Numerical simulation of blood flows through a porous interface

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Numerical simulation of blood flows through a porous interface

Miguel Angel Fernández, Jean-Frédéric Gerbeau, Vincent Martin

To cite this version:

Miguel Angel Fernández, Jean-Frédéric Gerbeau, Vincent Martin. Numerical simulation of blood flows through a porous interface. [Research Report] RR-6146, INRIA. 2007, pp.30. �inria-00136971v2�

a p p o r t

d e r e c h e r c h e

ISRNINRIA/RR--6146--FR+ENG

Thème BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Numerical simulation of blood flows through a porous interface

M.A. Fernández — J.-F. Gerbeau — V. Martin

N° 6146

Mars 2007

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]#i€A]N„l0\^psx{]^lcY‚s†

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(ui, pi)∈[H^k+1(Ωi)]ⁿ×H^k(Ωi), i= 1,2. ^4;7

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n‚=jm]?ƒ'jml{sn[o¥rqoqºN1l{o¥p=†sr¥rqp1­}oq†[t0€‚xc]?jcjcv[x{]’ƒio¸j{„ps†Dl{o¥†kv[o¥lcoq]Njal{Y[xcp=v[tsY'lcY‚].o¥†Dlc]Nxm¡¹=„]

γ^¦

$'&6$'&)( š ¿ 3) " ›iš1 ¿ ˜

`µ†·­}YS1la¡¢psrqr¥p1­~jN¤=­a]jmY‚sr¥rÂsj{jcv[\^]

Ω^l{p(nA] %©oq€‚j{„…Y[o6l{º´µ„p=†=l{o¥†kv[p=v‚jGƒ[ps\oq†—oq†

R^{n 4}n= 2 ^p=x 3^7;­}o¥lcY

.€ApsrqhkY[]Nƒix…rSnSp=v[†‚ƒ[sxch

∂Ωs†‚ƒp=vilz­asx{ƒJ€Apsoq†Dlcoq†[t†[p=xc\r

n^{¦ ž} ] jcY‚rqrSsrqjcp(=jcjcv[\^]lcY[] £koqj{„pDjmo¥lzh

µ

lcp(nS]~„ps†‚jml{s†=l?¦4%]l

{T^h}^0<h≤1ƒi]†‚plc]?j9x{]tsv‚rqsx/¡¹\^o¥rqh(ps¡Sl{xco¸†‚tsv[r¸1l{o¥p=†‚j/p¡AlcY[] ƒips\oq†

Ω4¢oq†0l{Y[]

jc]†‚jc]p¡ ؑ+7ƦB6[p=x#]N=„…Ylcx{oqs†[tsv[r¸1l{o¥p=†

T^h¤‚lcY[](jcv[n‚j{„x{oq€il

h∈(0,1]x{]¡¢]Nx{j}lcpl{Y[]rq]£s]Nr©p¡;x{]»‚†‚]\^]†Dl p¡lcY‚]lcx{oqs†[tsv[r¸1l{o¥p=†©¤k­}Y[o¸„…Yo¸j}ƒi]»‚†[]Nƒnkh

h= max

T∈Th

hT,

­}o¥lcY

hT

¤Al{Y[]Jƒ[oqs\J]lc]Nx ps¡

T^{¦ ž} ]0jcY‚rqr/sj{jmv[\^](¡¢psx~l{Y[]^jcs¶s]ps¡Gjmoq\^€[r¥o¸„o¥lzhlcY[]^uDv‚sjco¥´ v[†‚o6¡¢p=xc\^o¥lzh'ps¡

lcY‚]lcx{oqs†[tsv[r¸1l{o¥p=†©¤ ;l{Y[]x{]]#ioqjml}lz­apJ€Ap=jco¥lcoq£s].„p=†‚jzl…†Dl{j

Cmin

s†‚ƒ

Cmax

jcv‚„…YlcYS1l

CminhT ≤h≤CmaxhT, ∀T ∈ T^h.

ŒLp=xc]Np1£s]x?¤&¡¢psx^rqr

0 < h ≤ 1¤9lcY[]'lcx{oqs†[tsv‚rqlcoqps†

T^h o¸j^jmv[€‚€SpDjm]?ƒ¬lcp-nS]L„ps†i¡¢p=xc\^oq†[tŽ­}o¥lcYªl{Y[]

oq†=l{]xc¡¹s„]

γ^{¦ %©]l} G^h nS]l{Y[].jm]l~p¡&oq†Dlc]Nxm´¯]rq]\^]†Dl}nApsv[†‚ƒ‚x{o¥]?jps¡

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ϕ¤sl{Y[];yzv[\^€

JϕnKp1£s]x9s†]Nƒit=]

E∈ G^h o¸j9ƒi]»S†[]Nƒ

nkh

JϕnK(x) = lim

t→0⁺ ϕ(x+tn¹_E)n¹_E+ϕ(x+tn²_E)n²_E ,

­}Y[]Nxc]

x ∈ E^¤9†‚ƒ n¹_E 4¢x{]Njc€S]?„Ælcoq£s]Nr¥h

n²_E7.o¸jl{Y[]v[†[o¥l0†[p=xc\r£s]?„Ælcp=xl{p

E €Apsoq†=l{o¥†‚t¨psv[lz­asx{ƒ

Ω1

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Ω2

7Ʀ

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lcY‚](€[x{]Nj{jmv‚xc]l{Y‚1l#­}oqrqrnA]0ƒio¸jc„ps†Dlcoq†kv[psvSj}ps†l{Y[].¡¹s„]Nj#o¥†¨“ 4¹ps†lcY‚](]Nƒ[ts]Nj~o¥†¨ 27~p¡/lcY[]0o¥†Dl{]xc¡¹s„]

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V_h^k =

v_h∈ C⁰( ¯Ω) : ∀T ∈ T^h v_h|^T ∈(Pk)ⁿ ∩V, M_h^k = n

qh∈M˜ : p|^Ωi ∈ C⁰( ¯Ωi), i= 1,2, ∀T ∈ T^h qh|^T ∈Pk

o.

6o¥†Srqr¥h=¤i¡¢psx

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M_h^{k lcp} Ωi

¤

M_h,i^k =

qh|Ωi : qh∈M_h^k .

%©]l

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ku−SZ_h^kuk^0,Ω≤c0hkuk^1,Ω. ^4W‰;7

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ku−SZ_h^kuk^0,E ≤C0h^1/2kuk^1,Ω. ^4¯‘7

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T ∈ T^h¤[s†‚ƒ 0≤m≤2^¤ 2≤l≤k+ 1^:

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†Srqpstsp=v‚jcr¥h=¤DvSjmoq†[tJlcx…s„] oq†[]NuDv‚sr¥o¥lzhs¤k¡¢p=x

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ku−I_h^kuk^0,E ≤c2h^l−1/2 X

i=1,2

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6[p=x

i= 1,2^rq]l J_h,i^k :L²(Ωi)7→M_h,i^{k nA]l{Y[]} L²´¯€[x{pyz]N„lcoqps†Lp=†=l{p

M_h,i^k ¦XYkv‚jN¤‚¡¢p=x

0≤m≤1^¤ 1≤l≤k+ 1¤i­a]Y‚V£s]

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4zŠ?;7

ž ]lcY[]N†oq†Dlcx{piƒiv‚„]’l{Y[] 4¢t=r¥p=n‚r7ps€A]x…1lcp=x

J_h^k :L²(Ω)7→M_h^k¤[ƒi]»‚†[]Nƒnkh

(J_h^kp)|^Ωi=J_h,i^k (p|^Ωi), i= 1,2.

6[x{ps\ 4zŠN;7¤io¥l}lcY[]N†¡¢psrqr¥p1­~jal{Y‚1l}¡¢p=x

0≤m≤1^¤ 1≤l≤k+ 1^¤

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i=1,2

kpk^l,Ωi p∈L²(Ω), p|^Ωi ∈H^l(Ωi), i= 1,2.

4zŠ=Š7

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kJ∇v_h·nKk^0,E ≤c6h^−1/2kv_hk^1,T1∪T2 v_h∈V_h. ^4zŠ?“;7

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ku|^γk^0,γ ≤cγkuk^1,Ω u∈V.

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(uh, ph)∈V_h×Mh

jcv‚„…Y·l{Y‚1l

Bδ,R_γ (uh, ph),(vh, qh)

=Fδ (vh, qh)

, ∀(vh, qh)∈Vh×Mh, ^{4zŠN‹ 7}

­}o¥lcY

Bδ,R_γ (u, p),(v, q)

=(µ∇u,∇v)Ω+ (Rγu,v)γ−(p,^ƒio¥£ v)Ω+ (q, ^ƒioq£ u)Ω

+δ X

T∈Th

h²

µ (−µ∆u+∇p,∇q)_T

−δ X

E∈G_h

h

µ Jµ∇u·n−pnK+R_γu,JqnK

E,

Fδ (v, q)

=(ff,v)Ω+δ X

T∈T_h

h²

µ (ff,∇q)_T,

4zŠ %7

†Sƒ

δ >0o¸j}J€‚x…\^]l{]xoq†‚ƒi]€A]†Sƒi]†Dl#ps¡

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L²´µ„p=†=l{xcp=r[ps†

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jcoqjmlc]N†=l?¦9XY[o¸j}oqj}jml{lc]?ƒ·oq†¡¢psrqr¥p1­}oq†[t^w9x{ps€Ap=jco6l{o¥p=† :

š ÄÄ ˜! ¢™! Ä#" $'& & 0 +!*

(u, p)∈[H¹(Ω)]ⁿ×L²(Ω)^{,!1 #$ 4mŠ‹ 7 #0} ff+µ∆u− ∇p∈[L²(T)]ⁿ, ∀T ∈ T^h,

Jµ∇u·n−pnKE+R_γu∈[L²(E)]ⁿ, ∀E∈ G^h,

>

(uh, ph)∈Vh×Mh

!1 $ A 4mŠ‹;7 !1

Bδ,R_γ (u−uh, p−ph),(vh, qh)

= 0, ∀(vh, qh)∈Vh×Mh. ^4zŠ?‰;7

*F"#`¯loqj(ƒ[o¥x{]N„l„ps†‚jc]NuDv[]N†‚„]~ps¡©lcY‚]’jmlcx{ps†[t0„ps†Sjmo¸jzl{]†‚„h^p¡½lcY‚]’jml{s†‚ƒ[x…ƒ sr¥]Nxc¶koq†—\^]lcY‚pkƒ©¤=l{Y[]

¡¹s„l}lcY‚l}lcY[]jcpsrqvil{o¥p=†j{1l{oqjm»‚]Nj

f_f +µ∆u− ∇p= 0, ^oq† [D⁰(Ωi)]ⁿ, i= 1,2, Jµ∇u·n−pnKγ+Rγu= 0, ^o¥† [H^−1/2(γ)]ⁿ,

†Sƒ·l{Y[]

γ´¯„ps†i¡¢p=xc\^o¥lzhJps¡&l{Y[]lcx{o¸†[t=v[rqlcoqps†

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