Sensitivity analysis for the active manipulation of Helmholtz fields in 3D

Figures & data

Figure 1. Initial geometry for the sensitivity experiments with an almost non-radiating single source.

Figure 2. Different views of the surface field pattern on the actual source $\mathrm{\partial }{B}_{0.0105}\left(\mathbf{0}\right)$.

Figure 3. The near control region after an outward shift from the source.

Figure 4. $L^2$ norm of the source density $w_{\alpha }$ as a function of the distance between $D_1$ and $D_{a^{\prime }}$.

Figure 5. Relative supremum error in $D_1$ as a function of the distance between $D_1$ and $D_{a^{\prime }}$.

Figure 6. Absolute supremum error on $\mathrm{\partial }{D}_2$ as a function of the distance between $D_1$ and $D_{a^{\prime }}$.

Figure 7. Stability of the scheme as a function of the distance between $D_1$ and $D_{a^{\prime }}$.

Figure 8. The initial near control $D_1$ and an iterate $D_1^{\ast }$ after increasing the outer radius.

Figure 9. $L^2$ norm of the source density $w_{\alpha }$ as a function of the increments in $D_1$ outer radius.

Figure 10. Relative supremum error in $D_1$ as a function of the increments in $D_1$ outer radius.

Figure 11. Absolute supremum error on $\mathrm{\partial }{D}_2$ as a function of the increments in $D_1$ outer radius.

Figure 12. Stability measure as a function of the increments in $D_1$ outer radius.

Figure 13. The initial near control $D_1$ and an iterate $D_1^{\ast }$ after increasing both the inner and outer radii by equal increments.

Figure 14. $L^2$ norm of the source density $w_{\alpha }$ as a function of the $D_1$ radii increment.

Figure 15. Relative supremum error as a function of $D_1$ radii increment.

Figure 16. Absolute supremum error on $\mathrm{\partial }{D}_2$ as a function of $D_1$ radii increment.

Figure 17. Stability measure as a function of $D_1$ radii increment.

Figure 18. The initial geometry for the multi-region controls.

Figure 19. Different views of the surface field pattern on $\mathrm{\partial }{B}_{0.0105}$.

Figure 20. Illustration of two iterations showing the secondary region rotated around $D_a$.

Figure 21. $L^2$ norm of $w_{\alpha }$ as a function of the secondary region's angle of rotation.

Figure 22. Accuracy errors in $D_1$.

Figure 23. Supremum error in $D_2$ as a function of its angle of rotation.

Figure 24. Stability measure as a function of the secondary region's angle of rotation.

Figure 25. Sketch of the geometry where $D_1$ is acting as a near field obstacle to $D_2$.

Figure 26. Performance of the scheme in generating a null in $D_1$ and a plane wave in $D_2$.

Figure 27. Different views of the surface field pattern on $\mathrm{\partial }{B}_{0.0105}\left(\mathbf{0}\right)$.

Figure 28. Time snapshots of a cross section of the near field at kct values (left-right, top-down) $37\pi /2000,38\pi /2000,39\pi /2000,40\pi /2000,41\pi /2000,\dots ,45\pi /2000$.

Figure 29. (a) Time-averaged relative error in region $D_1$. (b) Time-averaged absolute error in region $D_2$ over one period.