As computational demands can be very advanced erstwhile doing simulations involving coupled physics – specified as is frequently the case for vibroacoustical transducers – this article looks at how to reduce the computational demands and associated cost utilizing symmetry schemes and dimensional reductions techniques, and the possible associated issues. This article was originally published in audioXpress, June 2024.
Any remedies to lessen this computational burden can improve the overall workflow, and plan changes can be evaluated faster, so it is always a good thought to search ways to simplify model complexities. Here, we will have a look at how to reduce computational costs utilizing symmetry schemes and dimensional reductions techniques, and possible issues associated with them.
All aspects of any transducer or any another product are fundamentally three-dimensional, and so to get the full image of its operation, any simulation should fundamentally be carried out utilizing the full geometry. However, in any instances symmetry conditions can be invoked to reduce the full number of degrees-of-freedom in the computational setup. There are respective considerations regarding whether the symmetry in geometry, boundary conditions, load/sources, and/or field, or only in a subset of these.
An example of this is shown in Figure 1, where a geometry has a symmetry plane, but applying a symmetry or “mirror” condition on the midline and solving this as an eigenmode problem utilizing only half the geometry in the simulation would not return all possible modes, as any will have anti-symmetry in the field. So, for each symmetry seen in a geometry, 1 must carefully consider if respective runs with different types of symmetry conditions are needed to scope the same full consequence as the full geometry would. This becomes especially crucial for any sub-physics specified as structural mechanics or microacoustics, where shear forces come into play.
Figure 1: 2 acoustic eigenmodes for a geometry with a single symmetry plane.
For structural mechanics there will typically be 1 degree-of-freedom per dimension, specified that for a three-dimensional problem, 1 needs to resolve 3 displacements per point in the geometry, so any computational costs saved here will benefit the user. For standard force acoustics, there is only 1 degree-of-freedom to consider, namely pressure, but while there may be little saved in general by invoking symmetries, the acoustics domain are frequently alternatively large and could be the bottleneck in the setup. In general, 1 must consider which types of physics are involved, and which kind of survey is being run, before naively applying symmetrical conditions based on geometry only.
Different Conditions
Different conditions can sometimes be utilized in akin situations where a repeating pattern is present in the geometry, the alleged periodic conditions. These can be simple mirror conditions, anti-symmetry, or more active specified a Floquet conditions for repeating patterns on a grid. These periodic conditions may be applicable to specified structures as those shown in Figure 2, but again this must be evaluated based on the physics, kind of study, boundary conditions, and loads in question. besides note that if optimization is added on top of the simulation, additional considerations must be made, as the optimization may have additional symmetries [1].
Figure 2: Geometries where periodic conditions should be considered on the red boundaries.
Utilizing symmetry planes can reduce the overall computational cost by only considering a subset of an first geometry, but the dimensions of the remaining subset will be the same as the original. A different approach to reducing complexity in a simulation is what can be called “dimensional reduction,” where a three-dimensional domain is modeled utilizing lower dimensions for any or all physics domains involved. This is commonly utilized in structural mechanics where “plane strain” or “plane stress” assumptions are made to reduce three-dimensional problems to two-dimensional ones, or in acoustics where for example a three-dimensional duct problem may be reduced to a one-dimensional one, under certain conditions. For transducers, where more than 1 physics is typically involved, respective different dimensional simplification combinations can be constructed, and this will be illustrated here with a conventional moving coil loudspeaker problem utilized as the prototypical example.
One Common Dimensional simplification Approach
One common dimensional simplification approach is to presume zero-dimensional behaviour for electromagnetism and structural mechanics, with the effect of the acoustics environment included to any degree in a zero-dimensional fashion locally. This is done via lumped components, see for example [2], where the electromagnetic effects are included via 1 or more resistors and ditto inductors, and a full moving mass, a full stiffness, and a full failure are found for the loudspeaker driver, typically via measurements. The stiffness comes mainly from the spider and the surround, while the mass comes from all moving parts. While these parts each have their own mechanical characteristics, here they are lumped into single components, specified that no modal behaviour another than the fundamental for a single degree-of-freedom spring-mass strategy is included in the model.
This is illustrated in Figure 3. The moving coil transductance rule is non-reciprocal and a coupling between the electromagnetics and structural mechanics is needed where the effort variable on the electromagnetic side voltage is coupled to the flow variable velocity on the structural mechanical side, and the flow variable current is coupled to the effort variable force, respectively. A alleged gyrator is utilized in the lumped circuit to get this coupling. Here, the zero-dimensional domain electromagnetics is coupled to the zero-dimensional domain via linking the mechanical velocity and the voltage u’ in the circuit as:
Blv = u’
Additionally, the force applied to the vibroacoustics is linked to the current as:
F = Bli
Figure 3: Standard lumped modeling with a [0D-0D-0D/Analytical] coupling.
The coupling between the mechanical and the acoustical domains is then implicitly included in that the impedance loading (typically denoted as “mass loading”) from the acoustical environment on to the structural mechanics domain is included via lumped components, specified as capacitor for the compliance from a rear enclosure, and simple radiation conditions included via an impedance. Analogously, the mechanical velocity found in the lumped circuit is assumed to be common for all parts, including the cone, and so this velocity is related to the acoustic excitation. However, the force of interest will almost always be at a certain distance distant from the transducer and this force cannot be found locally in the lumped circuit.
This issue is typically dealt with via assuming the transducer acting as simple monopole origin or a piston placed in a baffle and having an analytical expression for calculating the force at a distance. In the second case, the Rayleigh integral is employed and simplified:
The Rayleigh integral accurately gives the complex force for flat vibrating surfaces of arbitrary form and arbitrary velocity distribution sitting in an infinite baffle, but in most loudspeaker literature this is further simplified by assuming that the surface is circular with radius a, has a constant velocity v across it, and that the force p(P) of interest is on-axis and in the far-field. A two-way coupling is thus implicitly incorporated, as already described, but no modal behaviour is incorporated for this pistonic approach, and no diffraction effects can be calculated with the acoustical components lumped in this manner.
More Complicated Acoustic Environments
For more complicated acoustic environments, to include diffraction for the enclosure and possibly the inclusion of a complex rear chamber geometry, a different dimensional mapping can be utilized, where zero-dimensional components for the magnet strategy and structural mechanics part are used. These are then coupled to a three-dimensional acoustical domain, possibly with its own symmetry planes. The coupling takes place across the cone surface, and there are no moving parts in the finite elements analysis, although it may be advantageous to keep all explicit geometry parts to have the acoustics at the rear of the transducer better represented. This setup is shown in Figure 4, where the three-dimensional acoustic field is resolved with a coupling to the lumped mechanical components.
Figure 4: A [0D-0D-3D] coupling resolving the front and rear acoustics explicitly.
The lumped velocity is driving the acoustical velocity across the cone, surround, and dust cap, and here 1 needs to make choices about how this velocity is distributed and if the scalar velocity in the lumped model relates to a average or an axial direction.
The coupling can be two-way in that not only does the zero-dimensional velocity drive the acoustical energy input, but besides the force conditions across the cone can be coupled back to burden the electro-mechanical system, so that:
A simpler version of this coupling can be made for situations where the rear enclosure is included via a lumped component, and only the front acoustics is included, with or without its loading effect included. For any transducers, specified as MEMS microtransducers, this approach will not be accurate adequate as there can be intricate acoustic channels with microacoustical effects that request to be resolved, but for conventional loudspeakers, this may well be an approach worth pursuing.
The couplings for the moving coil transducers shown in Figure 5 are sequential in that there is an interface between the electromagnetics and the structural mechanics, and between the structural mechanics and the acoustics. The dimensional splitting can be done straight at those interfaces, but there are alternatives specified as splitting the dimension in between the acoustical parts, as just shown, but besides in between mechanical parts. This was described in the literature [3] as a “hybrid” model around the time any of us were besides doing it in the industry, and was years later described again [4], although the method was the same.
The thought is to include the vibrating surface, typically cone, dust cap, and surround, as explicit domains, and lumping together the remaining moving parts on the rear, specified as the spider and the voice coil (former plus windings), alongside a lumped magnet system. This setup is illustrated in Figure 6, where the cone, surround, and dust cap are now moving parts with structural mechanics physics applied, while the remaining mechanical parts are found in the lumped circuit. While not emphasized in the literature, it has already been mentioned here that it may be crucial to have the lumped parts inactive included as geometry, as to better capture the rear acoustics, where the parts may form somewhat closed off regions with distinct resulting force characteristics. Also, as the former, windings, and spider have their own distinct mechanical modal behavior, these can influence the overall displacement from a structural mechanics perspective, and hence affect the sound pressure, and these effects are excluded in this model.
Figure 5: A [0D-0D-3D] coupling with lumped rear acoustics and explicit front acoustics.
Figure 6: A [0D-0D-3D] mapping with lumped and explicit (grey) mechanical parts.
If all mechanical components are included explicitly, a setup is had, as illustrated in Figure 7, where only the electromagnetics are lumped, and the structural mechanics and acoustics are included in either 2D-axisymmetrically or three-dimensionally.
Figure 7: A [0D-2Daxi/3D] mapping of a tweeter with explicit mechanical parts in the 2Daxi or 3D domain.
Here, the zero-dimensional electromagnetics domain is coupled to a higher-dimensional mechanical domain via linking the three-dimensional velocity and a zero-dimensional voltage as:
where an average velocity of the windings is calculated in the axial direction, here z-direction, as:
There will typically be a two-way coupling, specified that the pressures on the moving part couple with the average stresses in the structural mechanics to have the acoustical loading included. This is simply a standard setup in most simulation software, but it will be computationally costly for many cases, since the acoustics environment will typically request 3 dimensions, and so the mechanical counterparts besides must be resolved in 3 dimensions, even if the transducer is inherently axisymmetric, at least in geometry, boundary conditions, and load.
Alternative Mapping
This leads to an alternate mapping that I have not seen described in the literature, shown in Figure 8. Here, the magnet strategy is included as lumped components, although it could be included in the 2D-axisymmetrical model for more elaborate magnet setups. The 2D-axisymmetrical domain includes all moving parts explicitly, and so modal behaviour is included, albeit excluding circumferential variation. These variations may not be crucial as these modes are not necessarily excited much, and/or do not contribute much to the sound field, although this must be evaluated on a case-by-case basis. The rear acoustics geometry may be highly intricate, and can for example include microacoustics effects, porous materials, ferrofluids, or another characteristics that are not easy lumped.
Figure 8: A [0D-2Daxi-3D] mapping of a tweeter with explicit mechanical parts in the 2Daxi domain.
Finally, the front acoustics geometry can be arbitrarily detailed, for example including enclosure diffraction, phase plugs, or surviving area or automotive environments. The transducer can thus be included in a very elaborate manner, while inactive saving crucial computational cost by only resolving it in 2D-axisymmetry, or 2D, depending on the transducer type, while the acoustic effects are captured near-perfectly. Different variations of this can be made, for example if the rear acoustics is simple adequate to go into the lumped model, or alternatively besides complex to be axisymmetric, specified that the same approach as for the front acoustics is applied to the rear utilizing 3 dimensions for all acoustics domains.
This kind of dimensional mapping is not trivial to set up, as the meridional velocity distribution on the transducer surface must be mapped to the corresponding meridional and circumferential points in the acoustics domain. It may be essential to besides have a two-way coupling incorporated. However, this overall strategy can take a simulation from requiring a high-performance cluster to solve, to solving locally on a more modest computer, and it is especially useful if additional studies specified as topology optimization [5] are applied on top of it.
Conclusions
For all dimensional mapping strategies shown here, 1 must take care of the scalar and vector aspects of each sub-physics. Voltages and currents in a lumped model are complex scalars in zero dimension but they have signs comparative to any arrow indications in the schematics. For any lumped model representing mechanical or acoustical parts, these signs relate to a peculiar direction, specified as average or axial, for underlying complex displacement vectors, or positive/negative aspects of scalar force fields.
Assuming standard acoustics, only average displacements make sound, but if microacoustics effects are to be included, both average and tangential displacements must be included. Also, single forces in a lumped circuit may be calculated from integral values coming from pressures in a three-dimensional domain. Getting all these signs and directions correctly implemented is crucial for the dimensional mapping to work.
For another transductance principles than the moving coil utilized as an example here, specified as electrostatics or piezoelectric, the dimensional mapping will be different, and 1 needs to consider, for example, that they may or may not be reciprocal in nature. I am planning an article series on transductance principles, to go into more of these details. aX
References
[1] R. Christensen, “Advancements in Acoustical Topology Optimization,” COMSOL Conference 2023, Munich, 2023.
[2] R. Christensen, “Lumped component Modeling of Transducers,” audioXpress, March 2023.
[3] M. Pasi and M. J. Herring, “A Hybrid Electroacoustic Lumped and Finite component Model for Modeling Loudspeaker Drivers,” Audio Engineering Society 51st global Conference, Helsinki, Finland, 2013.
[4] D. G. Nielsen, P. R. Andersen, J. S. Jensen and F. T. Agerkvist, “Estimation of Optimal Values for Lumped Elements in a Finite component – Lumped,” diary of Computational Acoustics, Volume 28, No. 2, 2020.
[5] R. Christensen, “Acoustic Topology Optimization for Vibroacoustics Applications,” audioXpress, April 2023.
This article was originally published in audioXpress, June 2024
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