#### Simulation Specialisms

### Acoustics

To understand, analyse and mitigate for all types of noise and acoustic issues, Xi uses a variety of modelling methods. Our approach depends on the specifics of the project, principally the size of the acoustic domain relative to the minimum wavelength which is being modelled. Another important consideration is understanding the nature of the output required and its subsequent use.

When the size of the acoustic domain is comparable to the wavelength of the sound being modelled then finite element (FE) analysis is favoured. An example of this modelling is the low-frequency response of a headphone speaker where the wavelength and acoustic domain are both centimetre scale.

In situations where the acoustic domain becomes large with respect to the wavelength, it is necessary to use other approaches to model the sound field. Boundary element method (BEM) provides a means to model large structures without requiring large numerical meshes, which are a requisite for FE. An example of the use of BEM is the modelling of the acoustic output of wind turbines with 200m tip heights and their effect on local communities.

Ray tracing and parametric equation approaches are also an efficient way to model the propagation of noise for point and linear sources, such as motorways and construction activity. Xi uses these approaches for modelling building acoustics, such as concert hall acoustics, and for modelling underwater noise and its impact on marine species.

A diffusion equation approach is a very effective way of modelling acoustics of coupled rooms within a building, the results of which can be used to improve office and residential environments.

###### Acoustics Modelling and Analysis

The laws of physics are typically described by partial differential equations (PDEs), where a dependent variable such as temperature, velocity or electric potential varies with respect to an independent variable, such as position or time. Analytical solutions to these equations are often only possible for the simplest geometries which, though mathematically rigorous, rarely reflect the reality of a practical design.

Modern computers provide a means of approximating solutions to these physical situations by employing numerical techniques, with the most common being Finite Element Analysis (FEA). A geometry of arbitrary complexity is spatially discretised by creating a mesh, containing nodes at which the dependent variables are solved. Basis functions are selected to represent a spatial variation in these variables, each scaled by piecewise weightings for a solution consistent with the underlying PDE. Boundary conditions are applied which represent the value of dependent variables at nodes or boundaries, for which a unique solution can then be derived.

The solution is an approximation and convergence will therefore depend on meeting a predefined criterion. A common approach is to define a maximum relative error in a dependent variable between solver iterations, of less than 0.1% for example. By using a finer mesh, or increasing the order of basis functions, the rate of convergence can be increased at the cost of computational time. In certain cases, it is necessary to refine the mesh where a steep gradient in the dependent variable is expected, for example when solving fluid velocity close to no-slip boundaries. A mesh refinement study can be conducted to gauge this effect in a similar manner to the relative error.

The aim of FEA software is to reduce the need for prototyping and experimentation in the design or optimisation of a device. COMSOL Multiphysics is the principal FEA modelling package used by Xi, for which Xi is a certified consultant and recognised internationally. COMSOL provides the capability to solve for stationary, frequency/time-dependent and modal problems across multiple physical domains including solid mechanics, acoustics and electromagnetics. By establishing an FEA model which reflects measured data, it is possible to develop a deeper understanding of a design or device, driving further optimisation and innovation.

###### Acoustics and Vibration Modelling and Analysis

Acoustics and vibration have a physical causal link which Xi can help explore via numerical modelling and analysis. Over a decade ago, Xi led the way in developing fully-coupled structural dynamics-acoustic models of large scale energy converting devices such as wind and tidal turbines. These initial models were focused on the identification of noise sources and the development of mitigations techniques.

Over the proceeding decade, the fully-coupled approach has been expanded to examine the performance of a range of products at all physical scales: from microscale transducer optimisation; to kilometre-scale offshore HVDC cable installation and its effect on the marine environment.

At all scales the central modelling principle is consistent: the structural dynamics are modelled; the surface acceleration of the structure imparts an acceleration on the fluid in an acoustic domain (e.g. air, water, etc); and sound is generated. Meanwhile, the mass and motion of the fluid domain also imparts a load on the structure thereby affecting its dynamics response. The model is therefore fully-coupled, i.e. the structure affects the acoustic domain and the acoustic domain affects the structure.

While the central fully-coupled premise is consistent across all models, the physical scale of the acoustic domain controls the modelling method. At physical scales where the wavelength is comparable to the size of the geometry, the acoustics can be modelled with a finite element analysis. However, when the physical scale is far greater that the wavelength of sound, other methods must be used, or couple with an FEA, such as boundary element method, ray trace or diffusion acoustics. At the small scale, such as in the case of hearing aid or mobile phone microphones, thermal and viscous acoustic losses should also be considered. Other physics can also be coupled to these models, such as electrostatics and fluid flow.

Xi has the expertise and experience to help research and development teams develop and implement the best simulation approach to acoustic and vibration modelling and analysis.

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