Useful techniques for running simulation

In this part, I want to briefly introduce some useful techniques when running simulation for the fluid-shell hybrid vibrational modes. In addition to the simulation setup that we have explained above, additional boundary condi- tions are needed for computing different mechanical modes, depending on their mode shape.

Techniques for getting wineglass modes: To obtain a wineglass mode more conveniently, since the mode shape is periodic in the azimuthal direc- tion, we can only simulate a small wedge portion of the OMFR. For example, for an M = 6 wineglass mode, the periodicity angle in the azimuthal direc- tion is 360◦/M = 60◦. Therefore when setting up the geometry, we can first draw a 2D geometry with the shape of a half cross-section of the OMFR and revolve it with the symmetry axis of the OMFR (Figure 3.8a) with an angle of 60◦. Next, with all the boundary conditions set as described above, we apply additional periodic boundary condition to both the fluid and solid do- main (Figure 3.8b). Such a periodic condition on the two sides of the wedge can help enforce the value of the solution to be the same on the periodic boundaries. The result is shown in Figure 3.8c.

Techniques for getting breathing modes: For a breathing mode, the periodic boundary condition is still valid with an arbitrary periodicity angle. But since the shell of the OMFR is only expanding and contracting, we can apply a roller boundary condition on the two sides of the solid part (Figure 3.9a). The result is shown in Figure 3.8b.

Evaluation of local properties through virtual particles: Further- more, as we will describe in later chapters for particle sensing, we often need to predict the frequency perturbations caused by particle transits at different locations. As we will see later in Chapter 5, this prediction can be made by evaluating the acoustic kinetic energy and potential energy of the unperturbed pressure field in the particle volume, which normally requires

Periodic boundary condition Periodic boundary condition

(b)

(c)

(a)

Figure 3.8: (a) A 2D geometry with the shape of half cross-section of the OMFR is first drawn and then revolved with respect to the revolving axis to get a wedge-like portion of the OMFR. (b) Periodic boundary condition is set to ensure the periodicity in the azimuthal direction for wineglass mode. The angle of the simulated wedge is set to be 360◦/M , where M is the azimuthal order. In this case, M = 6. (c) Simulation result for L = 2, M = 6 wineglass mode. Only half of the wedge is shown here.

Periodic boundary condition Roller boundary condition

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(b)

Figure 3.9: Periodic boundary condition is set on two sides of the fluid domain to ensure the periodicity in the azimuthal direction. Roller boundary condition is set on two sides of the solid domain to ensure the OMFR shell is only expanding and contracting in the radial direction. (b) Simulation result for L = 1, M = 0 breathing mode. Only half of the wedge is shown here.

exporting the simulation data and processing outside COMSOL. However, we can also do such integration in the simulation directly, by creating vir- tual particles at the desired radial location like shown in Figure 3.10. These virtual particles are simply entities that have the same simulation settings with the ambient fluid, the existence of which ideally should not affect our simulation results but provide us a easy ‘knob’ such that we can perform data manipulation easily within COMSOL.

Virtual particles

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Figure 3.10: (a) Multiple virtual particles can be created in the fluid domain, by creating particle-like entities and keeping the simulation settings in these particles the same with the ambient fluid. (b) The existence of the virtual particles doesn’t affect the numerically simulated result. But we can use these entities to evaluate the properties such as acoustic kinetic energy and acoustic potential energy inside an entity (entities) by doing an volume integration of these properties over the desired entity (entities), without exporting the pressure field data.

Effect of mesh settings: The mesh setting of the simulation also plays an important role. Not only does the mesh setting determine the computational power the simulation requires, it also is important for us to successfully get the right mode. What I usually do is to set the sequence type as ‘User- controlled mesh’, and under the element size settings, use calibration for ‘Fluid dynamics’ with predefined ‘Extremely coarse’ mesh size. Also, the

overlap between a virtual particle entity and a OMFR symmetry line can create an small volume that requires much finer mesh than usual during the simulation (Figure 3.11a), and causes problematic simulation results. We can solve this issue by moving the particle entity slightly, such that this small volume disappears (Figure 3.11b), or becomes large enough (Figure 3.11c).

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(b)

Small overlap between virtual particles and cylinder symmetry line

No overlap

Large overlap (c)

Figure 3.11: (a) Overlap between the virtual particle entity and the OMFR entity symmetry line causes problematic simulation result. Move the virtual particle entity slightly so that the overlap (b) disappears or (c) becomes large enough help solve the problem.

Scale of the eigenfrequency simulation is random: I would also like to note that the quantities associated with the mode shape of eigenfrequency study results using COMSOL, like the pressure field and the displacement

field, are at a random scale. This is because if you scale of the mode shape by any factor it is still a mode shape. Therefore, we cannot use the absolute simulated quantities like the pressure field or displacement field directly for our calculation when using eigenfrequency study. However, if we divide two quantities, which have the same scale, the scaling factor cancels. For example, term A and B introduced in Chapter 5 are defined as:

A = R Vs|p| 2 dV R Vc|p| 2 dV and B = R Vs|∇p| 2 dV k2 l R Vc|p| 2 dV . (3.9)

It is thus more appropriate for us to use these two normalized terms in our perturbation calculation.