A posteriori error analysis for Hydro-Mechanical couplings and implementation in Code_Aster

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A posteriori error analysis for Hydro-Mechanical

couplings and implementation in Code_Aster

Sébastien Meunier

To cite this version:

Sébastien Meunier. A posteriori error analysis for Hydro-Mechanical couplings and implementation

in Code_Aster. Mathematics [math]. Ecole des Ponts ParisTech, 2007. English. �pastel-00003314�

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m ' = h=+ g lh l= = l = . v} = ln=r = ' r= ly h = 8396 :;4561 o h = y= l h r= . g = l h ln. = h= p)) + g ' = lh w lh h h w w } = l ==r=y . ll h r==r= 8396 :;4561. h h~ l==l~ l ~g h h h = h+ p)) m = h= p)) . = l = l h = ly+ i = = l y~ <)659.,.+;9 )9)<,-5 oL l l ==r=y 8396 :;4561 + =ly l r= l h = w y= g l g I1A81 =/3>=1 ./4850/1 41< !"# $%&'#(+

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8396 :;4561 g = v} q} o l r= l=l l = h (. h= =l= = hl=* + h= .h= h r= h = h= yh~ yk= * + . y h+ =l + g h= hl=+ l g =r = =l= == = hl=+ + =*=+ .' = k=h+ = == = = h= rh== w 8396 :;4561 lh g g* iiiiir l l . = r = =hh , v u v -. g lhh= h l=l h ( o l l lh =. l z . l==r m ky = l+ l= , ii g - o l z = l = ly k~h h = jj + =l= k=h+ = h h & r w l==r M l==r lhh= ==h= g . l==r ==h= l=l h = . ly=r = * h r+ o r = hh h h= ,l=.= g r= . l.= = hh .l - l + ly=r = lhh= o g ' ly= lhh= o h l=l h ( w l =ryh =hh = h = h o lyh= =ly l h=+ r= h( l

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= h l=l w h = (ly lhh= l=l =* = h w ' = (r & k g = ( l= , . . - = , -. = l h==r , -. ly w h= py h*)k= + = h= = ,hh - = ( h== , -v . (ly = (ly h =r o (ly = ly=h = = = (ly lhh= o (ly h =r r h =r g w * l lhh= .= h z m ly=+ lhh= l r= l=l o = r= h == = y= ll h==r (ly lhh= = o= = *h K =h.l*g = v} = == =g l = . =l h r= h==r . == *= h = o l r h h = l = *h ' = m = ll (ly lhh= . = l = ' = 8396 :;4561 l= = k= w w =l lhh= , eii g - o v (* l= , = ly z hh= t =k K h=+ - h g = k= w h( r= h = y= ll (ly lhh= 9 *;5 +946.<+5 -8396 :;4561 = h= h yk+ hy ==* k == = w v} g . g l= h = w ~r t =l 7 = , t7- } . l g = + l = gr= r + l=l ' = ly l ' = + = = = *h . =ly l ' = h w = = =g r l(r =. =ll = g w l v = = + v} L =l= == gl w o h l + M =h= l= l= l . g h g h= ly+ h+ o + g h = l = =hh g= y+ l = = ly ly .= = == g = =g l ==r = hh h=+ o l h . l l | s* |m o o =l ~ h =l l w l l w v} z h=ly g h r= h g== = l =ll = + = o l f h o= g 6 0325/523 + =( , t 7 -.g= =hh ==h* h l h K = g w = g *g h l = kyh h= =l l lg lhh g== l h = o= g 96300665 ly y h== h l l. =h=* g= o g = l h = r l = } l g = = g w = w = o= g 216 l l w l = o w g .=~ l h . = g g h hh

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o =g= = = l y~ ~ = = g g h h=l = = 8396 :;4561 g = y+ =l+ = l= l y~ = l = = o l==l+ l* r= l + 8396 :;4561 h ll g + g ly= l=l = ly = v . = = lh r== r= l hl=h h L o g = 8396 :;4561 r= h l l = h ~r =g= g h( l l m l =. g h r l l l h =hh g( =ly l = y= = g M l l . l==r . l= . = l lC =g l = . r = l= w = 5 2;4-5 0 m =h = = l= l . h h = h+ p)) =g l h~ l=. r = = g f , - l . h= p) e } f } h w = =g l lh h hl=+ =+ o ly w l = h p)) = /91/52 6 =l h = 2 /216'= = h= z ly w = L ( l= l= h h = = z y=g= z y=i =L ( h=yh=+ h = l y~ = ly= = l= ==k / 012312 v . = h~ g p)) . r= /442 6 = . ' =g | = l= == + ==r . | = *)=hh l= == =r } . = hy r= h+ l = l or= 06 M=( h l = l= h+ l ~h ,=l h . - '=g= l h= w h h l ~h k lyl ,*lh= l = w h l = -. ly hh h ( lhh r= =l h or= 46 526 M = = r== = , l = w l g - ~h yh~ hl=+ h . = r= h . r= lg = v l ' r h == g g h h l =r = w hh . = = r = w | = } h lh h hl=+ l = h p)) =( lh = l(l rh== w , = z =h* z =k. h~ {=* l . o=r - o= g= yh~ yk= + r= h = = h~ lh w = w l h = e l = * = h w + = y h+ w h l l g= r

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l=l w l=g= r= ll = .= ' = h= ) == = = h p)) 8396 :;4561 6210 =1 100>.43562 =420 >2 <48<>8 =1<443562 74><1<48<>8 0>.>2 A4588471 7.60051. =.6531 <48<>8 0>.>2 A4588471 .42/ 4>36>.=1 84621 =1<443562+ o= h+ ~h p)) = h=~ = h l = 8396:;4561=g l =ryh s D o l g = = = h p)) = = l h = l l z y=i =. z y=i } = =l= . = h+ ~h p)) = === h l + = lg r l =ryh s D ,ly w l~ lg r l . g = l lg r l -. = ~ r= l=l ' = l= =* h l= = = m g+ = l l . = g= h =ly & = = h l ~h p)) = 8396 :;4561 = ( ly==r l=l ' hy h+ = + l = h ( + hy = g h (. h ly = ~h h yk= + z h=+ ly=h l y~ = =l + = = h l =r )k* l=+ = = = =l + + ~h r== w l =hh r ~ =l h ,l + w l = w h lh= l = w h l = + l l - t * ' = h=+ ln=r ly l= . h p)) r= h == = w = h = h+ v} lhh = w h =ly l l( h l = l= = ==r

(22)

TU z hh = h= h+ . = l=l p)) = w =l lh h yk+ + l o= l=l L ly o h 61161 = ' l = = ,= = h = = hy l = = =ryh -h= = hl g= o = w h=.l *'* = l = w =l = l=l . =g r z l ~g g . r = h+ . l= h= } = + =h = ~h . l r= h = l = w o= l l y~ = w l g h l= o= h= === h=.==k /0345612312l = h w l= ly+ l=l h= ' = + h = l=l ~h v l = ' h= / 012312.+ = g = w =l ~h = h= m g =h = h= h+ . h= /0345612312 f l==l+ g= M (= M h= h=L=r= , =l h - h . h= M h= lh h= l= w h =l l w *l = w h= l= . r=== =+ h= l= w g= = l=l h+ Mg= = h+ h= l = = = ' = h+ ~h o h= / 0345612312 ' r =ryh = h === = l= + h g h ,= h - l l=l o= l l h==r === = h=+ h ' = h h r ( w = rh = l l h h r = h h= =+ ' === = l= h h~ == hh = h g l= } = l= 8396 :;4561. h= / 0345612312 l * =g l h==r === ) &tq} . = h l rl l=* l v} h= g= r= h =l =g l = rl o l = g r h l l h ) & t q} ' = l h = lh~ sli r= * h & l* w h === h==r = ' = ) &tq} ~h hl=+ + =*=+ = = 8396 :;4561 & l~ ~l l= l r= ly & ly lh h hl=+ h== k u ly=r h hl=+ h y= = l & g + h==r =. ' = (r . = r & '=(r h==r = == h= h= = 8396 :;4561 = hl=+ + =*=+ & h=+ + h==r

(23)

D4588471 5253548+ I1A81 =1 .421A1234=43435 =1 A4588471 4.CD+ = = + = l g / 012312 = h ==k yk+ ~h } . = ~ = ~l =ly (~ h l= = ( ===. g = w =l g= <<54<6 5 2;)-+6,.+;9 =l ' ~h yk+ . r = ly h~ h = h= + = l h L = , -+ h~ . l *'* + =h o= h= l l l g=.+ lh h v = r= h =

(24)

' yky~ h(l=l lh + = h = = yk+ ~h } . h~ = g ==h~ + = h w h = + g w *hh = w h= o= l l ==h+ . + + .h ' r hl =r =( h= = w l ==* h+ l . + l= = g k~h . ' = = = w ly+ 01364 261464 z ly+ = = w h = g=r =* ng = o ~h g === = h w h= + = lyh , h= l= =l -. ykrr , (l= l lg yk* = + - =l + *h= , h= = h =l - } = w h =l= w = g=r ly o=h= r= h = l= h~ y+ = w g= w h = = k~h v h =h =l==l ,= . - = k~h yk+ ' = = llh= g= k~h = l==l y+ = = h = h+ v =hh h* r h~ g r= h ==h~ g= h o l = l =g= w { z { z j . . { . z . | y o l { z

+

l w p==r= p= j l* m = = . ly+ h= /0345612312. = {==ln v { v if . h r= h ==k h= == h~ = <<54<6 5 +6*<).+6,.+;9 ly h~ . l l= + l *l =h =. r=. + =l h g= or l l ly ly =ly l =.= = ' hy = w h= l=* + ,h (.g h (. l ( - l= h =l + = h~ m = w h . = = hy h (. = * ~h h~ =ly = lh= = l h = kh= z = w h= = h l + r ly ly ' h ' h } = l= =hy h (.= = h=* l= = hy ==k /0345612312 == o = L = =g= w {= n= q y { q j m ~h == . ( w + = * ~h g= ,lh = l = w h k s h= yhr~ h* ( -M

(

p g

u ∈ H

1 (Ω)

+

∇·φ(u) = f

=

Ω,

, f -

φ

= , ' g= = q

d

- =

f

=

L

2 (Ω)

t k= l =h h==r

{T

h

}

h>0

h=

Ω

. r =

u

h

= ~h , f -=ly= hy h ( = l= lh ,

u

h

∈ H

1 (Ω)

- h =g= w l y~ h= /0345612312 k . ' l = l l ~g h g = =r ly+ h= / 0345612312

(25)

0).1;5 (,< <)6+46 M z ly+ = = {= n= q y { q j u y u = g =g h w y= g m

v ∈ V

. (

ρ

= =ly

u

h

v

=

ρ(u

h

; v) =

X

T ∈T

h

ρ

T

(u

h

; v),

=g l l l=

ρ

T

(u

h

; v) = (f − ∇·φ(u

h

), v)

0,T

+ (φ(u

h

)·n

T

, v)

0,∂T

,

n

T

r = h= w ' = h=

T

m * h

R ⊂ Ω

s ≥ 0

.l

s,R

= l = l== =

H

s

_(R)

,

L

2 _(R)

s = 0

- ' = h =l K

F

=l h==r . l *'* + w w h=

T

1 T

2

=

T

h

=g l

F = T

1 ∩ T

2

& r =

n

1 n

2

h= w '

T

1 T

2

. lg h & ( = = lh= h=

φ(u

h

)

' =g

F

=

φ(u

h

)·n

= φ(u

h

)|

T

1 ·n

1 + φ(u

h

)|

T

2 ·n

2 .

, -m =l

F

= . h h

φ(u

h

)·n

= φ(u

h

)|

T

·n

T

h

T

h

F

=l z ( = ' =(r i m

T ∈ T

h

. ( h= l= ' = h=

T

=

F

n

T

1

2

QI1A81 =1 4<1 523/.51>.1 =1 A4588471+

η

T

= h

T

kf − ∇·φ(u

h

)k

0,T

+ h

1/2

T

k φ(u

h

)·n

k

0,∂T

,

, -

h

T

r =h~ = h=

T

& h=+ +

η

T

lh h == g + = , f -g= = ly=+ h= h == g = = = w =l =h= & = (= g=

k∇(u − u

h

)k

0,Ω

≤ c

1 



X

T ∈T

h

η

_T

2 



1/2

,

(26)

c

1

l= =

h

K = ~h o==l

φ(u) = −∇u

. h r= h = h= g= M w l=

c

2

. =

h

. +

T ∈ T

h

.

η

T

≤ c

2 (k∇(u − u

h

)k

0,∆

T

+ h

T

kf − f

h

k

0,∆

T

) ,

∆

T

h=l*h l =

T

f

h

=ll= = hl = w r= ' = hk

f

ly=+ h= 0).1;5 (,< 4,-+.) M z hy + h = w =g= w y y j { ln q ==ly { q i h h = w h= 2/ l*

ψ : H

1 _{(Ω) →}

q + = g + = = l(+ h r . lhh = w h hk =l h = *r h= yk+ hk l= =l K yky~ =+ = . h= g=

|ψ(u) − ψ(u

h

)| ≤ c

X

T ∈T

h

η

T

ω

T

,

+ =

η

T

( , - + =

ω

T

g = ~h = = w = h h= yk+ z ~h = r= h = l= == 0).1;5 1+)<,<*1+345 M z ly+ . = {=n B { B . l ' ly =l = w h=.

V

h

. = = rh = r kh = = = h h r= h==r z ly h ==l ' =l ly C l = .

V

_h

f

. ' =l = w h= ly

V

_h

e

= h

V

_h

e

= V

h

⊕ V

h

f

.

o h= g= l = ~h = w = =l ly C l = K l = yky~ ,=h* h yky~ = = + =h + =l

V

e

h

h = w h= = w =l +

V

h

-. h + g= l r= h= / 0345612312g =g= w { h=. v h= y { v l w } C sly } sie h + yky~ = = g( w =h= h l= } =g= w {=n B = ~h + k* h+ ( l = h ( o=r=r m

1

. = == g L o=h== =g= w l = l= ==.=g l h= /012312 /0345612312.= = t ly* ly=. t r =.{==r = tt { w h = h = h w = r ~h m r= h o ly h l ' ~h r= m g l l g. ) B yh y B ) hy lh h = + h ~h l= w o g=r t Dy & t& ii g =g h = h= / 0345612312 k y=ly+ 0).1;5 (,< 4 )34+-+A<)6 M K l h r+ . r h hy l = =l l C w =ly

ˆ

φ(u

h

)

= g r l

(27)

l= hh= =ly h l

f

=g l l o h= = = l=

φ(u

h

)

ˆ

φ(u

h

)

or l hy h = w =g= w m =r K kr mK j lh = l l C w = g r l w =l h h l z = l= h= lh h = o= g~ o=j . o=jj g r= h =g= w o= g~ o r oo f o=ly o= g~ = = = lh h ly+ == l l ly=h =h m ~h l=+ =l + =*=+ . l ' l . ' = = * l~ . l =l h *l= /924426.l *'* g(= w =l h l =h lh=+ =+ v . = w h= = hy h ( g( r= h = l l + = l= l~ = + =g l ly=r o= l l l =h l= l hy o lh h ( lhh l= = l= h ( = l= = = l l l* q l hh . g h l hy = g = l M = h== l= h=yh=+ . w ' l= ~h =r g h ly+ l l l= C w l = = l=l l w & = = w h l =g= w } k g } . s * ==h n q s q i . o l B yh y o B i . {= K ly { K i . sl= . B Dn B yh y s BB i . z ly * } y sl= z} sij v . K y= uy=n v K uij 0).1;5 5 +59+5+*<14 , -M o l = hy = l~ =( =ly l= h r= = w =l o= ly+ = = n Dl y j ~h =l = e }v = = w ~h e } = gl=l o) ==l { =+ = w ~h f } =l = gl=llh { i o h= l= w (lhh = =l r= l = w =l .

G

h

u

h

. r= = l~ = h=~ g=

η

T

= kG

h

u

h

− ∇u

h

k

0,T

,

T ∈ T

h

z ly+ = hl h g ' l = z =. r= h l= y+ h . hh (= = = {= z = { z ie } = 8396 :;4561. h= h= M h= == ==l } i = y h+ = {ie . h= = = = hl=+ + =*=+ } i h= k ~h =l e } } o h= / 0345612312 ~h == L =l * ~ h l= l = h g = =r r=hh =l= yk+ + yh~ . l h C lh=rh z =. l l L g h o= = h= / 0345612312 ~h ==+ ,= -

(28)

=g h o h =g= w r '=== h m = w h . y. s pyh sp i ==k / 0345612312 = l= = lyh= v hl h sly . K =g= u s K uii / 0345612312 ~h ==+ = m l= m l = l h~ ==k /0345612312 l= =l * h o h= =* yky~ lg + h==r h h.l + = = =+ = h =ryh === ==h~ l= w * h+ h u y u i r= h ==k /0345612312 ~h =g l* *=l == l = hy h ( =l * h o=ll = h= rh =g l * h= .l + l ' h= r= =l o =g= w { r=h. { = | y= {{ i l ' h= / 0345612312 =l * h h = = = h==r h.=g l (= h= h * = g =r === = h h==r } =ryh === h g = w h = sl= K = h s K i z y r z i o h= l l* ~h ==+ h= lg r = h=~ h= =l = = ' h

L

2 t

(H

x

1 )

,

L

2

h.

H

1

=l - g =ly .l hh = =n=n sly sif ==k = = o=nn =n=n o i . = = ly+ l l + h h= = h

L

∞

t

(L

2 x

)

,

L

∞

h

L

2

=l -=g l h= = lg r l z hy .+ =l == = . l lhh =l *= = l= /0345612312 = hy L l + = B y B y jf ==k / 012312 v = g ly=h = w = l .

U

. + g( e g= M = l

U − u

h

ly=h = w = = h+ g= 2/ h= /0345612312= l= == . = l

u − U

= l ly=h = w = == g h( + = ==+ r= . l h h

U − u

h

= = ~h l . l

u − U

=

U − u

h

} = =ly h= / 0345612312 = = h

L

∞

t

(L

2 x

)

~h ==+ g g . = .= =g= w v n y v . v pyh py j = ly+ =. = =g= {= n=. = { i = r= h. = =g= u y u + = ==+ + =*= } = ~h == . + l w K n =* r= = { = K { K i m ~h K n .{ = u y l~ l= h v hl l= == = h ( lh {ui o==k / 0345612312 h= = lg r l h= = = h

L

2 _t

(H

_x

1 )

,

L

2

h

H

1

=l - = g ==n== =n=n i l~ h*l= == = h ( = g h ( } h= h= h

L

∞

t

(L

2 x

)

L

∞

t

(H

x

1 )

= g h = = ly+ = l l K n

(29)

<<54<6 5 <)6;-4.+;9 o ' = h=ly ,= - = w =ryh =h g o h=ly = =+ = h + =g l h ( ly m = = . =ryh = h g g r= h l l= l~ lg r l l ' h+ = ~ r = h lg r o l~ lg r l r= lhh l l=l l v . h l=l =l h g =l h= o= rh = = =l h=ly . =hh ' =ly l ==~ . l h ' l ~h o= h= ly=h g r= l y~ + = = =g*= =l h h= = 8396:;4561 v =l . = l= = =l = + = L l= l y~ .= lg r l =ryh = TU > ? o l= L l l y~ ==k / 0345612312 k~h + = = w g = h w + = = + l .

u

. r g = + = + = .

p

.= + = k ==+ o=l= g k~h = =l , -*, e -. = h= ) . = l + =

−∇·σ

0 (u) + b∇p = ρ

ref

F

m

,

=

[0, T ] × Ω,

,

-∂

t

1 M

p + b∇·u

− ∇·(κ∇p) = −∇·(κρF

m

),

=

[0, T ] × Ω,

, j -h l h l =

σ

0 _{(u) = λ}

1 (∇·u)I

d

+ 2λ

2 ε(u)

l

Ω

h= =l

T

h ( h = m h( . * = ==k ' l h } ly yhr~ =l h . w ' l h w } ly *s h= = = l & h = = ly= e + = h = = ~h , -*, j -

t ∈ [0, T ]

.

sup

s∈[0,t]

ku(s, ·)k

1,Ω

+ sup

s∈[0,t]

kp(s, ·)k

0,Ω

+

Z

t

0 kp(s, ·)k

2 1,Ω

ds

1/2

.

, -z y=l f h l* l ' = h l=+ =l

L

∞

t

(H

x

1 )

.

L

∞

t

(L

2 x

)

L

2 _t

(H

_x

1 )

lg h & g = + = h = , -= l * == l= == =l h . = g= == = hk + ==+ == = l 6 366 6 5604 l= == o= l= ~h , -*, j - = = = lyh= v * | = n. l *'* = hy h ( lh =l lyh= v h* l h. l + l = ly w h+ l = 8396 :;4561 o w

(30)

==k ' l= h =

θ

*lyh= ~g = g = w ~h y+ } . = l= == . lhh = l h ( == = =l. ly = r

2

=l h r

1

= r= h r g =l h + = z ly w =L ( h=yh=+ h = ly= f & l = ==k / 0345612312 = w h= v * | = n * ~h , -*, j - yky~ + h==r l= h h l yky~ lg . == = h w = m ~h ) . ==k h=yh=+ = l= h==r = h l ly+ . = ly w = g h =l h lhh l .+ l = = l= ~h ==+ lhh + = =ly= & g = l = ly= } = = =+ . =r === ==h~ l= =* h l L g h l= ' h g ,~h = l h==r g== - o= * h=+ = l + 8396 :;4561 = = g=r = = l= l y~ } w ==k /0345612312 l*=~ o= h~ w = ~h , -*, j - l h = { r=h.{ = | y= {{ i + =* = ly= o h= / 0345612312 + lg =( l = h

L

2 _t

(H

_x

1 )

z =. = lg r l =l l h= *h=

L

∞

t

(H

x

1 )

=l h

L

∞

t

(L

2 x

)

z l h ==r + = = ly= = lg r l h= = =ly l *h= h

L

∞

t

(L

2 x

)

o= ly+ l l + o i h l ~h s = l l ==k '=h= ) =( g = w h= = lg r l h= = = ' =l h h

L

∞

t

(H

x

1 )

= h

L

∞

t

(L

2 x

)

TU ? o l hh lh ly= } = ly= e. + + h = =g ' = y h w w h = h ~h ) .l *'* + = l=+ k~h = =l = + =*=+ h = & = ' k~h + = lg= l =l h = + l = = rh== & e g ,== == - ~h h~ o =l h = g= =h r= ~ . =h h k~h l ) v (. l= == =( h( ==k h=yh=+ l = ly= g= =( r= ' } = ly= f. l ==k h=yh=+ ~h == == o ==k / 012312 / 0345612312 ~h == =

(31)

= ly= g= & h = l==l~ ~h l m . = l= ~h = hy h ( lh l ==k /012312.l + h w L l q + g h = l= == & ==k / 0345612312 = w ly+ o= h~ hy l = ly+ g ~h + . h = r== lg r l h= h= h= = h

H

1

o= l hy = =. h = l = yky~ h= =g l lg r l h= =l h h

H

1

m ly=l w hy . = (= h= = 7 + w h = h+ . = = 8396 :;4561.g ==k /012312 /0345612312 } = ly= . ==k h=yh=+ ~h == == & h = = ~h l ,r=== l = -+ h ( = l h=yh=+ ~h == & l ===k /012312= g = ly+ L l* + B y B y jf o==k /012312 = w h= v | = n ~h l= { = = = =.o =. = o e. o . po o ~h w =h l r~ h l= = + h { r=.l *'* + h = h g h = = , j - m = = .==k / 0345612312 = = hy l hy = l l + o=hy l l h = = ~h l h= (= =l h = = h

L

∞

_t

(H

_x

1 )

= = h

L

∞

t

(L

2 x

)

L

2 _t

(H

_x

1 )

z = = lg r l h=* =l =l h = = h

L

∞

t

(H

x

1 )

= = = h

L

∞

_t

(L

2 _x

)

= h= o= hy = l l + h l g l h= & ( = h+ = l= 8396:;4561g== ==k / 012312 = = h=l h= /0345612312 o ly= = ll l =g= y~ + + lg o= = = = w ly= f l = = l v ij h l= z =r= h 'g l l i =. i . ij=. ij