BIFURCATION ANALYSIS REVEALS SOLUTION STRUCTURES OF PHASE FIELD MODELS XINYUE EVELYN ZHAO LONG-QING CHEN WENRUI HAO AND YANXIANG ZHAO

2025-04-24 0 0 852.23KB 22 页 10玖币

侵权投诉

BIFURCATION ANALYSIS REVEALS SOLUTION STRUCTURES OF PHASE FIELD

MODELS

XINYUE EVELYN ZHAO, LONG-QING CHEN, WENRUI HAO∗, AND YANXIANG ZHAO

ABSTRACT. Phase ﬁeld method is playing an increasingly important role in understanding and predicting

morphological evolution in materials and biological systems. Here, we develop a new analytical approach based

on bifurcation analysis to explore the mathematical solution structure of phase ﬁeld models. Revealing such

solution structures not only is of great mathematical interest but also may provide guidance to experimentally

or computationally uncover new morphological evolution phenomena in materials undergoing electronic and

structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative

phase ﬁeld equations: Allen-Cahn equation, Cahn-Hilliard equation, and Allen-Cahn-Ohta-Kawasaki system.

The solution structures of these three phase ﬁeld equations are also veriﬁed numerically by the homotopy

continuation method.

1. INTRODUCTION

Phase ﬁeld approach is an important modeling tool for modeling interfacial evolution problems in

materials science and biological systems. It is rooted in the diffuse-interface description of interfaces for

ﬂuid interfaces proposed by van der Waals more than a century ago [22, 66]. The early applications of the

diffuse-interface to superconducting phase transitions and the compositional clustering and ordering in al-

loys led to the establishment of well-known time-dependent Ginzburg-Landau (TDGL) equations [34, 70],

the Cahn-Hilliard [9, 10], and Allen-Cahn equations [8] which form the basis for the evolution equations

in the phase-ﬁeld method. The term of “phase-ﬁeld” was coined in early applications of diffuse-interface

description to solidiﬁcation and dendrite growth [7, 27, 51, 52, 72]. The generalization of the phase-ﬁeld to

include both physical and artiﬁcial ﬁelds to distinguish different phases has led to wide-spread applications

of the phase-ﬁeld method to modeling morphological and microstructure evolution in a wide variety pro-

cesses beyond solidiﬁcation in materials science [14, 15, 69], biology [80],ﬂuid and solid mechanics [2, 6],

etc.

In the phase-ﬁeld method, one introduces a labeling function, called phase-ﬁeld, φwhich for a two-

phase system, is assigned a value (say, -1) for one phase, and another value (say, +1) for the other. In the

interfacial region, the phase ﬁeld labeling function φrapidly but smoothly transitions from -1 to 1. Mean-

while, the interface is tracked by a level set, typically the 0-level set, during the morphological evolution.

The main advantage of the phase ﬁeld approach is that it can predict the evolution of arbitrary morphologies

and complex microstructures without explicitly tracking interfaces, and thereby easily handle topological

changes of interfaces.

Since phase-ﬁeld method involves the numerical solutions to systems of partial differential equations in

time and space, there have been extensive efforts in developing numerical methods for solving the phase-ﬁeld

equations. For example, several classic methods such as explicit/implicit Euler methods, Crank-Nicolson

and its variant, linear multistep methods, and Runge-Kutta methods have been considered for the time dis-

cretization (See for example [13, 54, 25, 1, 68] and the references cited therein). For the spatial discretiza-

tion, methods such as ﬁnite difference, ﬁnite element, discontinuous Galerkin, and spectral approximation

are typical examples See the recent review article [23] and the reference cited therein for more detailed

discussion). Some stabilized schemes have recently been developed based on the inheriting the energy dis-

sipation law and phase ﬁeld gradient ﬂow dynamics, including the convex splitting [26], linearly-implicit

arXiv:2210.06691v1 [math.NA] 13 Oct 2022

2 XINYUE EVELYN ZHAO, LONG-QING CHEN, WENRUI HAO∗, AND YANXIANG ZHAO

stabilized schemes [74, 73], exponential integrator [20, 24], Invariant energy quadratization [77], and scalar

auxiliary variable schemes [67].

However, there are very limited studies to reveal the solution structure of phase ﬁeld models which

is critical to understanding the models from a mathematical point of view and to exploring the possible

formation of novel morphological patterns. Solution structures of nonlinear differential equations have been

well-studied by exploring bifurcations [64] and multiple solutions [4]. Existing theories and numerical

methods have contributed to a better understanding of these solution structures and the relationship between

solutions and parameters [36]. For example, the Crandall-Rabinowitz theorem has been used to theoretically

study the bifurcations of nonlinear differential equations such as free boundary problems [31, 28, 29, 78, 79].

Numerically, homotopy continuation method [47, 56] has been successfully employed to study parametric

problems such as bifurcation [64] and the structural stability [65]. Recently, several numerical methods

have been developed based on homotopy continuation methods for computing multiple solutions, steady-

states, and bifurcation points of nonlinear PDEs [40, 41]. These numerical methods have been also applied

to hyperbolic conservation laws [42], physical systems [44, 45] and some more complex free boundary

problems arising from biology [38, 39].

In this paper, we develop an analytical framework to study the solution structure of phase ﬁeld models

and apply it to three well-known phase ﬁeld equations. In particular, we study the solution structure of

Allen-Cahn equation in Section 3 and that of Cahn-Hillard equation in Section 4. Finally in Section 5, we

discuss the solution structure of Allen-Cahn-Ohta-Kawazaki system which is used to model the morphology

of diblock copolymer systems.

2. BIFURCATION ANALYSIS AND HOMOTOPY TRACKING

The goal of this paper is to compute global bifurcation diagram for various PDE models. Our ap-

proach combine analytical and numerical method. First, we analyze the bifurcations of various phase ﬁeld

models from the trivial steady-states by Crandall-Rabinowitz theorem, then we numerically compute the

global bifurcation diagram via homotopy tracking. Generally speaking, we consider the following nonlinear

operator

F(x, µ)=0,

where F(·, µ)is a Cpmap, p≥1from a real Banach space Xto another real Banach space Yand µ∈Ris

a parameter. The bifurcation of xwith respect to the parameter µcan be veriﬁed theoretically by Crandall-

Rabinowitz theorem [21].

Theorem 1. (Crandall-Rabinowitz theorem, [21])Let X,Ybe real Banach spaces and F(·,·)be a Cp

map of a neighborhood (0, µ0)in X×Rinto Y. Denote by DxFand DµxFthe ﬁrst- and second-order

Fr´

echet derivatives, respectively. Assume the following four conditions hold:

(I) F(0, µ)=0for all µin a neighborhood of µ0,

(II) Ker DxF(0, µ0)is one dimensional space, spanned by x0,

(III) Im DxF(0, µ0) = Y1has codimension 1,

(IV) DµxF(0, µ0)x0/∈Y1,

then (0, µ0)is a bifurcation point of the equation F(x, µ) = 0 in the following sense: in a neighborhood

of (0, µ0), the set of solutions F(x, µ) = 0 consists of two Cp−2smooth curves, Γ1and Γ2, which intersect

only at the point (0, µ0);Γ1is the curve (0, µ), and Γ2can be parameterized as follows:

Γ2: (x(), µ()),||small, (x(0), µ(0)) = (0, µ0), x0(0) = x0.

Although the bifurcation theory can help in some special cases, the in-depth study of solution structures

often requires numerical methods to derive bifurcation diagrams of nonlinear systems. Generally speaking,

BIFURCATION ANALYSIS REVEALS SOLUTION STRUCTURES OF PHASE FIELD MODELS 3

the nonlinear operator Fis approximated by Fhnumerically (hrefers to the mesh size of numerical dis-

cretization). Then the numerical solution xhis computed by solving the following discretized nonlinear

system:

(1) Fh(xh, µ) = 0,

where Fh:Rn×R→Rnand xhis the variable vector that depends on the parameter µ, i.e., xh=

xh(µ). Suppose we have a solution at the starting point, namely xh(µ0) = x0, various homotopy tracking

algorithms can be used to compute the solution path [37, 47, 49], xh(µ). If ∂xFh(xh, µ)is nonsingular, the

solution path xh(µ)is smooth and unique. However, when ∂xFh(xh, µ)becomes singular, the solution path

hits the singularity and different types of bifurcations are formed [46].

More speciﬁcally, the homotopy tracking algorithm consists of a predictor step and a corrector step to

solve the parametric problem. The predictor is to compute the solution at µ1=µ0+ ∆µby setting

Fh(x0+ ∆xh, µ0+ ∆µ) = 0,

which, at the ﬁrst order, yields an Euler predictor,

∂xFh(x0, µ0)∆xh=−∂µFh(x0, µ0)∆µ.

Then we apply Newton corrector to reﬁne the solution with an initial guess exh=x0+ ∆xh:

∂xFh(exh, µ1)∆xh=−Fh(exh, µ1),

and repeat exh=exh+ ∆xhuntil (exh, µ1)is on the path, namely, Fh(exh, µ1)=0.

3. BIFURCATION ANALYSIS OF ALLEN-CAHN EQUATION

In this and the following sections, we will consider two classical phase ﬁeld equations: Allen-Cahn

equation and Cahn-Hilliard equation. As a mathematical convention, we take φ=±1in two distinct

phases, respectively. This is in contrast to the Allen-Cahn-Ohta-Kawasaki equation in Section 5, in which

we rather take φ= 0 or 1in the two phases from physical perspective.

We consider Allen-Cahn equation

∂φ

∂t (x, t) = ∆φ(x, t)−1

W0(φ(x, t)),x∈Ω, t > 0.(2)

Here Ω = [−1,1]d, d = 1,2,3, and 0<  1is a parameter to control the width of the interface. φis a

phase ﬁeld labeling function which equals ±1in two distinct phases. The function W(φ) = 1

4(φ2−1)2is

a double well potential which enforces the phase ﬁeld function φto be equal to 1 inside the interface and -1

outside the interface. The Allen-Cahn equation (2) can be viewed as the L2gradient ﬂow dynamics for the

Ginzburg-Landau free energy functional

E(φ) = ZΩ



2|∇φ|2+1

W(φ)dx.(3)

In 1D case, Ginzburg-Landau free energy reduces to

−1



2(φx)2+1

4φ2−12dx,(4)

and the associated Euler-Lagrange equation (steady-state Allen-Cahn equation) becomes

(5) (−φxx +1

(φ3−φ)=0 −1<x<1,

φx(−1) = φx(1) = 0.

4 XINYUE EVELYN ZHAO, LONG-QING CHEN, WENRUI HAO∗, AND YANXIANG ZHAO

3.1. Bifurcation analysis. It is easy to verify that φ≡φ0=−1,0,1are three trivial solutions of the

steady-state system (5). Besides these trivial solutions, we are more interested in non-trivial steady states,

which can bifurcate from the zero trivial steady-state. More speciﬁcally, we consider the following shifted

system from φ0:

(6) (−φxx +1

2[(φ+φ0)3−(φ+φ0)] = 0 −1<x<1,

φx(−1) = φx(1) = 0.

We can verify that φ= 0 is always a solution to the system (6).

Next, we consider the following Banach space:

(7) Xl+α={φ(x)∈Cl+α[−1,1], φx(−1) = φx(1) = 0},

with the H¨

older norm

kukXl+α=kukCl([−1,1]) + max

|β|=l|Dβu|Cα([−1,1]),

where l≥0is an integer, 0< α < 1, and

|u|Cα([−1,1]) = sup

x6=y∈(−1,1)

|u(x)−u(y)|

|x−y|α.

Taking

(8) X=Xl+2+αand Y=Xl+α

in Crandall-Rabinowitz Theorem (Theorem 1), and deﬁning an operator Fas

(9) F(φ, ) = −φxx +1

2[(φ+φ0)3−(φ+φ0)],

where is the bifurcation parameter, we know that F(·, )maps Xinto Y.

Since φ= 0 is always a solution to the system (6), F(0, ) = 0 for every , and Condition I of Crandall-

Rabinowitz Theorem (Theorem 1) is satisﬁed. To verify other conditions, we need to compute the Fr´

echet

derivative of the operator F, which is given in the following lemma.

Lemma 3.1. The Fr´

echet derivative DφF(φ, )of the operator Fis given by

(10) DφF(φ, )[ξ] = −ξxx +1

2[3(φ+φ0)2ξ−ξ].

Proof. By taking φ, ξ ∈X, we have

F(φ+ξ, )− F(φ, )−DφF(φ, )[ξ]

2[(φ+φ0+ξ)3−(φ+φ0+ξ)−(φ+φ0)3+ (φ+φ0)−3(φ+φ0)2ξ+ξ]

2ξ2[ξ+ 3(φ+φ0)].

Therefore,

kF(φ+ξ, )− F(φ, )−DφF(φ, )ξkY

kξkX

=k1

2ξ2[ξ+ 3(φ+φ0)]kY

kξkX

≤1

2

kξk2

Xkξ+ 3(φ+φ0)kY

kξkX

→0,

as kξkX→0. Hence, DφF(φ, )is the Fr´

echet derivative of F.

BIFURCATION ANALYSIS REVEALS SOLUTION STRUCTURES OF PHASE FIELD MODELS 5

Given (10), we have by taking φ= 0 that

(11) DφF(0, )[ξ] = −ξxx +1

2(3φ2

0ξ−ξ).

For condition II of Crandall-Rabinowitz Theorem, we need to analyze the structure of KerDφF(0, )

which is given by DφF(0, )[ξ]=0,ξ∈X. By (11), it is equivalent to solve the following system

(12) (−ξxx +1

2(3φ2

0ξ−ξ) = 0,

ξx(−1) = ξx(1) = 0.

This system can be solved by using an eigenfunction ansatz, i.e.,

(13) ξ(x) = a0+∞

n=1

ancos(Lnx) + ∞

n=1

bnsin(Lnx),

where Lis to be determined. By taking the derivative, we have

(14) ξx(x) = −∞

n=1

anLn sin(Lnx) + ∞

n=1

bnLn cos(Lnx).

The two boundary conditions, ξx(1) = 0 and ξx(−1) = 0, yield that either an= 0 or bn= 0. If an= 0,

then Ln =π

2+ (n−1)π, hence

(15) ξ(x) = a0+∞

n=0

bnsin π

2+nπx;

if bn= 0, then Ln =nπ, and we have

(16) ξ(x) = a0+∞

n=1

ancos(nπx) = ∞

n=0

ancos(nπx).

Next, we will determine the bifurcations with respect to by solving (12)with different trivial solution

φ0.

3.1.1. Bifurcations around φ0= 0.

Theorem 2. For each integer n≥0,(1)

n=1

π/2+nπ ,(0, (1)

n)is a bifurcation point to the system (6)such

that there is a bifurcation solution (φn(x, s), n(s)) with

n(s) = (1)

n+s, φn(x, s) = ssin π

2+nπx+O(s2),where |s|  1.

Proof. We need to verify the four conditions of Crandall-Rabinowitz Theorem at (0, (1)

n). In this case, we

use the Fourier expansion of ξ(x)in (15). When φ0= 0, it follows from (11)that

DφF(0, )[ξ(x)] = −a0

2+∞

n=0

bnπ

2+nπ2−1

2sin π

2+nπx.

If =(1)

n=1

π/2+nπ , the term with sin (π

2+nπ)xdisappears while all the other terms remain. Hence,

we have

DφF0, (1)

nhbnsin π

2+nπxi=bnπ

2+nπ2−(1)

n−2sin π

2+nπx= 0,

DφF(0, (1)

n)[ξ(x)] = −a0

2+∞

k=0

k6=n

bkπ

2+kπ2−(1)

n−2sin π

2+kπx.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

BIFURCATIONANALYSISREVEALSSOLUTIONSTRUCTURESOFPHASEFIELDMODELSXINYUEEVELYNZHAO,LONG-QINGCHEN,WENRUIHAO,ANDYANXIANGZHAOABSTRACT.Phaseeldmethodisplayinganincreasinglyimportantroleinunderstandingandpredictingmorphologicalevolutioninmaterialsandbiologicalsystems.Here,wedevelopanewanalyticalapproachbas...

展开>> 收起<<

BIFURCATION ANALYSIS REVEALS SOLUTION STRUCTURES OF PHASE FIELD MODELS XINYUE EVELYN ZHAO LONG-QING CHEN WENRUI HAO AND YANXIANG ZHAO.pdf

共22页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

BIFURCATION ANALYSIS REVEALS SOLUTION STRUCTURES OF PHASE FIELD MODELS XINYUE EVELYN ZHAO LONG-QING CHEN WENRUI HAO AND YANXIANG ZHAO

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: