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

•

•

3

•

•

•

•

4

•

•

•

•

•

)log()log(

)(log)(log

εεΔε

εεΔερ

EEE ε

1ρ 3ρ 31 ρ

5

•

•

6

•

• •

•

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.

7

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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

•

⇒

•

⇒

◇ 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

•

< 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

•

•

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

•

•

18

•

•

•

•

< 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

•

•

•

< 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

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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

)(

230

)(

130

)(

230

)(

130130 )(3

1)

2

1(

3

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