8. Numerical Examples 89
8.3. Optimal Control of a Nonlinear Fluid-Structure Interaction Problem
8.3.2. Optimal Control of the FSI-2 Benchmark Example
Γin
Γf
Γf
Γf
Ωf Γout
Γi
+ + + +Γq
Γq
Γs A
Ωs
Figure 8.23.: Geometry for flow around cylinder with elastic beam and additional control boundary Γq.
We consider again the FSI-2 benchmark configuration presented in Section 8.2.1. We adapt a Neumann boundary control problem suggested by Becker in [15] for the Navier-Stokes bench-mark, on the FSI-2 benchmark. We add an additional boundary Γq on the upper and lower channel wall and control the mean pressure in the outflow condition on Γq, see Figure 8.23.
Thereby, we are able to control indirectly the in- and outflow at the boundary Γq. We use again the parameters from the FSI-benchmark given in Table 8.9 and the smoothly increasing parabolic inflow profile (8.2) with
vin(t) =
(0.5−0.5 cos(πt2) t≤2
1 t >2.
The simulation is computed on the time intervalI = (0; 15).
Optimal Control Configuration
We describe on the boundary Γq the outflow condition
(∇v+pId)n=qn on Γq×I,
whereby we control the mean pressure q here. For q = 0, the standard do-nothing outflow condition is enforced at Γq. The FSI-2 system is known to be highly dynamical. The beam starts to oscillate and sincerely influences the flow behavior. To be able to influence the system, we choose a time depend control variable
q∈Q:=L2((5;T))×L2((5;T)).
We only control the system after the time-point t > 5 as the system is very sensitive to numerical errors in the beginning. Thereby, we can increase the inflow profile smoothly and after a stable flow developed, we start to control the system.
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8.3. Optimal Control of a Nonlinear Fluid-Structure Interaction Problem
The control is chosen in such a way that the oscillations of the beam are reduced after the time pointt >8. Hence we minimize the tracking type functional
minq∈QJ(q,u) := 1 2
Z T
8 kuk2L2(Ωs) dt+α
2kqk2Q (8.9)
withL2-Tikhonov regularization of the control variable.
Numerical Example
The FSI-2 benchmark example is highly dynamical, therefore we need to compute with very small time steps even for the fractional-step theta time-stepping scheme. In addition to resolve the forces at the interface Γi, a very fine mesh has to be used. To improve the accuracy of the simulation, we apply the DWR estimator presented in Theorem 7.1 on the cost functional J(q,u). In addition, we equilibrate the control error estimator ηq, the time discretization error estimator ηk and the spatial discretization error ηh. We have seen in Section 8.2.5 that thereby, we can avoid ridiculous refinement and we can reduce the computational cost severely.
The control space is discretized by the space of continuous and piecewise linear functions. We use the fractional-step theta time-stepping scheme to discretize the optimal control problem in time and bi-linear finite elements in space. For the Tikhonov regularization the parameter α = 2·10−11 is chosen. The LBFGS algorithm terminates if the gradient could be reduced by a factor 10−3.
0 2 4 6 8 10 12 14
1 2 3
·10−2
timet
Figure 8.24.: Time step size km plotted over time t after 2 adaptive refinement cycles for optimal control of the FSI-2 benchmark.
After two refinement cycles the control space is discretized with dim(Qd) = 258 degrees of freedom. Furthermore we have M = 8500 time steps and a spatial discretization with N = 3895 degrees of freedom. We plotted in Figure 8.24 the resulting time step size and in Figure 8.25 the refined mesh. As we only have observation of the cost functional in the time interval (8,15), the adaptive algorithm mainly refines in this interval of interest. Furthermore the moment the algorithm starts to choose smaller time step sizes, correlates with the moment the oscillations in the flag set in.
Moreover the algorithm refines very locally around the flag and the circle. Only thereby the pressure field at the circle boundary and interface can be computed accurately. As the
8. Numerical Examples
Figure 8.25.: Adaptively refined mesh for optimal control of the FSI-2 benchmark.
boundary forces at the interface Γi severely influences the behavior of the solid motion, it is reasonable that the adaptive algorithm refines here. In addition, the algorithm refines very locally around the control boundary Γq. As we enforce a Neumann boundary condition on Γq and zero Dirichlet boundary conditions at Γf, we have a “corner” of angle 180◦, between a Neumann and Dirichlet boundary condition. Standard elliptic PDE theory predicts already for angles larger then 90◦ corner singularities. These singularities in the fluid velocity field and pressure can only can be resolved by local refinement, hence the adaptive algorithm refines very locally here.
0 5 10 15
0 2 4 6 8 ·10−2
timet
uncontrolled controlled
0 5 10 15
−0.1
−5·10−2 0 5·10−2 0.1
timet
uncontrolled controlled
Figure 8.26.: Flux at Γq (left) and displacement iny-direction at the tip of the flag A(right) plotted over time for optimal control of the FSI-2 benchmark. Plotted is the controlled vs uncontrolled solution.
We start the optimization algorithm with controlq(t) = 0. Thereby, we have in the uncon-trolled configurations a do-nothing outflow. The fluid can leave and enter the domain through the control boundary Γq. Despite the slight modification of the additional outflow boundary condition the flag starts to oscillate as for the standard FSI-2 benchmark. We plotted the displacement u at the tip of the flag A in y-direction in Figure 8.26 for the uncontrolled configuration on the two times refined mesh. If we compare the solution with the displace-ment in Figure 8.8, we can see that the amplitude and frequency correlate with the standard benchmark.
We plotted in Figure 8.26 in addition the flux R
Γqvkhndx leaving the lower boundary Γq. While the inflow velocity vin rises, the outflow is very strong, but after some time the flux reduces and the fluid flows across the boundary Γq. The displacements of the flag are very
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8.3. Optimal Control of a Nonlinear Fluid-Structure Interaction Problem
large in the FSI-2 benchmark configuration. The moment the tip of the flag reaches its minimal value, the gap between wall and elastic beam gets very thin. This enforces the fluid to leave the domain through the lower control boundary Γq and explains the oscillations in the flux profile.
0 5 10 15
−400
−200 0 200
timet
0 5 10 15
−200 0 200 400
timet
Figure 8.27.: Optimal control q at the upper (left) and lower (right) control boundary Γq plotted over time for optimal control of the FSI-2 benchmark.
In Figure 8.27 the optimal control is plotted over time at the upper and lower control boundary.
The control variable is still ragged as the control error estimator is much smaller, then the time and spatial discretization error. Hence, the adaptive algorithm only refines in space and time. Due to the almost symmetrical configuration the control on upper and lower control boundary Γq have the same values with opponent sign.
The optimal control sincerely influences the flow behavior at the boundary Γq, as we can see in Figure 8.26. The amplitude in the flux oscillations increases and the flux reduces at certain time-points so heavily such that we almost have inflow. In addition, we are able to reduce the amplitude of the oscillating flag slightly. The value of the cost functional can be reduced from J(qσ, uσ) = 2.63·10−5 toJ(¯qσ,u¯σ) = 2.48·10−5. To get an impression of the flow behavior we plotted in Figure 8.28 the velocity field in x-direction at the time points t= 14.74, t= 14.90 and t= 15.00. Here we can see again the large deformation of the solid beam, which influences the flow behavior sincerely. The outflow on the control boundary Γq
is relatively small with respect to the outflow at the outflow boundary Γout. Hence the flow behavior is very similar to the solution of the FSI-2 benchmark.