doctorat paristech l'‰cole nationale sup©rieure des mines de paris
TRANSCRIPT
MINES ParisTech CEMEF UMR CNRS 7635
1, rue Claude Daunesse - BP207 06904 Sophia Antipolis Cedex - France
présentée et soutenue publiquement par
M. Karim HITTI
le 07 décembre 2011
Direct numerical simulation of complex Representative
Volume Elements (RVEs): Generation, Resolution and Homogenization
~ ~ ~
Simulation numérique de Volumes Élémentaires Représentatifs (VERs) complexes : Génération, Résolution et Homogénéisation
Doctorat ParisTech
T H È S E
pour obtenir le grade de docteur délivré par
l’École nationale supérieure des mines de Paris
Spécialité “Mécanique Numérique”
Directeur de thèse : Thierry COUPEZ Co-encadrement de la thèse : Marc BERNACKI
Jury Pr. Houman BOROUCHAKI, LASMIS, Université de Technologie de Troyes Rapporteur
Pr. Olivier MILLET, LEPTIAB, Université La Rochelle Rapporteur
Pr. Patrice LAURE, Laboratoire J. A. Dieudonné, Université de Nice Examinateur
Pr. Nabil NASSIF, American University of Beirut Examinateur
Pr. Thierry COUPEZ, CEMEF, MINES ParisTech, Sophia Antipolis Directeur de thèse
Dr. Marc BERNACKI, CEMEF, MINES ParisTech, Sophia Antipolis Co-encadrant de thèse
Ecole doctorale n° 364 : Sciences Fondamentales et Appliquées
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Pi %+, Pj /()* .%,(( Ri %+, Rj; .#'0#$)(3#186 K#) R "# )*# .%,(&' -5 )*# +#/ 0%.)($1#
)- "# B#+#.%)#, %+, 1#)
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B = (Pi,−→u ,−→v ) $#+)#.#, (+ Pi; (' $-+').&$)#, /()*L
−→u =
−−→PiPj
‖−−→PiPj‖%+,
−→v = −→en ∧ −→u , @ 6 A
/()* ‖−−→PiPj‖ )*# M&$1(,(%+ +-.4 -5
−−→PiPj %+, ∧ )*# $.-'' 0.-,&$)6 =+ )*# +#/ "%'(' B; )*#
! !"#$%& '( )* &+,$&- $-&% .%/%&"$*+/
"##$%&'()*+ (u, v) #, )-* '*. /($)&"0* ($*1
u =
‖−−→PiPj‖2
+(Ri−Rj)(Ri+Rj+2R)
2‖−−→PiPj‖, v =
√(Ri +R)2 − u2
&, ‖−−→PiPj‖ ≤ Ri +Rj + 2R
u = Ri +‖−−→PiPj‖−Ri−Rj
2, v = 0 &, ‖−−→PiPj‖ > Ri +Rj + 2R
234 5
6' )-* 7 "(+*8 0*) T = (Pi, Pj, Pk) 9* )-* +*0*")*% ,$#') *0*:*')8 ( )$&(';0* ,#$:*% 9<
)-* )-$** +/-*$*+ .&)- "*')*$+ Pi8 Pj ('% Pk ('% $(%&& Ri8 Rj ('% Rk $*+/*")&=*0< ('% 0*) R9* )-* $(%&>+ #, )-* '*. /($)&"0* )# 9* ;*'*$()*%4 ?&$+)8 ( 0#"(0 9(+&+ B = (Pi,
−→u ,−→v ,−→w )8"*')*$*% &' Pi &+ "#'+)$>")*%8 .-*$*1
−→w =
−−→PiPj ×
−−→PiPk
‖−−→PiPj ×−−→PiPk‖
, −→u =
−−→PiPj
‖−−→PiPj‖('%
−→v = −→w ∧ −→u , 234@5
A-* "##$%&'()*+ (u, v) #, )-* '*. /($)&"0*8 &' )-* /0('* #, )-* )$&(';0* T 8 ($* "(0">0()*%
>+&'; )-* +<+)*:1
2uuj = (Ri −Rj)(Ri +Rj + 2R) + u2
j
2uuk + 2v.vk = (Ri −Rk)(Ri +Rk + 2R) + u2k + v2k
234B5
.-*$* uj =−−→PiPj.
−→u , uk =−−→PiPk.
−→u ('% vk =−−→PiPk.
−→v 4 C*'"*8 )-* "##$%&'()* w #, )-*
'*. /($)&"0* &+ ;&=*' 9<1
w =√(Ri + R)2 − u2 − v2 &, (Ri +R)2 ≥ u2 + v2 . 234D5
6' )-* "(+* .-*$* (Ri + R)2 < u2 + v28 )-* '*. /($)&"0* &+ '#) /0("*% >+&'; )-* +*0*")*%
,$#') *0*:*') ('% )-*$* ($* ).# /#++&90* "#'E;>$()&#'+1
• 6, )-*$* ($* +)&00 +#:* (")&=* *0*:*')+8 .-&"- -(=* '#) 9**' )*+)*% <*) )# /0("* )-*
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• 6, (00 )-* *0*:*')+ #, )-* ">$$*') ,$#') -(=* 9**' )*+)*% )# /0("* )-* '*. /($)&"0*8
.&)-#>) +>""**%&';8 )-*' )-* +*0*")*% ,$#') *0*:*') &+ '*"*++($&0< )-* E$+) (")&=*
*0*:*') &' )-* ">$$*') ,$#')4 6' )-&+ "(+*8 (:#'; )-* )-$** +*;:*')+ (Pi, Pj)8 (Pi, Pk)
('% (Pk, Pj)8 )-* +*;:*') (P1, P2) :(F&:&G&'; )-* H>(')&)< ‖−−→P1P2‖ − R1 − R28
.-*$* R1 ('% R2 ($* )-* $(%&& #, P1 ('% P2 $*+/*")&=*0<8 &+ +*0*")*%4 A-*' )-* '*.
/($)&"0* &+ /0("*% >+&'; )-* +*0*")*% +*;:*') (P1, P2)4 6'%**%8 ( '*. 0#"(0 9(+&+
B = (P1,−→u ,−→v ,−→en) &+ 9>&0)8 .-*$*1
−→u =
−−→P1P2
‖−−→P1P2‖, −→en = −→v ('%
−→v = −→en ∧ −→u . 234I5
A-*' )-* "##$%&'()*+ (u, v, w) #, )-* '*. /($)&"0*8 &' )-* '*. 0#"(0 9(+&+8 ($*1
! ! "#$%&% #'()*+, !
u = ‖−−−→P1P2‖2
+ (R1−R2)(R1+R2+2R)
2‖−−−→P1P2‖, v =
√(R1 +R)2 − u2,
w = 0 "# ‖−−→P1P2‖ ≤ R1 +R2 + 2R
u = R1 +R, v = 0, w = 0 "# ‖−−→P1P2‖ > R1 +R2 + 2R
$%&'(
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#0 12%$"21 S01 = S(c01, r1) $' 32%"2( c
01(x
01, y
01, z
01) +%1 (+1#*0 r1- :' z01 − r1 = 0& "/2 07/2(2
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E0 = S(c(x, y, z), r) ∈ SP\S01/z < z01 and (x0
1−x)2+(y01−y)2 < (r1+r)2. B=-<;C
./2% 42 02+(3/ '$( + S2 ∈ E0 )#%#)#D#%8 "/2 '*%3"#$% B022 E#8*(2 =-<F+C
f0(S) =
(z01 − z −
√(r1 + r)2 − ((x0
1 − x)2 + (y01 − y)2)
). B=-<<C
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01, y
01, r1), r1)- G502& "/2 7$0#"#$% $' c01 #0 *71+"21 6@
4(#"#%8 B022 E#8*(2 =-<F6C9
z11 = z2 +√(r1 + r)2 − ((x0
1 − x2)2 + (y01 − y2)2)
y11 = y01
x11 = x0
1
S11 = S(c11(x
11, y
11, z
11), r1)
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E#8*(2 =-<F69
x11 = x2 + (r1 + r2)cosϕcosθ
y11 = y2 + (r1 + r2)cosϕsinθ
z11 = z2 + (r1 + r2)sinϕ
B=-< C
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r1+r2
sinϕ =z11−z2
r1+r2
cosθ =x11−x2√
(x11−x2)2+(y1
1−y2)2
sinθ =y11−y2√
(x11−x2)2+(y1
1−y2)2
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M0 *+, */.)476/=.*06) =.*/0F@
M0 =
cosθ −sinθ 0sinθ cosθ 00 0 1
"#$%H'
I+, J/4* /6330); 4*,- 04 -,/76/=,8 C9 2-8.*0); *+, -640*06) 67 c11 C9 =680790); *+, :.32,
67 ϕ 0) *+, ),5 C.404$
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−→c2c = M−1
0−→c2c <=6DE?
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'4&%#5-?+ dist+ #% 2#A-1 ;BC
d1 =√x2 + z2 − (r1 + r2)
d2 =√d21 + y2
dist = r1 − d2
<=6DF?
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'( c11 )$- ';&)#1-5 ;B &)J#12 ϕ = 0 #1 KH%6<=6D ?C
x21 = x2 + (r1 + r2)cosθ
y21 = y2 + (r1 + r2)sinθ
z21 = z2
S21 = S(c21(x
21, y
21, z
21), r1)
<=6D!?
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/)A#12 &/- :)9#:)* ϕ′ A-$#(B#12C
ϕ′ ∈ [0, ϕ]
x21 = x2 + (r1 + r2)cosϕ
′cosθ
y21 = y2 + (r1 + r2)cosϕ′sinθ
z21 = z2 + (r1 + r2)sinϕ′
c21c = r1 + r
<=6DP?
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2)T2 − 2K1K3T +K2
3 −K22
K2(K3 −K1Ti) ≥ 07898:;
<+%2%6
K1 = 2(Wxcosθ +Wysinθ)
K2 = 2Wz
K3 = W 2x +W 2
y +W 2z + 1−W 2
7898=;
1(4
Wx =x−x2
r1+r2
Wy =y−y2r1+r2
Wz =z−z2r1+r2
W = r1+rr1+r2
78988;
>% <'$$ 4%(-&% solS &+% '(&%2,%3&'() ,-$.&'-( -/ &+% ?@,97898:; &- 78988;9 A+%,% %@.1&'-(,
31( 140'& :*= -2 8 ,-$.&'-(,9 B').2% 898=1 '$$.,&21&%, &+% C2,& 31,% <+%2% E2 6= ∅ D.& <+%2%
! !"#$%& '( )* &+,$&- $-&% .%/%&"$*+/
"#$ %&'("(&) &* c3 (' ()+&,%-"(./$ 0("# "#$ +&)1("(&) ϕ′ ∈ [0, ϕ] -)1 '& solS = ∅2 3(456$
7278. (//5'"6-"$' "#$ '$+&)1 +-'$ 0#$6$ &)/9 &)$ '&/5"(&) T1 (' %&''(./$ () -46$$,$)" 0("#
"#$ +&)1("(&) ϕ′ ∈ [0, ϕ] -)1 #$)+$: solS = T12 3()-//9: 3(456$ 7278+ (//5'"6-"$' "#$ "#(61+-'$ 0#$6$ "0& '&/5"(&)' T1 -)1 T2 -6$ %&''(./$2 ;) "#(' +-'$: &)/9 "#$ 5%%$6 %&'("(&) ('
+&)'$6<$1 -)1 '& solS = minT1, T22
!" #"
$"
!"#$% &'&() *%+$,-!." /0$ 1022!34% 204#5!0.2 0/ 5-% 672'8&'&9: 50 8&'&&: ;!5- $%"+$<2 50
5-% ,0.<!5!0. ϕ′ ∈ [0, ϕ] !. 5-% $044!." 1-+2% ('& !. 5-% => ?@A) 8+: .0 204#5!0.2 %B!25C
83: 0.4D 0.% 204#5!0. %B!252 +.< 8,: 5;0 204#5!0. %B!25
=&6$&<$6: /$" E3 = S ∈ E2/solS 6= ∅2 ;* E3 = ∅: "#$ +&&61()-"$' &* c11 -6$ 5%1-"$1
.9 '9'"$, >728?@ -)1 0$ 4& .-+A "& '"$% 82 B/'$: 0$ '$-6+# *&6 S3 ∈ E3 '5+# "#-"
solS3= minS∈E3
(solS) -)1 "#$ +&&61()-"$' &* c11 -6$ 5%1-"$1 .9 >'$$ 3(456$ 7277@C
! ! "#$%&% #'()*+, !
!"#$% &'&&( )*% $+,,!-" .*/0% 1'& !- 2*% 34 567( 2*% -%8 .+0!2!+- +9 c1 2/-"%-2 2+ c2/-: c3
ϕ′ = arccos(solS3)
x21 = x2 + (r1 + r2)solS3
cosθ
y21 = y2 + (r1 + r2)solS3sinθ
z21 = z2 + (r1 + r2)√1− sol2S3
S21 = S(c21(x
21, y
21, z
21), r1)
"#$#%&
'() *+ ,- ./)012)$
! "#$%&'() *+, *-%$ '(.$,/$%.'+( 01 '3124 -12/- % 0() 53 162 07854,169: ;2 (22) 15
<62<= ,3 162 -/6242 ,( ,1- (2; /5-,1,5( ,(124-2<1- 5(2 53 162 )590,(>- 30<2-$ ?2 @4-1 <62<=
,3 162 -/6242 ,(124-2<1- 162 30<2 Z = Zmin 162( ;2 <62<= 354 ,(124-2<1,5(- ;,16 162 51624
30<2-$ A3 162 -/6242 ,(124-2<1- 5(7B Z = Zmin: ,1 ,- /.1 ,( 0 30<2C15.<6,(8 /5-,1,5( 0()
16,- /5-,1,5( D2<592- @(07 0() *+ ,- ./)012)$ A( 162 51624 <0-2-: 162 4577,(8 <5(1,(.2-
03124 /.11,(8 162 -/6242 ,( 0 30<2C15.<6,(8 /5-,1,5($ A( 162 <0-2 ,77.-14012) DB E,8.42 #$#%:
162 -/6242 ,(124-2<1- 162 30<2 X = Xmax$F2(<2: ,1 -65.7) D2 /.1 ,( 0 30<2C15.<6,(8 /5-,1,5( 5(7B$ G6,- ,- )5(2 DB 95),3B,(8 ϕ′ 0-35775;-H
x21 = x2 + (r1 + r2)cosϕ
′cosθ = Xmax− r1, "#$# &
! !"#$%& '( )* &+,$&- $-&% .%/%&"$*+/
!"#$% &'&() *+,%$ ,-% $.//!0" 1,%21 ( .$ 34 5 651% 7-%$% ,-% 12-%$% .+ 6%0,%$ c1!0,%$1%6,!0" ,-% +56% X = Xmax
"#$%# &$'() *#+*
ϕ′ = arccos
(Xmax− x2 − r1(r1 + r2)cosθ
). ,!-!./
01*(2"+23)4 *#( %5523$6+*() 51 c21 +2( 753$8(3 9: "2$*$6& ,)(( ;$&<2( !-! +/=
x21 = Xmax− r1
y21 = y2 + (r1 + r2)cosϕ′sinθ
z21 = z2 + (r1 + r2)sinϕ′
,!-!>/
?6 523(2 *5 %56*$6<( *#( 25@@$6& A25%())4 + 8%*$*$5<) )A#(2( S3(cf3 , r
f3 )4 ):77(*2$% 51 S2 "$*#
2(&+23 *5 *#( A@+6( X = x214 $) %2(+*(3 ,)(( ;$&<2( !-! 9/- ;<2*#(2752(4 *#( $6*(2)(%*$56)
"$*# *#( 5*#(2 1+%() )#5<@3 9( %#(%B(3 +) "(@@ (C%(A* 152 Z = Zmax 9(%+<)( +@@ *#(
)A#(2() +2( 325AA(3 +63 $6*(2)(%*$6& *#$) A@+6( $) 65* A5))$9@(- ;52 + 9(**(2 <63(2)*+63$6&
$6 "#+* 15@@5")4 + A5))$9@( 8%*$*$5<) )A#(2( Sf3 (c
f3 , r
f3 ) "$@@ 9( +@)5 3(65*(3 +) + 2(&<@+2
)A#(2( S3(c3, r3)- ;$6+@@:4 +63 56%( +&+$64 $1 *#$) *2(+*7(6* $) 2(+@$D(34 S21 +63 EF +2(
<A3+*(3-
! "#$%&'()&*$+ &,- $-. /#)0/ )##'1*$0&- %2%&-34 G65"$6& *#( %(6*(2) c214 c2+63 c3 ,"#$%# %+6 9( *#( 56( 51 + 8%*$*$5<) )A#(2(/4 + 6(" %5523$6+*( ):)*(7 (Ω, B1 =(−→u1,
−→v1 ,−→w1)) $) %56)$3(2(3 "$*# Ω *#( 52*#5&56+@ A25H(%*$56 51 c21 56 [c2, c3] ,)(( ;$&<2(
!-!./- (−→u1,−→v1 ,−→w1) +2( &$'(6 9:=
! ! "#$%&% #'()*+, !
!" #"
!"#$% &'&() *+, -.% /0.%$% 12 3%45%$ c1 !/ $%5#$4%6 51 + 2+3%751#3.!4" 01/!5!14 +46 *8,5.% 314/5$#35!14 12 5.% 935!5!1#/ /0.%$% 12 3%45%$ cf3
−→u1 =−−→c2c3‖−−→c2c3‖
−→v1 = (0, 0, 1) + α−→u1
−→v1 .−→u1 = 0
−→w1 =−→u1 ×−→v1
"#$#%&
'() *+,-.-+/ +0 Ω -, 1-2)/ 34 .() )567.-+/8
−→c2Ω = −−→c2c3
−−→c2c
21.−−→c2c3
‖−−→c2c3‖2, "#$#9&
'() :++;<-/7.), +0 c21 -/ .() /)= :++;<-/7.) ,4,.)> 7;)8
0
Ωc21.cosβ
Ωc21.sinβ
"#$#?&
@/< .() .;7/,0+;>7.-+/ >7.;-A 3).=))/ .() :7/+/-: ,4,.)> 7/< .() /)= ,4,.)> -, =;-..)/8
M1 =
x−→u1x−→v1 x−→w1
y−→u1y−→v1 y−→w1
z−→u1z−→v1 z−→w1
. "#$!B&
!"#$%& '( )* &+,$&- $-&% .%/%&"$*+/
!"#$% &'&() *+% $,--!." /+01% 2'& !. 3+% 45 678) 3+% .%9 /,1!3!,. ,: c1 30."%.3 3, c20.; c3
! "#$$%&' ()*+, -!./ !" #$%&"'($" !"$" )* *)+),-$ .% *."# /0 1" 2$*. *"-$&! 3%$
.!" *#!"$"* )4."$*"&.)45 .!" 2&.).)%(* *#!"$" %3 &"4."$ Ω -4' $-')(* r1 + Ωc216 7% ,".
E4 = S(c, r) ∈ SP\S11 , S2, S3/Ωc < r1 + Ωc21 + r6 83 E4 6= ∅0 3%$ -,, .!" *#!"$"*
S(c(x, y, z), r) ∈ E40 1" *"-$&! 3%$ .!" %4"* )4."$*"&.)45 .!" 2&.).)%(* .%$(* %3 -9)*−−→c2c30
&"4."$ Ω0 *+-,, $-')(* r1 -4' :)5 $-')(* Ωc21 ;*"" <)5($" =6=>?6 !" &%%$')4-."* %3 c )4
.!" 4"1 :-*)*0 (x, y, z)0 -$" 5)@"4 :AB
−→Ωc = M−1
1
−→Ωc ;=6/C?
-4' ).* *)54"' ')*.-4&" .% .!" .%$(* ;1).! .!" &%4@"4.)%4 %3 #%*).)@" )4*)'" -4' 4"5-.)@"
%(.*)'"?0 dist0 )* 5)@"4 :AB
d1 =√y2 + z2 − Ωc21
d2 =√d21 + x2
dist = r1 − d2
;=6/=?
D". E5 = S ∈ E4/dist+ r > 06 83 E4 = ∅ %$ E5 = ∅0 .!" &%%$')4-."* %3 c21 -$" (#'-."'-* 3%,,%1* ;*"" <)5($" =6=E-?B
! ! "#$%&% #'()*+, !
x31 = xΩ − Ωc21.x−→v1
y31 = yΩ − Ωc21.y−→v1
z31 = zΩ − Ωc21.z−→v1
S31 = S(c31(x
31, y
31, z
31), r1)
"#$%%&
'() *+ ,- ./)012)$ 34 z31 ≤ min(z2, z3)5 62 78 90:; 18 -12/ < 6,1= S01 = S3
1 $ >?-2 ,4
z3 < z31 5 62 78 90:; 18 -12/ # 9@ A2/?0:,(7 S2 9@ S3$ 34 E5 6= ∅5 62 78 18 1=2 (2B1 A8??,(7
/=0-2$
!" #"
!"#$% &'&() *+% $,--!." /+01% &'2 !. 3+% 45 678) 90: ",!." ;0<= 3, 13%/ 2 0.> 9;: 3+%
1/+%$% ,? <%.3%$ c4 !1 ?,#.>
! "#$$%&' ()*+, -!-. C,A-15 β (22)- 18 92 :0?:.?012)$ D=2 :88A),(012- 84 c21 ,( 1=2
:88A),(012 -@-12E 84 :2(12A Ω5 )2(812) (a, b, c)5 0A2 7,F2( 9@G
−→Ωc21 = M−1
1
−→Ωc21 =
abc
. "#$% &
H2(:25 9@ :8E9,(,(7 1=,- 2I.01,8( 6,1= >I$"#$#J&5 β = arccos( bΩc1
).sign(c)$ D=2(5 62
(22) 18 -20A:= 48A 0 -/=2A2 84 E5 =0F,(7 0 E0B,E0? β′ F2A,4@,(7G
! !"#$%& '( )* &+,$&- $-&% .%/%&"$*+/
β′ ∈
[β, π] if β ≥ 0
[−π, β] if β < 0
x31 = 0
y31 = Ωc21.cosβ′
z31 = Ωc21.sinβ′
c21c = r1 + r
"#$%&'
() *+,,-./ T = cosβ′0 ,1+ 23*4,-3.2 -. T 35 ,1-2 2)2,+6 78+ ,1+ 833,2 35 ,1+ 2+93.: 38:+8
;3*).36-7*<
(K21 +K2
2)T2 − 2K1K3T +K2
3 −K22
K2(K3 −K1Ti) ≥ 0"#$%!'
=1+8+<
K1 = y
K2 =
z if β ≥ 0
−z if β < 0
K3 =x2+y2+z2+(Ωc2
1)2−(r1+r)2
2Ωc21
"#$%>'
?+ =-** :+.3,+ solS ,1+ -.,+82+9,-./ 23*4,-3. 35 ,1+ @A2$"#$%!' ,3 "#$%>'$ B1+2+ +A47,-3.2
97. 7:6-, C0D 38 # 23*4,-3.20 :+.3,+: T1 7.: T20 2-6-*78*) ,3 2,+; $ E. ,1+ 972+ =1+8+ ,=3
23*4,-3.2 78+ ;322-F*+0 ,1+ 4;;+8 ;32-,-3.2 -2 93.2+8G+: 7.: 23 solS = max(T1, T2)$ H+,
E6 = S ∈ E5/solS 6= ∅$ B1+.0 =+ 2+7891 538 S4 ∈ E6 2491 ,17, solS4= maxS∈E6
(solS)
7.: ,1+ 9338:-.7,+2 35 c21 78+ 4;:7,+: F)<
β′ =
arccos(solS4) if β ≥ 0
−arccos(solS4) if β < 0
x13 = 0
y13 = Ωc21.cosβ
′
z13 = Ωc21.sinβ
′
"#$%I'
J-.7**)0 ,1+ 9338:-.7,+2 35 c31 -. ,1+ 2,7.:78: F72-2 78+ /-G+. F) "2++ J-/48+ #$#!F'<
! ! "#$%&% #'()*+, !
−→Ωc31 = M1
−→Ωc31 , "#$%&'
()*+ S31 = S(c31(x
31, y
31, z
31), r1) ()* ,- ./ 01*(23*$
! "#$%&'() *+, *-%$ '(.$,/$%.'+( 01 45236 273 /231/ ! 86 9+ :3 )33* 28 ;73;< (=(.)
586 .)236/3;2.8)/ :.27 273 *8>(.)?/ 5(;3/$ @7./ /231 85 273 (A=86.27> ./ /.>.A(6 28 /231 B C02
:.27802 273 ;63(2.8) 85 ( D;2.2.80/ /17363 ()* CE ;8)/.*36.)= 273 5(;3F280;7.)= 18/.2.8) 85
273 /17363 (/ D)(A$ G) ;(/3 85 .)236/3;2.8)+ 273 /17363 S31 ./ 102 .) ( 5(;3F280;7.)= 18/.2.8)
CE >8*.5E.)= β′$ H.6/2 :3 /780A* :6.23 273 ;886*.)(23/ 85 c31 .) 273 /2()*(6* C(/./$ ,.);3
.2/ ;886*.)(23/ .) 273 )3: C(/./ 85 ;3)236 Ω (63 (0,Ωc31.cosβ′,Ωc31.sinβ
′)+ 273E (63 :6.223)
.) 273 /2()*(6* C(/./ (/I
x31 = xΩ + Ωc31(x−→v1cosβ
′ + x−→w1sinβ′)
y31 = yΩ + Ωc31(y−→v1cosβ′ + y−→w1
sinβ′)
z31 = zΩ + Ωc31(z−→v1cosβ′ + z−→w1
sinβ′)
"#$ J'
KE 2(<.)= 273 /(>3 3L(>1A3 (/ .) /231 B :7363 273 /17363 .)236/3;2/ 273 5(;3 X = Xmax+:3 8C2(.)I
x31 = xΩ + Ωc31(x−→v1cosβ
′ + x−→w1sinβ′) = Xmax− r1. "#$ M'
4/ .) /231 9+ 273 N(A03/ 85 β′ (63 8C2(.)3* CE /8AN.)= 273 /E/23> =.N3) CE OP$"#$%Q'
C02 :.27I
K1 = x−→v1
K2 =
x−→w1if β ≥ 0
−x−→w1if β < 0
K3 =Xmax−r1−xΩ
Ωc31
"#$ #'
45236:(6*/+ 273 ;886*.)(23/ 85 c31 =.N3) .) /E/23> "#$ J' (63 >8*.D3* 0/.)= OP$"#$ M'
()* 273 )3: N(A03 85 β′ ()* 273 /17363 ./ 273) .) .2/ D)(A 5(;3F280;7.)= 18/.2.8)+ Sfinal1 =
S(c31, r1) ()* ,- ./ 01*(23*$ 4A/8+ 273 .)236/3;2.8)/ :.27 273 82736 5(;3/ /780A* C3 ;73;<3*
(/ :3AA 3L;312 586 Z = Zmax C3;(0/3 (AA 273 /17363 (63 *68113* ()* .)236/3;2.)= 27./
1A()3 ./ )82 18//.CA3$
23! 4$/.'() .#$ /.-5'6'.71 G) 86*36 28 ;73;< 273 /2(C.A.2E 85 273 /17363 S1+ :3 136586>
273 ;A(//.;(A 23/2 R#%S 0/3* :.27 273 ;A(//.;(A *6811.)= ()* 68AA.)= (A=86.27>$ H.=063 #$#!
/78:/ 273 86278=8)(A 168T3;2.8)/+ Pc31+ Pc2 + Pc3 ()* Pc4 + 85 c
31+ c2+ c3 ()* c4 63/13;2.N3AE
8) 273 1A()3 z = z0 "z0 ;() C3 ;78/3) 6()*8>AE'$
! !"#$%& '( )* &+,$&- $-&% .%/%&"$*+/
!"#$% &'&() *+%,-!." /+% 0/12!3!/4 56 /+% 07+%$% 56 ,%./%$ c1
"#$ %&'()*)&+ ,- &#$ %.#$/$ S31 )% 0)1$2 (+ &#$ .,%)&),2 ,- Pc3
13 4- )& )% 5,2&')2$6 )2%)6$
&#$ &/)'20*$ Pc2Pc3Pc4 7 &#$ %.#$/$ )% %&'(*$3 42 &#)% 5,2809/'&),27 S31 ($5,:$% &#$ 82'*
.,%)&),2 ,- &#$ %.#$/$ '26 ;< )% 9.6'&$63 4- Pc31∈ Ωij7 =$ 0, ('5> &, %&$. ? =)&#
S2, S3 = Si, Sj3 4- Pc31∈ Ωi7 =$ 0, ('5> &, %&$. @ =)&# S2 = Si3
42 ,/6$/ &, 6$&$/:)2$ )- Pc31)% .,%)&),2$6 '& &#$ %':$ &#$ %)6$ ,- &#$ *)2$ (Pc2Pc3) '%
Pc4 -,/ $A':.*$7 =$ 5'*59*'&$ &#$ -,**,=)20 1'/)'(*$%B
−→u23 = M.−−−→Pc2Pc3
a23 =−→u23.
−−−→Pc2Pc4
b23 =−→u23.
−−−→Pc2Pc3
1
C@3 DE
=#$/$ M )% ' :'&/)A 0)1$2 (+B
M =
0 1 0−1 0 00 0 1
C@3 E
4- a23.b23 ≥ 07 Pc31'26 Pc4 '/$ ,2 &#$ %':$ %)6$3 4- 2,&7 &#$+ '/$ ,2 ,..,%)&$ %)6$%3 "#)%
5'*59*'&),2 )% .$/-,/:$6 &, 826 &#$ .,%)&),2 ,- Pc31=)&# /$0'/6 &, (Pc2Pc4) '26 (Pc3Pc4)
'% =$**3 4& )% )2&$/$%&)20 &, 926$/*)2$ &#'& &#)% $F5)$2& '26 5*'%%)5'* %&'()*)&+ &$%& -,/ '**
&#$ 6/,..)20 '26 /,**)20 '*0,/)&#:% 5'2 *$'6 &, .'/&)59*'/ 5'%$% =#$/$ &#$ 5,21$/0$25$ )%
2,& 5$/&')2 '26 &#'& &#)% 6)F59*&+ ='% 2$1$/ 6)%59%%$6 )2 &#$ *)&$/'&9/$3 426$$67 *$& 9%
5,2%)6$/ &#$ 5'%$ =#$/$ Pc31∈ Ω2 ,/ Pc3
1∈ Ω233 42 &#$%$ 5,2809/'&),2%7 /$'*)G)20 '0')2
&#$ 6/,..)20 '26 /,**)20 ./,5$69/$ 5'2 *$'6 &, '2 925#'20$ )2 &#$ 6$&$/:)2'&),2 ,- S27
S3 '26 S4 '26 %, ($ %+2,2+:,9% &, '2 )282)&$ *,,.3 42 &#$ 5,2&$A& ,- &#$ ./,.,%$6
! ! "#$%&% #'()*+, !
!"#$% &'&() * +,-% ./%$% Pc31∈ Ω23
"#$%&'()*+ ()",-. (% ()/ 0)%'0/ %1 ()/ ϕ′2',(/&3"# ', 4567!8 ",9 %1 ()/ β′2',(/&3"# ', 456:;8+
()/ .'(<"('%, Pc31∈ Ω2 0", ,/3/& %00<&6 =%>/3/&+ ', &"&/ ?"&('0<#"& 0%,@$<&"('%,.+ %<&
"#$%&'()* 0", #/"9 (% Pc31∈ Ω23 ",9 (% ", ',@,'(/ #%%? >)/, ()/ 9&%??',$ ",9 &%##',$
?&%0/9<&/ '. ?/&1%&*/9 "$"',6 A'$<&/ 565B '##<.(&"(/. %,/ %1 ()/./ 0"./.6
C .'*?#/ .%#<('%, (% %3/&0%*/ ()'. 9'D0<#(E '. ()/, (% *%9'1E ()/ β′2',(/&3"# ', 456:;8 FEG
β′ ∈
[−π, β] if β ≥ 0
[β, π] if β < 0456 ;8
H, ()/ (>%29'*/,.'%,"# 0"./+ ()/ IJC 0%,.'.(. "( @&.( ', .%&(',$ ()/ 9'.0.+ "00%&9',$
(% ()/ 9&%??',$ ",9 &%##',$ 9'&/0('%, 9/,%(/9 0+ 1&%* *','*<* (% *"K'*<*6 L)/,+
/"0) 9'.0 %1 0/,(/& ci '. 9&%??/9 <.',$ ()/ 9&%??',$ .(/? %1 ()/ "#$%&'()* 4.// A'$<&/
565!"86 C1(/&>"&9.+ '( '. &%##/9 ('## &/"0)',$ " ()'&9 9'.0 %1 0/,(/& ck6 L)/ &%##',$ .(/?
'. ?/&1%&*/9 <.',$ " .'*'#"& 3/&.'%, %1 .(/?. : ",9 F<( %,#E >'() ()/ <./ %1 %,/ ",$#/
>)'0) '. ϕ 4.// A'$<&/ 565!F86 H1 ()/ 9'.0 '. ,%( .("F#/+ >)'0) '. /M<'3"#/,( (% 3/&'1E',$
Py⊥(−→cicj).Py⊥(
−→cick) > 0+ >)/&/ Py⊥ 0%&&/.?%,9. (% ()/ ?&%N/0('%, %?/&"(%& %, ()/ y⊥
.?"0/+ '( 0%,(',</. '(. &%##',$ ?)"./ <,('# &/"0)',$ " .("F#/ ?%.'('%, 4.// A'$<&/ 565!086
! !"#$%& '( )* &+,$&- $-&% .%/%&"$*+/
!" #"
$"
!"#$% &'&() *++#,-$.-!/0 /1 -2% -3/45!6%0,!/0.+ 789 1/$ . 5!,: /1 :%0-%$ ci) ;.<5$/==!0"> ;?< $/++!0" .05 ;:< #0,-.?+% =/,!-!/0
"#$ %&'$($)' *+ '#$ ,*-./$0 1234256 7..(*7,# %) '* ,-8-/7'$ '#$ 7097&'7:$) *+ '#$
';* 8$'#*0)< '* ($).$,' 7 :%9$& (70%-) )'7'%)'%,7/ 0%)'(%=-'%*& 7&0 '* 7,#%$9$ 7 #%:#
0$&)%'> '#7&?) '* '#$ 123 7&0 '* ,*(($,' '#$ /*,7/ -&4.#>)%,7/ 0%).*)%'%*& *+ '#$ .7('%,/$)
;#%/$ %8.(*9%&: '#$ /*,7/ 0$&)%'> 7&0 ;%'#*-' 8*0%+>%&: '#$ (70%-) 0%)'(%=-'%*& '#7&?) '*
'#$ 97(%7&' *+ '#$ 0(*..%&: 7&0 (*//%&: 8$'#*0@ A-$ '* '#$ 7,'%*& *+ B0(*..%&:BC 7& $8.'>
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#**(4E)"#+)4$F
ρd ≃∑Nc
i=1 Fib
|Ω| , 8G, H;
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D$2 &4? b )& 6E/2 %&)$< +./ /C%#+)4$F
b =ρd|Ω|∑Nc
i=1 Fi
. 8G, K;
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:!98 ǫmin = 0.02 -1. ǫmax = 0.05 ;&(<
"#$ %&%'%() *$&+%', %+ '#$& -%.$& /, ρ0 ≃∑Nc
i=1Fiǫi
|Ω| , !"# $ %#$&&'( )!*' '+,$( "- ǫi + ǫj .-/
" - &'!0#1-/!&0 %'(() 2-/ 0/$!&)3 Ci $&4 Cj 2)'' 5!0,/' 678937
:."'/ $/4); !" !) '&-,0# "- <-4!.= '$%# ǫi 1= $ %-&)"$&" >$(,' b #!%# $((- ) /'$%#!&0"#' 4')!/'4 4'&)!"= #!(' %-&)'/>!&0 "#' #'"'/-0'&'-,) "#!%?&'))'); 1= ,)!&0@
ρd =
∑Nc
i=1 Fi(ǫi + b)
|Ω| ,
ρd = ρ0 +
∑Nc
i=1 Fib
|Ω| ,
b =(ρd − ρ0)|Ω|∑Nc
i=1 Fi
.
267A63
5!&$((=; B+7267C93 !) <-4!D'4 $)@
Vi = x ∈ Vi/d(x, Vi) ≥ ǫi + b. 267A83
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φ(x) = ±d(x,Γ), x ∈ ΩΓ = x ∈ Ω, φ(x) = 0 P=:GQ
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αij(x) =1
2‖ −−→SiSj ‖ −
−−→SiSj.
−−→SiX
‖ −−→SiSj ‖for 1 ≤ i, j ≤ N, j 6= i, P=:HQ
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'2+ %4-(+4, Si< /, '2+% 9/?+% 0)O
φi(x) = min1≤j≤Nj 6=i
(αij(x)). P=:>Q
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φglob(x) = maxφi(x), 1 ≤ i ≤ N. "#$#%
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∇.σ = 0, YL. Z
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∂t+∇.(ρv) = 0, YL.!Z
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σf = 2ηfǫ(v)−(2
3ηf∇.v + p
)I, )*+,-
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ǫ(v) =1
2
(∇v +∇v
T), "#$%&
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'(+5( +, /1- -() 5.,) +/ -(+, .004+5.-+1/ <)5.?,) -() .+* +, . ').@46 51;0*),,+<4) =?+9>
∇.v = 0 ./9 -() ,)51/9 -)*; 1/ -() *+8(- (./9 ,+9) 12 AB$"#$C& '+44 51/-.+/ 1/46 -()
0*),,?*)$ D6 '*+-+/8 AB$"#$C& '+-( . ;19+E)9 0*),,?*) p′ = p+ 23ηf∇.v ./9 *)04.5+/8 σf
+/ AB$"#$ &> ') 8)-F
2ηf∇. (ǫ(v))−∇p′ = 0. "#$#&
:1* . 4+/).*> )4.,-+5 ,14+9> -() ,-*),, -)/,1* σs +, 8+3)/ <6 -() 8)/)*.4+G)9 H11@)7, 4.'F
σs = 2µǫ(u) + λtrǫ(u)I, "#$I&
'()*) µ ./9 λ .*) -() -'1 J.;K 51)L5+)/-,> u +, -() 9+,04.5);)/-> ǫ(u) +, -() ,-*.+/
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λ = νE(1+ν)(1−2ν) ,
µ = E2(1+ν)
."#$Q&
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p = −σs
3, "#$T&
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K =2
3µ+ λ. "#$U&
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p = −Ktr(ǫ(u)). "#$ W&
X1*)13)*> -() <?4@ ;19?4?, +, 9)E/)9 <6>
K = −V ∂p
∂V, "#$ &
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'*+--)/F
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λ.tr(ǫ(u)) = −(p+
2
3µtr(ǫ(u))
). #$% !&
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,52./)01<
σs = 2µǫ(u)−(2
3µtr(ǫ(u)) + p
)I. #$% "&
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7, ?,/<
2µ∇. (ǫ(u))−∇p′ = 0. #$% C&
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∇.u+1
Kp = 0. #$% $&
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2µ∇. (ǫ(u))−∇p′ = 0 (a)
dudt= v (b)
∇.u+ 1K
(p′ − 2
3µ∇.u
)= 0 (c)
#$% 6&
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2ηf∇. (ǫ(v))−∇p′ = 0 (a)
∇.v + 1K
dp′
dt= 0 (b)
#$% I&
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E, 7+)//,1 .*<
∇.v +1
p′dp′
dt= 0. #$% K&
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ut = u
t−∆t + vt∆t. #$% Q&
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$6$(*4 3.810 ?* @-%((*& 2$A
2µ∆t∇. (ǫ(vt))−∇p′t = −2µ∇.(ǫ(ut−∆t)
)(a)
∇.vt + 1K− 2
3µ
dp′t
dt= 0 (b)
9:;!B=
C)* *+,-*$$%.& −2µǫ(ut−∆t) ./ ()* -%')(D)2&0 $%0* ./ E7;9:;!B92== %$ ()* *+(-2D$(-*$$ (*&$.-
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(. ()* /.-4A
∇.u = − 1
K − 23µp′ 9:;! =
C)*& -*,123%&' ∇.ut2&0 ∇.ut−∆t
?6 − 1K− 2
3µp′t 2&0 − 1
K− 2
3µp′t−∆t
-*$,*3(%F*16 %&A
∇.ut = ∇.ut−∆t +∇.vt∆t. 9:;!!=
I.-*.F*-> 8$%&' ()* *+,-*$$%.& ./ ()* ?81H 4.0818$ K %& E7;9:;J=> E7;9:;!B9?== 3.810 ?*
@-%((*& 2$A
∇.vt +1
λ
dp′t
dt= 0 9:;!K=
L%&2116> ()* '.F*-&%&' *782(%.&$ /.- ()* LGM ,-.?1*4 2-*A
!" #$% &'()
2ηf∇. (ǫ(vt))−∇p′t = 0
∇.vt + 1p′t
dp′t
dt= 0
9:;!"=
!" #$% *!+()
2µ∆t∇. (ǫ(vt))−∇p′t = −2µ∇.(ǫ(ut−∆t)
)
∇.vt + 1λdp′t
dt= 0
9:;!:=
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0 if α < −e
12
(1 + α
2
)if |α| ≤ e
1 if α > e
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Θ = H(α)Θf + (1−H(α))Θs. #"$!G&
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Ψ = H(α)ηf + (1−H(α))µ∆t, #"$!J&
75@
χ = H(α)1
p′+ (1−H(α))
1
λ, #"$!K&
=+32+ 634,; (; *+, B578 ;:;*,- .C .() 'LM 9).08,- =+,),/ C.) ;7N, .C 287)3*:/ p′t 3; @,5.*,@p 75@ v
t3; @,5.*,@ vF
2Ψ∇. (ǫ(v))−∇p = −2µ∇.(ǫ(ut−∆t)
)
∇.v + χdpdt= 0
#"$OP&
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d/ =+,), d 3; *+, ;972,
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L2(Ω) =q/
∫Ωq2dΩ <∞
H1(Ω) = u ∈ L2(Ω)/ ∇u ∈ L2(Ω)
H10(Ω) = u ∈ H1(Ω)/ u = 0 on ∂Ω)
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(u, v)Ω =
∫
Ω
u(x)v(x)dΩ. 6789 :
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2Ψ (ǫ(v) : ǫ(w))Ω − (p,∇.w)Ω = −2µ(ǫ(ut−∆t) : ǫ(w)
)Ω
− (∇.v, q)Ω −(
χ∆tpt, q
)Ω= −
(χ∆tpt−∆t, q
)Ω
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Qh = qh ∈ C0(Ω)/ qh|T ∈ P 1(T ), ∀T ∈ Th
Vh = vh ∈ (C0(Ω))d/ vh|T ∈ (P 1(T ))d, ∀T ∈ Th
Vh,0 = vh ∈ Vh/ vh|Γ = 0#$%&% P 1(T ) '( )$% *%+),& (-.+% ,/ 0&() 1%2&%% -,345,6'.3( 1%05%1 ,5 )$% ('6-3%7 T 8
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2Ψ (ǫ(vh) : ǫ(wh))Ω − (ph,∇.wh)Ω = −2µ(ǫ(uh
t−∆t) : ǫ(wh))Ω
− (∇.vh, qh)Ω −(
χ∆tpth, qh
)Ω= −
(χ∆tpt−∆th , qh
)Ω
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supvh∈Vh
(∇.vh, qh)Ω‖qh‖0‖vh‖1
≥ β ≥ 0, ?@8AKB
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V b = vb/vb|Ti∈ P 1(Ti) ∩H1
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Vh = Vh ⊕ V b. ?@8AQB
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vh|T =D∑
i
viTNi + vbT bT , 56789:
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∫
T
∂xkNi ∂xl
bTdΩ = 0 ∀(k, l, Ni) ∈ [0, d− 1]2 × P 1(T ). 5678":
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(v, ph) ∈ Vh ×Qh ./&* -*)- ∀(w, qh) ∈ Vh,0 ×Qh4
2Ψ (ǫ(v) : ǫ(w))Ω − (ph,∇.w)Ω = −2µ(ǫ(ut−∆t) : ǫ(w)
)Ω
− (∇.v, qh)Ω −(
χ∆tpth, qh
)Ω= −
(χ∆tpt−∆th , qh
)Ω
5678F:
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2Ψ (ǫ(vh) : ǫ(wh))Ω − (ph,∇.wh)Ω = −2µ(ǫ(uh
t−∆t) : ǫ(wh))Ω
2Ψ(ǫ(vb) : ǫ(wb)
)Ω−
(ph,∇.wb
)Ω= −2µ
(ǫ(ub
t−∆t) : ǫ(wb)
)Ω
− (∇.vh, qh)Ω −(∇.vb, qh
)Ω−
(χ∆tpth, qh
)Ω= −
(χ∆tpt−∆th , qh
)Ω
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0 Abb Atbq
Awq Abq Aqq
vh
vb
ph
=
Bw
Bwb
Bq
, #$%& (
E,-;-=
Aww(vh) = 2Ψ (ǫ(vh) : ǫ(wh))Ω , Abb(vb) = 2Ψ
(ǫ(vb) : ǫ(wb)
)Ω
,
Awq(vh) = − (∇.vh, qh)Ω , Abq(vb) = −
(∇.vb, qh
)Ω
,
Atwq(ph) = − (ph,∇.wh)Ω , At
bq(ph) = −(ph,∇.wb
)Ω
,
Aqq(ph) = −(
χ∆tph, qh
)Ω
, Bw = −2µ(ǫ(uh
t−∆t) : ǫ(wb))Ω
,
Bwb = −2µ(ǫ(ub
t−∆t) : ǫ(wh)
)Ω
, Bq = −(
χ∆tpt−∆th , qh
)Ω
.
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tbq ,
Bq = Bq − Abq.A−1bb .Bwb .
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2η(α) (ε(v) : ε(w))Ω − (p,∇.w)Ω = 0
(q,∇.v)Ω = 0:;<;D>
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2η(α)(ε(vb) : ε(wb)
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;4<*5+7- +*5
∂yvx(x, 0) = ∂yvx(x,H) = 0 L vz(x, y) = 0, ∂zp(x, y) = 0 L p(0, y) = P, p(L, y) = 0.#$%$M'
N% /01 67122<71 87+5)1*/ )2 )* /01 917/)3+, 5)713/)4* y. vx )2 *<,, 4* /01 1?/17*+, ;4<*5+7-
+*5
∂yvx(0, y) = ∂yvx(L, y) = 0 L vz(x, y) = 0, ∂zp(x, y) = 0 L p(x, 0) = P, p(x,H) = 0.#$%$O'
"% /01 67122<71 87+5)1*/ )2 )* /01 ,4*8)/<5)*+, 5)713/)4* z. vx +*5 vy +71 *<,,.
∂xvz(0, y) = ∂xvz(L, y) = 0 L ∂yvz(x, 0) = ∂yvz(x,H) = 0 L ∂zp(x, y) = P/L #$%&P'
D* =0+/ 34*317*2 5)24751715 Q;17 FGH 81*17+/)4*. =1 21*5 /01 71+517 /4 213/)4* N%!%"
)* /01 30+6/17 N =0171 =1 51/+),15 /01 >1/045 <215 /4 81*17+/1 /0)2 /-61 4: FGH2%
!! !"#$%& '( )*+*$% %,%-%+$ ", .,"$*/+0
!"!#!$ %&'()*+, Kr
"# $% $&'( () *)+,-(% (.% ,%/+%&0121(3 )4 516%/%'( !"#$% 70%/ &//&'8%+%'(# &# &
4-'*(1)' )4 )(.%/ ,&/&+%(%/# 0%3)'5 ,)/)#1(39 (.% +1*/)#(/-*(-/% '%%5# () 0% ,/),%/23
:-&'(17%5; <.% =1,2%3># Kr 4-'*(1)' ? @@9 @!A *&' 0% -#%5 () 516%/%'(1&(% 0%($%%'
/%8-2&/ &//&'8%+%'(#9 B)+,2%(%23 C,&(1&223 =&'5)+ DBC=E &//&'8%+%'(#9 &'5 *2-#(%/%5
,)1'( ,&((%/'#; <.1# 4-'*(1)' 1# 5%7'%5 &#F
Kr(r) =A
N2
N∑
k=1
Ik(r)
wk(r), DG;H E
$.%/% Ik(r) 1# (.% '-+0%/ )4 ,)1'(# 4)-'5 $1(.1' & 51#(&'*% r )4 (.% ,)1'( k9 N 1# (.%
()(&2 '-+0%/ )4 ,)1'(# 1' (.% &/%& )4 1'(%/%#( A9 &'5 wk(r) 1# & *)//%*(1)' 4&*()/ (&I1'8
1'() &**)-'( (.% 4&*( (.&( 1( 1# ,)##102% (.&( )'23 & ,&/( )4 (.% )0#%/J&(1)' &/%& πr2 4&22#
$1(.1' (.% &/%& )4 1'(%/%#( A; <.1# *)//%*(1)' 4&*()/ 1# *)+,-(%5 &# (.% ,/),)/(1)' )4
(.% ,%/1+%(%/ )4 (.% *1/*2% *%'(%/%5 1' k $1(. /&51-# r $.1*. 1# 1'#15% (.% &/%& )4 1'(%/%#(
? @GA; Kr(r) 5%#*/10%# *.&/&*(%/1#(1*# )4 ,)1'( ,&((%/'# &( +&'3 2%'8(. #*&2%#; <.% KrK
4-'*(1)' )4 & L)1##)' 51#(/10-(1)' 1# πr2 &'5 5/&$# & 51J151'8 21'% 0%($%%' & /%8-2&/ &'5
& *2-#(%/%5 ,&((%/'; M#(1+&(%# )4 Kr(r) &/% %N,%*(%5 () 0% #+&22%/ (.&' πr2 14 (.% ,)1'(#
4)/+ & /%8-2&/ ,&((%/'9 &'5 () 0% 2&/8%/ (.&' πr2 1' (.% ,/%#%'*% )4 *2-#(%/1'8 ? @HA;
O-/(.%/+)/%9 (.% %N(%'( )4 (.% 5%J1&(1)' )4 Kr(r) 4/)+ πr2 &'5 (.% 2%'8(. #*&2% &( $.1*.
#-*. 5%J1&(1)'# )**-/ 81J% #)+% &551(1)'&2 1'#18.( 1'() & +1*/)#(/-*(-/%; <.% LK,2)(9 &21'%&/1P%5 ,2)( )4 Kr(r) 5%7'%5 &#F L(r) =
√Kr(r)/π9 1# 4/%:-%'(23 -#%5 () #.)$ (.%#%
5%J1&(1)'# &'5 (.% 2%'8(. #*&2%# &( $.1*. (.%3 )**-/; <.% LK,2)( )4 & L)1##)' 51#(/10-(1)'
1# #1+,23 & #(/&18.( 21'% )4 !GK5%8/%% #2),% (./)-8. (.% )/181'; O18-/% G; Q #.)$# (.%
/%#-2(1'8 L(r)K4-'*(1)' &,,21%5 () (.% 70%/ 51#(/10-(1)'# 1' O18-/% R;!!; S( 1# %J15%'( (.&(
&( 2&/8% 51#(&'*%# &22 70%/ 51#(/10-(1)'# &,,/)&*. (.% BC= ,&((%/'; <.1# *)'*2-5%# (.&(
)-/ +%(.)5 *&' 8%'%/&(% /&'5)+ 51#(/10-(1)'# -#&02% 1' (.1# (3,% )4 #(-51%#;
!"! #$%&$'()*)+, -.&/0+'+).1
!"! !$ -./'0121&.3 .4 15) 1623,7)6,) ')6/)28&(&1* 1)3,.6
<.% (/&'#J%/#% ,%/+%&0121(3 1# *)+,-(%5 $1(. & 2%&#( #:-&/% +%(.)5; <.% *)+,-(&(1)'#
&/% ,%/4)/+%5 $1(. (.% 7/#( ($) #%(# )4 0)-'5&/3 *)'51(1)'# 4)/ RT C()I%# %:-&(1)'#
DG;!HE;
O1/#( &##-+1'8 (.&( (.% ,%/+%&0121(3 (%'#)/ K⊥ .&# (.% 4)/+
K⊥ =
(Kxx Kxy
Kxy Kyy
). DG;HRE
"'5 #%*)'59 (.&( k *)+,-(&(1)'# $%/% ,%/4)/+%5 $.1*. 81J%# 〈vi〉 &'5 〈∇pi〉f $1(. i =1, ..., k;
!"! #$%&$'()*)+, -.&#/+'+).0 .1 2)3.%2$%$2 1)($% '%%',3 !"
!" #"
$" %"
!"#$% &'()* +,% L(r) -#./0!1.2 1- 0,% 34%$ 5!20$!4#0!1.2 5%2/$!4%5 !. !"#$% 6'77
#$% &%'() (*+',% -%)$./ 0.1(2()( 21 -212-23214 )$% 5+10)2.1 J
J(Kxx, Kyy, Kxy) =k∑
i=1
(Kxx 〈∂xpi〉f +Kxy 〈∂ypi〉f + ηf 〈vx,i〉
)2
+(Kxy 〈∂xpi〉f +Kyy 〈∂ypi〉f + ηf 〈vy,i〉
)2
. 6"789:
#$% -212-+- .5 ;*76"789: 2( ,%'0$%/ <$%1 )$% =',)2'& /%,2>')2>%( .5 )$% 5+10)2.1 J
!" !"#$%& '( )*+*$% %,%-%+$ ", .,"$*/+0
#$%& '()*(+% %, %&( *('-(./$0$%1 +,-*,2(2%) .'( 3(', $4(4
∂J
∂Kxx
= 0 ,
∂J
∂Kyy
= 0 ,
∂J
∂Kxy
= 0 .
564"!7
8&$) 0(.9) %, %&( :,00,#$2; )1)%(- ,: (<=.%$,2)>
AKxx + CKxy = E ,
BKyy + CKxy = F ,
CKxx + CKyy + (A+B)Kxy = G ,
564"67
#$%&
A =k∑
i=1
(〈∂xpi〉f
)2
,
B =k∑
i=1
(〈∂ypi〉f
)2
,
C =k∑
i=1
〈∂xpi〉f 〈∂ypi〉f ,
E = −ηfk∑
i=1
〈∂xpi〉f 〈vx,i〉 ,
F = −ηfk∑
i=1
〈∂ypi〉f 〈vy,i〉 ,
G = −ηfk∑
i=1
(〈∂ypi〉f 〈vx,i〉+ 〈∂xpi〉f 〈vy,i〉
),
.29 ),?
Kxx =−BCG+ C2F − C2E + EAB + EB2
d,
Kyy =A2F − CAG+ C2E − C2F + ABF
d,
Kxy =BAG− BCE − FAC
d,
!"! #$%&$'()*)+, -.&#/+'+).0 .1 2)3.%2$%$2 1)($% '%%',3 !"
#$%& d = (A+B)(AB − C2)'() Kxy $* +**,-./ %0 1. 2.304 %&.54
Kxx =E
A,
Kyy =F
B.
() Kxy = 0 +5/ Kxx = Kyy4 + *6+7+3 %3+5*8.3*. 9.3-.+1$7$%: K 6+5 1. 60-9,%./ +*
K =E + F
A+B= −ηf
∑ki=1 〈∂xpi〉
f 〈vx,i〉+ 〈∂ypi〉f 〈vy,i〉∑k
i=1
(〈∂xpi〉f
)2
+(〈∂ypi〉f
)2
= −ηf∑k
i=1 〈vi〉 · 〈∇pi〉f∑ki=1 〈∇pi〉f · 〈∇pi〉f
. ;<'==>
!"! !# $%&'()*)+%, %- .%,/+)(0+,*. '12&1*3+.+)4
?&. 705@$%,/$5+7 9.3-.+1$7$%: $* 01%+$5./ 1: 60-9,%$5@ %&. *6+7+3 vz *07,%$05 0) %&.
3./,6./ A%0B.* .C,+%$05
ηf
(∂2vz∂y2
+∂2vz∂x2
)=
∂p
∂z05 Ωf , ;<'=">
+5/
K‖ = −ηf〈vz,i〉∂zp
. ;<'=D>
!"!# $%&'()*+, -'.%,/.
!"!5!6 781 9:;<= =+>1
?&. 3+5/0-5.** 0) %&. @.5.3+%./ E1.3 /$*%3$1,%$05* 7.+/* %0 + *6+%%.3 $5 %&. 60-9,%./
9.3-.+1$7$%:' F.56.4 +5 +8.3+@. 9.3-.+1$7$%: *&0,7/ 1. 6+76,7+%./ +705@ #$%& + /+%+ 8+3$G
+%$05 $5 03/.3 %0 1. *%+%$*%$6+77: 3.93.*.5%+%$8.' (5 %&$* #03B4 HI 3+5/0- -$630*%3,6%,3.*
#.3. @.5.3+%./ )03 .+6& 67+** 0) E1.3 /$*%3$1,%$05* 6&+3+6%.3$2./ 1: %&. *+-. φ +5/ σ ;*..
*.6%$05 H'!'J>' ?&. +8.3+@. 9.3-.+1$7$%: +5/ $%* *%+5/+3/ /.8$+%$054 σp4 +3. @$8.5 1:K
〈K〉 = 1
20
20∑
i=1
Ki, ;<'=L>
σp =1
19
√√√√20∑
i=1
(Ki − 〈K〉)2. ;<'"I>
!" !"#$%& '( )*+*$% %,%-%+$ ", .,"$*/+0
#$%&$'&%( )*& +&%,&-./0/)1 $2 3.&% 4/5)%/.6)/$75 /5 7$) $701 - 2678)/$7 $2 ,/8%$5)%689
)6%-0 +-%-,&)&%5 .6) -05$ - 2678)/$7 $2 )*& :;<=5 5/>& $% -05$ )*& 76,.&% $2 3.&%5 Nf
/7 )*& 4$,-/7? @7 &A)%&,&01 0-%B& 8$,+6)-)/$7-0 4$,-/7 /5 56%&01 %&+%&5&7)-)/'& $2 )*&
3.&% -%%-1 .6) )*/5 C/00 8-65& - 4%-5)/8 /78%&-5& /7 )*& 8$,+6)-)/$7-0 8$5) C*/8* ,-D&5
)*/5 )1+& $2 5)64/&5 E6-5/9/,+$55/.0&? @ 5/,/0-% -++%$-8* 2$% 4&)&%,/7/7B )*& :;<=5 5/>&
$2 )*& ;$%$7$F *$7&18$,.5 /5 +&%2$%,&4 2$% 374/7B )*& 5/>& $2 )*& 3.&% -%%-15= :;<? G7
)*/5 8-5& - 5)641 $2 )*& 5/>& &H&8) $7 〈K〉( 〈K‖〉 -74 φ C-5 +&%2$%,&4 -74 8$,+6)-)/$75
C&%& 8-%%/&4 $6) $7 &0&,&7)-%1 '$06,&5 $2 4/H&%&7) 5/>& &A)%-8)&4 2%$, ,/8%$5)%68)6%&5
B&7&%-)&4 C/)* σ = 0.06R -74 φ = 0.5 -5 /0065)%-)&4 /7 I/B6%& J? "?
!"#$% &'()* +,%-%./0$1 23,#-%4 %5/$06/%7 8$3- 0 10× 10 -!6$34/$#6/#$% !. 3$7%$ /37%/%$-!.% /9% 4!:% 38 /9% ;<+
K*& %&560)5 2$% )*& +&%,&-./0/)1 '-06&5 -%& 5*$C7 /7 I/B6%& J? L C*&%& )*&1 -%& -4/,&79
5/$7-0/5&4 -74 /7 I/B6%& J?MN 2$% )*& +$%$5/)1?
!"! #$%&$'()*)+, -.&#/+'+).0 .1 2)3.%2$%$2 1)($% '%%',3 !"
!"
#"
!"#$% &'()* +,% -!.% %/%01 23 〈K〉/R2' +,% 45%$ 6!-1$!5#1!23- 7%$% "%3%$81%6 7!1,
σ = 0.06R 836 φ = 0.51' +,% %$$2$ 58$- $%9$%-%31 1,% -18368$6 6%:!81!23- ±σp
#$%&'%(')*+, (-. */,.-0.1 )+ '2. 0($%., *3 〈K〉/R24 〈K‖〉/R2
(+1 〈φ〉 52.+ ( -.$(')0.$6
,7($$ +%7/.- *3 8/.-, ), &*+,)1.-.1 '2.+ '2. 0($%., -.(&2 ( 9$('.(% (3'.- Nf > 385: ;$,*4
!" !"#$%& '( )*+*$% %,%-%+$ ", .,"$*/+0
!"#$% &'()* +,% -!.% %/%01 23 1,% 42$2-!15
#$%&' Nf > 555 %(& #))*+,#%&- )%#.-#'- -&/,#%,*.) #'& '&-0+&-1 2) # +*.+30),*.4 #. 5678)
),9& *$ Nf > 555 ,) 0)&- ,. %(& '&:#,.-&' *$ %(,) +(#;%&'1
!"!#!$ %&'()'*+, *)'*-)(+-&
<& +*:;0%&- %(& /#30&) *$ %(& +*:;*.&.%) *$ %(& %'#.)/&')& ;&':&#=,3,%> %&.)*' K⊥4Kxx4 Kyy #.-Kxy4 ,. *'-&' %* )&& ,$ %(& )+#3#' %'#.)/&')& ;&':&#=,3,%> K +#. =& +*:;0%&-1
?. @,A0'& !1B %(& #/&'#A&- +*:;*.&.%) #'& +*:;0%&- #% -,C&'&.% ;*'*),%> 3&/&3) #.-
;3*%%&- #A#,.)% σ/R1 D(& /#30&) #% -,C&'&.% ;*'*),%,&) #'& ,330)%'#%&- ,. -,C&'&.% EA0'&)
$*' %(& )#F& *$ +3#',%>1
@,A0'& !1B )(*G) %(#% Kxx ,) &H0#3 %* Kyy #.- Kxy ,) .&A3,A,=3& *. #33 ;*'*),%> 3&/&3)
#.- #% #33 /#30&) *$ σ1 D(,) :&#.) %(#% %(& %'#.)/&')& ;&':&#=,3,%> ,) ,.-&;&.-&.% *$
%(& I*G -,'&+%,*. #.- %(#% %(& E=&' #''#>) #'& ,)*%'*;,+1 ?. %(,) +#)&4 # )+#3#' %'#.)/&')&
;&':&#=,3,%> ,) +*:;0%&- => 7H1J!1KKL1
!"!#!" ./- -0-,( '1 σ '2 K@,A0'& !1BB )(*G) %G* '&;'&)&.%#%,/& I*G ;#%() +*:;0%&- *. 567) #% φ = 0.7 #.- @,AM
0'& !1BN ,330)%'#%&) %(& /&3*+,%> +*:;0%&- $*' %(& =*0.-#'> +*.-,%,*.) *$ %>;& N J7H1J!1K"LL
*. %(& )#:& ;'&/,*0) 567)1 D(& -,C&'&.+& =&%G&&. %(& %G* :,+'*)%'0+%0'&) 3,&) ,. %(&
+(*,+& *$ σ Jσ = 0.03R ,. @,A0'&) !1BBJ#L #.- !1BNJ#L #.- σ = 0.4R ,. @,A0'&) !1BBJ=L
#.- !1BNJ=LL1
<& .*%,+& $'*: %(&)& %G* EA0'&) %(#% %(& :,+'*)%'0+%0'& (#) # A'&#% &C&+% *. %(&
I*G -,)%',=0%,*. ,. %(& ,.%&')%,%,#3 );#+&1 ?. G(#% +*.+&'.) %'#.)/&')& I*G) J=*0.-#'>
!"! #$%&$'()*)+, -.&#/+'+).0 .1 2)3.%2$%$2 1)($% '%%',3 !
!" #"
$"
!"#$% &'()* +,% -.%$-"%/ 0%$1%-2!3!45 671078%849 -4 /!:%$%84 .-3#%9 7; σ* <-=φ = 0.51> <2= φ = 0.7 -8/ <6= φ = 0.9
"#$%&'&#$( ) *$% +&,-./ !0))12 * 3*.,/. %/,.// #4 3#"*3 5/'/.#,/$/&'6 "*-(/% 76 *$ &$"./*(/
#4 σ ./(-3'( &$ * 8&%/. .*$,/ #4 9#8 :*'5(0 ;$ '5&( "*(/2 * 4/8 <*=#. 9#8 :*'5( />&('
85&3/ &$ '5/ "*(/ #4 σ = 0.03R * 3*.,/ $-<7/. #4 (<*33 :*'5( />&(' ,&?&$, * -$&4#.< 9#8
'5.#-,5 '5/ @7/. *..*60 ;$ 85*' "#$"/.$( 3#$,&'-%&$*3 9#8( A7#-$%*.6 "#$%&'&#$( B *$%
+&,-./ !0)B12 * 3*.,/. %/,.// #4 3#"*3 5/'/.#,/$/&'6 ./(-3'( &$ '5/ 4#.<*'&#$ #4 :#"C/'( #4
5&,5D(://% 9-&% *,*&$(' * <#./ -$&4#.< (://% 4#. (<*33 ?*3-/( #4 σ0 +.#< '5/(/ @,-./(2 &'
&( /?&%/$' '5*' @7/.( *' '5/ (*</ :#.#(&'6 "*$ />5&7&' %.*('&"*336 %&E/./$' :*''/.$( #4 9-&%
9#8(0 ;' 4#33#8( '5*' '5/ -(/ #4 :#.#(&'6 *3#$/ "*$$#' %/@$/ :.#:/.36 '5/&. :/.</*7&3&'60
;$ +&,-./ !0)F2 '5/ $-</.&"*336 "#<:-'/% 〈K〉 *$% 〈K‖〉 *' /*"5 :#.#(&'6 3/?/3 *./ :3#''/%
*,*&$(' σ/R0
;$ +&,-./( !0)FA*1 *$% !0)FA712 '5/ /$%&$, :#&$' &( '5/ :/.</*7&3&'6 #4 '5/ 5/>*,#$*3 *..*6
A@33/% ('*.12 4#. 85&"5 '5/ &$'/.D@7/. (:*"&$, &( #$36 3&$C/% '# '5/ :#.#(&'6 GHIJ0 +.#< '5/
:3#' &$ '5/(/ @,-./(2 '5/ /E/"' #4 σ/R #$ 〈K〉/R2*$% 〈K‖〉/R2
&( /?&%/$'0 K$ &$"./*(&$,
!" !"#$%& '( )*+*$% %,%-%+$ ", .,"$*/+0
!"
#"
!"#$% &'(() *+, -./01 2+$ 3+#45.$6 7+45!/!+41 ( .7$+11 #4!5!$%7/!+4.* 83%$ .$$.6) 9.:
φ = 0.7 ; σ = 0.03R .45 93: φ = 0.7 ; σ = 0.4R
σ #$ %&'# (#)*+, -$#( % .+*-#$( %$$%/ 0# %+ %$$%/ '1#2*+, 1*,13$ 43,$33' #- 4*'#$43$
$3'.&0' *+ %+ *+5$3%'3 #- 6#01 73$(3%6*&*0/ )%&.3'8 013 0$%+')3$'3 %+4 013 &#+,*0.4*+%&9
:13 $3'.&0' #- 〈K‖〉 %$3 *+ %,$33(3+0 2*01 01#'3 #- ;13+ %+4 <%7%01%+%'*#. =">? 21#
-#.+4 %&'# 01%0 %+ *+5$3%'3 #- 013 (*5$#'0$.50.$3 1303$#,3+3*0/ #$ % 435$3%'3 *+ 013 (3%+
!"! #$%&$'()*)+, -.&#/+'+).0 .1 2)3.%2$%$2 1)($% '%%',3 !"
!"
#"
!"#$% &'()* +,-.,#$/ ,0 1#!2 3%4,5!.6 0,$ 4,-"!.#2!-74 1,8/ 75$,// #-!2!$%5.!,-74 9:%$
7$$76* ;7< φ = 0.7 = σ = 0.03R 7-2 ;:< φ = 0.7 = σ = 0.4R
#$%&$'( )#($&*+,$& '-%.)#/0 δ10 $#1%#.$' 〈K‖〉2 31)' )' 45$ (6 (1$ 76&8%()6# 67 9%&/$ :6;
-%(1'2 <# (1$ 6(1$& 1%#40 65& &$'59(' .6#.$&#)#/ 〈K〉 %&$ 6--6')($ (6 (16'$ 67 =1$# %#4
>%-%(1%#%')65 ?@A $B.$-( 76& φ > 0.72 31$C $B-9%)# (1%( (1$ 4$.&$%'$ )# 〈K〉 )# (1$ &%#/$
0.45 < φ < 0.7 .68$' 7&68 (1$ -&$'$#.$ 67 #%&&6; /%-' ;1).1 &$45.$ (1$ -$&8$%,)9)(C
%#4 (1%( (1$ )#.&$%'$ 67 〈K〉 )# (1$ &%#/$ φ > 0.7 .68$' 7&68 (1$ 76&8%()6# 67 :6; -%(1'
;16'$ ')D$ )' .68-%&%,9$ (6 (1$ ')D$ 67 +,$& %//&$/%($'2 E# 65& .%'$0 ;$ $B-9%)# (1)'
)#.&$%'$ ,C (1$ ')8-9$ 7%.( (1%( 9%&/$ /%-' $#1%#.$ 〈K〉 %' (1$C $#1%#.$ 〈K‖〉2
!" !"#$%& '( )*+*$% %,%-%+$ ", .,"$*/+0
!"
#"
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60$;2<!=%4 23%$2"%4 /$2653%$5% >2? 264 <06"!/#4!62< >9? :%$;%29!<!/@' A<50 5,0B6 2$%
:%$;%29!<!/@ 32<#%5 10$ ,%C2"062< 2$$2@5 DEFG
!" #$%&'()*$%
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3")#'4 % 5*"'( (-(,(*' %*%-6#"# !%# 7((* &)(#(*'(+ '. #",8-%'( '!( 7"%9"%- $.,&)(##".*
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