8/13/2019 Numerical analysis of junctions between thin shell.pdf
1/39
ISSN0
249-6399
appor t
de r ec h er c he
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Numerical analysis of junctions between thin
shells, Part 2 : Approximation by finite element
methods
Michel Bernadou , Annie Cubier
N 2922
Juin 1996
THEME 4
8/13/2019 Numerical analysis of junctions between thin shell.pdf
2/39
8/13/2019 Numerical analysis of junctions between thin shell.pdf
3/39
N u m e r i c a l a n a l y s i s o f j u n c t i o n s b e t w e e n t h i n s h e l l s ,
P a r t 2 : A p p r o x i m a t i o n b y n i t e e l e m e n t m e t h o d s
M i c h e l B e r n a d o u
*
, A n n i e C u b i e r
* *
T h e m e 4 | S i m u l a t i o n
e t o p t i m i s a t i o n
d e s y s t e m e s c o m p l e x e s
P r o j e t M O D U L E F
R a p p o r t d e r e c h e r c h e n 2 9 2 2 | J u i n 1 9 9 6 | 3 6 p a g e s
A b s t r a c t : T h e p u r p o s e o f t h i s w o r k i s t o s t u d y t h e n u m e r i c a l a n a l y s i s o f j u n c t i o n s
b e t w e e n t h i n s h e l l s . W e d e s c r i b e t h e a p p r o x i m a t i o n b y a \ p s e u d o - c o n f o r m i n g " n i t e
e l e m e n t m e t h o d a s s o c i a t e d w i t h t h e A r g y r i s t r i a n g l e a n d t a k i n g i n t o a c c o u n t t h e
n u m e r i c a l i n t e g r a t i o n . U n d e r s u i t a b l e h y p o t h e s i s o n t h e i n t e g r a t i o n s c h e m e s a n d
o n t h e d a t a , w e p r o v e t h e c o n v e r g e n c e o f t h i s m e t h o d a n d w e d e r i v e a p r i o r i e r r o r
e s t i m a t e s .
K e y - w o r d s : T h i n s h e l l s . E l a s t i c j u n c t i o n . R i g i d j u n c t i o n . A r g y r i s T r i a n g l e .
N u m e r i c a l i n t e g r a t i o n . E r r o r e s t i m a t e . N u m e r i c a l s i m u l a t i o n s
( R e s u m e : t s v p )
*
P o l e U n i v e r s i t a i r e L e o n a r d d e V i n c i
* *
I N R I A R o c q u e n c o u r t
8/13/2019 Numerical analysis of junctions between thin shell.pdf
4/39
A n a l y s e n u m e r i q u e d e j o n c t i o n s d e c o q u e s m i n c e s ,
P a r t i e 2 : A p p r o x i m a t i o n s p a r d e s m e t h o d e s d ' e l e m e n t s
n i s
R e s u m e : L e b u t d e c e t a r t i c l e e s t l ' a n a l y s e n u m e r i q u e d u p r o b l e m e d e j o n c -
t i o n s d e c o q u e s m i n c e s . N o u s d e c r i v o n s u n e m e t h o d e d ' a p p r o x i m a t i o n " p s e u d o -
c o n f o r m e " u t i l i s a n t l ' e l e m e n t n i d ' A r g y r i s e t p r e n a n t e n c o m p t e l e s p h e n o m e n e s
l i e s a l ' i n t e g r a t i o n n u m e r i q u e . S o u s d e s h y p o t h e s e s c o n v e n a b l e s s u r l e s s c h e m a s
d ' i n t e g r a t i o n n u m e r i q u e s e t s u r l e s d o n n e e s , n o u s m o n t r o n s l a c o n v e r g e n c e d e c e t t e
m e t h o d e e t n o u s d o n n o n s d e s e s t i m a t i o n s d ' e r r e u r a p r i o r i t a n t p o u r l e p r o b l e m e d e
j o n c t i o n e l a s t i q u e q u e p o u r l e p r o b l e m e d e j o n c t i o n r i g i d e
M o t s - c l e : C o q u e s m i n c e s . J o n c t i o n e l a s t i q u e . J o n c t i o n r i g i d e . T r i a n g l e
d ' A r g y r i s . I n t e g r a t i o n n u m e r i q u e . E s t i m a t i o n s d ' e r r e u r . S i m u l a t i o n s n u m e r i q u e s .
8/13/2019 Numerical analysis of junctions between thin shell.pdf
5/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 3
1 I n t r o d u c t i o n
T h e n u m e r i c a l a n a l y s i s o f j u n c t i o n s b e t w e e n t h i n p l a t e s w a s s t u d i e d b y B e r n a d o u -
F a y o l l e - L e n e ( 1 9 8 9 ) w h i l e F a y o l l e ( 1 9 8 7 ) t h o r o u g h l y d e s c r i b e d t h e c o r r e s p o n d i n g
a p p r o x i m a t i o n b y c o n f o r m i n g n i t e e l e m e n t m e t h o d s . S u c h a n a p p r o x i m a t i o n i s
m u c h m o r e c o m p l i c a t e d i n c a s e o f j u n c t i o n s b e t w e e n t w o g e n e r a l t h i n s h e l l s . I n d e e d
t h e t r a n s m i s s i o n c o n d i t i o n s w h i c h a p p e a r i n t h e d e n i t i o n o f t h e a d m i s s i b l e s p a c e s
c a n n o t b e e x a c t l y s a t i s e d i n t h e a s s o c i a t e d d i s c r e t e s p a c e s . T h u s , w e c o n s i d e r
p s e u d o - c o n f o r m i n g n i t e e l e m e n t m e t h o d s w h i c h a r e c o n f o r m i n g e v e r y w h e r e e x c e p t
a l o n g t h e j u n c t i o n .
I n t h e r s t p a r t o f t h i s w o r k ( B e r n a d o u - C u b i e r , t o a p p e a r ) , w e a n a l y s e t h e c o n t i -
n u o u s p r o b l e m s o f j u n c t i o n s b e t w e e n t w o t h i n s h e l l s a s s o c i a t e d w i t h a n e l a s t i c o r a
r i g i d b e h a v i o u r o f t h e h i n g e . W e s t a r t b y g i v i n g t h e e q u i l i b r i u m e q u a t i o n s o f t h e s e
p r o b l e m s a n d t h e c o r r e s p o n d i n g v a r i a t i o n a l f o r m u l a t i o n s . W e s t u d y t h e n u m e r i c a l
p r o p e r t i e s o f t h e s e s e q u a t i o n s a n d s h o w t h e e x i s t e n c e a n d u n i q u e n e s s o f t h e s o l u -
t i o n . W e a l s o p r o v e t h a t t h e s o l u t i o n o f t h e e l a s t i c j u n c t i o n p r o b l e m c o n v e r g e s t o
t h e s o l u t i o n o f t h e r i g i d j u n c t i o n p r o b l e m w h e n t h e c o e c i e n t o f e l a s t i c s t i n e s s o f
t h e h i n g e b e c o m e s v e r y l a r g e .
I n t h i s s e c o n d p a r t o f t h e w o r k , w e s t a r t i n S e c t i o n 2 b y r e w r i t i n g t h e c o n t i -
n u o u s p r o b l e m s i n t e r m s o f m a t r i c e s a n d v e c t o r s , w h i c h a r e w e l l a d a p t e d t o t h e
a p p r o x i m a t i o n b y n i t e e l e m e n t m e t h o d s . I n S e c t i o n 3 , w e b u i l d t h e d i s c r e t e s p a c e s
w h i c h a r e a s s o c i a t e d w i t h t h e A r g y r i s t r i a n g l e , a n d w h i c h c o n t a i n t h e d i s c r e t e j u n c -
t i o n c o n d i t i o n s . T h e d i s c r e t i z a t i o n o f t h e t r a n s m i s s i o n c o n d i t i o n s i s b a s e d o n t h e
r e s u l t s o b t a i n e d b y Z e n i s e k ( 1 9 8 1 ) f o r n o n h o m o g e n e o u s b o u n d a r y c o n d i t i o n s . F r o m
S e c t i o n 4 , w e r e s t r i c t o u r a t t e n t i o n t o t h e e l a s t i c j u n c t i o n p r o b l e m ; w e g i v e t h e
m a i n r e s u l t s c o n c e r n i n g t h e r i g i d j u n c t i o n p r o b l e m i n S e c t i o n 6 . T h e n , w e s t a t e
t h e r s t d i s c r e t e p r o b l e m t a k i n g i n t o a c c o u n t t h e n i t e e l e m e n t a p p r o x i m a t i o n a n d
t h e n o n c o n f o r m i t y o f t h e m e t h o d a l o n g t h e h i n g e . W e i n t r o d u c e i n t h i s p r o b l e m a n
a d d i t i o n a l l i n e a r f o r m w h i c h i s v o i d f o r c o n t i n u o u s p r o b l e m a n d w h i c h t a k e s i n t o
a c c o u n t t h e n o n c o n f o r m i t y o f t h e m e t h o d . T h e n w e p r o v e t h e e x i s t e n c e a n d u n i q u e -
n e s s o f t h e s o l u t i o n . T h i s r e s u l t i s b a s e d o n t h e t r a n s m i s s i o n o f a c l a m p e d c o n d i t i o n
b y t h e d i s c r e t e j u n c t i o n c o n d i t i o n s d e s c r i b e d i n S e c t i o n 3 . A f t e r w a r d s , w e g i v e a n
a b s t r a c t e r r o r e s t i m a t e w h i c h i s r e d u c e d t o t h e u s u a l i n t e r p o l a t i o n e r r o r ; t h i s i s
a d i r e c t c o n s e q u e n c e o f t h e d e n i t i o n o f t h e d i s c r e t e p r o b l e m b y u s i n g a n a d d i t i o -
n a l l i n e a r f o r m a s m e n t i o n n e d b e f o r e . I n S e c t i o n 5 , w e s t u d y t h e s e c o n d d i s c r e t e
p r o b l e m w h i c h t a k e s i n t o a c c o u n t t h e a d d i t i o n a l e e c t o f t h e n u m e r i c a l i n t e g r a t i o n .
W e p r o v e t h e u n i f o r m e l l i p t i c i t y o f t h e b i l i n e a r f o r m s a n d t h u s t h e e x i s t e n c e a n d
u n i q u e n e s s o f t h e s o l u t i o n o f t h i s p r o b l e m . N e x t , w e g i v e c r i t e r i a o n t h e i n t e g r a t i o n
s c h e m e s a n d r e q u i r e d r e g u l a r i t y c o n d i t i o n s o n t h e d a t a s o t h a t t h e n i t e e l e m e n t
m e t h o d c o n v e r g e s . T h e s e c o n d i t i o n s l e a d t o a s y m p t o t i c e r r o r e s t i m a t e s o f t h e s a m e
o r d e r t h a n t h e i n t e r p o l a t i o n e r r o r . T h e s e r e s u l t s a r e b a s e d o n l o c a l e r r o r e s t i m a t e s
s t u d i e d b y B e r n a d o u ( 1 9 9 6 ) a n d C u b i e r ( 1 9 9 4 ) . I n S e c t i o n 7 , w e i l l u s t r a t e p r e v i o u s
r e s u l t s b y s o m e n u m e r i c a l t e s t s o n r i g i d a n d e l a s t i c j u n c t i o n s b e t w e e n a c y l i n d e r a n d
a s p h e r i c a l e n d c a p . I n o r d e r t o v a l i d a t e o u r r e s u l t s , w e c o n s i d e r a c o r r e s p o n d i n g
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
6/39
4 M i c h e l B e r n a d o u , A n n i e C u b i e r
t e s t u s i n g t h e t h r e e d i m e n s i o n a l e l a s t i c i t y m o d e l i n s t e a d o f a s h e l l m o d e l a n d w h i c h
r e p r e s e n t s a g o o d a p p r o x i m a t i o n o f a r i g i d j u n c t i o n . T h e r e s u l t o f t h i s t e s t i s v e r y
c l o s e d t o o u r s o b t a i n e d f o r r i g i d j u n c t i o n . M o r e o v e r , w e n d a g a i n n u m e r i c a l l y t h a t
t h e e l a s t i c j u n c t i o n b e c o m e s a l m o s t r i g i d w h e n t h e c o e c i e n t o f e l a s t i c s t i n e s s o f
t h e h i n g e b e c o m e s v e r y l a r g e .
N o t a t i o n s a n d r e f e r e n c e s : I n t h i s s e c o n d p a r t , w e m a k e m a n y r e f e r e n c e s t o t h e
n o t a t i o n s a n d t o t h e r e s u l t s o f P a r t 1 , j u s t b y a d d i n g " P a r t 1 " b e f o r e e a c h r e f e r e n c e .
2 V a r i a t i o n a l f o r m u l a t i o n s i n m a t r i x f o r m
I n t h i s s e c t i o n , w e g i v e n e w e x p r e s s i o n s i n t e r m s o f v e c t o r s a n d m a t r i c e s f o r t h e
b i l i n e a r a n d l i n e a r f o r m s w h i c h a p p e a r i n t h e v a r i a t i o n a l f o r m u l a t i o n s o f t h e e l a s t i c
a n d r i g i d j u n c t i o n p r o b l e m s ( P a r t 1 , ( 3 . 1 0 ) a n d ( 3 . 2 4 ) ) . T h e b i l i n e a r f o r m a ;
w h i c h r e p r e s e n t s t h e a d d i t i o n o f t h e s t r a i n e n e r g y o f b o t h s h e l l s , c a n b e w r i t t e n
a ( u ; u
) ; ( v ; v
) =
Z
T
U A V d
1
d
2
+
Z
T
U
A
V
d
1
d
2
; ( 2 . 1 )
w h e r e t h e c o l u m n m a t r i x V ] ( a n d s i m i l a r l y U ; U
; V
] ) i s g i v e n b y :
T
V
1 1 2
= v
1
v
1 1
v
1 2
v
2
v
2 1
v
2 2
v
3
v
3 1
v
3 2
v
3 1 1
v
3 1 2
v
3 2 2
; ( 2 . 2 )
a n d w h e r e t h e s y m m e t r i c a l 1 2 1 2 m a t r i x A ] ( r e s p e c t i v e l y A
] ) d e p e n d s o n l y o n t h e
s h e l l t h i c k n e s s e , o n t h e m e c h a n i c a l c h a r a c t e r i s t i c s o f t h e s h e l l a n d o n t h e r s t , s e -
c o n d a n d t h i r d p a r t i a l d e r i v a t i v e s o f t h e a p p l i c a t i o n :
!
S ( r e s p .
:
!
S
)
w h i c h m a p s ( r e s p .
) o n t o t h e m i d d l e s u r f a c e S ( r e s p . S
) . S u b s e q u e n t l y , w e a s -
s u m e t h a t 2 ( C
3
(
) )
3
a n d
2 ( C
3
(
) )
3
T h e s e c o n d b i l i n e a r f o r m b ; ] w h i c h a p p e a r s i n t h e v a r i a t i o n a l f o r m u l a t i o n o f
e l a s t i c j u n c t i o n b e t w e e n s h e l l s ( P a r t 1 , ( 3 . 1 0 ) ) i s a s s o c i a t e d w i t h t h e s t r a i n e n e r g y
o f t h e h i n g e . W e d e n e a 2 4 2 4 m a t r i x C ] w h i c h o n l y d e p e n d s o n t h e g e o m e t r y
o f t h e h i n g e , a n d a c o l u m n v e c t o r V V
2 4 1
w h i c h c o l l e c t s t h e v e c t o r s V ] a n d
V
] d e n e d b y r e l a t i o n ( 2 . 2 ) . T h u s , w e h a v e :
b ( u ; u
) ; ( v ; v
) =
Z
?
T
U U
C V V
d s ( 2 . 3 )
M o r e o v e r , w e i n t r o d u c e a n e w p a r a m e t e r i z a t i o n o f t h e h i n g e ? , a s t h e i m a g e o f
a o n e - d i m e n s i o n a l i n t e r v a l ! = 0 ; 1 b y a m a p p i n g , i . e . , : ! !
? . N o w , w e
c a n s u b s t i t u t e t h i s a p p l i c a t i o n i n t o r e l a t i o n ( 2 . 3 ) t o o b t a i n
b ( u ; u
) ; ( v ; v
) =
Z
!
T
U U
C V V
d ! ; ( 2 . 4 )
w h e r e t h e u n d e r l i n e d q u a n t i t i e s a r e o b t a i n e d b y c o m p o s i t i o n w i t h t h e m a p p i n g
a n d a r e d e n e d o n t h e i n t e r v a l ! . T h e e l e m e n t d ! i s a s s o c i a t e d t o t h e l i n e e l e m e n t
d s a l o n g t h e h i n g e ? t h r o u g h t h e m a p p i n g =
3
X
i = 1
i
( t ) e
i
:
d s =
h
( d x
1
)
2
+ ( d x
2
)
2
+ ( d x
3
)
2
i
1 = 2
;
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
7/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 5
s o t h a t
d ! =
h
(
0
1
( t ) )
2
+ (
0
2
( t ) )
2
+ (
0
3
( t ) )
2
i
1 = 2
d t
T h e l i n e a r f o r m ` ] w h i c h r e p r e s e n t s t h e w o r k o f t h e e x t e r n a l l o a d s c a n b e w r i t t e n
( P a r t 1 , ( 2 . 1 8 ) a n d ( 3 . 1 3 ) )
` ( v ; v
) =
Z
T
P V d
1
d
2
+
Z
T
P
V
d
1
d
2
+
R
1
T
L
s
V d +
R
1
T
L
s
V
d
;
9
>
>
>
=
>
>
>
;
( 2 . 5 )
w h e r e
T
P
1 1 2
=
p
a p
1
0 0 p
2
0 0 p
3
0 0 0 0 0 ] ( 2 . 6 )
a n d
T
L
s 1 1 2
=
q
a
( g
)
0
( g
)
0
N
1
+ b
1
M
0 0 N
2
+ b
2
M
0 0
N
3
M
1
M
2
0 0 0 ]
9
>
=
>
;
( 2 . 7 )
I n t h e a b o v e e q u a t i o n s , p
i
a n d N
i
d e n o t e r e s p e c t i v e l y t h e c o v a r i a n t c o m p o n e n t s o f
t h e b o d y f o r c e r e s u l t a n t , o f t h e r e s u l t a n t a n d o f t h e r e s u l t a n t m o m e n t o f t h e s u r f a c e
l o a d s w h i l e b
a n d a a r e t h e s e c o n d f u n d a m e n t a l f o r m a n d t h e d e t e r m i n a n t o f t h e
r s t f u n d a m e n t a l f o r m ; a l l t h e s e q u a n t i t i e s a r e r e f e r e d t o t h e m i d d l e s u r f a c e S a n d ,
b y d e n i t i o n , M = M
a
a
3
. T h e v e c t o r s P
; L
s
] a r e o b t a i n e d b y a n a l o g y .
3 C o n s t r u c t i o n o f t h e d i s c r e t e a d m i s s i b l e s p a c e s
F r o m n o w o n , w e s h a l l a s s u m e t h a t t h e d o m a i n s a n d
h a v e p o l y g o n a l b o u n -
d a r i e s . T h e n , w e c a n e x a c t l y c o v e r t h e s e d o m a i n s b y f a m i l i e s o f t r i a n g u l a t i o n s T
h
a n d T
h
. S u b s e q u e n t l y , w e a s s u m e t h a t t h e s e t r i a n g u l a t i o n s a r e c o m p a t i b l e a l o n g t h e
p a r t s a n d
o f t h e b o u n d a r i e s @ a n d @
: i n o t h e r w o r d s , t h e i r t r a c e s u p o n
a n d
a r e t h e i m a g e s o f a o n e d i m e n s i o n a l t r i a n g u l a t i o n T
h
o f t h e i n t e r v a l !
t h r o u g h t h e m a p p i n g s F =
? 1
?
a n d F
=
? 1
?
( F i g u r e 1 ) . F r o m n o w o n ,
f o r s i m p l i c i t y , w e n o t e
? 1
?
a n d
? 1
?
b y
? 1
a n d
? 1
. A l l t h e s e t r i a n g u l a t i o n s
a r e a s s u m e d t o b e r e g u l a r i n t h e s e n s e t h a t :
i ) T h e r e e x i s t s c o n s t a n t s a n d
s u c h t h a t
8 K 2 T
h
;
h
K
K
a n d 8 K
2 T
h
;
h
K
K
;
( 3 . 1 )
w h e r e h
K
= d i a m ( K ) ; h
K
= d i a m ( K
) ,
K
= s u p f d i a m ( S ) ; S i s a b a l l c o n t a i n e d i n K g
a n d
K
= s u p f d i a m ( S
) ; S
i s a b a l l c o n t a i n e d i n K
g
i i ) L e t h b e a r e a l n u m b e r d e n e d b y
h = s u p f m a x
K 2 T
h
h
K
; m a x
K
2 T
h
h
K
; m a x
K 2 T
h
h
K
g ; ( 3 . 2 )
w h e r e h
K
= d i a m ( K ) . T h e n , w e a s s u m e t h a t
h ! 0 ( 3 . 3 )
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
8/39
6 M i c h e l B e r n a d o u , A n n i e C u b i e r
1
2
a
0
a
1
a
3
a
2
F
1
!
S
?
?
0
S
e
3
e
1
1
1
s
0
= 0 s
1
s
2
s
3
= 1
s
F
?
1
e
2
2
a
0
?
1
?
1
1
0
a
3
a
2
a
1
F i g . 1 : D i s c r e t i z a t i o n o f t h e h i n g e i m a g e s
W i t h t h e t r i a n g u l a t i o n s T
h
a n d T
h
, w e a s s o c i a t e t h e n i t e e l e m e n t s p a c e s X
h
a n d X
h
c o n s t r u c t e d f r o m t h e A r g y r i s t r i a n g l e ( A r g y r i s - F r i e d - S c h a r p f ( 1 9 6 8 ) ) , w h o s e
d e n i t i o n i s r e c a l l e d i n F i g u r e 2 , a n d t h e s p a c e s V
h
a n d V
h
:
V
h
= V
h 1
V
h 1
V
h 2
; V
h
= ( X
h
)
3
w h e r e
V
h 1
= f v 2 X
h
; v = 0 a l o n g
0
g a n d V
h 2
= f v 2 X
h
; v = v
= 0 a l o n g
0
g
a n d w h e r e i s t h e o u t w a r d u n i t n o r m a l v e c t o r t o t h e b o u n d a r y
0
. T h e s e d e n i t i o n s
a n d t h o s e o f ( P a r t 1 , ( 3 . 7 ) a n d ( 3 . 8 ) ) l e a d t o t h e i n c l u s i o n s
V
h
V a n d V
h
V
( 3 . 4 )
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
9/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 7
a3a2
a1
b2
b1
b3
c1
c2
c
3
P
K
= P
5
( K ) d i m P
K
= 2 1
K
= f p ( a
i
) D p ( a
i
) ( a
i 1
? a
i
) D p ( a
i
) ( a
i + 1
? a
i
) 1 i 3
D
2
p ( a
i
) ( a
j + 1
? a
j 1
)
2
1 i j 3 D p ( b
i
) ( a
i
? c
i
) 1 i 3 g
F i g . 2 : T h e A r g y r i s t r i a n g l e
I n o r d e r t o b u i l d t h e d i s c r e t e a d m i s s i b l e s p a c e s , w e h a v e t o d i s c r e t i z e t h e j u n c t i o n
c o n d i t i o n s ( e q u a l i t y o f t h e d i s p l a c e m e n t s a n d e q u a l i t y o f t h e t a n g e n t i a l c o m p o n e n t s
o f t h e r o t a t i o n s a l o n g t h e h i n g e ) , i . e . , w e h a v e t o e x p r e s s t h e s e c o n d i t i o n s i n t e r m s o f
t h e d e g r e e s o f f r e e d o m . T h i s i s a d e l i c a t e s t e p i n t h e a p p r o x i m a t i o n o f t h e c o n t i n u o u s
p r o b l e m s a n d i t l e a d s t o t h e n o n - c o n f o r m i t y o f t h e m e t h o d f o r t h e a p p r o x i m a t i o n
o f t h e t r a n s m i s s i o n c o n d i t i o n s a l o n g t h e h i n g e . T h e e q u a l i t y o f t h e d i s p l a c e m e n t s
a l o n g t h e h i n g e i s a c o n d i t i o n w h i c h a p p e a r s f o r t h e e l a s t i c o r t h e r i g i d j u n c t i o n
p r o b l e m s a s w e l l . T h u s w e b e g i n b y s t u d y i n g t h e d i s c r e t i z a t i o n o f t h i s c o n d i t i o n .
3 . 1 T h e d i s c r e t e a d m i s s i b l e s p a c e f o r t h e e l a s t i c j u n c t i o n p r o b l e m
F i r s t , l e t u s r e c a l l t h e c o n d i t i o n o f c o n t i n u i t y o f t h e d i s p l a c e m e n t a l o n g t h e h i n g e
( P a r t 1 , ( 2 . 3 1 )
1
) :
u ( ) = u
(
) ; 8 2 ; 8
2
s u c h t h a t ( ) =
(
) ( 3 . 5 )
T h e e q u a t i o n ( 3 . 5 ) i s v e c t o r i a l . F o r i t s d i s c r e t i z a t i o n , w e h a v e t o u s e c o m p o n e n t s
o f d i s p l a c e m e n t s . T h e v e c t o r s u a n d u
a r e e x p r e s s e d u p o n t h e c o n t r a v a r i a n t b a s e s
f a
1
; a
2
; a
3
g a n d f a
1
; a
2
; a
3
g w h i c h d i e r a l o n g t h e h i n g e . T h e r e f o r e w e h a v e t o
w r i t e r e l a t i o n ( 3 . 5 ) u p o n o n e o f t h e s e b a s e s , f o r e x a m p l e f a
i
g :
u
i
(
) = A
j
i
( ;
) u
j
( ) ;
w h e r e A
j
i
( ;
) = a
i
(
) a
j
( )
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
10/39
8 M i c h e l B e r n a d o u , A n n i e C u b i e r
I n a d d i t i o n , w e i n t r o d u c e t h e t w o m a p p i n g s F : ! = 0 ; 1 ! a n d F
: ! !
w h i c h a r e a s s u m e d t o b e r e g u l a r . T h e c l o s e d i n t e r v a l ! i s s u b d i v i d e d i n t o n + 1 s e g -
m e n t s s
p
; s
p + 1
; f o r p = 0 ; : : ; n w i t h s
0
= 0 a n d s
n + 1
= 1 . T h u s F ( s
p
) = a
p
a n d
F
( s
p
) = a
p
w h e r e f a
p
g ; f a
p
g a r e t h e v e r t i c e s o f t h e t r i a n g l e s o f T
h
a n d T
h
l o c a t e d
a l o n g a n d
B y a n a l o g y w i t h Z e n i s e k ( 1 9 8 1 ) w h o c o n s i d e r e d t h e a p p r o x i m a t i o n o f n o n h o -
m o g e n e o u s b o u n d a r y c o n d i t i o n s , t h e a p p r o x i m a t i o n t h r o u g h A r g y r i s t r i a n g l e l e a d s
n a t u r a l l y t o i m p o s e t h e f o l l o w i n g c o n d i t i o n s :
u
h i
F
( s
`
) = A
j
i
u
h j
F ( s
`
) ;
d
d s
u
h i
F
( s
`
) =
d
d s
A
j
i
u
h j
F
( s
`
) ;
d
2
d s
2
u
h i
F
( s
`
) =
d
2
d s
2
A
j
i
u
h j
F
( s
`
) ;
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
( 3 . 6 )
f o r ` = 0 ; : : ; n + 1 a n d w h e r e w e h a v e s e t f o r c l a r i t y A
j
i
( s
`
) = A
j
i
( a
`
; a
`
) = A
j
i
( F ( s
`
) ; F
( s
`
) )
R e m a r k 3 . 1 . 1 : C o m p o n e n t s u
h i
a n d u
h j
a r e p i e c e w i s e t w o d i m e n s i o n a l v e d e g r e e
p o l y n o m i a l s . S i n c e w e h a v e s u p p o s e d t h a t a n d
a r e r e c t i l i n e a r , t h e m a p p i n g s F
a n d F
a r e a n e . T h u s , t h e c o m p o s e d m a p p i n g s u
h i
F a n d u
h i
F
a r e p i e c e w i s e
o n e d i m e n s i o n a l v e d e g r e e p o l y n o m i a l s .
N o w , w e h a v e t o r e w r i t e ( 3 . 6 ) i n t e r m s o f d e g r e e s o f f r e e d o m o f A r g y r i s t r i a n g l e
a n d t h u s t o e x p r e s s t h e s e c o n d i t i o n s o n t h e r e f e r e n c e d o m a i n s a n d
. T h e r e i s
n o p r o b l e m f o r ( 3 . 6 )
1
w h i c h c a n b e d i r e c t l y w r i t t e n o n t h e b o u n d a r i e s o r
. F o r
( 3 . 6 )
2
w e u s e t h e f o l l o w i n g e q u a t i o n :
d
d s
( u
h i
F ) ( s
`
) = D u
h i
( F ( s
`
) ) D F ( s
`
)
A u n i t t a n g e n t v e c t o r t o i s g i v e n b y : D F ( s
`
) ( s
`
) = D F ( s
`
) s o t h a t b y
s e t t i n g u
h i ;
( a
`
) = D u
h i
( a
`
) , w e o b t a i n
d
d s
( u
h i
F ) ( s
`
) = D F ( s
`
) u
h i ;
( a
`
) ( 3 . 7 )
W i t h s i m i l a r a r g u m e n t s a n d s i n c e F i s a n e , w e h a v e :
d
2
d s
2
( u
h i
F ) ( s
`
) = D F ( s
`
)
2
u
h i ;
( a
`
) ( 3 . 8 )
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
11/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 9
T h u s , t h e d i s c r e t e j u n c t i o n c o n d i t i o n s f o r e l a s t i c p r o b l e m a r e o b t a i n e d f o r ` =
0 ; : : : ; n + 1 b y s u b s t i t u t i n g ( 3 . 7 ) a n d ( 3 . 8 ) i n t o r e l a t i o n ( 3 . 6 ) :
u
h i
( a
`
) = A
j
i
( s
`
) u
h j
( a
`
) ;
u
h i ;
( a
`
) =
(
d
d s
A
j
i
) ( s
`
) u
h j
( a
`
) + D F ( s
`
) A
j
i
( s
`
) u
h j ;
( a
`
)
= D F
( s
`
) ;
u
h i ;
( a
`
) =
(
(
d
2
d s
2
A
j
i
) ( s
`
) u
h j
( a
`
) + 2 D F ( s
`
) (
d
d s
A
j
i
) ( s
`
) u
h i ;
( a
`
)
+ D F ( s
`
)
2
A
j
i
( s
`
) u
h j ;
( a
`
)
o
= D F
( s
`
)
2
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
( 3 . 9 )
T h u s , t h e a d m i s s i b l e d i s c r e t e d i s p l a c e m e n t s p a c e f o r t h e e l a s t i c j u n c t i o n p r o b l e m
i s d e n e d b y :
W
h e l
=
n
( v
h
; v
h
) 2 V
h
V
h
; s u c h t h a t r e l a t i o n s ( 3 . 9 ) a r e v e r i e d a t t h e
c o r r e s p o n d i n g v e r t i c e s f a
`
g a n d f a
`
g l o c a t e d o n a n d
o
9
>
>
=
>
>
;
( 3 . 1 0 )
3 . 2 T h e d i s c r e t e a d m i s s i b l e s p a c e f o r t h e r i g i d j u n c t i o n p r o b l e m
W e p r o c e e d b y s i m i l a r i t y f o r t h e r i g i d j u n c t i o n p r o b l e m w h i c h a m o u n t s t o d i s c r e -
t i z e t h e s e c o n d c o n d i t i o n ( P a r t 1 , ( 2 . 3 0 )
2
) r e l a t e d t o t h e e q u a l i t y o f t h e r o t a t i o n s .
T h e a p p r o x i m a t i o n t h r o u g h A r g y r i s t r i a n g l e l e a d s t o i m p o s e t h e f o l l o w i n g c o n d i -
t i o n s :
n
( u
h 3
+ b
u
h
) F ( s
`
) = ( t t
) n
( u
h 3
+ b
u
h
) F
( s
`
)
n
( u
h 3
+ b
u
h
) F ( q
|
) = ( t t
) n
( u
h 3
+ b
u
h
) F
( q
|
)
d
d s
f n
( u
h 3
+ b
u
h
) F g ( s
`
) =
d
d s
f ( t t
) n
( u
h 3
+ b
u
h
) F
g ( s
`
)
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
( 3 . 1 1 )
f o r ` = 0 ; : : ; n + 1 a n d | = 0 ; : : : ; n ; w h e r e n = n
a
i s t h e o u t w a r d u n i t n o r m a l v e c t o r
t o t h e j u n c t i o n ? i n t h e t a n g e n t p l a n e t o S a n d q
|
i s t h e m i d p o i n t o f s
`
; s
` + 1
I n o r d e r t o o b t a i n n o r m a l a n d t a n g e n t i a l d e r i v a t i v e s , w e u s e t h e r e l a t i o n
v
3
=
v
3
+
v
3
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
12/39
1 0 M i c h e l B e r n a d o u , A n n i e C u b i e r
w h e r e a n d a r e r e s p e c t i v e l y t h e u n i t t a n g e n t v e c t o r a n d t h e o u t w a r d u n i t n o r m a l
v e c t o r t o t h e t r i a n g l e o f T
h
w h i c h h a s a s i d e o n . R e l a t i o n s ( 3 . 1 1 ) g i v e
n
( u
h 3
+ b
u
h
) ( a
`
) = ( t t
) n
( u
h 3
+ b
u
h
) ( a
`
)
n
(
u
h 3
+
u
h 3
+ b
u
h
) ( b
|
) = ( t t
) n
(
u
h 3
+
u
h 3
+ b
u
h
) ( b
|
)
D F ( s
`
)
n
n
( u
h 3
+ b
u
h
) + n
(
u
h 3
+
u
h 3
+ b
u
h ;
+
b
u
h
)
o
( a
`
) = ( t t
) D F
( s
`
)
n
n
( u
h 3
+ b
u
h
) + n
(
u
h 3
+
u
h 3
+ b
u
h ;
+ b
u
h
)
o
( a
`
)
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
( 3 . 1 2 )
f o r ` = 0 ; : : ; n + 1 a n d | = 0 ; : : ; n ;
a r e t h e c o m p o n e n t s o f t h e u n i t t a n g e n t v e c t o r
u s e d i n ( 3 . 7 ) a n d b
|
i s t h e m i d p o i n t o f a
`
; a
` + 1
T h u s , t h e a d m i s s i b l e d i s c r e t e d i s p l a c e m e n t s p a c e f o r t h e r i g i d j u n c t i o n p r o b l e m
i s d e n e d b y :
W
h r i g
= f ( v
h
; v
h
) 2 V
h
V
h
; s u c h t h a t r e l a t i o n s ( 3 . 9 ) a n d ( 3 . 1 2 ) a r e v e r i e d
a t t h e c o r r e s p o n d i n g v e r t i c e s f a
`
g ; f a
`
g ; f b
`
g a n d f b
`
g l o c a t e d o n a n d
g
9
>
=
>
;
( 3 . 1 3 )
R e m a r k 3 . 2 . 1 . T h i s d i s c r e t i z a t i o n o f j u n c t i o n c o n d i t i o n s i n v o l v e s t h e n o n c o n f o r m i t y
o f t h e a p p r o x i m a t i o n , i . e . ,
W
h e l
= W
e l
a n d W
h r i g
= W
r i g
S i n c e t h e n o n c o n f o r m i t y j u s t a p p e a r s a l o n g t h e h i n g e w h i l e t h e m e t h o d r e m a i n s
c o n f o r m f o r a l l t h e o t h e r t e r m s d e n e d o n a n d
, w e s a y t h a t t h e a p p r o x i m a t i o n
m e t h o d i s p s e u d o - c o n f o r m i n g .
R e m a r k 3 . 2 . 2 . I n r e l a t i o n s ( 3 . 1 2 ) , t h e q u a n t i t i e s u
h 3
( b
|
) ; u
h
( b
|
) a n d t h e a s s o c i a t e d
q u a n t i t i e s o n S
, a r e n o t d e g r e e s o f f r e e d o m o f A r g y r i s t r i a n g l e , b u t t h e y c a n b e
e x p r e s s e d f r o m t h e m t h r o u g h t h e d e n i t i o n o f t h e i n t e r p o l a t i n g f u n c t i o n .
4 F i r s t d i s c r e t e p r o b l e m f o r e l a s t i c j u n c t i o n p r o b l e m
F r o m n o w o n , w e o n l y c o n s i d e r t h e e l a s t i c j u n c t i o n p r o b l e m . T h e r i g i d o n e c o u l d
b e c o n s i d e r e d s i m i l a r l y ; w e w i l l g i v e t h e c o r r e s p o n d i n g m a i n r e s u l t s i n S e c t i o n 6 .
4 . 1 D e n i t i o n o f t h e r s t d i s c r e t e p r o b l e m
T h e f o l l o w i n g v a r i a t i o n a l f o r m u l a t i o n t a k e s o n l y i n t o a c c o u n t t h e n i t e e l e m e n t
a p p r o x i m a t i o n ; t h e e e c t o f t h e n u m e r i c a l i n t e g r a t i o n w i l l b e a n a l y z e d i n S e c t i o n
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
13/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 1 1
5
F i n d ( u
k
h
; u
k
h
) 2 W
h e l
s u c h t h a t
a ( u
k
h
; u
k
h
) ; ( v
h
; v
h
) + k b ( u
k
h
; u
k
h
) ; ( v
h
; v
h
) = ` ( v
h
; v
h
) + f ( v
h
; v
h
) ;
8 ( v
h
; v
h
) 2 W
h e l
; k c o n s t a n t > 0 ;
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
( 4 . 1 )
w h e r e v
k
h
= v
k
h
F , v
k
h
= v
k
h
F
a n d w h e r e t h e s p a c e W
h e l
i s d e n e d b y r e l a t i o n ( 3 . 1 0 ) .
B y c o m p a r i s o n w i t h ( P a r t 1 , ( 3 . 1 0 ) ) w e h a v e i n t r o d u c e d a n e w l i n e a r f o r m f
w h i c h t a k e s i n t o a c c o u n t t h e n o n - c o n f o r m i t y o f t h e a p p r o x i m a t i o n a l o n g t h e h i n g e ,
i . e . ,
f ( v ; v
) =
Z
?
f N v ? N
v
g d ? =
Z
N v d +
Z
N
v
d
;
w h e r e N a n d N
a r e t h e r e s u l t a n t s o f t h e s u r f a c e l o a d . T h i s f o r m i s i d e n t i c a l l y z e r o
w h e n ( v ; v
) 2 W
e l
w h i l e i t i s g e n e r a l l y d i e r e n t f r o m z e r o w h e n a p p l i e d t o e l e m e n t s
( v
h
; v
h
) 2 W
h e l
. I n t h a t c a s e , w e r e w r i t e i n m a t r i x f o r m
f ( v
h
; v
h
) =
Z
T
N V
h
d +
Z
T
N
V
h
d
; ( 4 . 2 )
w h e r e t h e c o l u m n v e c t o r s V
h
] a n d V
h
] a r e d e n e d i n ( 2 . 2 ) , a n d w h e r e w e h a v e s e t
T
N
1 1 2
= N
1
0 0 N
2
0 0 N
3
0 0 0 0 0 ]
( a n d a s i m i l a r e x p r e s s i o n f o r N
] ) . T h e i n t r o d u c t i o n o f t h i s l i n e a r f o r m f ] i n ( 4 . 1 )
l e a d s t o a s i m p l i c a t i o n i n t h e a b s t r a c t e r r o r e s t i m a t e ( s e e P a r a g r a p h 4 . 3 ) .
4 . 2 U n i f o r m e l l i p t i c i t y
I n t h i s p a r a g r a p h , w e p r o v e t h e e x i s t e n c e a n d u n i q u e n e s s o f a s o l u t i o n f o r p r o -
b l e m ( 4 . 1 ) . T h a t l e a d s t o s h o w t h e u n i f o r m W
h e l
- e l l i p t i c i t y w i t h r e s p e c t t o h o f t h e
b i l i n e a r f o r m a ; + k b ;
F i r s t , l e t u s r e c a l l s o m e d e n i t i o n s i n t r o d u c e d i n P a r t 1 . L e t s p a c e E b e
E = ( H
1
( ) )
2
H
2
( ) ( H
1
(
) )
2
H
2
(
)
e q u i p p e d w i t h t h e n o r m
k ( v ; v
) k
E
= f k v
1
k
2
1
+ k v
2
k
2
1
+ k v
3
k
2
2
+ k v
1
k
2
1
+ k v
2
k
2
1
+ k v
3
k
2
2
g
1 = 2
T h e s p a c e W
h e l
, d e n e d i n ( 3 . 1 0 ) , i s a l i n e a r s u b s p a c e o f E a n d t h e a b o v e m a p p i n g
i s a n o r m o n W
h e l
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
14/39
1 2 M i c h e l B e r n a d o u , A n n i e C u b i e r
L e m m a 4 . 2 . 1 : T h e a p p l i c a t i o n ( v
h
; v
h
) 2 W
h e l
! k ( v
h
; v
h
) k
W
h e
i s a n o r m o n
W
h e l
w h e r e
k ( v
h
; v
h
) k
W
h e
= f a ( v
h
; v
h
) ; ( v
h
; v
h
) + k b ( v
h
; v
h
) ; ( v
h
; v
h
) g
1 = 2
I n t h i s e x p r e s s i o n , w e c o n s i d e r t h e d e n i t i o n o f t h e b i l i n e a r f o r m b ; g i v e n i n ( 2 . 3 ) .
P r o o f :
T h i s m a p p i n g i s c l e a r l y a s e m i - n o r m . T h u s w e j u s t h a v e t o s h o w t h a t :
k ( v
h
; v
h
) k
W
h e
= 0 ) ( v
h
; v
h
) = ( 0 ; 0 ) i n
T h e a s s u m p t i o n k ( v
h
; v
h
) k
W
h e
= 0 i m m e d i a t e l y i n v o l v e s :
i )
Z
e E
( v
h
)
( v
h
) +
e
2
1 2
( v
h
)
( v
h
)
p
a d
1
d
2
= 0 s o t h a t t h e b o u n -
d a r y c o n d i t i o n o n
0
g i v e s v
h
= 0 i n ;
i i ) t h e d i s c r e t e j u n c t i o n c o n d i t i o n s ( 3 . 9 ) i m p l y
v
h i
( a
`
) = v
h i ;
( a
`
) = v
h i ;
( a
`
) = 0 ( 4 . 3 )
f o r ` = 0 ; : : : ; n + 1 ; w h e r e f a
`
g i s t h e s e t o f v e r t i c e s l o c a t e d o n
S i n c e v
h i
a r e
v e d e g r e e p o l y n o m i a l s u p o n e a c h t r i a n g l e s i d e l o c a t e d o n
, r e l a t i o n s ( 4 . 3 ) i m -
p l y t r v
h i
= 0 o n
. M o r e o v e r , ( P a r t 1 , ( 2 . 1 5 ) , ( 3 . 1 2 ) ) a n d b ( v
h
; v
h
) ; ( v
h
; v
h
) =
0 l e a d s t o v
h 3 n
= 0 o n ? w h e r e n
= n
a
i s t h e u n i t o u t w a r d n o r m a l v e c t o r
t o ? l o c a t e d i n t h e t a n g e n t p l a n e t o S
H e r e , o u r p u r p o s e i s t o o b t a i n c l a m p e d c o n d i t i o n o n
, i . e . , v
h 3
= 0 o n
W e p o i n t o u t t h a t v
h 3 n
a n d v
h 3
a r e n o t t h e s a m e q u a n t i t i e s . I n d e e d , n
i s
d e n e d a l o n g ? w h e r e a s
i s d e n e d a l o n g
. T h e r e l a t i o n b e t w e e n t h e s e t w o
q u a n t i t i e s i s :
v
h 3 n
= n
v
h 3
= n
(
v
h 3
+
v
h 3
) ( 4 . 4 )
M o r e o v e r , n o t e t h a t v
h 3
= 0 o n
i m p l i e s v
h 3
= 0 o n
. T h u s , b y u s i n g
i n a d d i t i o n r e l a t i o n ( 4 . 4 ) a n d t h e a s s u m p t i o n v
h 3 n
= 0 o n ? a n d n o t i c i n g
t h a t t h e q u a n t i t y n
i s d i e r e n t f r o m z e r o , w e o b t a i n t h e r e q u i r e d c l a m p e d
c o n d i t i o n v
h 3
= 0 o n
. T h e n , w e o b t a i n v
h
= 0 i n
L e m m a 4 . 2 . 2 : U p o n t h e s p a c e W
h e l
, t h e n o r m s k ( v ; v
) k
E
a n d k ( v ; v
) k
W
h e
a r e
u n i f o r m l y e q u i v a l e n t w i t h r e s p e c t t o h .
P r o o f : T h e p r o o f o f t h i s l e m m a i s b a s e d o n t h e s a m e a r g u m e n t s t o t h o s e o f P a r t 1 ,
T h e o r e m 3 . 1 . 1 . T h e u n i f o r m - e l l i p t i c i t y c o m e s f r o m t h e i n c l u s i o n s ( 3 . 4 ) w h i c h a l l o w
u s t o c h o o s e t h e s a m e c o n s t a n t s t h a n f o r t h e c o n t i n u o u s p r o b l e m ( P a r t 1 , ( 3 . 1 0 ) ) .
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
15/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 1 3
T h e o r e m 4 . 2 . 1 : P r o b l e m ( 4 . 1 ) h a s o n e a n d o n l y o n e s o l u t i o n .
P r o o f : S i n c e t h e b i l i n e a r f o r m a ; + k b ; ] i s u n i f o r m l y W
h e l
- e l l i p t i c a n d u n i f o r m l y
c o n t i n u o u s w i t h r e s p e c t t o h , a n d s i n c e t h e l i n e a r f o r m ` + f ] i s c l e a r l y u n i f o r m l y
c o n t i n u o u s , w e h a v e j u s t t o a p p l y t h e L a x - M i l g r a m l e m m a t o c o n c l u d e .
4 . 3 A b s t r a c t e r r o r e s t i m a t e
T h e a b s t r a c t e r r o r e s t i m a t e i s u s e d i n p r a c t i c e t o o b t a i n a s y m p t o t i c e r r o r e s t i m a t e .
I n t h e f o l l o w i n g t h e o r e m , t h e e s t i m a t i o n i s r e s t r i c t e d t o t h e u s u a l a p p r o x i m a t i o n
t h e o r y t e r m i n f
( v
h
v
h
) 2 W
h e
k ( u
k
; u
k
) ? ( v
h
; v
h
) k
E
w h i c h i s k n o w n a s s o o n a s a n i t e
e l e m e n t a p p r o x i m a t i o n i s c h o s e n .
T h e o r e m 4 . 3 . 1 : L e t u s c o n s i d e r t h e d i s c r e t e p r o b l e m ( 4 . 1 ) f o r w h i c h t h e b i l i n e a r
f o r m a ; + k b ; i s u n i f o r m l y W
h e l
- e l l i p t i c , i . e . , t h e r e e x i s t s a c o n s t a n t > 0 ,
i n d e p e n d e n t o f h , s u c h t h a t :
a ( v
h
; v
h
) ; ( v
h
; v
h
) + k b ( v
h
; v
h
) ; ( v
h
; v
h
) k ( v
h
; v
h
) k
2
E
; 8 ( v
h
; v
h
) 2 W
h e l
( 4 . 5 )
W e s u p p o s e , m o r e o v e r , t h a t t h e r e e x i s t s a c o n s t a n t M > 0 , i n d e p e n d e n t o f h , s u c h
t h a t
a ( v ; v
) ; ( w ; w
) + k b ( v ; v
) ; ( w ; w
) M k ( v ; v
) k
E
k ( w ; w
) k
E
8 ( v ; v
) 2 W
e l
+ W
h e l
; 8 ( w ; w
) 2 W
e l
+ W
h e l
)
( 4 . 6 )
T h e n , t h e r e e x i s t s a c o n s t a n t C , i n d e p e n d e n t o f h , s u c h t h a t
k ( u
k
; u
k
) ? ( u
k
h
; u
k
h
) k
E
C i n f
( v
h
v
h
) 2 W
h e
k ( u
k
; u
k
) ? ( v
h
; v
h
) k
E
w h e r e ( u
k
; u
k
) ( r e s p . ( u
k
h
; u
k
h
) ) d e n o t e s t h e s o l u t i o n o f t h e c o n t i n u o u s p r o b l e m ( P a r t
1 , ( 3 . 1 0 ) ) ( r e s p . o f t h e d i s c r e t e p r o b l e m ( 4 . 1 ) ) .
P r o o f : L e m m a 4 . 2 . 2 i n v o l v e s t h a t r e l a t i o n ( 4 . 5 ) i s v e r i e d . L i k e w i s e , r e l a t i o n ( 4 . 6 )
i s a c o n s e q u e n c e o f c o n t i n u i t y p r o p e r t i e s o f t h e b i l i n e a r f o r m s a ; ] a n d b ; ] . L e t
( v
h
; v
h
) b e a n y e l e m e n t o f t h e s p a c e W
h e l
; w e c a n w r i t e b y u s i n g r e l a t i o n s ( 4 . 1 )
a n d ( 4 . 5 )
k ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) k
2
E
a ( u
k
; u
k
) ? ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
+ k b ( u
k
; u
k
) ? ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
? a ( u
k
; u
k
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) ? k b ( u
k
; u
k
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
+ ` ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) + f ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
16/39
1 4 M i c h e l B e r n a d o u , A n n i e C u b i e r
s o t h a t w i t h t h e c o n t i n u i t y p r o p e r t y ( 4 . 6 ) , w e o b t a i n :
k ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) k
E
M k ( u
k
; u
k
) ? ( v
h
; v
h
) k
E
+
s u p
( w
h
w
h
) 2 W
h e
a ( u
k
; u
k
) ; ( w
h
; w
h
) + k b ( u
k
; u
k
) ; ( w
h
; w
h
) ? ` ( w
h
; w
h
) ? f ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
N o w , l e t u s o b t a i n a n e w e x p r e s s i o n f o r a ( u
k
; u
k
) ; ( w
h
; w
h
) + k b ( u
k
; u
k
) ; ( w
h
; w
h
)
F o r t h a t w e c o m e b a c k t o t h e e q u i l i b r i u m e q u a t i o n s o f t h e j u n c t i o n p r o b l e m g i v e n
i n ( P a r t 1 , ( 2 . 2 7 ) - ( 2 . 2 8 ) ) . M a k i n g t h e p r o d u c t o f t h e s e e q u a t i o n s b y t e s t f u n c t i o n s
( w
h
; w
h
) 2 W
h e l
a n d u s i n g G r e e n ' s f o r m u l a , w e n a l l y g e t w i t h n o t a t i o n s i n t r o d u c e d
i n ( 4 . 1 )
a ( u
k
; u
k
) ; ( w
h
; w
h
) + k b ( u
k
; u
k
) ; ( w
h
; w
h
)
= ` ( w
h
; w
h
) +
Z
N w
h
d +
Z
N
w
h
d
= ` ( w
h
; w
h
) + f ( w
h
; w
h
) ;
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
( 4 . 7 )
s o t h a t t h e s e c o n d t e r m i n t h e a b o v e e s t i m a t i o n d i s a p p e a r s .
T o c o n c l u d e , i t r e m a i n s t o u s e t h e t r i a n g u l a r i n e g a l i t y a n d t o t a k e t h e m i n i m u m
w i t h r e s p e c t t o ( v
h
; v
h
) 2 W
h e l
R e m a r k 4 . 3 . 1 : T h e r e l a t i o n ( 4 . 7 ) a l l o w s u s t o c a n c e l t h e c o n s i s t e n c y t e r m t h a t
w e u s u a l l y n d i n t h e a b s t r a c t e r r o r e s t i m a t e a s s o c i a t e d w i t h n o n c o n f o r m i n g n i t e
e l e m e n t m e t h o d . H e r e , t h e n o n c o n f o r m i t y o n l y a p p e a r s a l o n g t h e h i n g e ? ; i t s e e c t
i s c i r c u m v e n t b y t h e i n t r o d u c t i o n o f t h e l i n e a r f o r m f ] i n ( 4 . 1 ) .
A n o t h e r d i s c r e t e p r o b l e m c o u l d a l s o b e c o n s i d e r e d b y d r o p p i n g t e r m f ] i n ( 4 . 1 ) . I t
s h o u l d b e d i e r e n t f r o m p r o b l e m ( 4 . 1 ) a n d w o u l d l e a d t o a m o r e c l a s s i c a l a b s t r a c t
e r r o r e s t i m a t e i n c l u d i n g a c o n s i s t e n c y t e r m b a s e d o n t h e l i n e a r f o r m f ] . T h e
s o l u t i o n o f s u c h a p r o b l e m w o u l d b e r e a l l y c l o s e d t o t h a t o f p r o b l e m ( 4 . 1 ) .
5 S e c o n d d i s c r e t e p r o b l e m : a d d i t i o n a l e e c t o f n u m e -
r i c a l i n t e g r a t i o n
5 . 1 D e n i t i o n o f t h e s e c o n d d i s c r e t e p r o b l e m
T h e i n t e g r a l s d e n e d o v e r t h e d o m a i n s a n d
h a v e t o b e e v a l u a t e d o v e r a l l
t h e t r i a n g l e s K 2 T
h
a n d K
2 T
h
a n d t h e y a r e s e l d o m e x a c t l y c o m p u t e d i n p r a c t i c e .
O n e r a t h e r u s e n u m e r i c a l i n t e g r a t i o n s c h e m e s . T h e n , l e t u s c o n s i d e r a n u m e r i c a l
i n t e g r a t i o n s c h e m e d e n e d o v e r a r e f e r e n c e t r i a n g l e
^
K ( f o r m o r e d e t a i l s s e e C i a r l e t
( 1 9 7 8 ) ) :
Z
^
K
^
( ^x ) d ^x
L
X
` = 1
^!
`
^
(
^
b
`
)
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
17/39
8/13/2019 Numerical analysis of junctions between thin shell.pdf
18/39
1 6 M i c h e l B e r n a d o u , A n n i e C u b i e r
N o w , w e c a n g i v e t h e e x p r e s s i o n o f t h e s e c o n d d i s c r e t e p r o b l e m w h i c h t a k e s i n t o
a c c o u n t t h e a d d i t i o n a l e e c t o f t h e n u m e r i c a l i n t e g r a t i o n .
F i n d ( u
k
h
; u
k
h
) 2 W
h e l
s u c h t h a t
a
h
( u
k
h
; u
k
h
) ; ( v
h
; v
h
) + k b
h
( u
k
h
; u
k
h
) ; ( v
h
; v
h
) = `
h
( v
h
; v
h
) + f
h
( v
h
; v
h
)
8 ( v
h
; v
h
) 2 W
h e l
; k c o n s t a n t > 0 ;
9
>
>
>
>
>
=
>
>
>
>
>
;
( 5 . 3 )
w h e r e w e h a v e s e t ( c o m p a r e w i t h ( 2 . 1 ) a n d ( 2 . 4 ) ) :
a
h
( u
h
; u
h
) ; ( v
h
; v
h
) =
X
K 2 T
h
L
X
` = 1
!
` K
f
T
U
h
A V
h
g ( b
` K
)
+
X
K
2 T
h
L
X
` = 1
!
` K
f
T
U
h
A
V
h
g ( b
` K
)
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
( 5 . 4 )
b
h
( ( u
h
; u
h
) ; ( v
h
; v
h
) =
X
K 2 T
h
L
X
` = 1
!
` K
f
T
U
h
U
h
C V
h
V
h
g ( b
` K
)
( 5 . 5 )
T h e l i n e a r f o r m s a r e d e n e d b y ( c o m p a r e w i t h ( 2 . 5 ) ) :
`
h
( v
h
; v
h
) =
X
K 2 T
h
L
X
` = 1
!
` K
f
T
P V
h
g ( b
` K
) +
X
K
2 T
h
L
X
` = 1
!
` K
f
T
P
V
h
g ( b
` K
)
+
X
K 2 G
1
L
X
` = 1
!
` K
f
T
L
s
V
h
g ( b
` K
) +
X
K
2 G
1
L
X
` = 1
!
` K
f
T
L
s
V
h
g ( b
` K
)
9
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
;
( 5 . 6 )
w h e r e G
1
a n d G
1
d e n o t e t h e s e t s o f t h e s i d e s o f t r i a n g l e s w h i c h a r e l o c a t e d u p o n
1
a n d
1
. M o r e o v e r ( c o m p a r e w i t h ( 4 . 2 ) ) :
f
h
( v
h
; v
h
) =
X
K 2 G
L
X
` = 1
!
` K
f
T
N V
h
g ( b
` K
) +
X
K
2 G
L
X
` = 1
!
` K
f
T
N
V
h
g ( b
` K
)
( 5 . 7 )
w h e r e G a n d G
d e n o t e a g a i n t h e s e t s o f t h e s i d e s o f t r i a n g l e s l o c a t e d o n a n d
5 . 2 A b s t r a c t e r r o r e s t i m a t e
T h e o r e m 5 . 2 . 1 : L e t u s c o n s i d e r a f a m i l y o f d i s c r e t e p r o b l e m s ( 5 . 3 ) f o r w h i c h t h e
b i l i n e a r f o r m s a
h
; + k b
h
; a r e W
h e l
- e l l i p t i c , u n i f o r m l y w i t h r e s p e c t t o h , i . e . ,
t h e r e e x i s t s a c o n s t a n t > 0 , i n d e p e n d e n t o f h , s u c h t h a t :
a
h
( v
h
; v
h
) ; ( v
h
; v
h
) + k b
h
( v
h
; v
h
) ; ( v
h
; v
h
) k ( v
h
; v
h
) k
2
E
; 8 ( v
h
; v
h
) 2 W
h e l
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
19/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 1 7
T h e n , t h e r e e x i s t s a c o n s t a n t C , i n d e p e n d e n t o f h , s u c h t h a t
k ( u
k
; u
k
) ? ( u
k
h
; u
k
h
) k
E
C i n f
( v
h
v
h
) 2 W
h e
n
k ( u
k
; u
k
) ? ( v
h
; v
h
) k
E
+ s u p
( w
h
w
h
) 2 W
h e
(
a ( v
h
; v
h
) ; ( w
h
; w
h
) ? a
h
( v
h
; v
h
) ; ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
+ k
b ( v
h
; v
h
) ; ( w
h
; w
h
) ? b
h
( v
h
; v
h
) ; ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
) )
+ C s u p
( w
h
w
h
) 2 W
h e
a ( u
k
; u
k
) ; ( w
h
; w
h
) + k b ( u
k
; u
k
) ; ( w
h
; w
h
) ? `
h
( w
h
; w
h
) ? f
h
( w
h
; w
h
)
k ( w
h
; w
h
) k
E
( 5 . 8 )
w h e r e ( u
k
; u
k
) ( r e s p . ( u
k
h
; u
k
h
) ) d e n o t e s t h e s o l u t i o n o f t h e c o n t i n u o u s p r o b l e m ( P a r t
1 , ( 3 . 1 0 ) ) ( r e s p . o f t h e d i s c r e t e p r o b l e m ( 5 . 3 ) ) .
P r o o f : T h e a s s u m p t i o n o f W
h e l
- e l l i p t i c i t y i n v o l v e s t h e e x i s t e n c e a n d u n i q u e n e s s
o f a s o l u t i o n ( u
k
h
; u
k
h
) f o r t h e d i s c r e t e p r o b l e m ( 5 . 3 ) . T h e n l e t ( v
h
; v
h
) b e a n y
e l e m e n t o f t h e s p a c e W
h e l
; w e c a n w r i t e :
k ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) k
2
E
a
h
( u
k
h
; u
k
h
) ? ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
+ k b
h
( u
k
h
; u
k
h
) ? ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
= a ( u
k
; u
k
) ? ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
+ k b ( u
k
; u
k
) ? ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
+ a ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) ? a
h
( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
+ k b ( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) ? k b
h
( v
h
; v
h
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
? a ( u
k
; u
k
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) ? k b ( u
k
; u
k
) ; ( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
+ `
h
( u
k
h
; u
k
h
) ? ( v
h
; v
h
) + f
h
( u
k
h
; u
k
h
) ? ( v
h
; v
h
)
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
20/39
1 8 M i c h e l B e r n a d o u , A n n i e C u b i e r
s o t h a t w i t h t h e c o n t i n u i t y p r o p e r t y ( 4 . 6 ) w e o b t a i n :
k ( u
k
h
; u
k
h
) ? ( v
h
; v
h
) k
E
M k ( u
k
; u
k
) ? ( v
h
; v
h
) k
E
+ s u p
( w
h
w
h
) 2 W
h e
(
a ( v
h
; v
h
) ; ( w
h
; w
h
) ? a
h
( v
h
; v
h
) ; ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
+ k
b ( v
h
; v
h
) ; ( w
h
; w
h
) ? b
h
( v
h
; v
h
) ; ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
)
+ s u p
( w
h
w
h
) 2 W
h e
a ( u
k
; u
k
) ; ( w
h
; w
h
) + k b ( u
k
; u
k
) ; ( w
h
; w
h
) ? `
h
( w
h
; w
h
) ? f
h
( w
h
; w
h
)
k ( w
h
; w
h
) k
E
T o c o n c l u d e , i t r e m a i n s t o u s e t h e t r i a n g u l a r i n e g a l i t y a n d t o t a k e t h e m i n i m u m
w i t h r e s p e c t t o ( v
h
; v
h
) 2 W
h e l
I n t h e e s t i m a t e ( 5 . 8 ) , i n a d d i t i o n t o t h e u s u a l a p p r o x i m a t i o n t h e o r y t e r m i n f k ( u
k
; u
k
) ?
( v
h
; v
h
) k , w e n d t w o a d d i t i o n a l t e r m s w h i c h m e a s u r e t h e c o n s i s t e n c y e r r o r b e t -
w e e n t h e b i l i n e a r f o r m s a ; ] a n d a
h
; , b ; ] a n d b
h
; ] ; t h e y t a k e i n t o a c c o u n t
t h e e r r o r d u e t o t h e n u m e r i c a l i n t e g r a t i o n . F i n a l l y , t h e l a s t t e r m c o m b i n e s t h e e r r o r
d u e t o b o t h a p p r o x i m a t i o n s , i . e . , n o n c o n f o r m i n g a p p r o x i m a t i o n a l o n g t h e h i n g e a n d
u s e o f t h e n u m e r i c a l i n t e g r a t i o n t e c h n i q u e s .
5 . 3 U n i f o r m e l l i p t i c i t y
T h e u n i f o r m W
h e l
- e l l i p t i c i t y i s b a s e d o n t h e l o c a l e r r o r e s t i m a t e t h e o r e m s g i v e n
b y B e r n a d o u ( 1 9 9 6 , p . 5 3 - 6 1 ) f o r a t r i a n g l e K a n d b y C u b i e r ( 1 9 9 4 , p . 7 6 - 8 7 ) f o r
a t r i a n g l e s i d e K
0
. T h e s e t h e o r e m s g i v e a g e n e r a l r e s u l t o f e r r o r e s t i m a t e ; t h e y
s p e c i f y c r i t e r i a o n t h e c h o i c e o f n u m e r i c a l i n t e g r a t i o n s c h e m e s i n o r d e r t o o b t a i n t h e
s a m e o r d e r o f a s y m p t o t i c e r r o r e s t i m a t e t h a n f o r e x a c t i n t e g r a t i o n .
T h e o r e m 5 . 3 . 1 : L e t T
h
a n d T
h
b e r e g u l a r f a m i l i e s o f t r i a n g u l a t i o n s o f t h e d o m a i n s
a n d
s a t i s f y i n g p r o p e r t i e s ( 3 . 1 ) t o ( 3 . 3 ) . L e t ( K ; P
K
;
K
) a n d ( K
; P
K
;
K
) b e
t w o a l m o s t a n e f a m i l i e s o f n i t e e l e m e n t s a s s o c i a t e d w i t h t h e A r g y r i s t r i a n g l e .
T h u s w e h a v e
P
K
= P
5
( K ) ; 8 K 2 T
h
a n d P
K
= P
5
( K
) ; 8 K
2 T
h
M o r e o v e r , a s s u m e t h a t t h e i n t e g r a t i o n s c h e m e o n t h e r e f e r e n c e t r i a n g l e
^
K s a t i s e s
t h e f o l l o w i n g c o n d i t i o n s :
i ) t h e i n t e g r a t i o n n o d e s
^
b
`
2
^
K ; 8 ` = 1 ; : : ; L ;
i i )
^
E ( ^' ) = 0 ; 8 ^' 2 P
8
(
^
K )
L i k e w i s e , t h e i n t e g r a t i o n s c h e m e o n t h e r e f e r e n c e s e g m e n t
^
K
0
v e r i e s
i i i ) t h e i n t e g r a t i o n n o d e s
^
b
0
`
2
^
K
0
; 8 ` = 1 ; : : ; L
0
;
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
21/39
8/13/2019 Numerical analysis of junctions between thin shell.pdf
22/39
2 0 M i c h e l B e r n a d o u , A n n i e C u b i e r
N o w , w e h a v e t o e s t i m a t e t h e t e r m b ( v
h
; v
h
) ; ( v
h
; v
h
) ? b
h
( v
h
; v
h
) ; ( v
h
; v
h
)
B y u s i n g d e n i t i o n ( 5 . 2 ) o f t h e e r r o r f u n c t i o n a l , w e o b t a i n
b ( v
h
; v
h
) ; ( v
h
; v
h
) ? b
h
( v
h
; v
h
) ; ( v
h
; v
h
)
X
K 2 T
h
2 4
X
I J = 1
E
0
K
( C
I J
V
h
V
h
I
V
h
V
h
J
)
I f w e d e n o t e
I
1
= f 1 ; : : ; 1 2 g f 1 ; : : ; 1 2 g ; I
2
= f 1 3 ; : : ; 2 4 g f 1 3 ; : : ; 2 4 g ;
I
3
= f 1 ; : : ; 1 2 g f 1 3 ; : : ; 2 4 g ; I
4
= f 1 3 ; : : ; 2 4 g f 1 ; : : ; 1 2 g ;
t h e n , w e h a v e t h e r e l a t i o n
X
K 2 T
h
2 4
X
I J = 1
E
0
K
( C
I J
V
h
V
h
I
V
h
V
h
J
) =
X
K 2 T
h
X
I J 2 I
1
E
0
K
( C
I J
V
h
I
V
h
J
)
+
X
K 2 T
h
X
I J 2 I
2
E
0
K
( C
I J
V
h
I
V
h
J
) +
X
K 2 T
h
X
I J 2 I
3
I
4
E
0
K
( C
I J
V
h
I
V
h
J
)
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
( 5 . 1 3 )
W i t h t h e h y p o t h e s e s m a d e i n t h e s t a t e m e n t o f t h e t h e o r e m , w e c a n a p p l y t o t h e s e
o n e d i m e n s i o n a l i n t e g r a t i o n t e r m s o f t h e r e l a t i o n ( 5 . 1 3 ) t h e s a m e k i n d o f t e c h n i q u e
a n d w e o b t a i n ( s e e C u b i e r ( 1 9 9 4 , p . 8 3 ) f o r d e t a i l s )
X
K 2 T
h
X
I J 2 I
1
E
0
K
( C
I J
V
h
I
V
h
J
) C h
X
K 2 T
h
(
X
I J 2 I
1
k C
I J
k
1 1 K
) k v
h
k
2
V ( K )
;
X
K 2 T
h
X
I J 2 I
2
E
0
K
( C
I J
V
h
I
V
h
J
) C h
X
K 2 T
h
(
X
I J 2 I
2
k C
I J
k
1 1 K
) k v
h
k
2
V
( K
)
;
X
K 2 T
h
X
I J 2 I
3
I
4
E
0
K
( C
I J
V
h
I
V
h
J
) C h
X
K 2 T
h
(
X
I J 2 I
3
I
4
k C
I J
k
1 1 K
) k v
h
k
V ( K )
k v
h
k
V
( K
)
;
s o t h a t t h e s u b s t i t u t i o n o f t h e s e i n e q u a l i t i e s i n t o r e l a t i o n ( 5 . 1 3 ) p r o v e s t h e e x i s t e n c e
o f a c o n s t a n t C , i n d e p e n d e n t o f h , s u c h t h a t
b ( v
h
; v
h
) ; ( v
h
; v
h
) ? b
h
( v
h
; v
h
) ; ( v
h
; v
h
)
C h (
2 4
X
I J = 1
k C
I J
k
1 1 !
) k ( v
h
; v
h
) k
2
E
C h k ( v
h
; v
h
) k
2
E
9
>
>
>
>
=
>
>
>
>
;
( 5 . 1 4 )
B y s u b s t i t u t i n g r e l a t i o n s ( 5 . 1 1 ) , ( 5 . 1 2 ) , ( 5 . 1 4 ) i n t o r e l a t i o n ( 5 . 1 0 ) , w e g e t
a
h
( v
h
; v
h
) ; ( v
h
; v
h
) + k b
h
( v
h
; v
h
) ; ( v
h
; v
h
) ( ? C h ) k ( v
h
; v
h
) k
2
E
a n d i t s u c e s t o t a k e =
2
a n d h
1
=
2 C
t o o b t a i n t h e e x p e c t e d e s t i m a t e ( 5 . 9 ) .
R e m a r k 5 . 3 . 1 : N u m e r i c a l i n t e g r a t i o n s c h e m e s e x a c t f o r p o l y n o m i a l o f d e g r e e e i g h t
c a n b e f o u n d i n D u n a v a n t ( 1 9 8 5 , p . 1 1 4 0 ) f o r t h e t r i a n g l e a n d i n Z i e n k i e w i c z - T a y l o r
( 1 9 8 9 , p . 1 7 3 ) f o r a n i n t e r v a l .
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
23/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 2 1
5 . 4 A s y m p t o t i c e r r o r e s t i m a t e
N o w , w e a r e a b l e t o e v a l u a t e t h e d i e r e n t t e r m s o f t h e a b s t r a c t e r r o r e s t i m a t e
( 5 . 8 ) a n d t o d e r i v e a n a s y m p t o t i c e r r o r e s t i m a t e k ( u ; u
) ? ( u
h
; u
h
) k b e t w e e n t h e
s o l u t i o n ( u ; u
) o f t h e c o n t i n u o u s p r o b l e m ( P a r t 1 , ( 3 . 1 0 ) ) a n d t h e s o l u t i o n s o f t h e
d i s c r e t e p r o b l e m s ( 5 . 3 ) . B e f o r e g i v i n g t h e a s y m p t o t i c e r r o r e s t i m a t e t h e o r e m , l e t u s
s p e c i f y s o m e n o t a t i o n s :
k ( p ; p
) k
( W
4 q
( ) )
3
( W
4 q
(
) )
3 = f
3
X
i = 1
k p
i
k
q
4 q
+
3
X
i = 1
k p
i
k
q
4 q
g
1 = q
;
k ( L
s
; L
s
) k
( W
5 s
(
1
) )
1 2
( W
5 s
(
1
) )
1 2
= f
1 2
X
I = 1
k L
s
I
k
s
5 ; s ;
1
+
1 2
X
I = 1
k L
s I
k
s
5 ; s ;
1
g
1 = s
;
w h e r e q ; s a r e t w o i n t e g e r n u m b e r s 1
T h e o r e m 5 . 4 . 1 : L e t T
h
a n d T
h
b e t w o r e g u l a r f a m i l i e s o f t r i a n g u l a t i o n s o f t h e
d o m a i n s a n d
s a t i s f y i n g t h e p r o p e r t i e s ( 3 . 1 ) t o ( 3 . 3 ) . L e t ( K ; P
K
;
K
) a n d
( K
; P
K
;
K
) b e t w o a l m o s t a n e f a m i l i e s o f n i t e e l e m e n t s a s s o c i a t e d w i t h t h e A r -
g y r i s t r i a n g l e . M o r e o v e r , w e a s s u m e t h a t t h e n u m e r i c a l i n t e g r a t i o n s c h e m e o n t h e
r e f e r e n c e t r i a n g l e
^
K s a t i s e s t h e f o l l o w i n g c o n d i t i o n s :
i ) t h e i n t e g r a t i o n n o d e s
^
b
`
2
^
K ; 8 ` = 1 ; : : ; L ;
i i )
^
E ( ^' ) = 0 ; 8 ^' 2 P
8
(
^
K )
L i k e w i s e , t h e n u m e r i c a l i n t e g r a t i o n s c h e m e o n t h e r e f e r e n c e i n t e r v a l
^
K
0
v e r i e s
i i i ) t h e i n t e g r a t i o n n o d e s
^
b
0
`
2
^
K
0
; 8 ` = 1 ; : : ; L
0
;
i v )
^
E
0
( ^' ) = 0 ; 8 ^' 2 P
8
(
^
K
0
)
A s s u m e t h a t
v ) t h e s o l u t i o n ( u
k
; u
k
) 2 W
e l
o f t h e c o n t i n u o u s p r o b l e m ( P a r t 1 , ( 3 . 1 0 ) ) b e l o n g s
t o t h e s p a c e K ( ) K
(
) = ( H
5
( ) )
2
H
6
( ) ( H
5
(
) )
2
H
6
(
) ;
v i ) A
I J
2 W
4 1
( ) ; A
I J
2 W
4 1
(
) f o r I ; J = 1 ; : : ; 1 2 ;
v i i ) C
I J
2 W
4 1
( ! ) f o r I ; J = 1 ; : : ; 2 4 ;
v i i i ) p
i
2 W
4 q
( ) ; p
i
2 W
4 q
(
) f o r i = 1 ; : : ; 3 ;
i x ) N
i
2 W
5 s
( ) ; N
i
2 W
5 s
(
) f o r i = 1 ; : : ; 3 ;
x ) L
s
I
2 W
5 s
(
1
) ; L
s I
2 W
5 s
(
1
) f o r I = 1 ; : : ; 1 2 ,
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
24/39
2 2 M i c h e l B e r n a d o u , A n n i e C u b i e r
w h e r e q ; s a r e i n t e g e r n u m b e r s 2 . T h e n , t h e r e e x i s t c o n s t a n t s C > 0 a n d h
1
> 0 ,
i n d e p e n d e n t o f h , s u c h t h a t f o r a n y h 2 0 ; h
1
, w e h a v e :
k ( u
k
; u
k
) ? ( u
k
h
; u
k
h
) k
E
C h
4
n
k ( u
k
; u
k
) k
K ( ) K
(
)
+ k ( p ; p
) k
( W
4 q
( ) )
3
( W
4 q
(
) )
3
+ h
1 = 2
k ( N ; N
) k
( W
5 s
( ) )
3
( W
5 s
(
) )
3 + k ( L
s
; L
s
) k
( W
5 s
(
1
) )
1 2
( W
5 s
(
1
) )
1 2
9
>
>
>
=
>
>
>
;
( 5 . 1 5 )
w h e r e ( u
k
h
; u
k
h
) i s t h e s o l u t i o n o f t h e d i s c r e t e p r o b l e m ( 5 . 3 ) .
P r o o f T h e c o n d i t i o n s f o r a p p l y i n g T h e o r e m 5 . 3 . 1 a r e s a t i s e d . H e n c e , t h e c o n d i -
t i o n o f t h e u n i f o r m W
h e l
- e l l i p t i c i t y i s v e r i e d a n d i t i s p o s s i b l e t o a p p l y T h e o r e m
5 . 2 . 1 . T h e r e f o r e , w e a r e g o i n g t o e v a l u a t e t h e d i e r e n t t e r m s o f t h e s e c o n d h a n d
m e m b e r o f t h e i n e q u a l i t y ( 5 . 8 ) . T h e p r o o f t a k e s v e s t e p s .
S t e p 1 : E s t i m a t e o f i n f
( v
h
v
h
) 2 W
h e
k ( u
k
; u
k
) ? ( v
h
; v
h
) k
E
L e t
h
b e t h e W
h e l
i n t e r p o l a t i o n o p e r a t o r o n t h e s p a c e W
e l
. W e d e n e
h
( v ; v
) = (
h
v ;
h
v
)
w h e r e
h
a n d
h
a r e t h e a s s o c i a t e d i n t e r p o l a t i o n o p e r a t o r s o n V
h
a n d V
h
. T h e n , w e
o b t a i n
( C i a r l e t ( 1 9 7 8 p . 1 2 4 ) )
i n f
( v
h
v
h
) 2 W
h e
k ( u
k
; u
k
) ? ( v
h
; v
h
) k
E
k ( u
k
; u
k
) ?
h
( u
k
; u
k
) k
E
C h
4
k ( u
k
; u
k
) k
K ( ) K
(
)
9
>
>
=
>
>
;
( 5 . 1 6 )
S t e p 2 : E s t i m a t e o f s u p
( w
h
w
h
) 2 W
h e
a
h
( u
k
; u
k
) ; ( w
h
; w
h
) ? a
h
h
( u
k
; u
k
) ; ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
B y u s i n g r e l a t i o n s ( 2 . 1 ) a n d ( 5 . 1 ) :
a
h
( u
k
; u
k
) ; ( w
h
; w
h
) ? a
h
h
( u
k
; u
k
) ; ( w
h
; w
h
)
X
K 2 T
h
1 2
X
I J = 1
E
K
( A
I J
h
U
k
I
W
J
) +
X
K
2 T
h
1 2
X
I J = 1
E
K
( A
I J
h
U
k
I
W
J
)
9
>
>
>
>
=
>
>
>
>
;
( 5 . 1 7 )
W e r e s t r i c t o u r a t t e n t i o n t o t h e r s t t e r m o f t h e s e c o n d h a n d m e m b e r o f i n e q u a l i t y
( 5 . 1 7 ) . T h e h y p o t h e s e s o f B e r n a d o u ( 1 9 9 6 , T h e o r e m 1 . 3 . 3 , p . 5 7 ) a r e v e r i e d . T h u s ,
w e o b t a i n t h e e x i s t e n c e o f a c o n s t a n t C > 0 , i n d e p e n d e n t o f h , s u c h t h a t
X
K 2 T
h
1 2
X
I J = 1
E
K
( A
I J
h
U
k
I
W
J
)
C
X
K 2 T
h
h
4
K
(
1 2
X
I J = 1
k A
I J
k
4 1 K
) ( k
h
u
k
1
k
2
5 K
+ k
h
u
k
2
k
2
5 K
+ k
h
u
k
3
k
2
6 K
)
1 = 2
k w
h
k
V ( K )
I N R I A
8/13/2019 Numerical analysis of junctions between thin shell.pdf
25/39
J u n c t i o n s b e t w e e n t h i n s h e l l s 2 3
w h e r e
h
u =
h
u
1
; (
h
u
1
)
1
; (
h
u
1
)
2
;
h
u
2
; (
h
u
2
)
1
; (
h
u
2
)
2
;
h
u
3
; (
h
u
3
)
1
; (
h
u
3
)
2
; (
h
u
3
)
1 1
; (
h
u
3
)
1 2
; (
h
u
3
)
2 2
T h e i n t e r p o l a t i o n o p e r a t o r
h 1
l e a v e s t h e s p a c e P
5
( K ) i n v a r i a n t , a n d w e o b t a i n
k
h
u
i
k
5 K
k u
i
k
5 K
+ k u
i
?
h
u
i
k
5 K
C k u
i
k
5 K
; i = 1 ; 2 ; 3
T h u s ,
X
K 2 T
h
1 2
X
I J = 1
E
K
( A
I J
h
U
k
I
W
h J
) C m a x
K 2 T
h
( h
K
)
4
(
1 2
X
I J = 1
k A
I J
k
4 1
) k u
k
k
K ( )
k w
h
k
V ( )
( 5 . 1 8 )
S i m i l a r l y f o r t h e s h e l l S
, w e c o u l d p r o v e
X
K
2 T
h
1 2
X
I J = 1
E
K
( A
I J
h
U
k
I
W
h I
) C m a x
K
2 T
h
( h
K
)
4
(
1 2
X
I J = 1
k A
I J
k
4 1
) k u
k
k
K
(
)
k w
h
k
V
(
)
( 5 . 1 9 )
B y c o m b i n i n g i n e q u a l i t i e s ( 5 . 1 8 ) , ( 5 . 1 9 ) a n d d e n i t i o n ( 3 . 2 ) , w e o b t a i n
a
h
( u
k
; u
k
) ; ( w
h
; w
h
) ? a
h
h
( u
k
; u
k
) ; ( w
h
; w
h
)
C h
4
s u p
8
>
>
>
>
>
=
>
>
>
>
>
>
>
;
( 5 . 2 0 )
S t e p 3 : E s t i m a t e o f s u p
( w
h
w
h
) 2 W
h e
b
h
( u
k
; u
k
) ; ( w
h
; w
h
) ? b
h
h
( u
k
; u
k
) ; ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
T h e r e s t r i c t i o n o f a n A r g y r i s t r i a n g l e t o o n e o f i t s s i d e i s a P
5
- o n e - d i m e n s i o n a l
n i t e e l e m e n t , s o t h a t w e d e n e t h e i n t e r p o l a t i n g f u n c t i o n :
h
( u
k
; u
k
) = (
h
u
k
F ;
h
u
k
F
)
B y u s i n g t h e m a t r i x e x p r e s s i o n s ( 2 . 4 ) a n d ( 5 . 5 ) o f t h e b i l i n e a r f o r m s b ; ] a n d b
h
;
, a s i m i l a r o n e - d i m e n s i o n a l s t u d y t o t h e p r e v i o u s o n e s g i v e s ( f o r d e t a i l s , s e e C u b i e r
R R n 2 9 2 2
8/13/2019 Numerical analysis of junctions between thin shell.pdf
26/39
2 4 M i c h e l B e r n a d o u , A n n i e C u b i e r
( 1 9 9 4 , T h e o r e m 5 . 2 . 4 , p . 8 3 ) )
b
h
( u
k
; u
k
) ; ( w
h
; w
h
) ? b
h
h
( u
k
; u
k
) ; ( w
h
; w
h
)
C h
4
(
2 4
X
I J = 1
k C
I J
k
4 1 K
)
h
k u
k
k
K ( )
k w
h
k
V ( )
+ k u
k
k
K
(
)
k w
h
k
V
(
)
i
s o t h a t , w e o b t a i n
b
h
( u
k
; u
k
) ; ( w
h
; w
h
) ? b
h
h
( u
k
; u
k
) ; ( w
h
; w
h
)
k ( w
h
; w
h
) k
E
C h
4
(
2 4
X
I J = 1
k C
I J
k
4 1 !
) k ( u
k
; u
k
) k
K ( ) K
(
)
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
( 5 . 2 1 )
S t e p 4 : E s t i m a t e o f
s u p
( w
h
w
h
) 2 W
h e
a ( u
k
; u
k
) ; ( w
h
; w
h
) + k b ( u
k
; u
k
) ; ( w
h
; w
h
) ? `
h
( w
h
; w
h
) ? f
h
( w
h
; w
h
)
k ( w
h
; w
h
) k
E
B y u s i n g r e l a t i o n ( 4 . 7 ) , w e o b t a i n t h e n e w e s t i m a t e :
s u p
( w
h
w
h
) 2 W
h e
a ( u
k
; u
k
) ; ( w
h
; w
h
) + k b ( u
k
; u
k
) ; ( w
h
; w
h
) ? `
h
( w
h
; w
h
) ? f
h
( w
h
; w
h
)
k ( w
h
; w
h
) k
E
s u p
( w
h
w
h
) 2 W
h e
` ( w
h
; w
h
) ? `
h
( w
h
; w
h
)
k ( w
h
; w
h
) k
E
+ s u p
( w
h
w
h
) 2 W
h e
f ( w
h
; w
h
) ? f
h
( w
h
; w
h
)
k ( w
h
; w
h
) k
E
I n t h i s i n e q u a l i t y , w e n d t h e t e r m f ( w
h
; w
h
) ? f
h
( w
h
; w
h
) w h i c h c o m -
b i n e s t h e e r r o r s d u e t o t h e n o n c o n f o r m i t y o f t h e n i t e e l e m e n t m e t h o d a n d t o t h e
u s e o f n u m e r i c a l i n t e g r a t i o n . T h u s , i t r e m a i n s t o s t u d y t h e f o l l o w i n g e s t i m a t e s
E s t i m a t e o f s u p
( w
h
w
h
)
Top Related