletime step of lengthδtn to calculatecn+1 from cn. Our local relative time discretization error estimate forthistime step is

en ≡ ||cn+1 − cn+1||max (||cn+1|| , ||cn+1||)

(35)

And the global relative time discretization error estimateis

en ≡ enδtn

(36)

The time step adaptivity is based on a target error toleranceeTOL and a maximum acceptable errortoleranceeMAX . If en < eMAX after time stepn is calculated, then the simulation continues with a newtime step size

δtn+1 ≡ δtn ·(

eTOL

en

)1/p

(37)

where herep is the global convergence rate of the time stepping algorithm being used; e.g.p = 2 for thetrapezoidal rule.

If it is found thaten ≥ eMAX after time stepn, or if the nonlinear solver fails, then that time step isrejected and recalculated with progressively halvingδtn. In practice on the Cahn-Hilliard problem, witheMAX = 1.2eTOL, rejections only commonly occur at the start of a simulation, as the algorithm finds anacceptable initial time step size.

5. Solution Algorithms

5.1. Parallel implementation

The parallel adaptive work described here extends and applies the capability of thelibMesh framework [13].In this finite element library, degrees of freedom on a finite element mesh are stored topologically, on degree-of-freedom objects corresponding to element vertices, edges, faces, and interiors. For each degree of freedom,the corresponding basis function has support on the set of elements which “contain” its degree-of-freedomobject, where an element is considered to contain its own vertices, edges, etc. This definition is independentof any internal edges and vertices which are part of a macroelement splitting. By treating any associateddegrees of freedom as ordinary element interior degrees, the software storing them does not need to be awareof subelements, and can operate independently of the element types in use on a mesh.

The parallel distribution of global vectors and matrices follows directly from the parallel subdivision ofthe mesh itself. A partitioner such as METIS assigns each element of the mesh to a particular partition, andall degrees of freedom which are contained by elements in only one partition are “owned” by the processorassociated with that partition. Degrees of freedom on the boundary of multiple partitions could be associatedwith any of their associated processors; inlibMesh such degrees of freedom are assigned to the processorwith the lowest index number. The degree-of-freedom numbering is performed partition-by-partition, so thateach processor owns a contiguous set of degree-of freedom indices. System residual vectors and Jacobian



matrices are assembled element by element, with each processor calculating the contributions of elementswithin its corresponding partition.

To enable parallel solutions of the algebraic systems generated by finite element code,libMesh links tothe PETSc solver library to perform sparse linear solves. The Krylov algorithms and linear preconditioneroptions available in PETSc are thus made available inlibMesh. For the Cahn-Hilliard problem, a widevariety of solvers for asymmetric systems provide good convergence with even simple preconditioners. Theresults below are obtained using the Generalized Minimal Residual Method, with Jacobi preconditioningwhich is easily parallelized or with Block Jacobi preconditioning along with an Incomplete LU factorizationof the diagonal block on each processor. The nonlinear problem is solved with a quasi-Newton loop, whichuses the exact Jacobian at each nonlinear step but inexactlysolves each linear system, decreasing lineartolerances for subsequent nonlinear steps.

5.2. Projection operator

When approximating data on the initial finite element space, or when transferring approximate solutions fromone grid to its adaptively refined/coarsened successor, an accurate projection operator is required. In transientcalculations, in particular, the local error at a time step includes both the error of the elliptic problem solvedat that time step and the projection error generated by the transfer to the subsequent mesh; this projectionerror should be minimal even when coarsening or when refiningin non-nested function spaces such as thoseconstructed from HCT triangles.

Other operator characteristics are also important. The projection operator should be computationallyefficient; ideally its cost should scale linearly with the number of elements in the mesh. The projectionoperator should be uniquely defined and parallelizable - when using parallel execution to operate onneighboring finite elements simultaneously, for instance,the traces of the projections computed should agreeat the neighbors’ interface so thatC1 continuity is preserved. Finally, the projection operatorshould be asindependent of finite element type as possible. There are many finite element types useful for constructingC1 function spaces, and it would be ideal to use the same algorithm for all of them.

Hilbert space projections are accurate and applicable to general finite element types, but there are severalHilbert-based operators to choose from, and each has tradeoffs involved. AnH2 projection would controlerror in all solution derivatives which appear in the weak Cahn-Hilliard equation. However, a globalH2

projection would require a global sparse system solve. Using a locally computable operator instead enablesmore efficient projections in general while reducing complexity in parallel. Using anH2 projection oneach element interior would be efficient, would run in parallel without interprocessor communication, andwould give an exact solution in the case of refinement using nested finite element spaces. However, forinitial-condition projections, element coarsening, and refinement on non-nested spaces, an element-wiseHs projection would not be uniquely defined, because the projections from neighboring cells could havedifferent function values or different fluxes along their shared side. A more complicated but similarly efficientalgorithm restores uniqueness by using only local solves and acting on these shared degrees of freedom first,as follows:

Start by interpolating degrees of freedom on coarse-element vertices. Holding these vertex values fixed,do projections of function values and gradients along each coarse-element edge. Because these projectionsinvolve only data from the original refined elements on that edge and not data from element interiors, theyare uniquely defined. In 3D, next project element faces whileholding vertex and edge data fixed. Finally,project element-interior degrees of freedom while holdingelement-boundary data fixed. Although this seriesof projections is more complicated than a single per-element projection, the number of degrees of freedom tobe simultaneously solved for at each stage is much smaller, and so the dense matrix inversions required arefaster.



6. Numerical Results

Algorithm performance is demonstrated here in two benchmarks: first, the evolution of a deforming interfacefrom cross-shaped initial conditions; second, the spinodal decomposition problem as modeled via randomlyperturbed initial conditions.

In the following results, a nondimensionalized potential well W = 1, a constant mobilityM(c) = 1, anda gradient coefficientǫ = 0.01 are used in the PDE. All examples use Backward Euler (θ = 1) or trapezoidalrule time integration.

Simulating a macroscopic domain with sufficient resolutionto approximate a problem with microscalematerial phase width and interface width length scales is often intractable. The model can be smoothedslightly by increasing the equilibrium interface widthδ while maintaining a physical surface energyσ, byadjusting equation parameters according to (12) and (13). There is, however, a limit to how much smoothingcan be done without impairing simulation fidelity. In practice, Cahn-Hilliard simulations use microscalesimulations on artificially truncated domains to find approximate bulk properties. The following spinodaldecomposition results also use truncated domains, either with periodic boundaries enforced via algebraicconstraints or with symmetry conditions enforced weakly. To enforce symmetry, it is sufficient to require that∂~nc = 0 and

(

M(c)∇(

f ′(c)− ǫ2∆c))

· ~n = 0 on the entire domain boundary∂Ω. The latter is a naturalboundary condition and is enforced weakly by substituting0 for the term in the corresponding boundaryintegral in F . Although constraints on∂~nc are familiar as natural boundary conditions in second-orderproblems, in this fourth-order problem constraints on the normal flux are essential boundary conditions. Here,these conditions are conveniently enforced approximatelyvia a penalty method as in (16). The followingexamples use the penalty valueεp = 10−10.

6.1. Benchmark Cross

To examine the interface contraction effects exhibited by Cahn-Hilliard phase evolution, this simulation usesa cross-shaped initial condition as suggested by [20]. Instead of using a discontinuous initial condition on asharp-cornered cross, however, we choose aC1 cubic initial interface function and a cross shape beveled bycircular arcs.

In Figure 1 one can see the solution and adaptive mesh behavior for a cross benchmark using themathematical free-energy model with a constant mobility. The adaptive refinement is controlled to maintainapproximately 1024 elements in the solution; in so doing it produces an interface resolution equivalent toa uniform mesh of 4096 elements. Although this algorithm is capable of tracking the moving boundarysuccessfully, it does not adapt to the decreasing length of the boundary, and so in later time solutions one cansee a few unnecessarily overrefined cells.

The interface first quickly diffuses from the arbitrary width specified in the initial conditions to theequilibrium interface width for the problem. Next, the process of free-energy minimization acts to reducethe interface length. For this initial condition, it is clear a priori that the minimum free energy will beachieved by a circular domain, and so this benchmark serves as a qualitative verification of the formulationand adaptive implementation. Some Cahn-Hilliard simulations can exhibit a non-physical “pinning” on coarsemeshes [35], but simulations here verify that even the extremely coarse phase-interior elements created byproper adaptivity do not cause such numerical artifacts.

6.2. Spinodal Decomposition

A classic application case for the Cahn-Hilliard problem isthe spinodal decomposition of a randomlyperturbed initial condition, as occurs when quenching a mixture below the temperature at which itscomponents completely interdiffuse. As the temperature ofthe mixture is lowered, the bulk free-energydensity diagram transitions from a single well to a double well, as shown in Figure 2, and mixtures with



Figure 1. Cross benchmark initial conditions and Cahn-Hilliard solutions att =

0.025, 0.050, 0.100, 0.200, 0.400. Hermite cubic elements are each plotted as four bilinear squares.

a concentration in between the two spinodal points of the diagram become unstable. For a mixture withconcentrationc for whichf ′′(c) < 0, any perturbation which causes the mixture phases to separate will tendto reduce the total bulk free energy, as

f(c+ δc) + f(c− δc) < 2f(c) (38)

Furthermore, for a partially separated mixture with concentrationsc1 < c2 for which f ′(c1) > f ′(c2),concentration flux from thec1 phase into thec2 phase will tend to decrease the total bulk free energy. Thisprocess continues until single-phase regions approach theequilibrium concentration values, the “binodalpoints” on the diagram.

For a mixture comprised of a finite number of volumesvi, each with distinct concentrationci, one candetermine equilibrium values by minimizing the total bulk free energy

E ≡∑

i

vif(ci) (39)

subject to fixed total concentration and volume constraints

G1 ≡∑

i

vici − C = 0 (40)

G2 ≡∑

i

vi − V = 0 (41)

Introducing Lagrange multipliers and taking partial derivatives ofΛ ≡ E + λ1G1 + λ2G2, one arrives at



the equations

f ′(ci) =f(ci)− f(cj)

ci − cj(42)

Thus the free energy density has constant derivative at the equilibrium points. For an asymmetric double-wellfunction, the two binodal points lie not at the minima of eachwell but on a double tangent line.

Figure 2. An arbitrary double-well bulk-free-energy density function, with spinodal points labeled by diamonds,and binodal points labeled by squares connected by a double tangent line

Because of the unstable initial concentration value, the short time transient behavior of a spinodal phasedecomposition is anti-diffusive, rapidly dividing much ofthe domain into two regions of nearly homogenousconcentration, one near each of the binodal points of the configurational free-energy function, reducing theconfigurational free energy. With a random initial perturbation, however, these regions are heavily interleavedand the interfaces between them are long and convoluted. Theinterpolated initial data and a short timeapproximate solution after ten time steps of lengthδt = 0.0001 are shown in Figure 3 for a square domain.The associated mesh is constructed from a Cartesian grid of 40x40 squares, each divided from lower left toupper right into two HCT triangles. The initial conditions are generated by small random perturbations ofnodal concentration values.

In the longer time transient behavior, the surface free energy is gradually reduced. The diffuse interfacesare shortened in an effect resembling that of surface tension on a sharp interface, as the material fronts moveto reduce their own curvature [36, 37]. The topology of the material regions simplifies as nearby regions ofmatching composition merge. Simulated spinodal decompositions using Clough-Tocher and Hermite tensorproduct elements demonstrate this behavior, as seen from the later stages of the Clough-Tocher experimentin Figure 4.

Finally, the Cahn-Hilliard system approaches a steady-state solution whose size depends on the totalconcentration and whose precise location depends on the particular concentration distribution in the initialconditions. The steady-state solution is not a unique solution to the stationary form of the Cahn-Hilliardequation, and it cannot be found efficiently by uniform time-stepping schemes with sufficiently small timesteps to resolve the initial transient behavior. Adaptive time stepping is necessary for efficient simulations tolong times.

In 3D, the behavior is similar, although computations at a similar grid resolution are significantly moreexpensive. A solution on a uniform32× 32× 32 Hermite tensor product hex mesh with 287,496 degrees offreedom is shown in Figure 5, after 800 time steps of lengthδt = 0.00025. Because of the high degree-of-freedom connectivity ofC1 tricubic elements, the Jacobian matrices for this problem have over 60 millionnon-zero entries, and on sixteen Xeon CPU cores this calculation can require minutes of clock time per timestep.



Figure 3. Left: initial conditions with random nodal values and zero edge fluxes. Right: a Cahn-Hilliard solutionat t = 0.001.

Qualitatively, this result exhibits some of the same behavior seen in the two dimensional studies; e.g.,as time progresses, interfacial effects smooth and shortenthe material interfaces. However, one of the mostimportant differences in the three dimensional problem is topological. In two dimensions, it is impossiblefor all four sides of a square domain to be connected by single-phase paths through each of two materialcomponents. For example, if there is a path of material A connecting the top and bottom sides of the domain,there can be no path of material B connecting the left and right sides. In three dimensions, because materialregions can pass unbroken over or under each other, it is not only possible but likely for all six sides ofa cubic domain to be joined by connected regions of both materials. When estimating average propertiesof the mixture based on distinct material properties of eachphase, this difference may have a large effecton the homogenized bulk values. When additional physics are added to the system to attempt to “pin”phase decomposition solutions into a self-assembled pattern, the additional solution connectivity in threedimensions can influence the final result. These effects are the subject of ongoing investigations.

The time evolution of the free-energy functional for this 3Dspinodal decomposition problem is plotted inFigure 6. As expected, the free energy of the discretizationis monotonically decreasing with time.

6.3. Discretization Error

In Figure 7, the time evolution of theL2 error for a typical transient spinodal decomposition problem isshown, for different spatial meshes and time steps with the same 1% initial perturbation conditions. Thediscretization sequence is generated by uniform refinementin space and time, and the finest discretization isused as a reference solution for error calculations. The coarsest solution is obtained with a uniform Hermitemesh with spacingh1 = 1/32 and time stepδt = 1/4000; the reference solution is obtained withh4 = 1/256andδt = 1/32000. The dissipation in the Cahn-Hilliard operator prevents exponential accumulation of theerror with increasing time. Instead, as seen in Figure 8, thecoarsest approximate solution develops topologicaldifferences to the reference solution, which lead to the slow but monotonic error growth in Figure 7. In thefiner two approximations, the solution becomes first topologically and then visually indistinguishable fromthe reference solution, and theL2 error varies but remains bounded as time increases.



Figure 4. Cahn-Hilliard solutions att = 0.005, 0.01, 0.02, 0.05. The interface widths are now effectively constant,and interface lengths progressively shorten.

6.4. Adaptive Mesh Refinement

In even a moderately developed Cahn-Hilliard system, the solution begins to be characterizable as a patternof single-phase regions with nearly constant concentration, separated by narrow interfaces of rapidly varyingconcentration. Efficiently approximating such solutions is a natural task for adaptiveh refinement andcoarsening, to allocate degrees of freedom in interfacial layers where they can have a significant effect onsolution accuracy and to remove degrees of freedom from region interiors where they can needlessly degradecomputational efficiency.

Maintaining a constant element count is a robust strategy when coupled with an imperfectly reliable errorindicator, and is efficient for time-dependent problems in which a constant element count corresponds toa roughly constant error value, but it may be inefficient for problems such as the spinodal decompositionwhere the solution complexity changes so drastically with progressing time. When a constant element countis maintained during a spinodal decomposition problem, theadaptive mesh may either be less refined than isdesirable for accuracy during the early evolution of the problem when material interfaces are numerous and



Figure 5. Concentration isosurfaces for a 3D spinodal decomposition problem att = 0.2. Red and blue isosurfacemanifolds distinguish different sides of the material interface.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Time

Glo

bal f

ree

ener

gy

Spinodal Decomposition Free Energy Evolution

Figure 6. Integrated free energy over the square domain for the Galerkin approximation to a 3D spinodaldecomposition problem.



0 0.005 0.01 0.01510

−4

10−3

10−2

10−1

100

Time

L 2 err

or

coarsemediumfine

Figure 7.L2 error compared to a reference solution, as a function of time, for a 2D spinodal decompositionapproximated on a sequence of uniformly refined discretizations

Figure 8. Solutions att = 0.015 onh1 andh4 grids. Theh2 solution is slightly perturbed from the referenceh4

solution, and theh3 solution is visually indistinguishable fromh4.

convoluted, or it may be more refined than is desirable for efficiency during the late evolution of the problemwhen material interfaces have coalesced and smoothed. The element count can be made dynamic to avoidthese problems.

As the interface location moves, the mesh must refine and coarsen appropriately to maintain a desiredaccuracy. This implies that the refinement and coarsening criteria should accomodate a more complexsituation than in typical stationary or single evolving layer cases where the layer location is fixed. One strategyis to specify conservative tolerances on the respective refine and coarsen actions so that the interface motion is“covered” by the fine mesh zone in a time step. The time step must be sufficiently small to accommodate this



requirement, and the conservative criteria should not be sogenerous that the mesh is not coarsened. In the veryearly evolution of a spinodal decomposition problem, thereis little benefit from coarsening and a fine uniformmesh is desired. At later times when single-phase domain regions have grown and coalesced, refinementand coarsening becomes increasingly useful, and in the latter stages of the process the desired mesh willbe quite coarse except in a relatively simple interface zone. A rigorous approach to guiding the AMR/Cbehavior toward these goals is needed and warrants further study. Figure 9 displays one time step from eachof two numerical experiments using adaptive mesh refinementand coarsening. The AMR/C strategy usedhere attempts to minimize the error at each time step while maintaining a target number of active elements,by trading element coarsening for refinement whenever the error indicator suggests that such a trade wouldimprove the final result.

Because of the narrow aspect ratio of the interface regions and because those interfaces move throughthe domain with advancing time, constructing an algorithm to automatically and efficiently track them withelement refinement and coarsening is difficult. Initial experiments based on heuristic rules using the Laplacianjump error indicator are capable of correctly locating interfaces and refining within the interfacial layers.Another theoretical concern is the effect of adaptive coarsening on the free-energy functional; an otherwisemonotonic scheme can see local increases in the free-energyfunctional when an element coarsening isperformed.

Figure 9. Left: Refined Hermite tensor mesh and solution tracking the evolving front on a cross benchmark. Right:Refined HCT mesh and solution in an adaptive transient Cahn-Hilliard spinodal decomposition simulation from

random initial conditions.

6.5. Adaptive Time Stepping

The effect of adaptive time stepping on efficiency is quite dramatic in the spinodal decomposition problem.For the example problem below, a steady state solution is notreached until roughlyt = 100, but time stepssmaller thanδt = 0.0001 are required to accurately model the growth of the initial random perturbation.Solving one million uniform time steps is unnecessarily expensive, when one can produce a time-accuratesolution up to the steady state in less than one thousand adaptive time steps.

In Figure 10, the time step lengthsδtn chosen by the adaptive algorithm are plotted against the time tn atwhich each time step ends. Both plots correspond to the same parameters and initial conditions, but in onesimulation the norm used in equation (35) is theL2 norm over the domain and in the other simulation theH2



norm is used.

10−10

10−8

10−6

10−4

10−2

100

102

104

10−10

10−8

10−6

10−4

10−2

100

102

Time

Tim

este

p Le

ngth

H2 Based TimestepsL

2 Based Timesteps

Figure 10. Adaptive time step sizes for a spinodal decomposition problemon a periodic domain.

Some of the more interesting points on this graph correspondto qualitative features of the system evolution.At approximatelyt ≈ 0.001, when the time step choices temporarily plateau or diminish, the strongest peaksof the concentration perturbation are beginning rapid growth. At t ≈ 0.01, when time steps begin to increaseagain, the concentration peaks have just reached the binodal values where they stop growing. Four sharpdownward spikes occur att ≈ 0.82, 3.16, 18.9, 25.9. Three of these correspond to the final “evaporation” of ashrinking droplet of single-phase material; the spike at3.16 corresponds to the connection of two approachinginterfaces as two nearby material regions merge into one. Finally, the time-step size selection levels off att ≈ 100 as the system reaches a steady state; at this point the error indicator is always satisfied, but time stepsizes are periodically reduced when the nonlinear solver fails to converge on larger time steps within a fewquasi-Newton iterations.

7. Conclusions

Cahn-Hilliard models for phase transitions in isothermal binary mixtures are characterized by diffuseinterfacial layers separating the material phases. These fine-scale layers are the result of a competitionbetween gradient and bulk terms in the free energy of the system. In many applications, diffuse-interfacemodels are better suited to describe the important underlying physical behavior than classical sharp-interfacemodels. Moreover, such phase-field models avoid the numerical complications commonly encounteredwith explicit front-tracking strategies on changing interface topologies. The inclusion of the fine-scaleinterface physics implies that steep solution gradients must be resolved, and by introducing adaptive meshrefinement/coarsening, this has been done in a more robust and efficient way. Using adaptive time integrationprovides similar improvements to performance by enabling the use of long time steps through most ofa simulation while still reliably capturing the more sensitive processes of phase separation and interfacetopology changes. The alternative of using a uniform fine mesh and uniform time stepping is simpler toimplement but computationally prohibitive. These aspectsare demonstrated in our adaptive results.



We use aC1 conforming finite element scheme based on the variational formulation of the conservativeform of the scalar Cahn-Hilliard model. This involves relatively complex macroelements in the adaptive2D studies with simplicial elements. The extension to tetrahedral macroelements with AMR/C is morecomplex; hence, the 3D results use AMR/C with tensor-product Hermite elements that are geometricallymore restrictive but suffice for the simulations here. Our work to enable AMR/C on complex elements fortransient fourth-order problems involved extensions of data structures and algorithms inlibMesh, a paralleladaptive finite element framework developed by CFDLab researchers.

Figure 1 illustrates the continuous accurate resolution ofa diffuse-interface layer by AMR/C as an initialcross interface evolves towards a circle. In the spinodal decomposition examples, the interfaces are numerousand tortuous as well as distributed over the domain through the full time interval of simulation consideredhere. A classic monotonic decay of the global free energy fora discretized Cahn-Hilliard problem is graphedin Figure 6.

The error incurred by these methods is plotted in Figure 7, which demonstrates the time-varyingL2

convergence of the solution on uniformly refined discretizations. To obtain convergence less expensively,adaptive discretizations in both space and time are successfully used. As the interfaces smooth and materialregions coalesce, the mesh must locally coarsen and refine nearby elements to efficiently approximate thedeveloping solution. This implies one has to use coarseningconservatively to maintain accuracy from aninitial fine mesh as applied here. On the other hand, the fact that there is constant “action” over the domainfor most simulation time steps implies that repeated parallel repartitioning is not needed. The simulationalgorithms described above are shown to be capable of adapting appropriately to both the moving materialinterfaces in space and the dramatically changing time steprequirements found in a phase decompositionsimulation. With the right choice of time integrator and theability to repeat time steps, solutions with largetime steps can be reliably obtained even when using the strongly nonlinear chemical free-energy model.In these ways, time-accurate solutions up to the steady state can be achieved with orders-of-magnitudereductions in computational cost over uniform methods.

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Approximation of Cahn-Hilliard diffuse interface models using …roystgnr/libmeshpdfs/roystgnr/chpap… · c 2008 John Wiley & Sons, Ltd. KEY WORDS: Cahn-Hilliard, diffuse interface,