Optimal Control of Flow in an Elastic Channel

We consider once again the configuration with fluid flow through a channel with elastic walls.

In comparison to the example in Section 8.1, we now use the nonlinear fluid-structure inter-action model. The geometry in Figure 8.18 is identical to the geometry in Section 8.1. We only renamed the right solid boundary to Γq.

Γin

Γ^D_s

Γ^D_s

Ωf Γout

Γ_q

Γ_q Γ_i

Γ_i

Ω_s

Ω_s

Γ^N_s

Γ^N_s

x

C = (3.75,0.5)

Figure 8.18.: Geometry for fluid flow through an elastic channel

As already stated in Section 4.3, to model arterial flow it is necessary to incorporate the whole arterial network. Very often reduced 0D and 1D models are used. We refer to [124] for an overview on reduced models and the different techniques available to couple 3D models with the reduced models at the outflow boundary. The coupling of fluid and elasticity equations in FSI enables the propagation of pressure waves in a channel. Therefore, the outflow condition has to be chosen carefully to let the pressure wave leave the system, see [145, 63]. As the outflow boundary is generated by artificially truncation of the arterial system, the outflow conditions cannot be physically deduced. This motivates to control the forces on the solid outflow boundary Γ_q and the mean pressure on Γ_out, instead of adapting parameters in a reduced model.

In this section, we only control the time dependent tangential force (Σsn)^Te2=q on I×Γq

on the solid boundary Γq with q∈L²(I) . For the fluid outflow we use again the do-nothing condition. The elastic wall is fixed inx-direction and free to move iny-direction at the control boundary Γ_q. The control should be chosen in such a way, that the energy induced in the

108

8.3. Optimal Control of a Nonlinear Fluid-Structure Interaction Problem

structure due to the pulsating inflow can leave the domain. Therefore, we minimize the kinetic and elastic energy in the vessel wall

JE(u) = 1 2

Z

I

(Σ_s, E_s)_s dt+1 2

Z

Ikρ_svk²_L²_(Ωs) dt.

To have a well-posed optimal control problem we add a Tikhonov regularization and minimize the functional

J(q,u) =JE(u) +α

2kqk²_L²_(I)

subject to the fluid-structure interaction model presented in Section 4.4.

We want to emphasize that the example is only motivated by hemodynamics. The configura-tion is not at all close to a real arteria. It is mainly thought to demonstrate various possibilities in this area. In addition, we use the configuration to test the dual-weighted residual error estimator for optimal control of the nonlinear fluid-structure interaction problem.

The inflow profile is identical to (8.1) in Section 8.1. We have again a parabolic inflow profile with v_in(t) given in Figure 8.2 and use a constant flow as initial condition, as in Section 8.1.

Due to the symmetry of the configuration we assume the control on the upper and lower control boundary to be identical with opposing sign. Furthermore, we use again similar parameters as in Section 8.1, see Table 8.1. Only for the fluid and solid density we use now the values ρs:= 1.2_cm^g3 and ρ_f := 1.0_cm^g3.

We apply the fractional-step theta time-stepping scheme and use bi-quadratic finite elements as well as a piecewise linear control space as suggested in Chapter 5.

Discussion of the Optimal Solution

The optimal control problem is solved on different space, time, and control grids by the LBFGS algorithm 6.1 presented in Chapter 6. As we can see in Table 8.14, the error in the cost functional reduces uniformly after refinement. As reference value Jref we thereby used the solution on the finest grid with N = 6305 degrees of freedom,M = 2880 time steps and dim(Q_d) = 2883.

Table 8.14.: Value and error of the optimized cost functionalJ(q_σ, u_σ) for different discretiza-tion levels for flow in an elastic channel

N M dim(Q_d) J(u_σ) Jref− J(u_σ) 117 360 363 1.082·10⁻¹ 1.125·10⁻² 425 720 723 1.178·10⁻¹ 1.654·10⁻³ 1617 1440 1443 1.190·10⁻¹ 4.403·10⁻⁴ 6305 2880 2883 1.194·10⁻¹

-Comparing the energy on the finest grid in the solid with and without control, we can see that we were able to reduce the elastic and kinetic energy from JE(u) = 3.8·10⁻¹ to JE(u) = 8.87·10⁻². If we take a closer look at the energy values plotted over time in Figure 8.19, only the elastic energy at the beginning could not be reduced.

8. Numerical Examples

Figure 8.19.: Elastic energy (left) and kinematic energy (right) in the solid plotted over time in the uncontrolled and controlled configuration for flow through an elastic channel.

If we regard the displacement at the point C plotted over time in Figure 8.20, we can see that the oscillations in the solid could be significantly reduced by the optimization algorithm.

The maximal values stay constant over time and do not increase anymore after every inflow pulse.

Figure 8.20.: Displacement at the pointC inx-direction (left) andy-direction (right) plotted over time in the uncontrolled and controlled configuration for flow through an elastic channel.

The resulting optimal control on the finest grid level plotted in Figure 8.21 is highly distributed in time. This justifies the use of the high-dimensional control space here. If we take a closer look at the computed control, we see a significant decent in the control variable a few seconds before the inflow profile rises. This decent causes the detour in the displacement profile of the pointCin Figure 8.20. The control induces a movement of the elastic layer, which counteracts the deformation due to the inflow.

To get a more realistic model, unphysical high values of the boundary forces can be avoided by the enforcement of control constraints. For vascular models it would be necessary to control the fluid outflow condition, too. The do-nothing outflow condition sets the outflow pressure to be zero. Thereby, the pressure wave is always reflected.

110

8.3. Optimal Control of a Nonlinear Fluid-Structure Interaction Problem

0 0.5 1 1.5 2 2.5 3 3.5 4

−20 0 20 40

timet

Figure 8.21.: Optimal controlqplotted over time for optimal control of flow through an elastic channel.

Adaptive Refinement Using A Posteriori Estimator

Now we would like to test if the dual-weighted residual error estimator works well for optimal control of the nonlinear fluid-structure interaction problem. We only consider the time and control discretization error estimator and compute in the following on a spatial grid with N = 1617 degrees of freedom. We refine either globally in time or use the a posteriori error estimator to refine locally the time and control time grid. The errors are plotted over the degrees of freedom of time and control discretization in Figure 8.22. Thereby, the discretization error of the cost functional reduces much faster using the adaptive algorithm.

10⁵ 10⁶ 10⁷ 10⁸

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

M·dim(Q_d)

est global err global est adaptive err adaptive

Figure 8.22.: Error in cost functional for optimal control of flow through an elastic channel plotted over degrees of freedom of the time and control discretization spaces (M·dim(Q_d))

We would like to emphasize that the number of degrees of freedom is here not directly

con-8. Numerical Examples

nected to the computational cost. As we have seen in Section 8.1, the number of optimization loops needed in the LBFGS algorithm is at least for the presented linear example independent of the control dimension.