Fourier Series Development for Solving a Steady Fluid Structure Interaction Problem

Int. J. Contemp. Math. Sciences, Vol. 8, 203, no. 2, 75-84 HIKARI Ltd, www.m-hikari.com Fourier Series Development for Solving a Steady Fluid Structure Interaction Problem Ibrahima Mbaye University of Thies, Department of Mathematics ibambaye2000@yahoo.fr Abstract This paper presents a new approximation of pressures on the interface between fluid and structure based on Fourier series development. This approximation gives easily an analytic solution to structure equation. Using the least squares method, we introduce an optimization problem to determine Fourier coefficients series and deduct the structure displacement, the fluid velocity and the fluid pressure. This present work aims to extend and improve the approximation method of pressures on the interface. Mathematics Subject Classification: 74F0, 65T40, 37M05 Keywords: Fluid-Structure, Fourier series development, Optimization Introduction Problems involving fluid structure interaction occur in a wide variety of engineering problems and therefore have attracted the interest of many investigations from different engineering disciplines. As a results, much efforts has gone into the development of general computational method for fluid structure systems [5], [6], [2], [5], [6],[3], [3], [9], [4], [0], [], [2]. We saw in a previous paper [3] that pressures on the interface was approximated by a linear combination of shape functions as well as by a development of Fourier sine series in []. In [3], fluid-structure interaction problem has been solved in the reference domain Ω F 0. Thus, this paper aims at showing a new approximation method of pressures on the interface between fluid and structure based on Fourier cosine series development. This appproximation gives easily an analytic solution to structure

76 I. Mbaye equation. The latter depends on Fourier coefficients series. Using the least squares method, we establish an optimization problem to compute Fourier coefficients and obtain the structure displacement, the fluid velocity and the fluid pressure. In this work, we note that the fluid structure interaction problem is solved in the moving domain Ω F u. To solve this optimization problem by the quasi-newton BFGS method we need the derivative of the cost function. Despite, it is a difficult task, the gradient of the cost function is computed in this work. In addition, the fluid is modeled by two dimensional Stoke equations for steady flow and the structure is represented by the one dimensional beam equation. 2 Fluid structure interaction problem We denote by Ω F u the two-dimensional domain occupied by the fluid, and Γ u the elastic interface between fluid and structure and Γ = Σ Σ 2 Σ 3 be the remaining external boundaries of the fluid as depicted Fig.. 2 u F u 3 2 Figure : Sets appearing to the fluid structure interaction problem Moreover, the reference domain is defined by Ω F 0 =[0,L] [0,H]. The coupled problem is to find (u, v, p) such that: ) ( (σ(p, v) n) e 2 +(u (x )) 2 for0 <x <L() EI d4 u = dx 4 u(0) = u(l) = du(0) dx = du(l) = 0 (2) dx μδv + p = f F, in Ω F u (3) v = 0, in Ω F u (4) v = g, on Σ (5) pi 2 n + μ v n = 0, on Σ 3 (6) v = 0, on Γ u (7) v 2 = 0, on Σ 2 (8) v x 2 = 0, on Σ 2 (9)

fluid structure interaction problem 77 Where, E is the young modulus, I is the moment of inertia, I 2 is the identity matrix, μ is the fluid viscosity, v first component of v, v 2 second component of v, the vector e 2 =(0, ), n is the unit outward normal vector, Σ 2 is the symetric axis, On Σ 2, we have the non penetration condition: v n = v 2 =0, On Σ 2, we have the continuity of Cauchy shear stress: σ n = v x 2 =0, g the velocity profil in inflow Σ. We assume that ( (σ(p, v) n) e 2 +(u (x )) 2 = p(x,h + u(x )) [3], [9]. As a consequence, equation () becomes EI d4 u = p(x dx 4,H + u(x )). 3 Fourier cosine series development We assume that on the interface between the fluid and the structure: Where ω = 2π T p(x,h+ u(x )) = α 0 + α n cos(nωx ). (0) n= is the frequency, α =(α 0,α,...,α N ) is Fourier cosine coefficients series. Now, to have numerical results, the partial sum is used: N p(x,h+ u(x )) α 0 + α n cos(nωx ). () n=

78 I. Mbaye Where N. Thus, this approximation gives the structure displacement u as follows: u(x )= ( x 4 N α 0 EI 24 + α n n= (nω) cos(nωx x 3 )+k 4 6 + k x 2 ) 2 2 + k 3x + k 4, (2) Thanks to equation (2) we determine constants k,k 2,k 3,k 4. Using the least squares method, we introduce the cost functional J to find Fourier coefficients. We have the following optimization problem: inf J(α) = ( L N 0 2 α n cos(nωx ) p(x,h+ u(x ))) dx, (3) n=0 subject to equations (2)-(2) and (3)-(9). To solve this optimization problem by the quasi-newton BFGS method we need the derivative of J with respect to α k, k {0,,...,N}. 4 Computation of gradient Let α k the k-th component of the vector α, we compute for each k in {0,,...,N} the partial derivative of J with respect to α k as follow: ( J L N )( = 2 a n cos(nωx ) p(x,h + u(x )) cos(kωx ) p(x ),H + u(x )) dx 0 n=0 To compute the term p(x,h+u(x )), we introduce two propositions [3], [9]. Proposition Find v (H (Ω F u ))2,v=g on Ω F u and p L2 (Ω F u )/R such that: { Ω μ v wdx F u Ω ( v)pdx = f F,w, w (H F u 0(Ω F u )) 2 Ω ( v)qdx = 0, q (4) F L2 (Ω F u u )/R have a unique solution. Thanks to the transformation T u : Ω F 0 Ω F u such that: x = x, ( x, x 2 ) Ω F 0 T u ( x, x 2 ) = (x,x 2 )= x 2 = H+u( x ) x H 2, ( x, x 2 ) Ω F 0 (5) we write the proposition in the reference domain Ω F 0 in order to partial derivatives under integral [3], [9]. After derivation, we use again the transformation T u to write equations in the moving domain Ω F u. Then, we deduct the following proposition.

fluid structure interaction problem 79 Proposition 2 Applications α R m v (H (Ω F u ))2 and α R m p L 2 (Ω F v u )/R are differentiable and there partial derivative (H (Ω F u ))2 and L 2 (Ω F u )/R verify the following system: p a F (α, v,w)+b F (α, w, p ) = c F (α, v, p), w (H (Ω F u ))2 Where, a F (α, v,w) = v Ω μ F u wdx b F (α, w, p ) = p Ω w F u dx c F (α, v, p) = 2 i= Ω F u b F (α, v,q) = 0 q L 2 (Ω F u )/R, a a f F i w idx (6) ( a w + μ ( a 2 a a Ω F u a x a a 2 2 ) w ) pdx x 2 2 i= Ω F u a = H + u(x ) (x,a 2 = u ) H H x 2, a = H Remark To compute the gradient, we need to solve: the structure equation ()-(2), to find u, the fluid equation (3)-(9), to find v and p, ( a v i w i μ ( a 2 a a a x x a a 2 2 ) v ) i w i dx x x 2 (7) u(x,α), a 2 = u (x,α) x 2 H the system (6), to find p(x,h+u(x )). Remark 2 In [3], author solves fluid-structure problem in reference domain Ω F 0. In the paper, we solve the fluid-structure problem in the moving domain Ω F u. 5 Numerical results We assume that the velocity on the boundary fluid domain is: { x 30( 2 2 ), (x v (x,x 2 ) = H 2,x 2 ) Σ 30, (x,x 2 ) Σ 2 v 2 (x,x 2 ) = 0, (x,x 2 ) Γ u The parameter values of the fluid and the structure are:

80 I. Mbaye Parameter related to fluid: The fluid viscosity is μ =0.035 g, the channel length is L =3cm, the channel width is H =0.5cm, the volume force cm s of the fluid f F =(0, 0). Paramater related to structure: The structure thickness h = 0.cm, Young s modulus is E =0.75 0 6 g, the moment of inertia is I = h3. cm s 2 2 We use the P 2 Lagrange finite element to approach velocities and P Lagrange finite element is used to approach the pressure. FreeFem++ [8] is used for the numerical tests. Table : Optimal values with BFGS method after 0 iterations, the starting point α initial =0. N ω J(α initial ) J(α optimal ) J CPU 37.524.30957 6.70 0 4 9.58s 2 37.524.30872 4.8 0 4 2.29s 3 37.524 0.69343 6.970 0 3 4.9s 4 37.524 0.6446 4.03 0 4 7.28s 6 37.524 0.040627 4.747 0 4 20.44s Table 2: α optimal with BFGS method after 0 iterations, the starting point α initial =0. N α optimal (9.83274, 7.68247) 2 (9.83582, 7.67294, 0.039202) 3 (9.76943, 7.82079, -0.05078,.27328) 4 (9.777, 7.80926, -0.092857,.25828, 0.02329) 6 (9.75304, 7.86072, -0.4766,.30454, 0.0567803, 0.40388, 0.0649477)

fluid structure interaction problem 8 25 pressure Fourier approximation 25 pressure Fourier approximation 20 20 5 5 0 0 5 5 0 0-5 0 0.5.5 2 2.5 3-5 0 0.5.5 2 2.5 3 Figure 2: Pressure and approximation case Figure 3: Pressure and approximation case N=2. N=4. 25 pressure Fourier approximation 20 5 25 20 Pressure N=2 Pressure N=4 Pressure N=6 Fourier N=2 Fourier N=4 Fourier N=6 0 5 5 0 0 5-5 0 0.5.5 2 2.5 3 0-5 0 0.5.5 2 2.5 3 Figure 4: Pressure and approximation case N=6. Figure 5: case N=2,4 and 6. The above figures display the pressure on the interface and the Fourier series approximation. When the number of Fourier coefficients N is increased, the approximation increases as well. It follows, therefore, that the approximation of the pressure on the interface by Fourier cosine series is becoming more accurate that greater the number of coefficients taken and tending to the pressure itself.

82 I. Mbaye Figure 6: velocity case N=2. Figure 7: Pressure case N=2. Figure 8: velocity case N=4. Figure 9: Pressure case N=4. Figure 0: velocity case N=6. Figure : Pressure case N=6. The above figures display the structure displacement, the fluid velocity and the pressure.

fluid structure interaction problem 83 6 Conclusion In this work, we introduce Fourier series development to solve fluid structure interaction problem. This approximation enable us to determine easily an analytic solution to structure equation. Using least squares method, we establish an optimization problem to compute Fourier coefficients α, the displacement u, the velocity v and the pressure p. Fourier series gives good results for N=6. It can be noted, if Fourier coefficients N is increased then J goes to zero and the coupled problem becomes equivalent to the optimization problem. This article is coming also with a view to extend and improve approximation methods of pressures on the interface. References [] S. Deparis, M.A. Fernandez, L. Formaggia. Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions. M2AN Math. Model. Numer. Anal., 2003. [2] L. Formaggia, J.F. Gerbeau, F. Nobile, A. Quarteroni. On the coupling of 3D and D Navier-Stokes equations for flow problems in compliant vessels. Comput. methods Appl. Mech. Engrg. 9(6-7) 56-582, 200. [3] M.A. Fernandez, M. Moubachir. An exact Block-Newton algorithm for solving fluid-structure interaction problems. C. R. Acad. Sci. Paris, Ser.I 336 68-686, 2003. [4] M.A. Fernandez, M. Moubachir. A Newton method using exact jacobians for solving fluid-structure coupling. Computers and Structures, vol.83, num.2-3, 2005. [5] J.F. Gerbeau, M. Vidrascu. A quasi-newton Algorithm on a reduced model for fluid-structure interaction problems in blood flows. Math. Model. Num. Anal., 2003. [6] V. Girault, P. A. Raviart. Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 986. [7] C. Grandmont, Y. Maday. Existence for an unsteady fluid-strucrure interaction problem. M2AN Math. Model. Numer. Anal. 34(3) 609-636, 2000. [8] F. Hecht, O. Pironneau. A finite element Software for PDE: FreeFem++. www-rocq.inria.fr/frederic.hecht, 2003.

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