Numerical experiment

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3. FSI models and decoupled algorithms

3.3 Numerical experiment

In this subsection, we present numerical experiments to demonstrate the conver- gence and stability performance of the multirateβ-scheme (60)–(61) for coupling a Stokes flow with a thin-walled structure by using a benchmark model. The model

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Figure 3.

An illustration of a multirate time stepping technique.

consists of a2-D rectangular fluid domainΩ_f ¼½0,LŠ ½0,RŠwithL¼6 andR¼0:5 and a1-Dstructure domainΓ¼½0,LŠ Rthat meanwhile also plays the role of the fluid–solid interface, as shown inFigure 2. The displacement of the interface is assumed to be infinitesimal and the Reynolds number in the fluid is assumed to be small (Figure 4). All the quantities will be given in terms of the centimeter-gram- second (CGS) system of units.

The physical parameters are set as follows:ρ_f ¼1:0,ρ_s ¼1:1,μ¼0:035;

L_sd,d_

¼c1∂²

xdþc0d, wherec1 ¼_{2 1þð}^Eε_νÞ,c0 ¼_R²ð^Eε_{1 ν}2Þwithε¼0:1, the Poisson ratio ν¼0:5 and the Young’s modulusE¼0:7510⁶. A pressure-wave

P tð Þ ¼P_maxð1 cos 2tπ=Tð ^∗ÞÞ=2 with P_max ¼210⁴, (63) is prescribed on the fluid inlet boundary forT^∗ ¼510 ³(seconds). Zero traction is enforced on the fluid outlet boundary and no-slip condition is imposed on the lower boundaryy¼0. For the solid, the two endpoints are fixed withd¼0atx ¼0 and x¼6.

The first experiment is set up to compare the Robin-Neumann scheme with theβ- scheme, the two stable decoupled methods recently developed for the benchmark model.Figure 5displays the displacements computed by the Robin-Neumann scheme and theβ-scheme, together with the coupled implicit scheme for reference, where the

Figure 4.

Geometrical configuration.

Figure 5.

Comparisons of the numerical results obtained by the coupled implicit scheme, the RN scheme, and theβ-scheme under the setting: h¼0:05andΔts¼10 ⁴.

mesh size and the time step size areh¼0:05 andΔt¼10 ⁴. It is clearly seen that both decoupled schemes converge as well as the coupled implicit scheme. Moreover, little difference is observed between the two decoupled schemes numerically. This suggests we focus on theβ-scheme for investigating the multirate time-step technique.

We then conduct numerical experiments to investigate the effects of the time-step ratior.Figure 6illustrates that a larger time step size in the fluid part results in a more accurate numerical solution than that obtained by using a larger step size in the structure part. In the test, we fixh¼0:1 andΔt¼10 ⁵. In addition, we observe that, when^Δt_Δt^s

f is further increased to be^Δt_Δt^s

f ¼5 or^Δt_Δt^s

f ¼10, there are substantial numerical instability. This screens out the possibility of using a larger time step size in the structure part.

To examine how the stability and approximation are affected when the time step in the fluid region is too large, we fix the time stepΔt_sandhwhile varying the time-step ratior¼1, 5, 10, 20, 50.Figure 7displays the computed displacements att¼0:015 with the structure time step sizeΔt_s ¼10 ⁵, the mesh sizeh ¼0:1 (left) andh ¼0:01 (right). In the left figure, we observe that the structure displacements computed by usingr¼1, 2, 5, 10 approximate very well to that by using the coupled implicit scheme. To further investigate the stability and the convergence of the multirateβ- scheme, in the right part ofFigure 7, we reduce the mesh size to beh¼0:01 while fixing the time step size. The numerical results confirm that the multirateβ-scheme is still stable even the time-step ratio is reasonably large.

InFigure 8, we present the numerical results of the fluid pressure distribution at t¼0:005, 0:01, 0:015. From the top to the bottom, numerical results are: a reference solution by the coupled implicit scheme, the numerical solution by theβ-scheme, and the solution by the multirateβ-scheme withr¼10. By comparing the results, we see that the multirateβ-scheme provides a very good approximation.

In order to examine the order of convergence, we start withh¼0:1 andΔts ¼ 0:0001, and then refine the mesh size by a factor of 2 and the time step size by a factor of 4. The space–time size settings are:

Figure 6.

Comparison of theβ-scheme and the multirateβ-schemes with two different time-step ratios (h¼0:1and Δt¼10 ⁵are fixed).

h,Δts

f gⁱ ¼n0:1ð0:5Þⁱ, 0:0001ð0:25Þⁱo

, i¼0, 1, 2, 3, 4: (64) We compare the numerical solutions of the multirateβ-scheme with the reference solution. The reference solution is computed by using the coupled implicit scheme with a high space–time grid resolutionh¼3:12510 ³,Δt¼10 ⁶

as that in [14].

In the multirate scheme,r¼1 andr¼10. The relative errors of the primary variables (u_f,p_f andd) att¼0:015 are displayed inFigure 9. From the comparisons, we see that the numerical errors are approximately reduced by a factor of 4 as the mesh size

Figure 7.

Numerical displacements under the settings: h¼0:1(left) h¼0:01(right) andΔts¼10 ⁵.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 8.

Fluid pressure distribution at t¼0:005, 0:010, 0:015obtained by the coupled implicit scheme (top), the multirate β-scheme with r¼1(middle) and r¼10(bottom) with h¼0:01andΔts¼0:00001. (a) t = 0.005, (b) t = 0.010, (c) t = 0.015, (d) t = 0.005, (e) t = 0.010, (f) t = 0.015, (g) t = 0.005, (h) t = 0.010, (i) t = 0.015.

and the time step size are refined once. Therefore, the multirateβ-scheme is of a second order inhand a first order int.

Finally, in order to demonstrate the advantage of the multirateβ-scheme, we

compare inTable 3the CPU times of the concerned numerical algorithms under various settings. We fixedΔts ¼10 ⁵and varying the mesh sizes ash ¼₁₀¹ , ₂₀¹ , ₄₀¹ , ₈₀¹ , ₁₆₀¹ . From the table, it is observed that the multirateβ-scheme takes much less computational cost than that of the coupled implicit scheme, particularly whenris large.

Figure 9.

Relative errors of primary variables with the spacing h and time step sizeΔtsin (64). (a) Relative error of uf, (b) Relative error of pf, (c) Relative error of d.

Implicit scheme Multirateβ-schemer= 1 Multirateβ-schemer= 10

h¼₁₀¹ 14.90 4.02 0.74

h¼₂₀¹ 48.64 16.00 2.82

h¼₄₀¹ 179.83 66.67 11.6

h¼₈₀¹ 797.76 297.96 49.23

h¼₁₆₀¹ 3165.26 1270.30 206.32

Table 3.

CPU times (in seconds) for the coupled implicit scheme and the multirateβ-scheme (with r¼1or10) under different settings of mesh sizes (Δts¼10 ⁵ is fixed).