In this paper, we propose a method to solve coupled problem. Our computational method is mainly based on conjugate gradient algorithm. We use finite difference method for the structure and finite element method for the fluid. Conjugate gradient method gives suitable numerical results according to some papers.
Problem involving fluid structure interaction occurs in a wide vatiety of engi- neering problems and therefore has attracted the interest of many investiga- tions from different engineering disciplines. As a result, much efforts has gone into the development of general computational method for fluid structure systems [
Amongst the computational methods for fluid structure interaction problem, we cite the fixed point method, the Newton method, the Quasi-Newton method, the fictitious domain method. In this work we present a method based on the conjugate gradient algorithm. In effect, the fluid interaction problems occur in biomedical fluids areas for example blood flow interaction with elastic veins. Thus, this paper aims at showing that, we can combine the finite difference method, the finite element method and the conjugate gradient method to solve fluid structure interaction problem. On the one hand, we use finite difference method to approximate the structure model in order to have a linear systems, On the other hand, we solve the stokes equation by the finite element method. Moreover, conjugate gradient method will be intruduced to compute the displacement of the structure. Thus, the velocity v and the pressure p of the fluid are done in the deformed domain. In addition, the fluid represented by the blood is modelled by two dimensional Stokes equation for steady flow and the structure represented by the body vessel is modelled by the one dimensional beam equation.
The fluid domain noted Ω F u ⊂ ℝ 2 is represented in the
Where, the border ∂ Ω F u = Σ 1 ∪ Σ 2 ∪ Σ 3 ∪ Γ 2 .
- Γ 2 is the interface between the fluid and the elastic structure
- Σ 1 is The inflow
- Σ 2 is a rigid border
- Σ 3 is the outflow
- L is the domain length
- H is the domain height
- u is the displacement of the structure
The fluid is considered to be Newtonian, incompressible and its state is describ- ed by the velocity v = ( v 1 , v 2 ) and the pressure p . The balance equations are
− μ Δ v + ∇ p = f F , in Ω F u (1)
∇ ⋅ v = 0 , in Ω F u (2)
v = G , on Σ 1 ∪ Σ 2 (3)
v = 0 , on Γ 2 (4)
− p I d n + ∂ v ∂ n = 0 , on Σ 3 (5)
- μ : the fluid viscosity
- I d the identity matrix
- f F = ( f 1 F , f 2 F ) the volume force of the fluid
- n is a unit normal vector
- G = ( G 1 , G 2 ) the velocity profil in Σ 1
The structure is assumed by elastic beam. We note u : [ 0 , L ] → ℝ the displace- ment of the structure, it is modelled by the beam equation
D u ( 4 ) ( x ) = p ( x , H + u ( x ) ) , ∀ x ∈ [ 0 , L ] , (6)
with the boundary conditions,
u ( 0 ) = u ( L ) = 0 (7)
u ′ ( 0 ) = u ′ ( L ) = 0 (8)
where,
• D = E × h 3 12 ( 1 − ν 2 )
• E is the Young modulus
• h elastic structure thickness
• ν the Poisson's coefficient
Remark: In Equation (6) we assume that only the pressure force is acting on the interface and also u is the transversal displacement [
The coupled problem is to find ( u , v , p ) such that:
{ D u ( 4 ) ( x ) = p ( x , H + u ( x ) ) ∀ x ∈ [ 0 , L ] u ( 0 ) = u ( L ) = 0 u ′ ( 0 ) = u ′ ( L ) = 0 − μ Δ v + ∇ p = f F , in Ω F u ∇ ⋅ v = 0 , in Ω F u v = G , on Σ 1 ∪ Σ 2 v = 0 , on Γ 2 − p I d n + ∂ v ∂ n = 0 , on Σ 3
In order to solve this coupled problem, we transform its continuous problem into a discreet problem by using finite difference method and finite element method.
Assumption: We consider u as a small displacement.
Thus, the Taylor formula gives
p ( x , H + u ( x ) ) ≈ p ( x , H ) + u ( x ) ∂ p ( x , H ) ∂ y , (9)
the Equation (6) becomes:
D u ( 4 ) ( x ) − u ( x ) ∂ p ( x , H ) ∂ y = p ( x , H ) , (10)
we pose α ( x ) = − ∂ p ( x , H ) ∂ y , finally we have,
D u ( 4 ) ( x ) + α ( x ) u ( x ) = p ( x , H ) . (11)
To discretize the Equation (11), we introduce a space step Δ x = L N + 1 . We denote by u i the value of the discrete solution at x i = i × Δ x for i ∈ { 0 , 1 , ⋯ , N + 1 } . We must also discretize The boundary conditions . A centred formula gives
u 1 = u − 1 et u N + 2 = u N (12)
and the boundary conditions u ( 0 ) = u ( L ) = 0 become
u 0 = u N + 1 = 0 (13)
we rewrite the Equation (11) in the discreet form
D u i − 2 − 4 u i − 1 + 6 u i − 4 u i + 1 + u i + 2 Δ x 4 + α i u i = P ( x i , H ) i = 1 , 2 , 3 ⋯ N (14)
Then, the continuous problem becomes the following algebraic equation A U = P , where
A = ( 7 D Δ x 4 + α 1 − 4 D Δ x 4 D Δ x 4 0 0 ⋯ 0 − 4 D Δ x 4 6 D Δ x 4 + α 2 − 4 D Δ x 4 D Δ x 4 0 ⋯ 0 D Δ x 4 − 4 D Δ x 4 6 D Δ x 4 + α 3 − 4 Δ x 4 D Δ x 4 ⋯ 0 ⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ 0 ⋯ D Δ x 4 − 4 D Δ x 4 6 D Δ x 4 + α N − 2 − 4 D Δ x 4 D Δ x 4 0 ⋯ 0 D Δ x 4 − 4 D Δ x 4 6 D Δ x 4 + α N − 1 − 4 D Δ x 4 0 ⋯ 0 0 D Δ x 4 − 4 D Δ x 4 7 D Δ x 4 + α N )
P = ( p 1 ( x , H ) p 2 ( x , H ) ⋮ p N ( x , H ) ) et U = ( u 1 u 2 ⋮ u N )
Proposition 1. Note that A is symmetric positive definite under this assumption α ( x ) ≥ 0 .
Proof. We will prove that α ( x ) ≥ 0 for all x ∈ [ 0 , L ] .
For all ( x , y ) ∈ [ 0 , L ] × [ 0 , H ] we have ∂ p ( x , y ) ∂ y = f 2 F ( x , y ) + Δ v 2 ( x , y ) , for y = H and x ∈ [ 0 , L ] we have v 2 ( x , H ) = 0 , then ∂ p ( x , H ) ∂ y = f 2 F ( x , H ) . By choosing f 2 ( x , H ) = − g × ρ s × h [
Since A is symmetric positive definite, so we can use the conjugate gradient method to solve the following coupled problem. Find ( v , p ) ∈ ( H 0 1 ( Ω F u ) ) 2 × L 2 ( Ω F u ) and U so that
{ A U = P μ ∫ Ω F u ∇ v : ∇ w d x − ∫ Ω F u p div ( v ) d x = ∫ Ω F u f F w d x , ∀ w ∈ ( H 0 1 ( Ω F u ) ) 2 ∫ Ω F u div ( v ) q d x = 0 , ∀ q ∈ L 2 ( Ω F u )
To solve numerically the coupled problem we use the following conjugate gradient algorithm.
Proposition 2. Let A be a symmetric positive definite matrix, and u 0 ∈ ℝ . Let ( u k , r k , z k ) be three sequences defined by the induction relations
z 0 = r 0 = P − A u 0 , and for k ≥ 0 { u k + 1 = u k + λ k z k r k + 1 = r k − λ k A z k z k + 1 = r k + 1 + α k z k
with
α k = ‖ r k + 1 ‖ 2 ‖ r k ‖ 2 and λ k = ‖ r k ‖ 2 ( z k , A z k )
Then, ( u k ) 0 ≤ k ≤ k 0 + 1 is the sequence of approximate solutions of the the conjugate gradient method [
Descriptionofthecomputationalmethod
Step 1: It computes in the initial field the velocity and the pressure.
Step 2: It uses the conjugate gradient algorithm to find the structure deformation u .
Step 3: It computes again the pressure and the velocity in the deformed domain
Let the real noted test defined by test = ‖ P ‖ 2 × ε . We define the stopping criterion of iterations for the conjugate gradient algorithm by rhoold > test and k < n where, k the number of iterations and rhoold = ‖ z 0 ‖ 2 .
We assume that the velocity on the boundary fluid domain is [
G = { G 1 ( x 1 , x 2 ) = v 1 = 30 ( 1 − x 2 2 H ) and G 2 ( x 1 , x 2 ) = v 2 = 0 , on Σ 1 G 1 ( x 1 , x 2 ) = v 1 = 30 on G 2 ( x 1 , x 2 ) = v 2 = 0 , on Σ 2 G 1 ( x 1 , x 2 ) = v 1 = 0 and G 2 ( x 1 , x 2 ) = v 2 = 0 , on Σ 3
We take parameters for fluid and structure in [
The
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| Paramerters | Values |
|---|---|
| 0.1 cm | |
| 0.3 | |
| Paramerters | Values |
|---|---|
| 3 cm | |
| 0.5 cm | |
| Paramerters | Values |
|---|---|
| 10 | |
In this paper, we present a method to solve a steady coupled problem. Our method is based namely on the conjugate gradient algorithm, it takes simulta- neously into account three unknown parameters so that each of them depends on the others. To get the results it is necessary to solve the fluid in the initial domain with the finite element method in order to determine the displacement of the structure by the conjugate gradient method and finally to deduce the velocity and the pressure. The velocity, the pressure and the displacement profile compared to [
Diop, M. and Mbaye, I. (2017) Conjugate Gradient Method to Solve Fluid Structure Interaction Problem. Applied Mathematics, 8, 444-452. https://doi.org/10.4236/am.2017.84036