Smoothing topology optimization results using pre-built lookup tables

Abstract

Topology optimization techniques are typically performed on a design domain discretized with finite element meshes to generate efficient and innovative structural designs. The optimized structural topologies usually exhibit zig-zag boundaries formed from straight element edges. Existing techniques to obtain smooth structural topologies are limited. Most methods are computationally expensive, as they are performed iteratively with topology optimization. Other methods, such as post-processing methods, are applied after topology optimization, but they cannot guarantee to obtain equivalent structural designs, as the volume and geometric features may be changed. This study presents a new method that uses pre-built lookup tables to transform the shape of boundary elements obtained from topology optimization to create smoothed structural topologies. The new method is developed based on the combination of the bi-directional evolutionary structural optimization (BESO) technique and marching geometries to determine structural topologies and lookup tables, respectively. An additional step is used to ensure that the generated result meets a target volume. A variety of 2D and 3D examples are presented to demonstrate the effectiveness of the new method. This research shows that the new method is highly efficient, as it can be directly added to the last step of topology optimization with a low computational cost, and the volume and geometric features can be preserved in smoothed topologies. Finite element models are also created for original and smoothed structural topologies to show that the structural stiffness can be significantly enhanced after smoothing.

Full text

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Se(i)=Sk
e(i)+Sk−1
e(i)
2U9V
h?2 i`;2i pQHmK2 7Q` i?2 M2ti Bi2`iBQM- Vk+1-
/2T2M/b QM i?2 +m``2Mi pQHmK2- Vk- M/  bT2+B}2/
2pQHmiBQM`v `iBQ- ert- r?2`2
Vk+1 =Vk(1 ±ert)U8V
LQi2 i?i  i?`2b?QH/ b2MbBiBpBiv MmK#2`- Sth - Bb
+H+mHi2/ #b2/ QM Vk+1 mbBM; i?2 K2i?Q/ /2@
b+`B#2/ BM (je)X 1H2K2Mib i?i TQbb2bb i?2 b2MbBiBp@
Biv MmK#2` ?B;?2` M/ HQr2` i?M Sth `2 i?2M T`2@
b2`p2/ M/ `2KQp2/- `2bT2+iBp2HvX h?2 QTiBKBxiBQM
T`Q+2bb Bb `2T2i2/ mMiBH `2+?BM; i?2 i`;2i pQHmK2
M/ biBb7vBM; i?2 +QMp2`;2M+2 +`Bi2`B (Re)X
kXkX aKQQi?BM; T`2T`iBQM
PM+2 i?2 2H2K2Mi@#b2/ bi`m+im`H iQTQHQ;v Bb
Q#iBM2/ mbBM; i?2 K2i?Q/ /2b+`B#2/ BM a2+iBQM
kXR- i?2 MQ/H b2MbBiBpBiv }2H/- Sn- Bb +H+mHi2/ 7Q`
bKQQi?BM; T`2T`iBQMX h?Bb +M #2 +?B2p2/ #v
2KTHQvBM;  }Hi2` QM 2H2K2Mib (jd- j3)- b b?QrM BM
6B;m`2 R- r?2`2 i?2 }Hi2` +Qp2`b  +B`+mH` /QKBM
ΩprBi?  +2Mi2` Q7 MQ/2 pX h?2 }Hi2`BM; b+?2K2 Bb
;Bp2M b
Sn(p)=Nn
q=1(rmin −rpq )ˆ
Se(q)
Nn
q=1(rmin −rpq )UeV
r?2`2 Sn(p)Bb i?2 b2MbBiBpBiv MmK#2` Q7 MQ/2 pX
NnBb i?2 MmK#2` Q7 2H2K2Mi +2Mi2`b +Qp2`2/ #v
i?2 }Hi2`X rmin Bb i?2 }Hi2` `/Bmb- M/ rpq Bb i?2
/BbiM+2 #2ir22M i?2 +2Mi2` Q7 2H2K2Mi qM/ MQ/2
pX Ai Bb MQi2/ i?i }Hi2`BM; HH 2H2K2Mib ;Bp2b  MQ/H
b2MbBiBpBiv }2H/- SnX
6BMHHv- M BbQ@pHm2- Siso- Bb bT2+B}2/ #2ir22M
i?2 KtBKmK M/ KBMBKmK Sn- i?i Bb- rBi?BM i?2
MQ/H b2MbBiBpBiv }2H/- 7Q` 7m`i?2` bKQQi?BM; T`Q+2@
/m`2b b /2b+`B#2/ #2HQrX
Mesh elements
Filter:
Element centers
Selected centers
p
:
p
min
r
6B;m`2 R, P#iBMBM; i?2 MQ/H b2MbBiBpBiv }2H/ #v 2KTHQvBM;
 }Hi2` QM K2b? 2H2K2MibX h?Bb };m`2 b?Qrb i?i i?2 }Hi2`
+Qp2`b  +B`+mH` /QKBM ΩprBi?  +2Mi2` Q7 MQ/2 p- r?B+?
?b  `/Bmb Q7 rminX
kXjX aKQQi?BM; k. bi`m+im`H iQTQHQ;B2b
h?Bb bm#@b2+iBQM /2b+`B#2b  K2i?Q/ iQ /B`2+iHv
i`Mb7Q`K i?2 b?T2 Q7 2H2K2Mib ;2M2`i2/ 7`QK
iQTQHQ;v QTiBKBxiBQM- bQ i?i bKQQi? k. bi`m+@
im`H iQTQHQ;B2b +M #2 +QMp2MB2MiHv +`2i2/X 6Q`
 k. bi`m+im`H iQTQHQ;v- Bib MQ/H b2MbBiBpBiv }2H/
7Q`Kb  H2p2H@b2i 7mM+iBQM (k8- jN)X aT2+B7vBM; Siso
/2i2`KBM2b i?2 bQHB/ /QKBM ΩS- pQB/ /QKBM ΩV
M/ #QmM/`v Γ- b b?QrM BM 6B;m`2 kU+VX 6Q` MQ/2
pBM i?2 /2bB;M /QKBM- Bib (x, y, z)TQbBiBQM- T- `2@
Hi2/ iQ Siso +M #2 /2b+`B#2/ b





p∈ΩV, if Sn(p)>S
iso
p∈Γ, if Sn(p)=Siso
p∈ΩS, if Sn(p)<S
iso
UdV
St(p)Bb BMi`Q/m+2/ iQ /2i2`KBM2 r?2i?2` Sn(p)
+Q``2bTQM/b iQ bQHB/- pQB/- Q` #QmM/`v mbBM; y M/
R QMHv- r?2`2
St(p)=0, if Sn(p)>S
iso
1, if Sn(p)≤Siso
U3V
h?Bb 2[miBQM `2T`2b2Mib i?i pQB/ TQBMib `2 i?Qb2
rBi? St(p)=0- M/ #Qi? bQHB/ M/ #QmM/`v TQBMib
TQbb2bb St(p)=1X
hQ i`Mb7Q`K i?2 b?T2 Q7 bi`m+im`H 2H2K2Mib-
i?Bb bim/v mb2b HQQFmT i#H2b iQ T`2@/2}M2 HH
TQbbB#H2 i`Mb7Q`KiBQM `2bmHib Q7  mMBi 2H2K2MiX
6Q` [m/`M;mH` 2H2K2Mib b?QrM BM 6B;m`2 kUV-
i?2 +Q``2bTQM/BM; HQQFmT i#H2 +M #2 #mBHi #v
KQ/B7vBM; i?2 J`+?BM; a[m`2b UJaV H;Q`Bi?K
(9y- 9R)X JQ`2 bT2+B}+HHv- i?2 BbQ@HBM2b BM i?2
j

Q`B;BMH K`+?BM; b[m`2b `2 `2TH+2/ rBi? T`2@
/2}M2/ i`BM;mH` Q` [m/`M;mH` K2b?2b iQ `2T`2@
b2Mi bQHB/ /QKBMb- b b?QrM BM 6B;m`2 kU#VX lbBM;
1[miBQM 3- 2H2K2MiH MQ/2b +M #2 bbB;M2/ rBi?
St(p)=1Q` 07Q` BMi2`MH U`2/V M/ 2ti2`MH U;`2vV
TQBMib- `2bT2+iBp2HvX //BiBQMH #QmM/`v UQ`M;2V
TQBMib `2 i?2M +`2i2/ QM i?2 KB/TQBMi #2ir22M BM@
i2`MH M/ 2ti2`MH TQBMibX Ji?2KiB+HHv- i?2`2
`2 Re TQbbB#H2 i`Mb7Q`KiBQM `2bmHib Q7  [m/`M@
;mH` 2H2K2Mi- +H+mHi2/ mbBM;
Cs(i)=
Ni

j=0
St(nj)2jUNV
r?2`2 Cs Bb i?2 +b2 MmK#2`- NiBb i?2 MmK#2` Q7
MQ/2b Q7 2H2K2Mi i- M/ St(nj)`2T`2b2Mib i?2 0/1
bii2 Q7 ji? MQ/2 BM 2H2K2Mi ic i?2 MmK#2` Q7 TQbbB@
#H2 +b2b Bb +H+mHi2/ b 2Ni=2
4= 16X >Qr2p2`-
i?2`2 `2 +imHHv R3 K`+?BM; b[m`2b HBbi2/ BM i?2
HQQFmT i#H2- b b?QrM BM 6B;m`2 kU#VX
Ai b?QmH/ #2 MQi2/ i?i i?2 +H+mHi2/ Cs pHm2
Kv +Q``2bTQM/ iQ KQ`2 i?M QM2 K`+?BM; ;2QK@
2i`v (9k)X am+? bT2+BH +b2b Q++m` r?2M i?2 /B;Q@
MH +Q`M2`b `2 /2i2`KBM2/ b BMi2`MH U`2/V TQBMib-
rBi? i?2 BMi2`K2/Bi2 `2;BQM bm``QmM/2/ #v #QmM/@
`v UQ`M;2V TQBMib #2BM; 2Bi?2` }HH2/ Q` p+Mi-
`2bmHiBM; BM +QMM2+i2/ M/ /Bb+QMM2+i2/ bQHB/ /Q@
KBMb- `2bT2+iBp2HvX AM i?2 T`2b2Mi 2tKTH2- bT2+BH
+b2b `2 Cs =5M/ Cs = 10- rBi? 2+? +QMiBMBM;
irQ TQbbB#H2 K`+?BM; b[m`2bX h?2`27Q`2- i?2 iQiH
MmK#2` Q7 TQbbB#H2 K`+?BM; b[m`2b Bb 16+ 2 = 18X
AM i?Bb bim/v- i?2 b2H2+iBQM Q7 K`+?BM; b[m`2b 7Q`
bT2+BH +b2b +M #2 bBKTHv /2i2`KBM2/ #b2/ QM
i?2 p2`;2/ 2H2K2MiH b2MbBiBpBiv- ˆ
Se(i)Ub22 1[m@
iBQM 9VX aT2+B}+HHv- ˆ
Se(i)≤Siso M/ ˆ
Se(i)>S
iso
+Q``2bTQM/ iQ +QMM2+i2/ M/ /Bb+QMM2+i2/ bQHB/ /Q@
KBMb- `2bT2+iBp2HvX
L2ti- HH 2H2K2Mib BM i?2 /2bB;M /QKBM `2 `2@
b?T2/ mbBM; i?2 KQ/B}2/ Ja HQQFmT i#H2X 6B@
MHHv- 7Q` 2+? #QmM/`v UQ`M;2V TQBMi t#2ir22M
M BMi2`MH U`2/V TQBMi AM/ M 2ti2`MH U;`2vV
TQBMi B- Bib (x, y, z)+QQ`/BMi2 Bb `2HQ+i2/ iQ 
M2r TQbBiBQM tnew mbBM; i?2 7QHHQrBM; HBM2` BMi2`@
TQHiBQM K2i?Q/- b b?QrM BM 6B;m`2 kX
tnew =A+Siso −Sn(A)
Sn(B)−Sn(A)(B−A)URyV
r?2`2 M/ "`2 i?2 (x, y, z)HQ+iBQMb Q7 i?2 BM@
i2`MH TQBMi AM/ 2ti2`MH TQBMi B- `2bT2+iBp2HvX
Sn(A)M/ Sn(B)`2 }Hi2`2/ MQ/H b2MbBiBpBiv MmK@
#2`b Q7 TQBMib AM/ B- `2bT2+iBp2HvX Ai Bb MQi2/
i?i `2HQ+iBM; HH #QmM/`v TQBMib ;Bp2b  bKQQi?
#QmM/`vX
6m`i?2`KQ`2- i?2 BMi2`b2+iBQMb #2ir22M i?2 ;2M@
2`i2/ bi`m+im`H iQTQHQ;v M/ i?2 /2bB;M /QKBM
`2 mM+?M;2/ 7i2` bKQQi?BM;X h?Bb Bb #2+mb2 i?2
2H2K2Mib HQ+i2/ QM i?2 #QmM/`B2b Q7 i?2 /2bB;M
/QKBM TQbb2bb St =1QM i?2B` MQ/2bX
hQ bmKK`Bx2/- 1[miBQM e +`2i2b  MQ/H b2M@
bBiBpBiv }2H/- Sn- 7Q`  k. bi`m+im`H iQTQHQ;vX h?2
bT2+B}2/ BbQ@pHm2- Siso- /2i2`KBM2b i?2 +QM/BiBQM
Q7 2+? MQ/2- r?2`2 St =y Q` R- b /2b+`B#2/ BM
1[miBQMb 3X AM /QBM; bQ- 2H2K2Mib +M #2 `2b?T2/
mbBM; i?2 KQ/B}2/ Ja HQQFmT i#H2 #b2/ QM 1[m@
iBQM NX h?2M- i?2 b?T2b Q7 i?2 i`Mb7Q`K2/ 2H2@
K2Mib `2 7m`i?2` mT/i2/ mbBM; HBM2` BMi2`TQH@
iBQM- b b?QrM BM 1[miBQM RyX h?Bb k. bKQQi?BM;
Bb /2KQMbi`i2/ KQ`2 +H2`Hv BM 6B;m`2 kX
kX9X aKQQi?BM; j. bi`m+im`H iQTQHQ;B2b
j. bi`m+im`H iQTQHQ;B2b +M HbQ #2 bKQQi?2/
#b2/ QM /Bz2`2Mi HQQFmT i#H2b- /2T2M/BM; QM i?2
K2b? 2H2K2Mi ivT2X h?2 irQ KQbi +QKKQM HQQFmT
i#H2b `2 J`+?BM; *m#2b UJ*V (9j) M/ J`+?BM;
h2i`?2/` UJhV (99) 7Q` ?2t?2/`H M/ i2i`?2@
/`H K2b? 2H2K2Mib- `2bT2+iBp2HvX *QKT`2/ rBi?
i?2 Q`B;BMH J* M/ Jh K2i?Q/b- i?2 j. i`Mb@
7Q`KiBQM +b2b `2 T`2@/2}M2/ #v +HQb2/ bm`7+2
K2b?2b iQ `2T`2b2Mi bQHB/ /QKBMb- b b?QrM BM 6B;@
m`2 jUV M/ U#VX Ai Bb rQ`i? TQBMiBM; Qmi i?i i?2 3@
MQ/2b ?2t?2/`H 2H2K2Mi +M #2 i`Mb7Q`K2/ BMiQ
key TQbbB#H2 +QM};m`iBQMb- #mi 6B;m`2 jUV QMHv
b?Qrb Rd mMB[m2 +b2bc i?2 `2KBMBM; 260−17 = 243
+b2b +M #2 Q#iBM2/ #v `QiiBM; M/ KB``Q`BM;
i?2b2 Rd +b2bX KQM; i?2b2 Rd mMB[m2 +b2b-
Cs = 65 Bb i?2 bT2+BH +b2- r?2`2 Bi +QMiBMb irQ
TQbbB#H2 K`+?BM; +m#2b- ?B;?HB;?i2/ BM 6B;m`2 jUVX
aBKBH` iQ i?2 T`2pBQmb k. 2tKTH2- +QMM2+i2/ M/
/Bb+QMM2+i2/ bQHB/ /QKBMb 7Q` i?2 T`2b2Mi j. +b2
`2 b2H2+i2/ r?2M ˆ
Se(i)≤Siso M/ ˆ
Se(i)>S
iso-
`2bT2+iBp2HvX
h?2 2MiB`2 bKQQi?BM; T`Q+2bb 7Q` j. bi`m+im`H
iQTQHQ;B2b Bb bBKBH` iQ i?2 rQ`F~Qr /2p2HQT2/ 7Q`
k. +b2bX 6B`bi- SnBb +H+mHi2/ 7Q` HH MQ/2b mbBM;
1[miBQM eX a2+QM/- Siso Bb bT2+B}2/ iQ bbB;M HH
MQ/2b rBi? St =y Q` R #b2/ QM Sn- b /2b+`B#2/ BM
3X h?B`/- HH 2H2K2Mib `2 `2b?T2/ 7`QK i?2 HQQFmT
i#H2 ++Q`/BM; iQ St mbBM; 1[miBQM NX 6Qm`i?- i?2
b?T2 Q7 i`Mb7Q`K2/ 2H2K2Mib `2 7m`i?2` KQ/B}2/
mbBM; 1[miBQM RyX .Bz2`2Mi iQ k. bKQQi?BM;- M
//BiBQMH bi2T Bb `2[mB`2/ BM j. +b2bX
9

Linear
interpolation
Replace
(a) (b) (c)
A
B
t
A
B
Boundary
condition
Design
domain
Loading
condition
Structural
topology
Smoothed
structural
topology
Internal points External points
Boundary points
0
0Cs
1
32
1Cs 2Cs 3Cs 4Cs
6Cs 7Cs 8Cs
9Cs 10Cs 11Cs 12Cs
13Cs 14Cs 15Cs
Modified Marching Squares
S
:
Solid:
V
:
Void:
*
Boundary:
t
new
5Cs
ˆ( )
e iso
S
i Sd
ˆ( )
e iso
S
i S!
ˆ( )
e iso
S
i Sd
ˆ( )
e iso
S
i S!
6B;m`2 k, aKQQi?BM; i?2 bi`m+im`H iQTQHQ;v Q7  k. +MiBH2p2`X UV AMBiBH b2imT Q7 i?2 QTiBKBxiBQM T`Q#H2K- r?2`2 i?2 /2bB;M
/QKBM Bb /Bb+`2iBx2/ rBi? [m/`M;mH` 2H2K2MibX U#V h?2 HQQFmT i#H2 Bb #mBHi mbBM; i?2 KQ/B}2/ J`+?BM; a[m`2b H;Q`Bi?K
i?i ?b R3 TQbbB#H2 +QM};m`iBQMbX U+V h?2 i`Mb7Q`KiBQM `2bmHi Bb Q#iBM2/ i?`Qm;? `2b?TBM; HH Q`B;BMH 2H2K2MibX
b b?QrM BM 6B;m`2 9- i?Bb bim/v `2KQp2b HH BM@
i2`MH 7+2b Q7 j. bi`m+im`H iQTQHQ;B2bX Ai b?QmH/
#2 ?B;?HB;?i2/ i?i HH i`Mb7Q`K2/ 2H2K2Mib `2
+HQb2/ bm`7+2 K2b?2bX h?2v 7Q`K  bKQQi?2/ +QM@
};m`iBQM i?i BM+Hm/2b +QBM+B/2Mi 7+2b- b b?QrM
BM 6B;m`2 9UVX h?2b2 +QBM+B/2Mi 7+2b `2 B/2MiB@
}2/ b BMi2`MH 7+2bX _2KQpBM; HH BMi2`MH 7+2b
;Bp2b j. bm`7+2 K2b?2b- b b?QrM BM 6B;m`2 9U#V-
r?B+? +M #2 +QMbB/2`2/ b *.@+QKTiB#H2 KM@
B7QH/ K2b?2bX
kX8X am`7+2 `2K2b?BM;
h?2 T`Q/m+ib Q7 i?2 #Qp2 k. M/ j. bKQQi?@
BM; T`Q+2bb2b `2 bm`7+2 K2b?2b (98)X b b?QrM BM
6B;m`2 8- i?2 [mHBiv Q7 i?2b2 bm`7+2b +M #2 BK@
T`Qp2/ i?`Qm;? dzbm`7+2 `2K2b?BM;ǴX Ai b?QmH/ #2
MQiB+2/ i?i KMv `2K2b?BM; i2+?MB[m2b `2 /2p2H@
QT2/ #b2/ QM KMB7QH/ *. KQ/2Hb HBF2 bm`7+2
K2b?2b (9e)X
h?Bb bim/v mb2b i?2 BM+`2K2MiH `2K2b?BM; H;Q@
`Bi?K /2b+`B#2/ BM (98- 9d- 93)X h?Bb H;Q`Bi?K +M
#2 7QmM/ BM KMv *. bQ7ir`2- bm+? b _?BMQ dX
h?2 `2K2b?BM; H;Q`Bi?K `2T2i2/Hv KQ/B}2b i?2
K2b? 2/;2b M/ p2`iB+2b #b2/ QM HQ+H QT2`iBQMb
mMiBH HH 2/;2 H2M;i?b `2 +HQb2 iQ i?2 T`2b+`B#2/ i`@
;2i pHm2c i?2 HQ+H QT2`iBQMb BM+Hm/2 bTHBiiBM; HQM;
2/;2b- +QHHTbBM; b?Q`i 2/;2b- ~BTTBM; K2b? 2/;2b-
M/ `2HQ+iBM; p2`iB+2bX b  `2bmHi Ub22 6B;m`2 8V-
i?2 2H2K2Mib QM i?2 bm`7+2 +M #2 `2@/Bbi`B#mi2/
mMB7Q`KHv mbBM; i`BM;mH` 2H2K2Mib iQr`/ i?2 i`@
;2i 2/;2 H2M;i?X Ai +M HbQ #2 b22M i?i i?2 ;2Q@
K2i`B+ 72im`2b- bm+? b +Q`M2`b M/ b?`T 2/;2b-
`2 T`2b2`p2/ /m`BM; i?2 `2K2b?BM; T`Q+2bbX
kXeX S`2b2`pBM; `2 Q` pQHmK2
h?2 bKQQi?2/ +QM};m`iBQMb Q#iBM2/ 7`QK i?2
#Qp2 K2i?Q/b +M #2 mM/2`biQQ/ b BbQ@b2MbBiBpBiv
KQ/2Hb- b i?2v `2 /2i2`KBM2/ #b2/ QM i?2 Siso
8

(b)
(a)
01
5
4
3
2
6
7
Modified Marching Cubes
1
0
2
3
Modified Marching Tetrahedra
58Cs 67Cs 74Cs
105Cs
51Cs
50Cs
5Cs 3Cs
1Cs
113Cs 165Cs 177Cs 178Cs
255Cs
0Cs
1Cs
2Cs 3Cs 4Cs 5Cs 6Cs 7Cs
8Cs 9Cs
10Cs
11Cs
12Cs 13Cs 14Cs 15Cs
ˆ( )
e iso
S
i S!
0Cs
65Cs
ˆ( )
e iso
S
i Sd
6B;m`2 j, JQ/B}2/ j. HQQFmT i#H2bX h?2 `2/ MmK#2`b `2T`2b2Mi i?2 BM/B+2b Q7 i?2 2H2K2MiH MQ/2bX UV Rd mMB[m2 +b2b BM i?2
KQ/B}2/ J`+?BM; *m#2b K2i?Q/ Q7  i`Mb7Q`K2/ ?2t?2/`H 2H2K2MiX U#V HH Re TQbbB#H2 +b2b BM i?2 KQ/B}2/ Jh K2i?Q/
Q7  i`Mb7Q`K2/ i2i`?2/`H 2H2K2MiX
pHm2X JQ`2Qp2`- +?M;BM; Siso +M /B`2+iHv BM~m@
2M+2 i?2 }MH `2 M/ pQHmK2 Q7 k. M/ j. BbQ@
b2MbBiBpBiv KQ/2Hb- `2bT2+iBp2HvX AM Q`/2` iQ K22i
 i`;2i `2 Q` pQHmK2- i?Bb bim/v i`2ib Siso b
 p`B#H2 M/ BMi`Q/m+2b i?2 #Bb2+iBQM K2i?Q/ iQ
KQ/B7v Siso miQKiB+HHvX
Siso =Sup +Slo
2URRV
r?2`2 Slo =0M/ Sup =2×Sth `2 i?2 BMBiBH
HQr2` M/ mTT2` #QmM/- `2bT2+iBp2HvX h?2 Sth Bb i?2
i?`2b?QH/ b2MbBiBpBiv MmK#2` K2MiBQM2/ BM a2+iBQM
kXRX h?2 #QmM/b `2 mT/i2/ mbBM;
Slo =Siso, if V > V ∗
Sup =Siso, if V ≤V∗URkV
r?2`2 VBb i?2 +m``2Mi pQHmK2f`2 Q7 i?2 BbQ@
b2MbBiBpBiv KQ/2H- M/ V∗Bb i?2 i`;2i pQHmK2f`2X
e

(a) (b)
Faces to be
removed
6B;m`2 9, _2KQpBM; +QBM+B/2Mi 7+2b Q7 j. 2H2K2MibX UV HH +QBM+B/2Mi 7+2b `2 B/2MiB}2/ b BMi2`MH 7+2b- r?B+? `2 `2[mB`2/
iQ #2 `2KQp2/X U#V _2KQpBM; HH BMi2`MH 7+2b ;Bp2b  j. +HQb2/ bm`7+2 K2b?X
Remeshing
6B;m`2 8, AKT`Qp2 i?2 bm`7+2 [mHBiv mbBM; i?2 BM+`2K2MiH
`2K2b?BM; H;Q`Bi?K bQ i?i i?2 2H2K2Mib QM i?2 bm`7+2 +M
#2 `2@/Bbi`B#mi2/ mMB7Q`KHv #b2/ QM  i`;2i 2/;2 H2M;i?
M/ T`2b2`p2 i?2 ;2QK2i`B+ 72im`2bX
h?2 #B@b2+iBQMH Bi2`iBp2 T`Q+2bb Bb `2T2i2/ mM@
iBH |V−V∗|
VD
≤Vtol URjV
r?2`2 VDBb i?2 iQiH pQHmK2f`2 Q7 i?2 /2bB;M /Q@
KBM- Vtol =0.1% Bb i?2 HHQr#H2 iQH2`M+2X h?Bb
2[miBQM bBKTHv `2T`2b2Mib i?i i?2 }MH bi`m+im`H
iQTQHQ;v rBHH TQbb2bb i?2 iQiH pQHmK2f`2 +HQb2
iQ i?2 i`;2i pHm2 rBi? M ++2Ti#H2 /Bz2`2M+2X
h?2 2MiB`2 bKQQi?BM; T`Q+2bb i?i +QMbB/2`b T`2@
b2`pBM; `2 Q` pQHmK2 T`Q+2bb Bb bmKK`Bx2/ BM
i?2 ~Qr+?`i b?QrM BM 6B;m`2 eX
jX LmK2`B+H MHvbBb
 MmK2`B+H MHvbBb K2i?Q/ Bb /2p2HQT2/ iQ
2tKBM2 i?2 +T#BHBiB2b Q7 i?2 #Qp2 bKQQi?BM;
K2i?Q/bX JQ`2 bT2+B}+HHv- i?2 Q#iBM2/ "1aP
bi`m+im`H iQTQHQ;B2b `2 +QKT`2/ rBi? i?2B` +Q`@
`2bTQM/BM; bKQQi?2/ `2bmHib iQ Q#b2`p2 Mv p`B@
iBQMb BM b?T2 M/ bi`m+im`H T2`7Q`KM+2X
h?Bb bim/v mb2b i?2 +QKK2`+BH }MBi2 2H2K2Mi
MHvbBb U61V bQ7ir`2- #[mb- M/  Svi?QM
b+`BTi iQ T2`7Q`K bi`m+im`H MHvbBb M/ "1aP
iQTQHQ;v QTiBKBxiBQM- `2bT2+iBp2Hv- rBi? /2iBHb
7mHHv /2b+`B#2/ BM (je)X PM+2 i?2 QTiBKH bi`m+im`H
iQTQHQ;B2b `2 Q#iBM2/-  *O +Q/2 Bb mb2/ iQ 2t2@
+mi2 i?2 #Qp2 H;Q`Bi?Kb 7Q` ;2M2`iBM; bKQQi?2/
KQ/2Hb BM i?2 _?BMQ d *. bQ7ir`2X h?2 Q`B;B@
MH M/ bKQQi?2/ QTiBKH bi`m+im`H iQTQHQ;B2b `2
i?2M BKTQ`i2/ BMiQ #[mb iQ T2`7Q`K 61 bi`m+@
im`H MHvbBb iQ +QKT`2 i?2B` biBzM2bb T2`7Q`@
KM+2X AM Q`/2` iQ ?p2  7B` +QKT`BbQM- i?2B`
61 K2b?2b `2 b2i iQ ?p2 i?2 bK2 K2b? ivT2 M/
i`;2i 2/;2 H2M;i?X aT2+B}+HHv- k. M/ j. +b2b
`2 MHvx2/ mbBM; i`BM;mH` M/ i2i`?2/`H 2H2@
K2Mib- `2bT2+iBp2Hv- rBi? i?2 i`;2i 2/;2 H2M;i? b2i
iQ #2 i?2 bK2 pHm2 b mb2/ BM i?2 `2K2b?BM; T`Q@
+2bbX LQi2 i?i 2[mBpH2Mi iQTQHQ;B2b `2 /Bb+`2iBx2/
mbBM; i?2 bK2 K2b?BM; H;Q`Bi?K 2K#2//2/ BM i?2
+QKK2`+BH #[mb bQ7ir`2 iQ +`2i2 61 KQ/2Hbc
K2b?BM; H;Q`Bi?Kb Q7 61 KQ/2Hb `2 MQi BM+Hm/2/
BM i?2 b+QT2 Q7 i?Bb bim/vX
h?2 Ki2`BH mb2/ BM i?Bb bim/v Bb bbmK2/ iQ #2
BbQi`QTB+ M/ HBM2` 2HbiB+- rBi? uQmM;Ƕb KQ/mHmb
Q7 E=1JS M/ SQBbbQMǶb `iBQ Q7 v=0.3X lM@
H2bb Qi?2`rBb2 bii2/- "1aP iQTQHQ;v QTiBKBxiBQM
T`K2i2`b `2, ert = 2%-p=3- M/ rmin =3
KKX
h?2 j. bKQQi?BM; T`Q+2bb Bb }`bi i2bi2/ mbBM;
 +MiBH2p2` 2tKTH2- b b?QrM BM 6B;m`2 dX qBi?
`272`2M+2 iQ 6B;m`2 dUV- i?2 /2bB;M /QKBM Q7 i?2
d

Topology
optimization
Topology
optimization
Topology
optimization
Surface
Remeshing
Iso-sensitivity
model
Iso-sensitivity
model
Nodal
sensitivity field
Nodal
sensitivity field
No
Yes
No
Final design (smoothed)Final design (smoothed)Final design (smoothed) Yes
V
ˆ
e
S
up iso
SS
up iso
SS
lo iso
SS
lo iso
SS
*
tol
D
VV V
V


*
tol
D
VV V
V


2
up lo
iso
SS
S


2
up lo
iso
SS
S


*
VV
*
VV
n
S
iso
S
Surface
mesh
6B;m`2 e, h?2 ~Qr+?`i Q7 i?2 2MiB`2 bKQQi?BM; T`Q+2bbX h?2 T`Q+2bb Q7 T`2b2`pBM; `2 Q` pQHmK2 rBHH 2Mbm`2 i?i i?2 bKQQi?2/
`2bmHi K22ib  i`;2i `2 Q` pQHmK2X
+MiBH2p2` Bb ey KK ×Ry KK ×9y KKX  F=−1
L TQBMi HQ/ Bb TTHB2/ i i?2 +2Mi2` Q7 i?2 7`22
2M/X  }t2/ #QmM/`v +QM/BiBQM Bb bbB;M2/ #2?BM/
i?2 r?QH2 +MiBH2p2`X h?2 Q#D2+iBp2 pQHmK2 7+iBQM
Q7 "1aP Bb b2i b R8WX aKQQi?2/ +QM};m`iBQMb
`2 Q#iBM2/ 7`QK bi`m+im`H iQTQHQ;B2b rBi? /Bz2`@
2Mi K2b? bBx2b- b b?QrM BM 6B;m`2b dUV@U#V M/
U+V@U/VX 6Q` Q`B;BMH "1aP bi`m+im`H iQTQHQ;B2b-
bQHB/ +m#B+ 2H2K2Mib i?i ?p2 M 2/;2 H2M;i? Q7
R KK M/ yX8 KK `2 mb2/- b b?QrM BM 6B;m`2b
dUV M/ U+V- `2bT2+iBp2HvX 6Q` bKQQi?2/ bi`m+im`H
iQTQHQ;B2b- Q`B;BMH 2H2K2Mib `2 `2b?T2/ mbBM; i?2
KQ/B}2/ J* HQQFmT i#H2 M/ i?2M i?2 bb2K#HB2b
`2 `2K2b?2/ iQ  bm`7+2 K2b? rBi? i`BM;mH` 2H2@
K2Mib i?i ?p2 M p2`;2 2/;2 H2M;i? Q7 yX8 KK-
b b?QrM BM 6B;m`2b dU#V M/ U/VX HH bi`m+im`H
iQTQHQ;B2b `2 /Bb+`2iBx2/ mbBM; bQHB/ i2i`?2/`H 2H@
2K2Mib rBi? M 2/;2 H2M;i? Q7 yX8 KK BM 61- bQ
i?i i?2 +QKTHBM+2 pHm2b- C- `2 +QKT`#H2X
AM 6B;m`2 d- Bi +M #2 b22M i?i i?2 xB;@x; #QmM/@
`B2b Q7 Q`B;BMH bi`m+im`H iQTQHQ;B2b ?p2 #22M bm+@
+2bb7mHHv bKQQi?2/X Ai +M HbQ #2 b22M i?i i?2
;2QK2i`B+H 72im`2b ?p2 #22M T`2b2`p2/- r?2`2
i?2 b?T2b Q7 i?2 Q`B;BMH M/ bKQQi?2/ bi`m+@
im`H iQTQHQ;B2b `2 HKQbi B/2MiB+HX JQ`2Qp2`- i?2
bi`m+im`H pQHmK2b `2 +?M;2/ rBi?BM M HHQr#H2
iQH2`M+2X hQ;2i?2`- Bi +M #2 +QM+Hm/2/ i?i i?2
T`QTQb2/ bKQQi?BM; K2i?Q/ +M ;2M2`i2 2[mBp@
H2Mi bi`m+im`H iQTQHQ;B2b iQ i?2 Q`B;BMH iQTQHQ;v
QTiBKBxiBQM `2bmHibX
6m`i?2`KQ`2- i?2 +QKTHBM+2 Q7 6B;m`2b dUV@U/V
`2 8Xjje LKK- 9X388 LKK- 8Xyey LKK- M/ 9XNdj
LKK- `2bT2+iBp2HvX Ai Bb b22M i?i i?2 bKQQi?2/
bi`m+im`2b `2 biBz2`- +Q``2bTQM/BM; iQ HQr2` +QK@
THBM+2X JQ`2 bT2+B}+HHv- i?2 +QKTHBM+2 Q7 6B;@
m`2b dUV M/ U+V `2 `2/m+2/ #v NXykW M/ RXdjW
7i2` bKQQi?BM;- `2bT2+iBp2HvX h?2 2M?M+2/ bi`m+@
im`H T2`7Q`KM+2 Kv #2 ii`B#mi2/ iQ i?2 bi`2bb
#2BM; KQ`2 mMB7Q`KHv /Bbi`B#mi2/ BM i?2 bKQQi?2/
KQ/2Hb- i?mb `2bmHiBM; BM HQr2` +QKTHBM+2- /2KQM@
bi`i2/ KQ`2 +H2`Hv BM 6B;m`2b dUV@U#VX >Qr2p2`-
i?2 BKT`Qp2K2Mi Bb H2bb MQiB+2#H2 mbBM;  }M2`
K2b?- b i?2 `2bQHmiBQM Q7 i?2 Q`B;BMH bi`m+im`H
iQTQHQ;v Bb H`2/v ?B;?- K2MBM; i?i i?2 b?T2
Bb bBKBH` iQ Bib +Q``2bTQM/BM; bKQQi?2/ 7Q`K- b
b?QrM BM 6B;m`2b dU+V@U/VX L2p2`i?2H2bb- i?Bb 2tK@
TH2 biBHH /2KQMbi`i2b i?i M 2M?M+2/ bi`m+im`H
T2`7Q`KM+2 +M #2 Q#iBM2/ i?`Qm;? bKQQi?BM;X
9X .Bb+mbbBQM
9XRX _2;mH` M/ B``2;mH` k. K2b?2b
h?Bb bm#b2+iBQM 2tKBM2b i?2 bi`m+im`H T2`7Q`@
KM+2 Q7 bKQQi?2/ iQTQHQ;B2b Q#iBM2/ 7`QK `2;m@
H` M/ B``2;mH` k. [m/`M;mH` K2b?2b (9N)- b
b?QrM BM 6B;m`2b 3UV M/ U#V- `2bT2+iBp2HvX
6B;m`2 3UV b?Qrb  k9y KK ×9y KK J"" #2K
BMBiBHHv K2b?2/ rBi? R KK b[m`2 2H2K2MibX 
F=−1L TQBMi HQ/ Bb TTHB2/ i i?2 KB/TQBMi
Q7 i?2 iQT 2/;2X 6Q` iQTQHQ;v QTiBKBxiBQM- i?2
Q#D2+iBp2 pQHmK2 7+iBQM Bb b2i b 8yW- M/ rmin
Bb +?M;2/ iQ k KKX .m`BM; i?2 bKQQi?BM; T`Q@
+2bb- i?2 KQ/B}2/ Ja HQQFmT i#H2 Bb mb2/ iQ Q#iBM
i`Mb7Q`K2/ [m/`M;mH` 2H2K2Mib- M/ i?2M i?2v
3

S, Mises
Max
Min
(a) (b)
(c)
C = 5.336 Nmm
F
60
10
40
Smoothed
(d)
C = 5.060 Nmm
C = 4.855 Nmm
C = 4.973 Nmm
Original
6B;m`2 d, 61 `2bmHib Q7 Q`B;BMH M/ bKQQi?2/ bi`m+im`H iQTQHQ;B2bX UV@U#V b?Qrb i?i +m#B+ 2H2K2Mib i?i ?p2 M 2/;2 H2M;i?
Q7 R KK `2 bKQQi?2/ BMiQ  bm`7+2 K2b? rBi? i`BM;mH` 2H2K2Mib i?i ?p2 M p2`;2 2/;2 H2M;i? Q7 yX8 KKX U+V@U/V b?Qrb
i?i +m#B+ 2H2K2Mib i?i ?p2 M 2/;2 H2M;i? Q7 yX8 KK `2 bKQQi?2/ BMiQ  bm`7+2 K2b? rBi? i`BM;mH` 2H2K2Mib i?i ?p2
M p2`;2 2/;2 H2M;i? Q7 yX8 KKX UV M/ U+V `2 Q`B;BMH bi`m+im`H iQTQHQ;B2bc U#V M/ U/V `2 bKQQi?2/ bi`m+im`H iQTQHQ;B2bX
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Ry

F’
F
(a) (b)
F
Non-design domain Design domain
Structural topology
Smoothed structural topology
C = 94.465 Nmm
C = 89.998 Nmm
C = 88.729 Nmm
C = 86.133 Nmm
Problem definition
Regular Irregular
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F’
F
C = 61.231 Nmm C = 57.235 Nmm
Non-design domain
Design domain
Symmetry
Fixed support
(a) (b) (c)
Structural
topology
Smoothed
structural
topology
Tetrahedral
meshes
(d)
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