A computational approach to fractures in crystal growth

rystal growth. Notadi Matteo Novaga ed Emanuele Paolini.

Abstra t. { In the present paper, we motivate and des ribe a

nu-meri al approa h in order to dete t the reation of fra tures in a fa et of

a rystal evolving by anisotropi mean urvature. The result appears to

be in a ordan e with the known examples of fa et-breaking. Graphi al

simulationsarein luded.

KeyWords: Nonlinearpartialdierentialequationsofparaboli type;

Crystalgrowth; Non-smoothanalysis.

Riassunto. { Un appro io numeri o alle fratture nella res ita di

ristalli. Inquestolavoro,presentiamoedis utiamounappro ionumeri o

alproblemadiindividuarelanas itadifrattureinunafa iadiun ristallo

he si evolve per urvatura media anisotropa. I risultati sono in a ordo

on gli esempi noti no ad ora di frattura di fa e. Sono inoltre in luse

al unesimulazionigra he.

0. Introdu tion

We model rystal growth in R 3

as an anisotropi evolution by mean

urvature when the ambient spa e is endowed with a onvex

one-homogeneousfun tionwhoseunitball(usually alledtheWulshape)

is a polyhedron. Su h evolutions are onsidered in materials s ien e

and phasetransitions when thevelo ity of theevolvingfrontdepends

on the orientation of the normal ve tor and has a nite number of

preferred dire tions. This problemhas been widely studied; we refer

among othersto [10℄, [12℄, [7℄, [9℄, [3℄.

This motionhas a variationalinterpretation as \gradient ow" of

an asso iated rystalline energy dened on the subsets of R 3

having

nite perimeter, see [12℄, [2℄and Remark 1.2 below.

The simplest situation of motion by rystalline urvature is the

bidimensional ase. In this ase short time existen e and uniqueness

ofthe evolutionhas been established,aswellasanin lusionprin iple

between the evolving fronts. Moreover, if the initialset E is a

poly-gon ompatible with the Wul shape, its edges translate parallely to

themselvesduringtheevolution,sothatnoedge{breakingo urs[10℄,

[12℄, [1℄. It is interesting to observe that the situation be omes quite

dierentin presen e of a spa e dependent for ing term [8℄, sin e new

ommonand,even startingfromapolyhedralset whose shape isvery

lose to the Wul shape, some fa ets an break or bend and urve

portions of the boundary an appear (see [4℄, [11℄, [13℄ for related

ex-amples). For what on erns the qualitative behaviourof the evolving

interfa es,tothebestknowledgeoftheauthors, theproblemof

under-standingifand wherefa et{breaking phenomenatake pla ehas not a

deniteanswer.

Here we try to answer to this question by means of a numeri al

approa h. More pre isely, with a formalargument we show that the

rystalline urvatureofanevolvingset (whi hisequaltothe velo ity)

is expe ted tosolvea minimumproblem onea h fa et.

This orresponds to look for the anoni al element of the

subdif-ferential of the rystalline energy (restri ted to the fa et), whi h is a

ommonpro edure inmultivalued onvex analysis ( ompare [9℄). We

then determine this minimum with a numeri al approa h, using a

-nite element method. The dis ontinuities of the rystalline urvature

orrespond tothe fra tures appearingin the fa et, morepre isely the

fa etwill not break orbend if and only if the rystalline urvature is

onstant onthe fa et.

The plan of the paper is the following. In Se tion 1 we givesome

notation and we re all, following [3℄, the general denitions of

-regularset and -regular ow. Observe that thesedenitions on ern

Lips hitz surfa es,and we donot restri tourselvesto polyhedral

sur-fa esinviewoftheexamplesdis ussedinSe tion3. Wealsointrodu e

theminimumproblem(5),whi hwewillsolvenumeri ally,inorderto

hara terize the velo ity of the evolvingset. In Se tion 2 we present

the algorithmwhi hweuse tond su ha minimum, andin Se tion3

we dis ussthe results of the algorithmappliedtothe examplesof [4℄.

Numeri alsimulationare in a ordan ewith the resultof [4℄.

Moregraphi sandthesour e odeoftheprogram(writteninC++

language) are availableon http://ro k.sns.it/ rystals /.

WethankG.BellettiniandM.Paoliniformanyusefuldis ussions.

1. Notation and main definitions

In the followingwedenoteby
the standard eu lidean s alarprodu t

inR 3

, and by j
j the eu lidean normof R 3

R , we denote by jAj the Lebesgue measure of A. H , k =1;2;3will

denote the k-dimensional Hausdor measure.

We indi ateby  :R 3

! [0;+1[ a onvex fun tion satisfyingthe

properties

()jj; (a)=a();  2R 3

; a0; (1)

for a suitable onstant  2 ℄0;+1[, and by  o : R 3 ! [0;+1[,  o (  ):=supf 

 : ()1gthe dual of . We set

F  :=f  2R 3 : o (  )1g; W  :=f 2R 3 :()1g: F  and W 

are onvex sets whoseinteriorparts ontainthe origin. In

this paper we shall assume that  is rystalline, i.e. W

 (and hen e F  ) is a onvex polyhedron. F 

is usually alled the Frank diagram

and W

the Wul shape.

Let T o :R 3 !P(R 3

) be the dualitymapping dened by

T o (  ):= 1 2  ( o (  )) 2 ;   2R 3 ; whereP(R 3

)isthe lass ofallsubsetsofR 3

and  denotesthe

subd-ierentialinthesenseof onvexanalysis. T o

isamultivaluedmaximal

monotone operator, and

T o (a  )=aT o (  ); a0:

One an show that

 
 = o (  ) 2 =() 2 ;   2R 3 ; 2T o (  ): Given E R 3 and x2, we set dist  (x;E):= inf y2E (x y); dist  (E;x):= inf y2E (y x); d E  (x):=dist  (x;E) dist  (R 3 nE;x):

Atea h point whered E

is dierentiable, there holds [5℄

 o (rd E  )=1:

The next denition generalizes to the rystalline ase the notion of

Denition 1.1. Let E R andn

:E !R be Borel{measurable.

We say that the pair (E;n

)is -regular if

1. the set E is ompa t and Lips hitz ontinuous;

2. n  (x)2 T o rd E  (x)
H 2 a:e: x2E;

3. there is an open set A  E su h that for a.e. y 2 A there

exists a unique (x;s) 2 E R so that y = x +sn

 (x) and letting n e  (y):=n  (x) there holds n e  2L 1 (A;R 3 ); div n e  2L 1 (A); 4. div n e 

admits a tra e on E, whi h we denote by div

 n  2 L 1 (E).

We say that E is -regular if there exists a fun tion n

: E ! R 3

su h that (E;n

) is a -regular pair.

Su h a generality (asinDenition 1.4below) is needed in viewof

the se ondexample of Se tion3(see alsoRemark 1.5below). We

no-ti ethat, ifE isaplane, thendiv

isthe usualtangentialdivergen e

in the sense of distributions.

In the rystallineliterature n

isusually alledthe Cahn-Homan

ve tor eld. Given a -regular pair (E;n

), we dene the -mean

urvature  of E atH 2 -almostevery x2E as   :=div  n  : (2)

Remark 1.2. We observe that the ve tor eld 

 n

in Denition

1.1 has a variational interpretation as \gradient ow" of the

rys-talline energy A! Z A  o ( A )dH 2 ; where  A

isthe exterior unit normal to A, see [4℄ for a formal

om-putation in this dire tion.

Remark 1.3. Let E bea polyhedralset havingthe followingproperty:

givenanyvertexv ofE, the interse tion ofT o (rd E  ) jQ

overallfa ets

Q ontaining v is non empty. Then there exists a ve tor eld n

 2

Lip(E;R 3

) su h that (E;n

If t 2 [0;T℄ 7! E(t)  R 3

is a parametrized family of subsets of R 3 , we let d E(t)  (x):=dist  (x;E(t)) dist  (R 3 nE(t);x):

Wheneverno onfusionis possible, we set d

(x;t):=d E(t)

 (x).

We now dene a -regular ow (E(t);n

(
;t)), for t 2 [0;T℄, as

a -regular evolution of a boundary movingwith velo ity, in the n

-dire tion, equalto 

(see [4℄).

Denition 1.4. A -regular ow is a pair (E(t);n

 (
;t)), n  (
;t) : E(t)!R 3

, whi h satises the following properties:

(1) (E(t);n

(
;t)) is -regular for any t2[0;T℄;

(2) thefun tion d

isLips hitz ontinuousonR 3

[0;T℄, dierentiable

for a.e. t2[0;T℄ and for H 2

a.e. x2E(t), and su h that

d  t (x;t)=  (x;t); a.e. t2[0;T℄; H 2 a.e. x2E(t): (3)

In ase no new fa et reates, the previous evolution law on ides

with the evolutionlaw onsidered in[12℄ and [9℄.

Remark 1.5. We observe that the lass of polyhedral subsets of R 3

seems not to be stable under the evolution dened by (3), i.e. some

polyhedral set may develop urve portions of the boundary during the

evolution (see [4℄, [11℄, [13℄ and Se tion 3 below).

Letnow(E;n

)bea-regularpair. AssumethatEisapolyhedron

andletF beafa etofE. Wenoti ethatby[9,Lemma9.2℄,thetra e

of n 
 F is well{dened in L 1 (F) ( F

is the exterior unit normal

to F lying in the plane ontaining by F). For H 1

-a.e. s 2 F, we

denote su h a tra e by

F (s). Letalso v F := 1 jFj Z F F (s) dH 1 (s)= 1 jFj Z F   (x)dH 2 (x):

Noti e that, sin e the fa etF isplanar, for any x2int(F) itis well{

dened the onvex set T o (rd E  (x))  W  , and it is independent of x2int(F).

Using [9, Lemma 9.2℄, one an get the following result (see also

of E. Then, the fun tion

F 2 L

1

(F) depends only on E and is

independent of the hoi e of n

. More pre isely, for H 1 -a.e. s 2 F there holds F (s)= ( sup fn
 F (s): n2T o (rd E  )g ifE is lo ally onvex in s; inf fn
 F (s): n2T o (rd E  )g ifE is lo ally on ave in s: (4)

Observethatexpression(4)isinagreementwiththe orresponding

denition given in[12℄.

We now try to hara terize the rystalline urvature dened in

(2) as minimum of a variational problem. This formal argument is

intended tomotivate the numeri alapproa h dis ussed inSe tion2.

Let (E(t);n

(
;t)), t 2 [0;℄, be a -regular ow, and assume for

simpli ity that E := E(0) is a polyhedron. Then we expe t that

n

(
;0)solves the followingminimum problemonE

min n Z E div  
2  o ( E )dH 2 (x): 2L 1 (E;R 3 ); (5) div   2L 2 (E);  2T o (rd E  ) H 2 a:e: onE o :

Noti e that, by Proposition 1.6, problem (5)is equivalent to

min n Z F div  
2 dH 2 (x): 2L 1 (F;R 3 ); div  2L 2 (F);  2T o (rd E  ) H 2 a:e: inF; 
 F = F on F o : (6)

for any fa etF E.

Wexafa etF E andweassumethat E(t)isthe graphof a

Lips hitz ontinuous fun tion u:[0;℄ !R, inaneighbourhood

of F. LetP bethe proje tion ofF on. Then equation(3)be omes

8 < : u t 2  o

( ru;1)  (u); a:e: (x;t)2[0;℄;

u(x;0)=u 0 (x); x2; u(x;t)=v(x;t); (x;t)2[0;℄; (7) where u 0

2 Lip() represents the set E as graph over , v(
;t) 2

Lip(), t 2 [0;℄, represents the set E(t) as a graph over  and

:H 1 ()L 2 ()!R is dened as (u) := Z  o ( ru;1)dx:

Here  is the subdierential in L () in the sense of onvex

analy-sis [9℄.

By [9,Lemma 9.1℄, there holds

f 2 (u) () f =div ; for some  2  L 1 lo ()  2 , with  2   o

( ru;1) a.e. in . Noti e

that this onstraint on implies that 2  L 1 ()  2 .

Ontheanalogyoftheresultsdis ussedin[6℄formaximalmonotone

operators, we an expe t that u

t =  o ( ru;1)  (u)
0 for a.e. t2[0;℄, where  o ( ru;1) (u)
0 solves min  jj o ( ru;1) fjj L 2 () : f 2 (u) ; (8)

whi hin turn gives (5).

Thefollowingresult(see[4℄)isusefulinorderto hara terizewhi h

fa ets of E donot break during the evolution.

Proposition 1.7. Let (E;n

) be a -regular pair. Assume that E

is a polyhedron and let F be a fa et of E. If F does not instantly

break or bend during theevolution starting from E, then thefollowing

ondition holds: v P :=jPj 1 Z P P (s) dH 1 (s)v F ; (9)

for any set P F with Lips hitz ontinuous boundary,where

P (s):= ( F (s) if s2P \F; sup fn
 P (s): n2T o (rd E  )g otherwise :

Proof. From (3) we have 

(x)=v

F

for any x2 F. Moreover, if

we integrate   overP, we get jPjv F = Z P   (x)dH 2 (x)= Z P n  (s)
 P (s) dH 1 (s) Z P P (s) dH 1 (s); whi himplies (9). 2

We expe t that ondition (9) is alsosuÆ ient for a fa et F E

In order to give numeri al examples of fa et{breaking, we onsider a

singlepolygonalfa etF E (weshall onsider F asasubsetof R 2

)

and minimizethe fun tional

E  ():= Z F (div(x)) 2 dx; (10)

overall fun tions 2L 1

(F;R 2

)su h that div 2L 2

(F),


F =

F

assigned, and  2W a.e. inF, where W is the orthogonal proje tion

onthe plane ontaining F of the onvex set T o (rd E  jF ).

We observe that, following [9, Se tion 9. ℄, the fun tional (10)

admits a minimum  min 2L 1 (F;R 2

) under the previous restri tions;

moreovertwodierentminimaof(10) have thesame divergen e inF.

By (6), we expe t that if  is a minimum of (10) then div 

represents the velo ity (along the Cahn-Homan ve tor eld) of the

pointsofF,sothatpossiblefra tureswillappearin orrespondan eof

dis ontinuities of div  while, if div  is ontinuousbut non onstant,

the fa et F willbend.

Here isabriefexplanation of thenumeri alalgorithmusedtond

the minimaof (10). Weuse a niteelement method: given a

triangu-lationofF withverti esx

1

;:::;x

N

,we onsider thenitedimensional

spa e of pie ewise linear fun tions

= N X i=1  i  i ; with  i 2 R 2 and  i

: F ! R being ontinuous fun tions whi h

are linear on every single triangle and su h that 

i (x j ) = Æ ij (hat

fun tions). ItisnotdiÆ ulttondthegradientofthe fun tional(10)

restri ted to this spa e of fun tions:

E   i ()= X j M ij  j ; (11) where M ij

are the 22matri es given by

M ij =2 Z F r i r j dx:

foraminimumof E

lettingevolve following(minus) thegradientof

E

given by (11). After ea h step of the evolution, we proje t  onto

the onstraint W, in order to guarantee that  is always admissible

for (10).

We begin with a huge triangulationof F and, as initialguess, we

onsider  :=

F 

F

on F and  := 0 inside F. Then we ompute

thematrixM

ij

andwestartthesteepest des entminimizingevolution

for E

. When we are lose enough to a minimum of E

for the given

triangulation we thi ken the triangulation by adding new triangles

wherediv  has morevariation,and thenwerestartthe minimization

pro ess with the new triangulationand the new initialguess.

3. Examples and on lusions

In this se tion we show the numeri al results obtained in two simple

examples for whi h it is possible to nd expli itly the velo ity eld

(see [4℄, [13℄).

In the rst examplewe onsider the rystalline norm onR 3 given by (x):=maxfjx 1 j;jx 2 j;jx 3

jg,and weletE :=F [0;1℄,where F is

a non onvex polygon whi h isthe union of a square and are tangle,

see Figure 1.

In Figure 1 it is shown the adaptive triangulation onstru ted by

thealghorithmduringthe minimizationpro ess. Wenoti ethatsmall

trianglesare reated alongthesegmentseparatingthesquarefromthe

re tangle, where the velo ity has a great os illation. In Figure 2 the

ontour urvesof the resultingvelo ity are shown. We noti ethat on

the segment between the square and the re tangle the ontour urves

a umulate, while elsewhere the resulting normal velo ity is nearly

onstant. This is in agreement with the theoreti al result predi ting

the fra ture of F along the segment [4℄.

In the se ondexamplewe onsider a rystallinenorm whoseWul

shapeis an orthogonal prism entered at the origin, having a regular

hexagon as basis. We let E := F [0;1℄, where F is the hexagon

shown in Figure3.

Wenoti ethatthe omputedvelo ityhas greatos illationsin

or-respondan eoftheshortedgesofF. Morepre isely,thevelo ityseems

toin rease approa hing su h edges. In Figure4the ontour urvesof

sents the value of div  onevery triangle.

in [4℄, one expe ts that the velo ity is ontinuous on F (in reasing

neartheshortedges),sothatthefa etF shouldbendinsideE during

Figure3: The adaptive triangulation. The brightness of olors

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M. Novaga, E. Paolini: S uola Normale Superiore, Piazza dei