8. Numerical Examples 89
8.2. A posteriori Error Estimation for a Nonlinear FSI Problem
8.2.5. Flapping Test
In [73] an alternative test configuration was suggested. The fluid in a channel of length 8cm and height 1.61cm flows through the gap between two vertical elastic beams of width 0.0121cm and height 0.7cm (see Figure 8.10), whereby the fluid flow induces a flapping of the two beams.
At the inflow boundary Γin we enforce a parabolic inflow profile v(0, y) :=y(1.61−y) 4
1.612vin(t).
Motivated by applications in hemodynamics, we choose for the mean inflow velocity vin(t) the periodic profile given in Figure 8.11. This configuration was developed in [73] to test alternative methods to the ALE mapping used in this thesis. For large inflow velocities, the algorithm fails as the fluid-structure interaction equation cannot be solved anymore due to the large displacement of the thin beam. Hence, the maximal value of parabolic inflow profile is chosen in such a way, that the mesh motion can still be handled by the ALE method.
As in the FSI benchmark, we enforce zero Dirichlet boundary conditions at the outer boundary Γf and Γsand a do-nothing outflow condition at the outflow boundary Γout. The computations
102
8.2. A posteriori Error Estimation for a Nonlinear FSI Problem
0 1 2 3 4
0 2 4
timet
vin(t) 0.5vin(t)
Figure 8.11.: Inflow velocities vin(t) and 0.5vin(t) plotted over time t
are carried out on the time interval I = (0,3.6) and the material parameter values are given in Table 8.9.
0 1 2 3 4
0 0.2 0.4 0.6
time t
0 1 2 3 4
0 0.1 0.2 0.3 0.4
time t
Figure 8.12.: Solution of the flapping test. Displacementuat the pointB inx-direction (left) and displacement u at the point B in y-direction (right) plotted over time t.
To visualize the behavior of the space-time discretized FSI model in Problem 5.5 for the given configuration, we plotted the displacement of the flag at the point B = (2.0,0.91), the drag around the two beams, and the vorticity over time in Figure 8.12 and 8.13. For the simulation, we use bi-quadratic finite elements with N = 5537 degrees of freedom and the fractional-step theta time-stepping scheme with M = 184320 time steps. Due to the varying inflow, the beams are at rest most of the time. The moment the inflow profile increases the gap between the beams increases as well. In addition the change in the fluid velocity induces large changes in the drag and vorticity values.
As in the previous configurations, we evaluate a functional of interest to compare the time and spatial discretization error on different time and spatial grids. Here, we choose the vorticity value on the domain Ωf given by the functional J4(u) defined in (8.5).
8. Numerical Examples
0 1 2 3 4
0 1,000 2,000 3,000
time t
0 1 2 3 4
0 20 40
timet
Figure 8.13.: Solution of the flapping test. Drag value (left) and vorticity value (right) plotted over timet.
Adaptive Refinement in Time
First, we solve the configuration on different globally refined time grids. Then, we use the DWR error estimator presented in Theorem 7.2 to refine the time grid adaptively. We plotted again the error in vorticity functional over the number of time steps M. The discretization error is computed using as reference value Jref the vorticity functional computed on a fine grid. The reference solution is solved on the same spatial grid with N = 5537 degrees of freedom and M = 184320 time steps. Furthermore we use bi-quadratic finite elements. As we can see in Figure 8.14, the error in the functional of interest reduces much faster if we use the adaptive algorithm in comparison to the error by global refinement of the time grid.
103 104 105
10−3 10−2 10−1
number of timestepsM err global err adaptive
Figure 8.14.: Error in the vorticity functionalJ4(u) for the flapping test plotted over number of time stepsM.
If we take a closer look on the resulting time step size in Figure 8.15 we see that the adaptive algorithm chooses locally very small time steps. The adaptive algorithm refines especially in the areas of large inflow velocity, but also sees the kinks in the inflow profile and refines there.
104
8.2. A posteriori Error Estimation for a Nonlinear FSI Problem
Figure 8.15.: Time step size km plotted over time t after 1 (left), 3 (middle) and 6 (right) adaptive refinements for the flapping test.
Space-Time Adaptive Refinement with Equilibration
One of the major advantages of dual-weighted residual error estimators is the possibility to equilibrate the spatial and time discretization error. To demonstrate the efficiency of such an approach, we refine globally in space and time and compare the error in the vorticity functional with the discretization error we obtain by refining adaptively. The adaptive algorithms only refines in space and time, if the error indicatorsηkandηhhave the same order of magnitude.
In comparison to the previous sections, we use bi-linear finite elements here. The reference solution is computed on a fine grid with bi-quadratic finite elements withN = 20273 degrees of freedom andM = 23040 time steps. In addition, we reduced the inflow velocity to 0.5vin(t) as given in Figure 8.11.
Table 8.12.: Effectivity indices of spatial and time discretization error estimator using constant Young’s modulusEm (left) and modified distributed Young’s modulus Em in the mesh motion equation for the flapping test
Em const Em distributed
To be able to handle the large deformations we weight the Young’s modulus Em in the mesh motion equation with the inverse determinant of the deformation gradient J−1. Nevertheless the regularity of the ALE transformation is disturbing. The volume of the cells in the trans-formed mesh reach a critical value and the angles in the cells around the tip of the flag are either extremely small or very large. If we set the mean inflow slightly larger, the Newton algorithm will not converge anymore.
8. Numerical Examples
The lack of regularity of the mesh motion around the tip causes the dual weighted residual error estimator to rise on the critical cells. However, as we can see in Table 8.12 the error estimator overestimates the discretization error heavily. We give in Table 8.12 the effectivity indicesIeff of the DWR error estimator of the combined error of time and spatial discretization on the adaptively refined grid. As the mesh deteriorates in the area around the tip of the flag, the determinant of the deformation gradient J = det(F) converges to zero. The neglected remainder terms in the a posteriori error estimator, including the derivatives of F−1, cannot be assumed small anymore. This effect was already predicted in [128].
Figure 8.16.: Mesh after 6 refinement steps with constant Young’s modulus (left) and after 6 refinement steps with modified distributed Young’s modulus (right). Zoom into the area of the beam.
As the error estimator is especially located in the area around the tip of the two flags the adaptive algorithm only refines very locally in this region. The mesh after 6 refinement steps is given in Figure 8.16. However, the refinement of the mesh around the flag does not increase the regularity of the ALE transformation. That is why the quality of the a posteriori error estimator does not improve with local refinement.
In Figure 8.17, we plotted the discretization error in the vorticity functional over the degrees of freedom of the space-time discretized fluid-structure interaction problem (N ·M). The error inJ4(u) converges using the adaptive algorithm much faster then by global refinement in space and time. However, the overestimation of the error results in local refinement around the tip of the flag, which does not contribute to a further decrease in the discretization error, as we can see in Figure 8.17.
To improve the regularity of the ALE transformation, we use now a distributed value for the Young’s modulus Em in the mesh motion equation. The chosen distribution thereby is motivated by the results of the optimal control problem in Section 8.3.3, see Figure 8.32.
Thereby, the mesh properties of the cells around the tip can be conserved.
The effectivity indices in Table 8.12 are now very close to the desired value one. Only on the finest level the cell distortion influences the error estimator. In comparison to the mesh generated by solving a linear elasticity equation with constant Young’s modulus, the adaptive algorithm refines now in a much wider region and not only at the tip of the flag as we can see in Figure 8.16.
106
8.2. A posteriori Error Estimation for a Nonlinear FSI Problem
105 106 107 108
10−2 10−1 100 101 102
N·M err global
err adaptive (Em dist.) err adaptive (Em const)
Figure 8.17.: Error in the vorticity functional for the flapping test plotted over the degrees of freedom on the space-time cylinder (N·M) using global and adaptive refinement strategies.
Altogether we can see in Figure 8.17 an extreme fast reduction of the discretization error using the adaptive algorithm in comparison to global refinement in space and time. For the identical number of degrees of freedom of spatial and time discretization we can compute the vorticity functional 103-times more accurate by applying the equilibrated adaptive approach.
As the error estimates of the time discretization error, given in Table 8.13, are much smaller than the spatial discretization error the algorithm only refines in space. Hence, this example shows an additional advantage of error estimators in space and time, because they indicate which errors dominate and we can avoid unnecessary refinement.
Table 8.13.: Error estimators in space ηh and time ηk for the adaptive refinement algorithm with equilibration
N M ηh ηk η
351 1440 −3.84·101 4.43·10−2 −3.84·101 733 1440 −2.93·101 3.10·10−2 −2.93·101 1633 1440 −1.29·101 1.81·10−2 −1.29·101 3777 1440 −3.70·100 9.90·10−3 −3.69·100 8885 1440 −5.23·10−1 9.19·10−3 −5.13·10−1
Nevertheless, this example also reveals the limits of dual-weighted residual error estimators in space for FSI. For small solid displacements the method works excellent, but if the ALE transformation gets too irregular the error estimator tend to overestimate the error and local mesh refinement does not contribute to a smoothing of the ALE transformation.
8. Numerical Examples