Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

5.1 Introduction

The use of a distributed Lagrange multiplier for the modeling and approximation of interface problems has a long history within approaches based on fictitious domain techniques [6].

The applications of this methodology for fluid-structure interaction problems has been rediscovered and discussed in recent research [1, 4] originating from the immersed boundary method [2, 9]. The theoretical properties of this approach are quite good, showing unconditional stability for a semi-implicit time discretization and inf-sup stability for the global saddle point problem under suitable conditions on the underlying meshes. Our formulation and some of the main results about it will be summarized in Sect. 5.2.

In this paper we present a series of numerical tests performed with the help of FreeFem++ [7]. All results are collected in Sect. 5.3. In all tests, the agreement with the analytical solution (if known) or with solutions present in the literature is quite good.

One of the main difficulties in the implementation of any fluid-structure interaction model, consists in the appropriate treatment of the exchange of information between fluid and solid. In our formulation the fluid mesh is fixed, while the solid mesh is defined on a reference configuration and mapped to the actual solid domain via the (unknown) transformation which defines the position of the body. It turns out that some terms in our variational formulation need to combine quantities defined on the fluid and solid meshes. Actually, FreeFem++ has a built in function that allows the computation of such terms. Our codes are listed in Appendix 1 and some comments are provided in Appendix 2.

5.2 Problem Setting

The model introduced in [4] can deal with co-dimension zero (thick) or co-dimension one (thin) bodies immersed in a fluid of two or three space dimensions. Our numerical tests involve thick bodies in two space dimensions; we recall the related formulation.

Let Ω be a bounded domain in \(\mathbb {R}^2\) with Lipschitz continuous boundary. We assume that the domain is partitioned into a fluid part Ω f and a solid part Ω s (both subdomains are time dependent). The solid domain Ω s is the image of a reference domain . More precisely, the mapping associates to each point its image x = X(s, t) ∈ Ω s at time t. We denote by ρ f and ρ s the fluid and solid densities, respectively, by ν the fluid viscosity, and by λ and μ the Lamé constants.

The problem considered in [4] is the following one: given an initial velocity \({\mathbf {u}}_0\in (H^1_0(\varOmega ))^2\), an initial body position , find velocity and pressure \(({\mathbf {u}}(t),p(t))\in (H^1_0(\varOmega ))^2\times L^2_0(\varOmega )\), body position , and a Lagrange multiplier λ(t) ∈ Λ such that for almost every t ∈ ]0, T[ is holds

(5.1)

where \({{\mathbb {D}_t}}\) is the total derivative and D is the symmetric gradient.

The constants c 1, c 2 and the space Λ are crucial for the definition of the model and the subsequent numerical scheme: in our computations we consider both c 1 and c 2 positive and different from zero (H 1-based Lagrange multiplier), so that the space Λ is .

5.2.1 Numerical Approximation

The time semi-discretization of Problem (5.1) is constructed as follows: in the first equation the total derivative is approximated by the Galerkin-characteristic method (see [10]); the second derivative ∂ tt X(t) in the second equation is approximated by (X n+1 − 2X n + X n−1)∕δt 2; ∂ t X is approximated by (X n+1 −X n)∕δt; all other quantities are evaluated implicitly at time n + 1 with the following exception. Clearly, there is a problem when a term involving \(\hat {\mathbf {u}}({\mathbf {X}})\) has to be integrated on . Treating this term fully implicitly would imply the use of the mapping X n+1 which is not yet available; for this reason we use a semi-implicit scheme where \(\hat {\mathbf {u}}({\mathbf {X}}^n)\) is used, instead.

In [4, Prop. 3] it has been shown that the resulting semi-discrete scheme is unconditionally stable with respect to the time step δt. The proof is based on a discrete energy estimate which is analogous to the stability estimate for the continuous problem:

where the energy E is defined in terms of the energy density \(W(\mathbb {F})\) (\(\mathbb {F}\) being the deformation gradient)

The numerical approximation of Problem (5.1) is based on a set of four finite element spaces: \(V_h\subset (H^1_0(\varOmega ))^2\) and \(Q_h\subset L^2_0(\varOmega )\) are inf-sup stable finite element spaces, while and Λ h ⊂ Λ are finite elements defined in the solid domain. More precisely, V h and Q h are finite elements defined according to a triangulation of Ω, while S h and Λ h are finite elements defined on a mesh of the reference solid configuration .

5.3 Numerical Tests

In all tests the solid is hyper-elastic with Young modulus E, Poisson ratio κ and density ρ s. The shear modulus is then given by μ = E∕(1 + κ)∕2.

The fluid is Newtonian incompressible with density ρ f and viscosity ν.

5.3.1 Disk Falling in a Liquid

A disk of diameter d is at rest initially centred at x c = W∕2, y c = H − h in a rectangular channel of width W and height H. Only the disk is subject to gravity g, not the fluid. No slip conditions are applied on the walls of the channel.

This test was proposed by Zhao et al. in [12] and more recently by Wang et al. in [11]. Here we chose Wang’s values for the parameters:

$$\displaystyle \begin{aligned} \begin{array}{rcl}&\displaystyle &\displaystyle W=2,~d=0.125,~h=0.5,~H=4, \\ &\displaystyle &\displaystyle \rho^s=1.2,~\kappa=0.3,~\mu=10^8,~\rho^f=1,~\nu=1,~g=981 \end{array} \end{aligned} $$
(5.2)

The asymptotic vertical velocity is known to be − 0.3567. Figure 5.1 shows the evolution of the vertical velocity versus time for four meshes: a coarse mesh with 1182 vertices, a middle mesh with 4693 vertices and a fine mesh with 18,661 vertices. The corresponding time steps are 0.02, 0.01, 0.005.

Fig. 5.1
figure 1

Left: vertical velocity of the solid versus time computed with three body fitted mesh and one non-body fitted mesh. Convergence seems monotone towards a limit curve for the first three meshes; the coarse mesh is the highest, the middle mesh is in the middle and the finest mesh is below. Right: area of the solid divided by πd 2∕4, as a function of time

Full size image

For these three cases the fluid mesh is modified at each time step to include the fluid-structure interface as an interline curve made of edges of the triangulation. Computation is also made on a fourth mesh with 4693 and the initial fluid-solid interface at time zero but not changed with time. On this fourth mesh precision degrades with time, probably due to mesh interpolations. On the three previous body fitted mesh there is convergence to a limit curve, but the asymptotic value seems to be − 0.3788 rather than − 0.3567. It could be due to the fact that the fluid model is extended in the solid leading to an error proportional to ρ s − ρ f. But it could also be the effect of interpolation needed to computed variables earlier defined on the mesh before motion. We are currently intersecting meshes to reduce this interpolation error and preliminary tests (to be published later) point to the direction of a more accurate falling velocity. On the other hand, mass is remarkably preserved as shown on Fig. 5.1-right.

A pressure map at t = 0.7 is given on Fig. 5.2-left. Next the same simulation is done with a very soft material having μ = 10. The shape of the solid at t = 0.7 is given with a color map of the yy component of the stress on Fig. 5.2-right. The computing time for this last test is 434” on a Core i7-2.5GHz on a single core.

Fig. 5.2
figure 2

Left: pressure map at t = 0.7 close to the solid disk falling in a liquid. Right: the yy component of the stress inside a very soft disk falling in a liquid displayed at t = 0.7. The shape is also the result of the numerical simulation

Full size image

For these two simulations the influence of the coefficients c 1 and c 2 are small, as long as c 2 is not zero. Here both are set at 1. The influence of the degree of the finite element spaces is also surprisingly small. Both the P2/P1 element for velocity pressure or the P1-bubble/P1 element gave the same results. Changing P1 into P2 for the Lagrangian coordinates also didn’t make a difference. It seems that the precision of the method is entirely driven by the quadrature formula used for the mixed integrals involving a function on the fluid mesh times a function on the transformed solid mesh.

Some integrals in the variational formulation involve piecewise polynomial functions defined on different meshes. We have developed a special quadrature formula in FreeFem++ (see [7]) to handle them. For instance let u be defined on mesh \(T^u_h\) and v be defined on mesh \(T^v_h\) obtained from \(T^u_h\) by convecting the vertices \(q^i\in T^u_h\) with X, namely X(q i) is a vertex of \(T^v_h\). Then for a triangle T of \(T^v_h\) the integral on T of u ∘ X ⋅ v is approximated by \(\sum _{j=1}^J u(X(\xi _j))v(\xi _j)\omega _j\) where ξ j, ω j are a valid set of quadrature points and coefficients for a quadrature on T (shown in the FreeFem++ code by a parameter in the integral like int2d( Ths,qft=qf9pT,mapu=[Xo,Yo]).

5.3.2 Validation with a Rotating Disk

The purpose of this test is to compare the numerical solution with a semi-analytical solution which can be computed to any desired accuracy.

A cylinder contains a fixed rigid cylindrical rod in its center, a cylindrical layer of hyperelastic material around the rod and the rest is filled with a fluid (see Fig. 5.3). First the system is at rest and then a constant rotation is given to the outer cylinder. This cause the fluid to rotate with an angular velocity which depends on the distance r to the main axis; in turn, because the friction of the fluid at the interface the hyperelastic material will be dragged into a angular velocity ω which is also only a function of r and time . Due to elasticity ω will oscillate with time until numerical dissipation and fluid viscosity damps it.

Fig. 5.3
figure 3

A fluid-structure system inside a rotating cylinder (giving a constant angular velocity to the fluid outer boundary) with a fixed rod in its center. Left: sketch of the system. Right: a 2d calculation showing the velocity vectors at time 0.85 for the coarser mesh

Full size image

In a two dimensional cut perpendicular to the main axis, the velocities and displacements are two dimensional as well. Hence the geometry is a ring of inner and outer radii, R 0 and R 1, with hyperelastic material between R 0 and R and fluid between R and R 1. Because of axial symmetry, R is constant, so the geometry does not change.

In this test R 0 = 3, R = 4, R 1 = 5. The solid is an hyperelastic material with μ = 100 and λ = 2κμ∕(1 − 2κ) with κ = 0.3 and ρ s = 10. The Newtonian fluid has ν = 1, ρ f = 1. The velocity of the outer cylinder has magnitude 3. As everything is axisymmetric the computation can be done in polar coordinates r, θ, and the fluid-solid system reduces to

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle \rho{\partial}_t v - \frac{1}{r}{\partial}_r[\xi^f r{\partial}_r v+\xi^s r{\partial}_r d] =0, \\ &\displaystyle &\displaystyle \quad {\partial}_t d = v,~r\in(R_0,R_1),~~v_{|R_0}=0,~v_{|R_1}=3, \end{array} \end{aligned} $$
(5.3)

with ρ = ρ s 1 r≤R + ρ f 1 r>R, ξ s = μ 1 r≤R, ξ f = ν 1 r>R, and with d(r, 0) = 0.

In all 2d computations c 1 = c 2 = 10 and δt = 0.005. We have verified that a smaller time step does not improve the precision.

Comparison between this one dimensional approach and the numerical solution of system (5.1) is given on Fig. 5.3—right, at T = 0.5 and a coarse mesh with 505 vertices. Then the same is computed on a finer mesh having 1986 vertices and finally with a mesh with 7433 vertices. Results are displayed on Fig. 5.4.

Fig. 5.4
figure 4

Rotating cylinder. Left: Evolution of the L 2 error versus time for the three meshes. Right: velocities normal to the ray at θ = π∕4 versus r − 3, computed on the coarsest meshes shown in green with continuous line and crosses. The “exact” solution of the one dimensional equation is shown in blue

Full size image

This test has two qualities: (a) the exact solution is easy to compute to any precision; (b) the geometry does not change and quadrature errors are due only to quadrature for integrals involving functions on the same domain but with two different triangulations.

5.3.2.1 Flow Past a Cylinder with a Flagella Attached

This test is known as FLUSTRUK-FSI-3 in [5]. The geometry is shown on Fig. 5.5. The inflow velocity is \(\bar U=2\), μ = 2106 and ρ s = ρ f. After some time a Karman-Vortex alley develops and the flagella beats accordingly. Results are shown on Figs. 5.5 and 5.6 with a mesh of 9692 vertices and a time step size of 0.0015; the first one displays a snapshot of the velocity vector norms and the second the y-coordinate versus time of the top right corner of the flagella.

Fig. 5.5
figure 5

FLUSTRUK-FSI-3 Test. Color map based on the norm of the fluid and solid velocity vectors

Full size image
Fig. 5.6
figure 6

FLUSTRUK-FSI-3 Test. Vertical position of the upper right tip of the flagella versus time shown up to t = 5

Full size image

These numerical results compare reasonably well with those of [5]. The frequency is 5.6 compared to 5.04 and the maximum amplitude 0.018 compared with 0.032. Amplitude is very sensitive to the method (see [8]).