Practical Aspects of Modeling Acoustic Systems with FEA

The following paragraphs describe some practical considerations when mod-eling acoustic systems with Finite Element Analysis (FEA).

Acoustic wavelength. The acoustic wavelength in a media is related to the speed of sound and the excitation frequency by the following equation

λ = c₀

f , (2.10)

where λ is the acoustic wavelength, c0is the speed of sound in the media, and f is the excitation frequency. It is vitally important to consider the acoustic wavelength when meshing the acoustic and structural models, as this will affect the accuracy of the results.

Mesh density. The finite element method can be useful for low-frequency problems. However, as the excitation frequency increases, the number of nodes

20 2. Background TABLE 2.2

Advantages and Disadvantages of Displacement-Formulated Elements

Advantages Disadvantages

The set of equations to be solved in a general fluid–structure inter-action analysis are symmetric.

Displacement boundary condi-tions and applied loads have the same physical meaning as those used for standard structural ele-ments.

Energy losses can be included in the displacement element via a fluid viscosity parameter as well as the standard techniques with solid elements.

There are 3 displacement DOFs per node, which can result in a model with a large number of DOFs.

Modal analyses can result in a large number of (near) zero fre-quencies associated with shear-ing of the fluid elements.

The acoustic pressure at a point in the fluid cannot be expressed in terms of a known incident pressure and an unknown scat-tered pressure.

The shape of the elements should be nearly square for good results.

and elements required in a model increases exponentially, requiring greater computational resources and taking longer to solve. A general rule-of-thumb is that acoustic models should contain at least 6 elements per wavelength as a starting point [150, p. 5-1]. For better accuracy, it is recommended to use 12 elements per wavelength for linear elements (i.e., FLUID29 and FLUID30) and 6 elements per wavelength for quadratic elements (i.e., FLUID220 and FLUID221). Accurate models can still be obtained for lower mesh densities;

however, caution should be exercised. At regions in a model where there is a change in the acoustic impedance, for example where the diameter of a duct changes, at a junction of two or more ducts, or at the opening of the throat of a resonator into a duct, a complex acoustic field can exist with steep pressure gradients. It is important to ensure that there is sufficient mesh density in these regions to accurately model a complicated acoustic field.

Mean flow. Many finite element software packages with acoustic finite el-ements require that there is no mean flow of the fluid, which is a significant limitation. When there is mean flow of fluid, a different formulation of the wave equation is required, which modifies the propagation of the acoustic dis-turbance (due to “convection”), depending on whether the flow is rotational or irrotational. However, it is still possible to conduct finite element modeling

2.6. Practical Aspects of Modeling Acoustic Systems with FEA 21 for low-speed fluid flow, where the compressibility effects of the fluid are neg-ligible, using “no flow” FEA software packages, but some assumptions that underpin the analysis will be violated. When there is mean flow in a duct, aero-acoustic phenomena might be important. For example, consider the sit-uation of mean flow in a duct where the throat of a Helmholtz resonator attaches to the main duct, or over a sharp edge. It is possible that as air flows over the edge of the throat, noise will be generated, similar to blowing air over the top of a glass soda bottle. In some situations the flow over the structure might cause vortex shedding. Standard finite element models, such as those in ANSYS finite element packages, are not able to model these effects.

If the flow speed is significant or it is expected that there will be aero-acoustic phenomena, consider the use of Computational Fluid Dynamic (CFD) software to analyze the problem. However this software also has limitations for the analysis of acoustic problems. Alternatively, some Boundary Element Analysis software packages are able to model acoustic systems with mean flow, but are not able to model noise generation from shedding type phenomena.

Rigid or Flexible Boundaries. Acoustic finite element models have rigid-wall conditions at boundaries where no elements are defined. This assumption is valid in situations where it is not expected that the motion of the bound-ary is likely to have any significant effect on the acoustics of the system.

However, consider an automobile cabin comprising flexible sheet metal pan-els. Depending on the stiffness of these panels, acoustic excitation within the enclosure can cause the panels to vibrate, which in turn will affect the acous-tic mode shapes and resonance frequencies of the enclosure. As highlighted above, modeling fluid–structure interaction can be computationally complex and can require substantial computer resources to solve. Hence careful consid-eration is required to decide whether the fluid–structure interaction should or must be modeled. A second subtle point is the consideration of re-radiation of structures in a different part of the acoustic model. Consider a duct with two Helmholtz resonators attached to a duct to reduce sound radiated from its exit as shown in Figure 2.3. A simple acoustic model could be constructed assuming rigid-walls. However if parts of the system are in fact flexible, for example the wall dividing the two resonators, then high sound levels in the first resonator would vibrate the dividing wall and reduce its effectiveness and

Helmholtz

Duct with two Helmholtz resonators with a flexible dividing wall.

22 2. Background would re-radiate sound into the second Helmholtz resonator. For further dis-cussion see Refs [55, 76]. Alternatively, if the entire system were made from lightweight sheet metal, then vibrations could be transmitted along the duct work and result in the re-radiation of sound into the main duct.

Results and Frequency Range. The results from acoustic analyses are usu-ally the acoustic pressure at discrete locations. Sometimes this level of detail is required but often it is not; instead an indicative global sound pressure level or total sound power level may be required for assessment, which will require post-processing of the results from the analysis. This can sometimes be per-formed within ANSYS or may require exporting data and post-processing in another software package such as MATLAB. For higher-frequency problems, statistical energy analysis methods may be more appropriate and significantly faster in obtaining a solution.