Steady Fluid-Structure Interaction Using Fictitious Domain

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Steady Fluid-Structure Interaction Using Fictitious Domain

Andrei Halanay, Cornel Murea

To cite this version:

Andrei Halanay, Cornel Murea. Steady Fluid-Structure Interaction Using Fictitious Domain. 26th

Conference on System Modeling and Optimization (CSMO), Sep 2013, Klagenfurt, Austria. pp.128-

137, �10.1007/978-3-662-45504-3_12�. �hal-01286405�

Steady fluid-structure interaction using fictitious domain

Andrei Halanay^1? and Cornel Marius Murea²

1 Department of Mathematics 1, University Politehnica of Bucharest 313 Splaiul Independent¸ei, RO-060042, Bucharest, Romania

halanay@mathem.pub.ro

2 Laboratoire de Math´ematiques, Informatique et Applications Universit´e de Haute Alsace

4-6, rue des Fr`eres Lumi`ere, 68093 Mulhouse Cedex, France cornel.murea@uha.fr

http://www.edp.lmia.uha.fr/murea/

Abstract. We present a formulation for a steady fluid-structure inter- action problem using fictitious domain technique with penalization. Nu- merical results are presented.

Keywords: fluid-structure interaction, fictitious domain

1 Setting for a steady fluid-structure interaction problem

LetD⊂R² be a bounded open domain with boundary∂D. LetΩ^S₀ be the un- deformed structure domain, and suppose that its boundary admits the decom- position∂Ω^S₀ =ΓD∪Γ0, whereΓ0 is a relatively open subset of the boundary.

OnΓ_Dwe impose zero displacement for the structure. We assume thatΩ^S₀ ⊂D.

Suppose that the structure is elastic and denote byu= (u1, u2) :Ω₀^S →R² its displacement. A particle of the structure with initial position at the point X will occupy the position x = ϕ(X) = X+u(X) in the deformed domain Ω^S_u =ϕ Ω^S₀

.

We assume that Ω^S_u ⊂ D and the fluid occupies Ω^F_u = D \Ω^S_u. We set Γ_u=ϕ(Γ₀), then the boundary of the deformed structure is∂Ω^S_u =Γ_D∪Γ_uand the boundary of the fluid domain admits the decomposition∂Ω_u^F =∂D∪Γ_D∪Γ_u. The fluid-structure geometrical configuration is represented in Figure 1.

Generally, the fluid equations are described using Eulerian coordinates, while for the structure equations, the Lagrangian coordinates are employed. The gradi- ents with respect to the Eulerian coordinatesx∈Ω_u^S of a scalar fieldq:D→R or a vector field w = (w₁, w₂) : D → R² are denoted by ∇q and ∇w. The divergence operators with respect to the Eulerian coordinates of a vector field w = (w1, w2) :D →R² and of a tensorσ= (σij)_1≤i,j≤2 are denoted by∇ ·w and∇ ·σ.

?Supported by Grant ID-PCE 2011-3-0211, Romania.

Γ Γ 0

Γu

D

ΩF ΩS

u u

D

Fig. 1.Geometrical configuration.

When the derivatives are with respect to the Lagrangian coordinatesX = ϕ⁻¹(x)∈Ω₀^S, we use the notations:∇_Xu,∇_X·u,∇_X·σ.

IfAis a square matrix, we denote by detA,A⁻¹,A^T its determinant, the in- verse and the transposed matrix, respectively. We write cofA= (detA) A^−1T the co-factor matrix ofA. We writeA^−T = A^−1T

.

We denote byF(X) =I+∇Xu(X) the gradient of the deformation and by J(X) = det F(X) the Jacobian determinant, whereIis the unit matrix.

We introduce the tensor (w) = ¹₂

∇w+ (∇w)^T

and we assume that the fluid is Newtonian and the Cauchy stress tensor is given by σ^F(v, p) =

−pI+2µ^F(v), whereµ^F >0 is the viscosity of the fluid andIis the unit matrix.

We assume that the structure verifies the linear elasticity equation, under the assumption of small deformations. The stress tensor of the structure written in the Lagrangian framework isσ^S(u) =λ^S(∇ ·u)I+ 2µ^S(u), whereλ^S, µ^S >0 are the Lam´e coefficients.

The problem is to find the structure displacementu : Ω^S₀ →R², the fluid velocityv:Ω^F_u →R² and the fluid pressurep:Ω^F_u →Rsuch that:

−∇X·σ^S(u) =f^S, inΩ₀^S (1)

u= 0, onΓ_D (2)

−∇ ·σ^F(v, p) =f^F, in Ω_u^F (3)

∇ ·v= 0, inΩ_u^F (4)

v= 0, on∂D (5)

v= 0, onΓD (6)

v= 0, onΓ_u (7)

ω σ^F(v, p)n^F

◦ϕ=−σ^S(u)n^S, onΓ0 (8) wheref^S :Ω₀^S→R²are the applied volume forces on the structure andn^Sis the structure unit outward vector normal to∂Ω₀^S. Similarly, we definef^F :Ω_u^F →R² andn^F the fluid unit outward vector normal to∂Ω_u^S.

Fluid-structure interaction using fictitious domain 129 We point out that the stress tensor of the structure is defined on the un- deformed structure domain Ω₀^S, while the Cauchy stress tensors of the fluid is defined in the deformed domainΩ^F_u.

We have used the notationω(X) =

JF^−Tn^S R²=

cof (F)n^S

R²forXon

∂Ω₀^S, which is a kind of Jacobian determinant for the change of variable formula for integral over surface.

The equations (1), (2) concern the structure, while (3)-(6) concern the fluid.

The equations (7), (8) represent the boundary conditions on the moving fluid- structure interface. The fluid and the structure domainsΩ_u^F,Ω_u^S depend on the structure displacement uwhich is unknown.

2 Parametrization and regularization of the characteristic function

The regularization of the characteristic function of the deformed structure do- main is necessary in order to prove the continuity of the solution with respect to the structure displacement, [3].

Denote byk·k_1,∞,Ω the usual norm of the Sobolev space W^1,∞(Ω) and by k·k_m,Ω the usual norm ofH^m(Ω),m≥0 with the conventionH⁰(Ω) =L²(Ω).

For every 0< δ <1, there exists 0< η_δ <1 such that

1−δ≤det (I+∇u)≤1 +δ, a.e.x∈Ω₀^S (9) for allu∈ W^1,∞(Ω₀^S)2

that satisfykuk_1,∞,ΩS 0 ≤η_δ. We define

Bδ ={u∈W^1,∞(Ω₀^S)²;||u||_1,∞,ΩS

0 ≤ηδ, u= 0 onΓD}. (10) Letj∈W^1,∞(D) be a parametrization ofΩ^S₀ ⊂D, i.e. :

j(x)>0, x∈Ω₀^S, j(x)<0, x∈D\Ω^S₀, j(x) = 0, x∈∂Ω₀^S. The parametrization is not necessarily unique.

Let u ∈ Bδ be a given structure displacement. Denote, as before, Ω_u^S = ϕ(Ω₀^S), where ϕ(X) = X+u(X). Then ϕ : Ω^S₀ → Ω^S_u is bijective and bilips- chitzian and

ju(y) =







j(x), y=ϕ(x)∈Ω^S_u

0, y∈∂Ω_u^S

−dist(y, Ω^S_u),y∈/ Ω^S_u is a parametrization ofΩ_u^S, ju∈W^1,∞(D).

IfH is the Heaviside functionH :R→ {0,1}, H(r) =

1, r≥0 0, r <0

thenH(ju(·)) is the characteristic function ofΩ_u^S.

Forε >0, letΩ₀^ε⊂⊂Ω₀^S. Sincej:D→Ris Lipschitz continuous andj >0 in Ω₀^S, there isµε>0 such thatj(x)≥µε>0, for allx∈Ω₀^ε. Consequently,

µε≤ min

y∈Ω^ε_u

ju(y), ∀u∈Bδ. Then we takeH^µε, the Yosida regularization ofH

H^µε(r) =







1, r≥µε r

µ_ε 0≤r < µε

0, r <0

and we setHeu(x) =H^µε(ju(x)) for allx∈D, which is a Lipschitz regularization of the characteristic function ofΩ_u^S. We have constructedHeu:D→R, Lipschitz onD, 0≤Heu(x)≤1, for allxin D such that

Heu(x) =

0,x∈D\Ω^S_u

1,x∈Ω_u^ε (11)

whereΩ_u^ε⊂⊂Ω_u^S.

3 Weak formulation using fictitious domain technique with penalization

We assume thatD andΩ₀^S are Lipschitz. Let us introduce the bi-linear forms a_S u,w^S

= Z

Ω^S₀

λ^S(∇ ·u) ∇ ·w^S

+ 2µ^S(u) : w^S dX

aF(v,w) = Z

D

2µ^F(v) :(w)dx b_F(w, p) =−

Z

D

(∇ ·w)p dx and the Hilbert spaces

W^S =n

w^S ∈ H¹ Ω₀^S2

; w^S= 0 onΓ_Do , W = H₀¹(D)²

,

Q=L²₀(D) ={q∈L²(D) ; Z

D

q dx= 0}.

We assume thatf^F ∈ L²(D)2

,f^S ∈ L²(Ω₀^S)2

.

Fluid-structure interaction using fictitious domain 131

Weak fluid formulation using fictitious domain

For a given u ∈ Bδ, we define: fluid velocity vε ∈ W and fluid pressure pε∈Q, as the solution of the following problem:

aF(vε,w) +bF(w, pε) +1

ε Z

D

Heu(vε·w+∇vε:∇w)dx= Z

D

f^F·wdx,∀w∈W (12) bF(vε, q) = 0,∀q∈Q (13)

Weak structure formulation

For givenu∈B_δ,v_ε∈W andp_ε∈Q, we define the structure displacement u_ε∈W^S as the solution of

aS uε,w^S

= Z

Ω₀^S

f^S·w^SdX+ Z

Ω₀^S

J σ^F(vε, pε)◦ϕ

F^−T :∇Xw^SdX +1

ε Z

Ω₀^S

JHeu(ϕ) (vε◦ϕ)·w^S+ (∇vε◦ϕ)F^−T :∇Xw^S dX

− Z

Ω₀^S

J f^F ◦ϕ

·w^SdX, ∀w^S ∈W^S(14)

whereϕ(X) =X+u(X),F(X) =I+∇Xu(X),J(X) = detF(X).

Remark 1. From the structure equation−∇ ·σ^S(uε) =f^S, inΩ₀^S using Green’s formula, we obtain for allw^S = 0 onΓD that

aS uε,w^S

= Z

Ω₀^S

f^S·w^SdX+ Z

Γ0

σ^S(uε)n^S·w^SdS.

We can prove (see [3]) that the sum of the last three terms in (14) is equal to the fluid forces acting on the structure which is also equals toR

Γ0σ^S(u_ε)n^S·w^SdS.

In fact, from (14) and the above weak formulation of the structure, we can get that the boundary condition at the interface concerning the continuity of the stress (8) is verified in a weak sense (see [3]). The second boundary condition at the interface is the continuity of the velocity (7). This is obtained by using the penalization term in the structure domain in (12).

For each i ∈ N^∗, there exists an unique eigenvalue λ_i > 0 and an unique eigenfunctionφⁱ∈W^S, solution of

aS φⁱ,w^S

=λi

Z

Ω^S₀

φⁱ·w^SdX, ∀w^S ∈W^S (15) such that

Z

Ω^S₀

φⁱ·φ^jdX=δij, (16)

see [8], chap. 6. We assume thatλ1≤λ2≤. . .. The set{φⁱ, i∈N^∗} forms an orthonormal basis of L²(Ω₀^S). Letm ∈N^∗ be given. Let u^m_ε be the orthogonal projection of uε on span φⁱ, i= 1, . . . , m

in L²(Ω₀^S), so u^m_ε = Pm i=1αiφⁱ, αi∈R.

We define B_δ^m=n

u∈ W^1,∞(Ω₀^S)²

; u= 0 onΓD, kuk_1,∞,ΩS 0 < ηδ, u=

m

X

i=1

αiφⁱ, αi∈R )

. (17) We haveB_δ^m⊂B_δ. For eachu∈B_δ, we define the nonlinear operator

T_ε^m(u) =u^m_ε .

A solution of the penalized fluid-structure interaction problem will be, by definition, a fixed point ofT_ε^min B_δ^m.

Remark 2. As in [3], we can prove the existence of a solution of the penalized fluid-structure interaction problemT_ε^m(u^m_ε ) =u^m_ε using the Schauder fixed point theorem. In order to obtain supplementary regularity of the Stokes equations, a non linear penalization term ¹_εR

DHe_usgn(v_ε)|v_ε|^α−1·wdx was employed in [3], whereα >2. In the present paper, we use a linear penalization term in (12), but we are working only with a finite number of eigenfunctions of the structure equations. The behavior ofu^m_ε whenεgoes to zero will be studied in [4].

4 Fixed point iterations

We replaceHe_uin (12) and (14) byχ^S_u the characteristic function ofΩ_u^S in order to simplify the computation. The regularization of the the characteristic function was necessary to obtain continuity of the solution with respect to the structure displacement.

Under the assumption of small displacements for the structure, we can ap- proach the Jacobian determinantJ by 1 and the gradient of the deformationF by the unit matrixI.

Algorithm 1 by fixed point iterations

Step 1. Given the initial displacement of the structure u^m,0 = Pm i=1α⁰_iφⁱ, compute the characteristic functionχ^S_u0, putk:= 0.

Step 2. Find the velocity v^k_ε ∈ H¹(D)2

, v_ε^k = g on ∂D and the pressure p^k_ε∈Qby solving the fluid problem

aF v_ε^k,w

+bF w, p^k_ε +1

ε Z

D

χ^S_uk

ε v^k_ε·w+∇v_ε^k:∇w dx=

Z

D

f^F·wdx, ∀w∈W bF v^k_ε, q

= 0, ∀q∈Q.

Fluid-structure interaction using fictitious domain 133 Step 3.Find the new displacement of the structureu^m,k+1_ε =Pm

i=1α^k+1_i φⁱ by solving

α^k+1_i λi= Z

Ω^S₀

f^S−f^F

·φⁱdx

+ Z

Ω^S₀

2µ^F v^k_ε

: φⁱ dx−

Z

Ω₀^S

∇ ·φⁱ p^k_εdx

+ 1 ε

Z

Ω₀^S

v^k_ε◦ϕ^k_ε

·φⁱ+ ∇v^k_ε◦ϕ^k_ε :∇φⁱ

dx, i= 1, . . . , m

whereϕ^k_ε(X) =X+u^m,k_ε (X).

Step 4.Stopping test: if

u^m,k_ε −u^m,k+1_{ε 0,Ω}S

0

≤tol, thenStop.

Step 5. Compute the characteristic function χ^S

u^m,k+1_ε , put k :=k+ 1 and Go to Step 2.

5 Least squares approach

The previous algorithm converges if the operatorT_ε^m is a contraction. But, the Algorithm 1fails for some physical parameters. For this reason, we introduce a second algorithm which is more robust.

Forα= (α₁, . . . , α_m)∈R^m, we can defineβ= (β₁, . . . , β_m)∈R^mby

T_ε^m

m

X

i=1

α_iφⁱ

!

=

m

X

i=1

β_iφⁱ.

We set the cost function J(α) = ¹₂Pm

i=1(αi−βi)² and now the problem to be solved is inf_α∈R^mJ(α).In order to solve the optimization problem, we employ the quasi-Newton iterative method called Broyden, Fletcher, Goldforb, Shano (BFGS) scheme (see for example [2], chap. 9).

Algorithm 2 by the BFGS method

Step 0.Choose a starting pointα⁰∈R^m, anm×msymmetric positive matrix H0. Setk= 0.

Step 1.Compute∇J(α^k).

Step 2.If

∇J(α^k)

< tolStop.

Step 3.Setd^k =−Hk∇J(α^k).

Step 4. Determine α^k+1 = α^k +θkd^k, θk > 0 by means of an approximate minimization

J(α^k+1)≈min

θ≥0J(α^k+θd^k).

Step 5.Computeδk =α^k+1−α^k.

Step 6.Compute∇J(α^k+1) andγk =∇J(α^k+1)− ∇J(α^k).

Step 7.Compute Hk+1=Hk+

1 + γ^T_kHkγk

δ_k^Tγk

δkδ^T_k δ_k^Tγk

−δkγ_k^THk+Hkγkδ^T_k δ^T_kγk

Step 8.Updatek=k+ 1 and go to theStep 2.

The matricesHkapproach the inverse of the Hessian ofJ. For the inaccurate line search at theStep 4, the methods of Goldstein and Armijo were used. If we denote byg: [0,∞)→Rthe functiong(θ) =J(α^k+θd^k), we determineθ_k >0 such that:g(0) + (1−λ)θ_kg⁰(0)≤g(θ_k)≤g(0) +λθ_kg⁰(0), whereλ∈(0,1/2).

In this paper, we compute∇J(α) by the Finite Differences Method _∂α^∂J

k(α)≈ (J(α+∆αkek)−J(α))/∆αk, whereek is thek-th vector of the canonical base ofR^mand∆αk >0 is the grid spacing.

Concerning the convergence rate, the fixed point algorithm is slower than the BFGS Method. If the starting point is not sufficiently close to the solution, the fixed point algorithm diverges. On the contrary, the BFGS Method is less sensitive to the choice of the starting point and, in general, it is convergent to a local minimizer from almost any starting point. This is the main advantage, (see [6]).

6 Numerical results. Deformation of a tall building under the action of wind

We have performed numerical simulations using a 2D model adapted from [1]

(see Figure 2).

Σ 1

Σ

Σ Σ

2

3 4

D

Γ Γ 0

Γu

D

ΩF Ω

S u

u

Fig. 2.Geometrical configuration for the numerical results

The dimensions of a rectangular tall building are: heightH = 180m, length L = 30 m. The computational domain of the fluid D is a rectangle of height

Fluid-structure interaction using fictitious domain 135 H1= 3H and lengthL1 =L+ 4H, its left bottom corner is at (0,0). We shall allow nonhomogeneous Dirichlet data in the numerical experiments.

The distance between the left side of the fluid and the left side of the structure isH. We denote byΣ1,Σ3the left and the right vertical boundaries and byΣ2, Σ4the bottom and the top boundaries, respectively.

The mechanical proprieties of the building assumed to be an elastic structure are: Young modulusE^S = 2.3×10⁵N/m², Poisson’s ratioν^S = 0.25, the applied volume forces on the structure f^S :Ω₀^S →R², f^S = (0,0) N/m³. If the Young modulus is E^S = 2.3×10⁸ N/m² as in [1], the displacements of the structure are very small.

The fluid is the air with: dynamic viscosity µ^F = 7.03×10⁻² N ·s/m², the applied volume forces on the fluid f^S : D → R², f^F = (0,0) N/m³. The inflow velocity profile isg(x₁, x₂) = 100 ^x_H²0.19

m/s.The considered boundary conditions for the fluid are more natural from the point of view of applications and differ slightly compared with the previous sections. We impose: v=gon Σ₁∪Σ₄, v= 0 onΣ₂ andσ^F(v, p)n^F = 0 onΣ₃.

Fig. 3.The fixed mesh of the fluid domain (left). The finer zone [H−L, H+ 2L]× [0, H+L] of the fluid mesh covers the structure mesh which occupies initially the rectangle [H, H +L]×[0, H] (zoom, right). The fluid and structure meshes are not compatible, for example, a vertex on the structure boundary is not necessary a vertex on the fluid mesh (right).

The numerical tests have been produced using the softwareFreeFem++[5].

For the approximation of the fluid velocity and pressure we have employed the triangular finite elements P1+bubble and P1 respectively on a mesh of 34871 triangles and 17550 vertices. The finite elementP1was used in order to solve the structure problem on a mesh of 192 triangles and 125 vertices. The characteristic function was approached byP0finite element.

We have performed the simulation using theAlgorithm 2described in the previous section. We have used the initial displacementα⁰ = 0 at the Step 0 and the tolerance tol = 0.0001 for the stopping criterion at the Step 2. The penalization parameter is ε = 0.001 and the number of the eigenfunctions is

m = 5. The stopping criterion holds after 6 iterations of the BFGS algorithm, the initial value of the cost function is 34.30 and the final value is 5.9×10⁻¹³. The maximal structural displacement is 0.148m.

Fig. 4.Velocity (left) and pressure (right) of the fluid around the deformed structure.

The fluid velocity is almost zero in the deformed structure domain, more preciselykvεk_1,ΩS

uε =q R

Dχ^S_uε(vε·v+∇vε:∇vε)dx= 0.00555.

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