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Ph.D Defence
쉘 구조물 해석을 위한 연속체 역학 기반 범용유한요소 개발
심사위원 교수님
이 필 승
이 병 채
윤 정 환
김 도 년
정 현
2
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3
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4
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)log()log(
)(log)(log
εεΔε
εεΔερ
EEE ε
1ρ 3ρ 31 ρ
5
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6
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• •
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Shell problems Asymptotic behavior (ρ) Dominant element behavior
Fully clamped plate Bending-dominated (ρ=3.0) Bending (Regular or distorted mesh)
Free cylindrical shell Bending-dominated (ρ=3.0) Bending (Regular mesh) Membrane (Distorted mesh)
Clamped cylindrical shell Membrane-dominated (ρ=1.0) Bending (Regular or distorted mesh)
Free hyperboloid shell Bending-dominated (ρ=3.0) Bending (Regular mesh) Membrane (Distorted mesh)
Clamped hyperboloid shell Membrane-dominated (ρ=1.0) Bending (Regular or distorted mesh)
< Hyperboloid shell > < Cylindrical shell > < Regular vs. Distorted mesh >
◇ Lee PS and Bathe KJ. Comput Struct 2002:80;235-55. ◇ Lee PS and Noh HC. journal of KSCE 2007:27(3A);277-89.
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◇ AHMAD et al. Analysis of thick and thin shell structures by curved finite elements. Int J for Numer Meth Eng, 1970:2;419-51.
8
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(a) Regular mesh (b) Distorted mesh (a) Regular mesh (b) Distorted mesh
Convergence studies
9
t/L Regular mesh Distorted mesh
1/100 4.48696E-7 4.30054E-7
1/1,000 4.44367E-4 8.69754E-5
1/10,000 4.43945E-1 1.52094E-3
Order of change ~(t/L)3 ~(t/L)2
Free
Free
Locking
Shell element
Shear locking
Membrane locking
Thickness locking
Degenerated shell
√ √ -
Solid shell √ √ √
Flat shell √ √ -
10
•
⇒⇒
⇒
⇒
⇒
•
⇒
⇒
◇ Belytshcko and Tsay. Comp Meth Appl Mech Eng 1994:115;277-86.
◇ Belytshcko and Leviathan. Comp Meth Appl Mech Eng 1994:113;321-50.
◇ Rankin and Nour-Omid. Comp Struct 1988:30;257-67.
◇ Simo and Rifai. Int J numer Meth Eng 1990:29;1595-638.
◇ Fox and Simo. Comp Meth Appl Mech Eng 1992:98;329-43
◇ Ibrahimbegovic et al. Int J numer Meth Eng 1990:30;445-57.
◇ Taylor. Proc Math FEM 1987:191-203.
◇ Choi and Paik. Comp Meth Appl Mech Eng 1994:2;17-34.
◇ Koschnick et al. Comp Meth Appl Mech Eng 2005:194;2444-463. < warped element >
11
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⇒
•
⇒
◇ Kim JH, Kim YH and Lee SW. Int J numer Meth Eng 2000:47;1481-97.
◇ Sze KY and Chan WK. Finite Elem Anal Design 2001:37;639-55.
◇ Hong WI, Kim JH, Kim YH and Lee SW. Int J numer Meth Eng 2001:52;747-61.
◇ Kim CH, Sze KY and Kim YH. Int J numer Meth Eng 2000:57;2077-97.
◇ Dvorkin EN and Bathe KJ. Eng Comput 1984:1;77-88.
◇ Lee Y, Lee PS and Bathe KJ. Comput Struct 2014:138;12-23.
12
13
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< MITC4 element >
)()( )1(2
1)1(
2
1~ B
rt
A
rtrt esese
)()( )1(2
1)1(
2
1~ D
st
C
stst erere
•
< MITC3+ element >
14
•
< Linear shell elements> < Quadratic shell elements>
15
•
⇒⇒
⇒⇒
Low-order shell element
Shear locking
Membrane locking
Thickness locking
Remark
3 node degenerated shell
√ - -
Accurate element
4 node degenerated shell
√ √ -
No satisfactory solution to
membrane locking
6 node solid shell √
-
√
No satisfactory solution to shear
and thickness locking
8 node solid shell √ √ √ Accurate elements
Motivation
16
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•
Free
Free
< Problem for convergence study >
“No answer regarding which is better, quadrilateral or triangular element”
17
•
Towards improving finite elements for analysis of general shell structures
Improving 4-node quadrilateral degenerated shell finite element
Improving 6-node triangular solid-shell finite element
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•
18
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•
< Mesh used for the patch test>
•
•
•
< General distorted elements >
•
◇ Irons BM, Razzaque A. Experience with the patch test, 1972.
◇ Lee PS et al. Comput Struct 2004:82;945-62.
◇ Kim DN et al. Comput Struct 2009:87;1451-60.
19
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•
•
< Test of in-plane bending >
•
•
•
’
•
◇ Cook RD. ASCE J Struct Div 1974:100;1851-63.
< Test of in-plane shear >
◇ MacNeal RH. Finite elements: their design and performance. 1994.
◇ Belytshcko and Leviathan. Comp Meth Appl Mech Eng 1994:113;321-50.
◇ Abaqus theory manual, V.6.14
20
•
•
•
•
< Distortion pattern for the convergence studies>
< Cylindrical shell problem > < Hyperboloid shell problem > < Plate problem >
◇ Bathe KJ et al. Comput Struct 2003:81;477-89.
◇ Chapelle D et al. Comput Struct 1998:66;19-36,711-2.
21
•
Benchmarks Boundary condition
Asymptotic behavior
t/L
Strain energy ratio (%)
Bending Membrane Transverse
shear
Plate Clamped Bending-
dominated
1/100 1/1,000 1/10,000
99.94 100.0 100.0
0.00 0.00 0.00
0.00 0.00 0.00
Cylindrical shell
Clamped Membrane-dominated
1/100 1/1,000 1/10,000
1.94 0.32 0.08
98.02 99.68 99.92
0.03 0.00 0.00
Free Bending-
dominated
1/100 1/1,000 1/10,000
99.77 99.93 99.98
0.22 0.07 0.02
0.01 0.00 0.00
Hyperboloid shell
Clamped Membrane-dominated
1/100 1/1,000 1/10,000
4.16 1.16 0.35
95.78 98.84 99.65
0.06 0.00 0.00
Free Bending-
dominated
1/100 1/1,000 1/10,000
99.11 99.99 100.0
0.83 0.01 0.00
0.06 0.00 0.00
◇ Bucalem et al. Int J numer Meth Eng 1993:36;3729-54.
22
•
Benchmarks Boundary condition
Asymptotic behavior
t/L
Strain energy ratio (%)
Bending Membrane Transverse
shear
Plate Clamped Bending-
dominated
1/100 1/1,000 1/10,000
99.94 100.0 100.0
0.00 0.00 0.00
0.00 0.00 0.00
Cylindrical shell
Clamped Membrane-dominated
1/100 1/1,000 1/10,000
1.94 0.32 0.08
98.02 99.68 99.92
0.03 0.00 0.00
Free Bending-
dominated
1/100 1/1,000 1/10,000
99.77 99.93 99.97
0.22 0.07 0.03
0.01 0.00 0.00
Hyperboloid shell
Clamped Membrane-dominated
1/100 1/1,000 1/10,000
4.16 1.16 0.34
95.78 98.84 99.66
0.06 0.00 0.00
Free Bending-
dominated
1/100 1/1,000 1/10,000
99.10 99.99 99.99
0.83 0.01 0.01
0.06 0.00 0.00
◇ Bucalem et al. Int J numer Meth Eng 1993:36;3729-54.
23
24
•
⇒
•
–
•
◇ Choi and Paik. Comp Meth Appl Mech Eng 1994:2;17-34.
◇ Koschnick et al. Comp Meth Appl Mech Eng 2005:194;2444-463.
Membrane locking
mechanism
Retaining membrane behaviors
Start
25
•
•
• ⇒
• ⇒
2
2
3
2
2
3
2
2
3
2
3
22
2
3
2
2
3
2
2L
s
s
u
L
r
r
u
L
s
s
u
L
s
L
r
sr
u
L
r
r
u
L
uLL
ε
◇ Prathap G. The finite element method in structural mechanics 1993.
mmm
rrsru u
xx
21
3 ),(
32
2
1
1
,,2
1),( ubuuuusr ijijijijjiij βΓΓγ 0),(),(
2
1srCsr klijklij γγ
03
21
2
u
rr Prevention of pure bending ⇒ Membrane locking
Locking-causing part ⇒ bi-linear term
26
•
Membrane part
Bi-linear term
Membrane locking
Reduced integration
- -
QMITC √ √
Area Coordinate
Method
√
√
◇ Dvorkin EN et al. Eng Comput 1989:6;217-24.
◇ Chen XM et al. Comp Struct 2004:82;35-54.
rsrh ),(1
ssrh ),(2
srsrh 1),(3
)(4
1),(1 rssrsrh
)(4
1),(2 rssrsrh
)(4
1),(3 rssrsrh
)(4
1),(4 rssrsrh
•
Locking mechanism
Start
Directly adopt 3-node shape function to membrane strain
< Subdivision of quadrilateral into triangular domains>
Solution
27
•
4
1
4
1
),(2
),(),,(i
i
nii
i
ii srhat
srhtsr Vxx
)(),(2
),(),,( 1
4
1
2
4
1
i
i
i
i
i
ii
i
ii srhat
srhtsr VVuu
)()( )1(2
1)1(
2
1~ B
rt
A
rtrt esese
)()( )1(2
1)1(
2
1~ D
st
C
stst erere
221 b
ij
b
ij
m
ijij etetee
i
m
j
b
j
m
i
b
i
b
j
m
j
b
i
mb
ijrrrrrrrr
euxuxuxux
2
11
i
b
j
b
j
b
i
bb
ijrrrr
euxux
2
12
4
1
),(i
iim srh xx
4
1
),(i
iim srh uu bm uuu
bm xxx
28
•
•
3
1
),(),,(i
ii srhtsr xx
3
1
),(),,(i
ii srhtsr uu
ijji
m
ijrrrr
euxux
2
1
Base Triangular Quadrilateral
Notation
seereeeeeee Am
ij
Bm
ij
Cm
ij
Dm
ij
Dm
ij
Cm
ij
Bm
ij
Am
ij
m
ij )(2
1)(
2
1)(
4
1~ )()()()()()()()(
))(( l
j
k
i
m
kl
m
ij ee gggg
•
•
i
ir
xg
i
ir
xg
29
Assumed membrane strain Details
MITC4+
Choi and Paik’s element
Discrete Strain Gap
seeeee AmBmAmBmm )(2
1)(
2
1~ )(
11
)(
11
)(
11
)(
1111
seereeeeeee Am
ij
Bm
ij
Cm
ij
Dm
ij
Dm
ij
Cm
ij
Bm
ij
Am
ij
m
ij )(2
1)(
2
1)(
4
1~ )()()()()()()()(
◇ Choi and Paik. Comp Meth Appl Mech Eng 1994:2;17-34.
◇ Koschnick et al. Comp Meth Appl Mech Eng 2005:194;2444-463.
reeeee CmDmCmDmm )(2
1)(
2
1~ )(
22
)(
22
)(
22
)(
2222
)(4
1~ )(
12
)(
12
)(
12
)(
1212
DmCmBmAmm eeeee
4
1
4
1
12121 1
~
k
s
sl
r
r
mlkm dsdrer
h
s
he
k l
4
1
22221
~
k
s
s
mkm dses
he
k
4
1
11111
~
k
r
r
mkm drer
he
k
•
New assumed membrane strain field is linear in r, s direction for all components ⇒ unique idea
30
4
1
5
i
iiuu γ
4
1
5
i
iixx γ
•
]3
1
3
1
3
10[
2
1]
3
10
3
1
3
1[
2
1
21
2
21
1
4321AA
A
AA
A
]3
1
3
10
3
1[
2
1]0
3
1
3
1
3
1[
2
1
43
4
43
3
AA
A
AA
A
Center point is located at the ‘average’ of two centroids of triangles
31
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Free
Free
32
•
Accuracy ⇒ MITC9 < MITC4+
33
• •
Pass of Membrane Patch test
Treat
Maintain the subdivision of mid-surface,
but interpolate on the whole quadrilateral
domain
< Subdivision of quadrilateral into triangular domains>
Solution
1q 2q< Area coordinates ( and ) >
34
Center point is located at the ‘mid-point’ in shortest line segment joining two diagonals ⇒ Essential to satisfy zero energy mode, isotropy and patch tests, (with “modified” ACM)
35
•
AAAG CA /)(1 AAAG DA /)(2
AAAG BD /)(3 AAAG CB /)(4
DCBA AAAAA
pGqG
h 313
12
pGqG
h 424
22
pGqG
h 111
32
pGqG
h 222
42
)1)((4
1131 rsGGsrq )1/(
2
)()()(2)(3 4231
3142224131
2
2
2
1 GGGGGGGG
qGGqGGqqp
◇ Chen et al. Comp Struct 2004:82;35-54.
◇ Cen et al. Int J Num Meth Eng 2009:77;1172-200.
DCBA AAAAA
36
•
•
•
ijji
m
ijrrrr
euxux
2
1
4
1
),(),,(i
ii srhtsr xx
4
1
),(),,(i
ii srhtsr uu
Base (r,s) (0,0)
Notation ),0,0( tri
c
i
xg),,( tsr
ri
i
xg
))(( lc
j
kc
i
m
kl
m
ij ee gggg
seeree
eeeee
Am
ij
Bm
ij
Cm
ij
Dm
ij
Dm
ij
Cm
ij
Bm
ij
Am
ij
m
ij
)(2
1)(
2
1
)(4
1~
)()()()(
)()()()(
•
Element Performances Membrane patch test
MITC4+
Equivalent
Approximate
MITC4+N Exact
•
37
• •
Bending performance & Membrane
Patch test
Treat
< Optimal sampling points >
Solution
Construct new assumed membrane strain 1) Use five optimal points 2) identical to displacement-based
element for flat geometry
))0,0()0,1()0,1()1,0()1,0( 125223222111110* ececececece
iir xA4
1 iis xB
4
1 iiisr xC
4
1
< Locking-causing part > *e
◇ Kulikov et al. Int J Num Meth Eng 2010:83;1376-406. ◇ Choi and Paik. Comp Meth Appl Mech Eng 1994:2;17-34.
38
•
•
•
dr
m sr
xxx
ds
m rs
xxx
dr
m sr
uuu
ds
m rs
uuu
)1)(1(4
1),( sηrξsrh iii 11114321 ξξξξ
11114321 ηηηη
4
14
1
i
iir ξ xx
4
14
1
i
iis η xx
4
14
1
i
iiid ηξ xx
sr
sr
xx
xxn
i
jr
rδ
j
i xm 0nm ir
39
Optimally-converging element Displacement-based element
•
2
bil.lin.con.seseee m
rs
m
rr
m
rr
m
rr
2
.billin.con.rereee m
rs
m
ss
m
ss
m
ss
rsesereee m
rs
m
ss
m
rr
m
rs
m
rs bil.lin..lincon. 2
1
2
1
rr
m
rre ux con. ss
m
sse ux con.
rssr
m
rse uxux 2
1
con.
rddr
m
rre uxux lin. sdds
m
sse uxux lin.
dd
m
rse ux .bil
: Locking-causing term (rs bilinear term)
: Terms that should be consistently changed to pass patch test
seeeseeeee m
rr
m
rs
m
rr
Bm
rr
Am
rr
Bm
rr
Am
rr
m
rr lin.bil.con.
)()()()(
2
1
2
1
reeereeeee m
ss
m
rs
m
ss
Dm
ss
Cm
ss
Dm
ss
Cm
ss
m
ss lin.bil.con.
)()()()(
2
1
2
1
con.
)()()()()(
4
1 m
rs
Em
rs
Dm
rs
Cm
rs
Bm
rs
Am
rs
m
rs eeeeeee
m
rr
m
rr ee
ˆ m
ss
m
ss ee
ˆ sereee m
ss
m
rr
m
rs
m
rs lin.lin. 2
1
2
1ˆ
◇ Choi and Paik. Comp Meth Appl Mech Eng 1994:2;17-34. ◇ Roh and Cho. Comp Meth Appl Mech Eng 2004:193;2261-99.
2
.bil.bilˆ seeee m
rs
m
rs
m
rr
m
rr
2
.bil.bilˆ reeee m
rs
m
rs
m
ss
m
ss
rseee m
rs
m
rs
m
rs .bil
ˆ
: Terms that should be consistently changed
40
rr xu
ss xu
dd xu
•
•
lin.lin.con.bil.con.bil.con..bil
~ m
ss
m
rr
m
rs
m
rs
m
ss
m
rs
m
rr
m
rs eEeDeCeeBeeAe
◇ Kulikov et al. Int J Num Meth Eng 2010:83;1376-406.
rr au
ss au
0u ddd xu
0u r
0u s
For flat geometry ⇒ distortion vector has only in-plane parts … (*)
The new term should be same as displacement-based term for following in-plane modes
2 stretching 2 bending 1 shearing
s
s
dr
r
dd xmxxmxx )()(
nnxxmxxmxx )()()( ds
s
dr
r
dd
.bil
~m
rse dd
m
rse ux .bil
rasa
with arbitrary constant vectors and
41
•.bil.bil
~ m
rs
m
rs ee 0nxd
0~.bilm
rse rr
m
rs
m
rr ee ax bil.con.
rssr
m
rse axax 2
1
con. ss
m
rs
m
ss ee ax bil.con. rd
m
rre ax lin.
sd
m
sse ax lin.
02/2/ sdsrrdsr EBCDCA axxxaxxx
0xxx dsr DCA 2/ 0xxx dsr EBC 2/
dcA r /2 dcB s /2 dccC sr /2 dcD r / dcE s /
r
drc mx s
dsc mx d
lin.lin.con.bil.con.
2
bil.con.
2
.bil
2~ m
sssm
rrrm
rssrm
rs
m
sssm
rs
m
rrrm
rs ed
ce
d
ce
d
ccee
d
cee
d
ce … (**)
42
ddrr
m
rs
m
rr ee xxxx .bilcon.
ddss
m
rs
m
ss ee xxxx .bilcon.
ds
m
sse xx 2lin.
dr
m
rre xx 2lin.sr
m
rse xx con.
dd
m
rse xx .bil
~dr
m
rre xx lin.
ds
m
sse xx lin.
dd
m
rs
m
rr ee xx .bilcon.
dd
m
rs
m
ss ee xx .bilcon.
•.bil.bil
~ m
rs
m
rs ee 0nxd
0con.
m
rse
122 sr ccd
ddssrrssrrssrr cccccc xxxxxxxx )(2)()(
.bil.bil
~ m
rsdd
m
rs ee xx
43
•
)(2)(2)(2 12212
1221
2
1~ Cm
ssC
Bm
rrBB
Am
rrAA
m
rr esaesasaesasae
)(2)(2 11 Em
rsE
Dm
ssD esaesa
)(2)(2)(2 2212
111~ Cm
ssCC
Bm
rrB
Am
rrA
m
ss eraraeraerae
)(2)(2 12212
1 Em
rsE
Dm
ssDD eraerara
)()()( 44
14
4
14
4
1~ Cm
ssC
Bm
rrB
Am
rrA
m
rs ersasersarersare
)()( 144
1 Em
rsE
Dm
ssD ersaersas
d
cca rr
A2
)1(
d
cca rr
B2
)1(
d
cca ss
C2
)1(
d
cca ss
D2
)1(
d
cca sr
E
2
•
r
drc mx
s
dsc mx
Distortion measured by in-plane vector i.e. in-plane distortion
Distortion measured by in-plane vector i.e. in-plane distortion
rm
sm
1))(())(( 1342 r
e
r
e
s
e
s
ed mxmxmxmx Distortion of pair of element edges
44
•
•
Shell elements Zero energy mode test
Isotropic test Patch test
MITC4 Pass Pass Pass
MITC4+ Pass Pass Pass bending
and shear
MITC4+N Pass Pass Pass
New MITC4+ Pass Pass Pass
E=1.0, ν=1/3
45
•
E=1.0×103, ν=0.0
E=2.0×105, ν=0.0
46
•
Shell elements Remark Drawback
MITC4 Widely used -
New MITC4+ Present study -
S4 ABAQUS -
S4R ABAQUS Artificial (Stabilization) parameter and
Displacement projection
Nx6N mesh
E=2.9×107, ν=0.22, L=12, b=1.1, t=0.32 or 0.0032
47
•
NxN mesh
: Performances of MITC4, New MITC4+ and ABAQUS S4 are nearly identical !
NxN mesh
E=6.825×107, ν=0.3, R=10, Φ0=18º, t=0.04
E=3.0×106, ν=0.3, R=300, L=600, t=3
48
•
•
•
ref
ref
T
shref dΩ
ΩΔΔ τεuu2
2
2
sref
shref
hEu
uu
NLLL N :...:2:1:...:: 21
NLh / NRh / L R
49
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Clamped
Clamped
50
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Clamped
Clamped
51
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Free
Free
52
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Clamped
Clamped
P
53
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Free
Free
P
54
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Clamped
P
55
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Free
P
56
•
57
•
Convergence behavior in Regular mesh distorted mesh
8x8 mesh
12x12 mesh E=6.825×107, ν=0.3, R=10, Φ0=18º, t=0.04
58
•
Convergence behavior in Regular mesh distorted mesh
t/L=1/100
t/L=1/10000
t/L=1/1000
E=2.1×106, ν=0.0, R=10, L=20, θ=30º, M=M0×t3
59
•
In-plane load
Out-of-plane load
E=2.9×107, ν=0.22, L=12, b=1.1, t=0.0032
60
61
•
⇒
⇒
•
◇ Klinkel S et al. Comp Meth Appl Mech Engrg 2006:195;179-208.
◇ Hauptmann R and Schweizerhof K. Int J Numer Meth Engrg 1998:42;49-69.
Thickness locking
mechanism
Bending, membrane behaviors
Start
62
•
•
•
•
•
•
◇ MacNeal RH. Int J Numer Meth Engrg 1987:24;1793-99.
◇ Sze KY and Yao LQ. Int J Numer Meth Engrg 2000:48;545-64. < Assumed geometry >
< Assumed displacement >
◇ Betsch P and Stein E. Comm Numer Meth Engrg 1995:11;899-909.
◇ Bischoff M and Ramm E. Int J Numer Meth Engrg 1997:40;4427-49.
◇ Nguyen NH. ACOMEN 2008.
63
•
•
64
eeeesre UB33330330330330 ))1,0()0,1()0,0((3
1),(~
Assumed Strain method ⇒ alleviate shear and curvature thickness locking
: comes from previous treatments
: comes from MITC3+
e
CCAA rceeeee UB23
)(
230
)(
130
)(
130
)(
230230 )31(~
3
1)(
3
1)
2
1(
3
2~
e
CCBB sceeeee UB23
)(
230
)(
130
)(
230
)(
130130 )13(~
3
1)(
3
1)
2
1(
3
2~ )(
230
)(
230
)(
130
)(
130~ EFDF eeeec
•
: comes from previous treatments
eijji
t
j
t
i
enh
ije ΛGuggu )(2
1 plane-in
,
plane-in
,0
)(2
121
plane-inVVu βαthb
3
thickness
2
1Vu γhq
)1(27 srrshb
21 thq
: comes from MITC3+
e
CbubCbubAbubAbubenh eeeee ΛG23
)(
230
)(
130
)(
130
)(
230230 )(3
1)
2
1(
3
2
e
CbubCbubBbubBbubenh eeeee ΛG13
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230
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130130 )(3
1)
2
1(
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2
e
tenhe ΛGug 33
thickness
3,3330
65
Enhanced Assumed Strain method ⇒ alleviate shear and Poisson thickness locking
T
e γβα ][Λ
•
•
Shell elements Zero energy mode test
Isotropic test Patch test
MITC-S6 Pass Pass Pass
MITC-S8 Pass Pass Pass
SC6R in ABAQUS
Pass Pass Pass
Sze et al. Pass Pass Pass
◇ Sze KY et al. Fin Elem Anal Des 2001;37:639-55.
E=3.0×106, ν=0.3, R=300, L=600, t=3
66
•
67
•
Convergence behavior in (a) Regular mesh (b) distorted mesh
Problem definition
Free
Free
: just shear locking treatment
: Both shear and thickness locking treatment
P
68
69
Development of the accurate 4-node degenerated shell finite element
Development of the accurate 6-node solid shell finite element
•
•
•
•
Improving 4-node quadrilateral degenerated shell finite element
Improving 6-node triangular solid-shell finite element
•
•
•
70
71
Relative error
where
: the reference solution obtained by a very fine mesh (a mesh of 72x72 MITC9 shell elements)
: the solution of the finite element discretization with NxN meshes (N = 4, 8, 16, 32 and 64)
: One-to-one mapping
: For optimal convergence behavior for low-order shell elements
h is the element size, C must be constant, k=2
ref
ref
T
shref dΩ
ΩΔΔ τεuu2
2
2
sref
shref
hEu
uu
href εεε Δ href τττ Δ )( href xx Π
refu
hu
Π
k
h ChE
72
Use projected displacements
where
Matrix R for node I is
originalPuu
T1T RR)R(RΙP
100
010
001
0)(
0)(
)(0
CICI
CICI
CICI
I
xxyy
xxzz
yyzz
R
z
I
y
I
x
I
z
I
y
I
x
I
CICI
CICI
CICI
I
VVV
VVV
xxyy
xxzz
yyzz
111
222
0)(
0)(
)(0
R
6 D.O.F. per node 5 D.O.F. per node
73
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