0.+1$10,-.*%#%2#34.&1#.5#$%&%'%()#%2#paulino.ce.gatech.edu/conferences/presentations/09...0 0.002...
37
Transcript of 0.+1$10,-.*%#%2#34.&1#.5#$%&%'%()#%2#paulino.ce.gatech.edu/conferences/presentations/09...0 0.002...
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! !"#$%&%'%()#%&*+,-.*%"/#&.0.+1$10,-.*%"#%2#34.&1#."5#$%&%'%()#%2#
$41#513,("#4.3#611"#$0.5,*%".'')#7.00,15#%8$#%"#8",2%0+#(0,53##
! 9%":1"*%".'#.&&0%.7413#831#8",2%0+#+13413#7%"3,3*"(#%2#;.(0."(,."<
$)&1#=",$1#1'1+1"$3#>1?(?#',"1.0#@8.53A#$%#3,+&',2)#5%+.,"#5,3701*-.*%"#
."5#$41#.".')3,3#0%8*"1#
! B3#.#0138'$#%2#$4,3#74%,71/#"8+10,7.'#.0*2.7$3#3874#.3#$41#C1''<D"%C"#
7417D106%.05#&.$4%'%()#."5#%"1<"%51#7%""17*%"3#+.)#.&&1.0#
E#
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! !"#$%&%'%()#%&*+,-.*%"/#&.0.+1$10,-.*%"#%2#34.&1#."5#$%&%'%()#%2#
$41#513,("#4.3#611"#$0.5,*%".'')#7.00,15#%8$#%"#8",2%0+#(0,53##
!"#$%#&'()&*+,
-.#/.(*#,"0.1#2+,
F#
! 9%":1"*%".'#.&&0%.7413#831#8",2%0+#+13413#7%"3,3*"(#%2#;.(0."(,."<
$)&1#=",$1#1'1+1"$3#>1?(?#',"1.0#@8.53A#$%#3,+&',2)#5%+.,"#5,3701*-.*%"#
."5#$41#.".')3,3#0%8*"1#
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'G#
! B#&0%6'1+#'133#%H1"#"%$15#,3#$4.$#$41#7%"3$0.,"15#(1%+1$0)#%2#3$087$8015#
(0,53#7."#7.831#6,.3#,"#$41#%0,1"$.*%"#%2#+1+6103/#'1.5,"(#$%#+134<
51&1"51"$#386<%&*+.'#513,("3#
9=,=C*'?'D*C=E0*'FGBBGH'8"+#=)':3)0"I5'J##$':3E0'4.%K%'I#
! B#&0%6'1+#'133#%H1"#"%$15#,3#$4.$#$41#7%"3$0.,"15#(1%+1$0)#%2#3$087$8015#
(0,53#7."#7.831#6,.3#,"#$41#%0,1"$.*%"#%2#+1+6103/#'1.5,"(#$%#+134<
51&1"51"$#386<%&*+.'#513,("3#
1.0
V = 0.3|Ω|
F4.0
5.5
3&('4#5,*#6.07(.+, 80$"#44/9:;#,2(4<7(.+,
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'J#
! !2#$41#(1%+1$0,7#.K0,68$13#%2#$41#+134#.01#$%%#013$0,7*"(/#710$.,"#
74.0.7$10,3*7#&.K10"3#%2#$41#%&*+.'#3%'8*%"#+.)#61#1L7'8515#20%+#$41#
=".'#513,("#
=(4<7(.,')2#*,(.,),29&<$9<&#*,5#2",>09",?@?A,BC,#4#5#.92,
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'M#
! !"#$4,3#C%0D/#C1#1L.+,"1#$41#831#%2#,001(8'.0#+13413#7%"3,3*"(#%2#7%":1L#
&%')(%"3#,"#$%&%'%()#%&*+,-.*%"#$%#.550133#$41#.6%:1+1"*%"15#,33813##
=(4<7(.,')2#*,(.,),D(&(.(0,5#2",>09",EFFFF,#4#5#.92,
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'N#
! !"#.55,*%"#$%#&%33133,"(#4,(410#51(011#%2#(1%+1$0,7#,3%$0%&)/#$4131#
+13413#.''%C#2%0#(01.$10#O1L,6,',$)#,"#5,3701*-,"(#7%+&'1L#5%+.,"3#C,$4%8$#
38P10,"(#20%+#"8+10,7.'#,"3$.6,',*13?#
=(4<7(.,')2#*,(.,),D(&(.(0,5#2",=(4<7(.,')2#*,(.,),BC,5#2",
! Q%*:.*%"#
! R%')(%".'#+134#(1"10.*%"#
! S,",$1#1'1+1"$#3741+1#
! T8+10,7.'#0138'$3#
! 9%"7'85,"(#01+.0D3#
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'U#
GA'L*#?':"0$3?'M"$-.I3KF@AANHO'8"+#=)3K6J*I3I'8*P*$'-.I'Q.RK-5)K=E)=K3'4.%*.33K*.%'
#,?1?/#1.74#71''#7%"3,3$3#%2#&%,"$3#7'%310#$%#$41#7%0013&%"5,"(#3115#$4."#
.")#%$410#&%,"$#,"#$41#31$#
! ;1$#########################51"%$1#.#31$#%2#&%,"$3#,"#5%+.,"#####?#V41#W%0%"%,#5,.(0.+#
%2######7%0013&%"53#$%#.#&.0**%","(#%2#$41#5%+.,"#,"$%#71''3X###
P = pini=1 Ω
P
Vi =
∀j,j =i
x ∈ Ω : δ(x, pi) ≤ δ(x, pj) , i = 1, · · · , n
! B#&%')(%".'#5,3701*-.*%"#%2######7."#61#(1"10.$15#20%+#######6)#,"7'85,"(#
.55,*%".'#&%,"$3#3874#$4.$#$41#0138'*"(#W%0%"%,#5,.(0.+#,"7%0&%0.$13#."#
.&&0%L,+.*%"#$%#$41#6%8"5.0)##
Ω
∂Ω
P
GG'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
#Y$1&#ZX#V41#,"$10,%0#%2######,3#&%&8'.$15#C,$4#.#513,015#"8+610#%2#(1"10.*"(#
31153?#[1#51"%$1#$4,3#&%,"$#31$#6)#
ΩPint
Pint
Ω
G@'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
#Y$1&#E/FX#V41#,"$10,%0#&%,"$3#.01#01O17$15#.6%8$#$41#15(13#%2#$41#5%+.,"#
."5#W%0%"%,#5,.(0.+#%2##
Paux
P = Pint ∪ Paux
G>'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
#Y$1&#GX#B#&%')(%".'#5,3701*-.*%"#%2#$41#5%+.,"#,3#(,:1"#6)#$41#71''3#
.33%7,.$15#$%##Pint
V(Pint)
GS'T=?'2-U3K?'V=.,U=K%3KO'FGBBBH'9QJ:'
! B#W%0%"%,#$1331''.*%"#,3#71"$0%,5.'#,2#1.74#(1"10.*"(#&%,"$#7%,"7,513#C,$4#
$41#71"$0%,5#%2#$41#7%0013&%"5,"(#W%0%"%,#71''X#
pi = pi ∀i = 1, · · · , n where pi :=
Vi
xµ(x)dx
Viµ(x)dx
! B'$10".*:1')/#9WV#7."#61#$4%8(4$#%2#.3#+,",+,-103#%2#1"10()X#
! ;'%)5\3#.'(%0,$4+#7%"3$087$3#9WV3#6)#01&'.7,"(#$41#31153#6)#$41#71"$0%,53#
%2#$41#7%0013&%"5,"(#71''3#
E (P ) =n
i=1
Vi
|x− pi|2µ(x)dx
GN'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! ]."5%+#31153#>%\3A#."5#71"$0%,53#%2#$41#71''3#>L\3AX#
GW'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! BH10#%"1#,$10.*%"#>,?1?/#5,.(0.+#(1"10.$15#6)#71"$0%,53A#
GX'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! BH10#$1"#,$10.*%"3/#$41#5,.(0.+#,3#"1.0')#71"$0%,5.'#
G;'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! V41#;'%)5\3#.'(%0,$4+#,3#'%7.'')#7%":10(1"$#,"#1"10()#
0 2 4 6 8 10
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
Number of Lloyd’s algorihm iterations
En
erg
y
GB'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! ]."5%+#
0 0.002 0.004 0.006 0.008 0.01 0.0120
0.05
0.1
0.15
0.2
0.25
0.3
element area
rela
tive
fre
qu
en
cy
µ = 3.7× 10−3
σ = 1.6× 10−3
20 40 60 80 100 120 140 160 1800
0.02
0.04
0.06
0.08
0.1
0.12
0.14
angle
rela
tive fre
quency
σ = 26.5µ = 120
1'1+1"$#.01.#
,"$10,%0#."('13#
7?%?:#%2#15(1#'1"($4#
0 0.002 0.004 0.006 0.008 0.01 0.0120
0.05
0.1
0.15
0.2
0.25
0.3
element area
rela
tive
fre
qu
en
cy
µ = 3.7× 10−3
σ = 0.6× 10−3
20 40 60 80 100 120 140 160 1800
0.02
0.04
0.06
0.08
0.1
0.12
0.14
angle
rela
tive
fre
qu
en
cy
µ = 120
σ = 16.8
@A'M"$-.I3K?'9-*)"'FGBB;HO'4.%'2K-E)':3E0'
! ^8.3,<0."5%+#1'1+1"$#.01.#
7?%?:#%2#15(1#'1"($4#
,"$10,%0#."('13#
0 0.002 0.004 0.006 0.008 0.01 0.0120
0.05
0.1
0.15
0.2
0.25
0.3
element area
rela
tive
fre
qu
en
cy
µ = 3.7× 10−3
σ = 0.2× 10−3
20 40 60 80 100 120 140 160 1800
0.02
0.04
0.06
0.08
0.1
0.12
0.14
angle
rela
tive
fre
qu
en
cy
σ = 7.7µ = 120
@G'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! 9WV#1'1+1"$#.01.#
7?%?:#%2#15(1#'1"($4#
,"$10,%0#."('13#
@@'13K55".?'9)K-.%'F@AASH'9QJ:'Y3P'
! R'.71+1"$#."5#01O17*%"#%2#31153#7."#61#7.00,15#%8$#(1"10,7.'')#83,"(#
."#,+&',7,$#51370,&*%"#5%+.,"#(1%+1$0)##
! !"#&.0*78'.0/#$41#_3,("15#5,3$."71`#28"7*%"/#########/#7."#61#8315#d(x)
x = x− 2d(x)n
x
of d(x)
n =∇d(x)|∇d(x)|
domain boundary
level set|d(x)|
@>'
! Y%+1#3,+&'1#(1%+1$0,13#."5#31$#%&10.*%"3X#
Circle: d(x) = |x− xo|− r
Half-plane x1 ≤ a: d(x) = x1 − a
Union: dA∪B(x) = min (dA(x), dB(x))
Difference: dA\B(x) = max (dA(x),−dB(x))
Intersection: dA∩B(x) = max (dA(x), dB(x))
13K55".?'9)K-.%'F@AASH'9QJ:'Y3P'
@S'
! 9%"3,510#$41#2%''%C,"(#5%+.,"X#
y = cos(x)
y = 5
2x
5π
4
− 5
Domain boundary is the zero level set of f(x)
x2 = cos(x1)
x2 = 5
2x
5π
4
− 5
13K55".?'9)K-.%'F@AASH'9QJ:'Y3P'
@N'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! V41#5,3$."71#28"7*%"#,3#%6$.,"15#6)#3%':,"(#"%"',"1.0#3)3$1+#."5#6.3,7#
31$#%&10.*%"3X#
S%0#1.74#6%8"5.0)X##d(x) = |x− xp|× sgn(f(x))
@W'!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'
! V41#=".'#9WV#+134X#
V%#7.&$801#$41#7%0"103/#31153#.01#01O17$15#.6%8$#6%$4#6%8"5.0,13#
9=C=+-K?'!-U-KK-3*F@AASH'Q.)<'Z=+3K:3)0"I5'4.%''EM#
! V41#=",$1#1'1+1"$#3741+1#2%0#&%')(%".'#1'1+1"$3#,3#%6$.,"15#6.315#%"#
"%*%"3#%2#".$80.'#"1,(46%03#."5#".$80.'#"1,(46%0#28"7*%"3#
hi
siqip
! S%0#.#7%":1L#&%')(%"/#$41#;.&'.71#,"$10&%'."$#.$#."#,"$10,%0#&%,"$#,3#(,:1"#6)X#
φi(x) =wi(x)ni=1(x)
where wi(x) =si(x)hi(x)
V)&,7.'#34.&1#28"7*%"X#
EN#
! a)#7%"3$087*%"/#$41#;.&'.71#28"7*%"3#.01#"%"<"1(.*:1/#6%8"515#."5#
3.*32)#&.0**%"#%2#8",$)X#
0 ≤ φi(x) ≤ 1,n
i=1
φi(x) = 1
! #V4131#28"7*%"3#.01#',"1.0')#7%+&'1$1/#$483#7%":10(1"71#%2#$41#b.'10D,"#
+1$4%5#2%0#317%"5#%0510#&.0*.'#5,P101"*.'#1@8.*%"3#,3#1"38015X##n
i=1
xiφi(x) = x
! Q%01%:10/#$41)#.01#',"1.0#%"#$41#6%8"5.0)#%2#$41#1'1+1"$#."5#3.*32)#$41#
c0%"17D10<51'$.#&0%&10$)X#
φi(xj) = δij
9=C=+-K?'!-U-KK-3*F@AASH'Q.)<'Z=+3K:3)0"I5'4.%''
EU#
! V41#.d"1#+.&&,"(#51="15#6)#$41#34.&1#28"7*%"3#7."#01&0131"$#.")#
7%":1L#&%')(%"/#."5#$483#$41#W%0%"%,#+13413#7."#61#38&&%0$15#6)#$41#31$#
%2#012101"71#01(8'.0####<(%"3?##
ξ2
ξ1
φi(ξ)
x1
x2
Parent domain Physical domain
n
! V41#0138'*"(#=",$1#1'1+1"$3#2%0##############."5##############.01#,51"*7.'#$%#&%&8'.0#
7%"3$."$#3$0.,"#$0,."('13#."5#6,',"1.0#@8.50,'.$10.'/#013&17*:1')?##
n = 3 n = 4
9=C=+-K?'!-U-KK-3*F@AASH'Q.)<'Z=+3K:3)0"I5'4.%''
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'Fe#
! 9%"3,510#9%%D\3#&0%6'1+#7%"3,3*"(#%2#.#$.&1015#
3C1&$#&."1'#386f17$15#$%#8",2%0+#341.0#'%.5,"(X#
48
44
16 p = 1
E = 1ν = 1/3
C
! ^8."*$)#%2#,"$1013$#,3#$41#*3O17*%"#.$#+,5<
51&$4#%2#$41#&."1'#*&#>&%,"$######A#C
! V41#0138'$3#83,"(#@8.50,'.$10.'#>&0%(0133,:1#01="1+1"$A#."5#&%')(%".'#
>,"51&1"51"$#01="1+1"$A#5,3701*-.*%"3#.01#7%+&.015#
G#1'1+1"$3# ZJ#1'1+1"$3# JG#1'1+1"$3# ZJ#1'1+1"$3#
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'FZ#
! R%')(%".'#1'1+1"$3#.01#"%$#.3#3*P#.3#$41#@8.50,'.$10.'#1'1+1"$3#."5#
&0%5871#+%01#.7780.$1#0138'$3/#13&17,.'')#C,$4#7%.0310#+13413#
101
102
103
8
10
12
14
16
18
20
22
24
26
Number of Elements
Deflection a
t P
oin
t C
4!gons (Q4)n!gonsReference
'[3/*.5C*?'\0"=?'Y",P-.&'FGBBSH'Q.)<':3E09E*'FE#
! V41#0138'$3#.01#&0131"$15#2%0#+,",+8+#7%+&',."71#&0%6'1+#6.315#%"#Y!QR#
2%0+8'.*%"#."5#QQB#.3#$41#3%':10##
V = 0.5|Ω|1.0
3.0
Fg13,("#5%+.,"X#
Q,741''<$)&1#3%'8*%"X#
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'FF#
! ]138'$3#6.315#%"#5,P101"$#'1:1'3#%2#+134#01(8'.0,$)#
]."5%+#+134#S,".'#7%+&',."71#h#ZNN?ZU###
^8.3,<0."5%+#+134#S,".'#7%+&',."71#h#ZNN?EI#
9WV#+134#S,".'#7%+&',."71#h#ZNM?Fe#
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'FG#
D(&(.(0,5#2"+,G?,5#2"+,
ZEee#1'1+1"$3/##FMJZ#"%513#
EMee#1'1+1"$3/##NFGZ#"%513#
ZFFJ#1'1+1"$3/##EJMG#"%513#
ENJU#1'1+1"$3/#IMGe#"%513#
GNee#1'1+1"$3/##ZGMEZ#"%513# IeJE#1'1+1"$3/##ZeZEJ#"%513#
V=35)?'1K3P"5)?'M3$&)5E0C"F@AASHO'Q.)<'Z=+3K:3)0"I5'4.%''FI#
! ]138'$3#83,"(#&0%f17*%"#+1$4%5X#
ZEee#1'1+1"$3#S,".'#7%+&',."71#h#EJe?IJ#
EMee#1'1+1"$3#S,".'#7%+&',."71#h#EIM?GF#
GNee#1'1+1"$3#S,".'#7%+&',."71#h#EIM?UU###
!"#"$"%&'(#)*+*,-)*".'/*)0'1"$&%".-$'2*.*)3'4$3+3.)5'6'789:(6;'6'<=.3'>?'@AAB'FJ#
! Q,741''#7."*'1:10#61.+#C,$4#7,078'.0#38&&%0$#
T8+610#%2#"%513##h#ZUUMF#T8+610#%2#1'1+1"$3#h#Zeeee#
D(&(.(0,5#2"+, BC,5#2"+,
T8+610#%2#"%513##h#ZMGGZ#T8+610#%2#1'1+1"$3#h#NING#
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