The modeling of dissipative silencers is often considered as a tedious task by the practitioners of acoustics, in relation on the one hand to the high number and the great variety of the phenomena having to be taken into account, and on the other hand in relation to the difficulty to be in a position of using calculation means adapted to a dimensioning context which must (quite often), as far as possible, be carried out with little effort and within a short timeframe (e.g. when the context is that of a project for which limited financial means are allocated, and that is with an ambitious timeline).
Different approaches coexist :
 evaluations (which some at least) consider to be rough or even simplistic or simply inappropriate (when the backgrond is sufficiently known), often based on the compilation of measurement results that are not necessarly reproductible (sometimes: under conditions that would deserve to be all specified, and which are not always as favorable as desirable, even when carried out in a laboratory rather than in the field): the deduction of the acoustic performance for configurations (e.g. geometry of the silencer, nature of the lining, presence of a fluid circulating in a direction equal to or opposite to the transmission of sounds, thermodynamic conditions) which has not been the exact (and therefore: often limited) subject of a dedicated metrology is not always eligible (unanimously) for the rank of "good practice ”(no more than the use of formulas explicitly involving the absorption coefficient at normal incidence of the acoustic structure considered to evaluate the propagation loss, or the transmission loss, or insertion loss  if the distinction is made when such evaluations are conducted )
 methods involving finite elements (FEM ^{[1]}) or by boundary elements (BEM ^{[2]}) : the fact that they (sometimes) require the use of very expensive tools to purchase and to upgrade version, extensive training for their use by personnel with important prerequisites, whether (quite often) of a complex use for the modeling of a silencer (requiring the creation of a geometric model, then a mesh before calculations strictly speaking  which calculations are long even with the unusual means of calculations which are made necessary ) and the fact that they offer tempting output data (maps with fully configurable shimmering colors) does not always constitute, however, a warranty that the relevance of the results  as to their substance  reaches the pinnacle of perfection in terms of prediction of the acoustic performance of dissipative silencers (would it not open to question when the absorbent lining is taken into account by means of incomplete models, when a surface impedance relating to a locally reacting absorber being planar i.e. without curvature radius is used for the calculation of silencers of circular section, or when the acoustic performance indicator is not the one corresponding to the propagation of plane waves, a case however often favored by many acousticians because considered to be the most conservative in terms of acoustic performance, and  this is not nothing  (solely) making the simulation results comparison with standardized laboratory measurements possible ^{[3] }?)
This is why, in the context of the search for a good compromise between on the one hand versatility of use, reliability and precision, and on the other hand the cost (in initial investment, in training, and in use e.g. in relation to the power of the computers required), the Module 1 of the SILDIS^{®} software often occupies a prominent place for the prediction of the acoustic and aeraulic (aerodynamic) performance of silencers.
Regarding the lining, the materials are taken into account with sophisticated and robust models (always: by considering the thickness of each layer, its interactions with the adjacent layers, and being able to use for the simulation of materials with properties referenced in the software libraries or that the user can freely select):
 for porous media:
 applicable to wools (e.g. mineral or polyesterbased) and foams
 taking into account up to 5 parameters (resistivity, porosity, tortuosity, thermal and viscous characteristic lengths)
 for cloths :
 applicable to unwovens, fabrics, membranes (for the calculation of resonators)
 taking into account 2 parameters (resistance to the passage of air, mass density) unless the considered layer is modeled as a porous medium (cf. above) or else as an elastic plate (in which case Young's modulus, density, Poisson's ratio, loss factor, dimensions and installation conditions are taken into account)
 for perforated protections :
 applicable to sheets (e.g. for the calculation of resonators) ^{[4]}
 taking into account the geometry of the holes (e.g. round, squared, slitshaped) their dimensions, the perforation rate, the mass density, unless the layer considered is modeled as a porous medium (see above)
The extrapolation of the parameters of the lining materials measured under laboratory conditions (at room temperature) to the context of each simulation (e.g. at high temperature in the case of exhaust gas mufflers) is carried out by the software which also evaluates the physical parameters of the air at the thermodynamic conditions to be considered (unless the user prefers to enter such input data himself, which is necessary in the case of taking into account another fluid than air).
There are 3 conditions for sound propagation inside the lining material that can be modeled: local reaction material (i.e. no longitudinal sound propagation in the absorber or in the splitters), isotropic material, or (finally) anisotropic material i.e. with different properties longitudinally and transversely.
The lining considered is not necessarily monolithic (homogeneous), which may consist of a number at most equal to 4 of subassemblies each consisting of a porous medium, a cloth and a perforated protection.
The flow of a fluid in the silencer (for what relates to flow rate and flow direction with respect to the direction of sound propagation, depending on whether it is a suction or a discharge) are taken into account, not only in terms of self noise, but also for the calculation of the propagation loss (this is the component of the acoustic performance which is proportional to the length of the noise attenuator); a limitation of this proportionality of the acoustic performance according to the length (due to unwanted sound paths) can be considered, being then based on the compilation of results of laboratory measurements.
In the case of the presence of obstacles (for silencers with splitters), the reflection loss (at the inlet and at the outlet of the silencer) is duly taken into account.
It is of course possible to simulate the performance of silencers of different geometries, in particular according to the cross section to be considered :
 rectangular : with splitters i.e. baffles (if applicable)
 circular : with splitters i.e. baffles (if applicable) being either transverse or concentric (when it is not a unique central absorbent core)
 square

Fig. 1 Sound transmission loss of a silencer with a rectangular cross section simulated with the Module 1 of the SILDIS^{®} software. It comes to a double resonator consisting (from rear to front) of an air gap 0.05 m thick with a circular perforation plate with diameter 0.0003 m (perforation rate 1%), with an air gap 0.06 m thick with a circular perforation plate with a diameter of 0.0003 m (perforation rate 1.77%), with an open area ratio of 33.3%, with a length 1.2 m. For ambiant temperature, for zero flow speed, the transmission loss reaches 10.6 dB (i.e. 8.8 dB/m) for the frequency 1kHz. 
In terms of acoustics, with regard to the calculation of the propagation loss, it is proceeded to the numerical resolution of the (analytical) sound propagation equations, both in the layers of absorbing materials and in the channels allowing the circulation of the fluid, without it being necessary  therefore  to indicate any particular limitation (of any kind) of the input data to be considered (since it is not called upon frozen databases).
When the surface impedance of an absorbent layer is involved in a calculation, its flatness or the existence of a radius of curvature (which is not negligible in the case of cylindrical silencers) is taken into account (surface impedance and absorption coefficient at normal incidence, which are not indicators of silencer quality, can nevertheless be the subject of an additional display of output data (Module 1+)  like all the others: in the form of graphs and tables, with fineband values, per 1/3 octave band, per 1/1 octave band, and overall Aweighted values with respect to a userselected acoustic power spectrum . The export of surface admittance data (i.e. the inverse of the surface impedance, which is a complex quantity) is possible in the form of a file that can be used for calculations with other tools (FEM, BEM).
The calculations are carried out for plane waves ^{[5]}, for which there is generally a consensus in the profession of acoustics, both with respect to the consequences in terms of safety of the dimensioning (it is the most often admitted that it is the least attenuated mode of acoustic propagation in ducts and the modal distribution of the acoustic power of the noise sources to be reduced is  except in the theoretical exceptional case ?  never known), and in terms of possibility of comparison of the simulation results with those obtained by a standardized laboratory metrology ^{[1]}, which is also possible, for the output data (for single frequencies cf. fig. 1, per 1/3 or 1/1 octave band or in terms of overall value in dB(A) with respect to a reference spectrum) of the SILDIS^{®} software, with regard (besides insertion loss) to the self noise and the aerodynamic performance (pressure drop evaluation).
One of the last refinements in terms of prediction of the acoustic performance of dissipative silencers (i.e. whose efficiency has to do with the presence of soundabsorbing materials) with Module 1 of the SILDIS^{®} software consisted in completing the resolution of transcendent equations allowing the evaluation of the wave number (a complex quantity i.e. with a real part & with an imaginary part) by the means of expansions based on truncated continuous fractions (which preexisted) by an iterative method, all the more daring as far as its implementaion is concerned in the case of circular sections, because the calculations are then involving Bessel and Neumann functions with complex arguments (ranging from infinitely small to infinitely large, or almost), the evaluation of which requires significant resources in terms of computation means.
The expected consequence is an improvement in the precision of the simulation results, in particular when the insertion loss of the dissipative stage of a silencer must be cumulated (e.g. by means of Module 1B of the SILDIS^{®} software, for which the output data of Module 1 then constitute input data) with the effects due to the presence of reactive elements (e.g. changes of section, not only due to the presence of separators or an absorbing lining installed  in whole or in part  in the periphery of a duct) by a method based on the multiplication of transfer matrices (for what the alternative consisting in the addition of the corresponding insertion losses does not always constitute what is best to do in a perspective of optimization of the quality of the prediction in terms of acoustics) as is often appropriate for the sizing of exhaust noise attenuators for heat engines.
Another recent development that can be noted is the extension of the choice (possible for the user) of simulation modalities for perforated protections (facing sound absorbing materials) to less sophisticated models than those (constituting the ultimate?) already implemented in the SILDIS^{®} software ; the expected consequence is to facilitate  under the best possible conditions, because it can now be based on the same assumptions, even if imperfect  comparisons of simulation results obtained with Module 1 of the SILDIS^{®} software with diversified and more or less recent bibliographic data, what may sometimes be of special interest(in a context of Research and Development in acoustics, or in the context of a project requiring the sizing of a silencer for the noise reduction of any hardware, machine or process).
The number of development and validation ^{[6]} hours devoted, over the last 30 years, by the human resource of ITS, to Module 1 of the SILDIS^{®} software is counted in the thousands, which makes it an acoustic and aeraulic sizing tool, being unique by the richness of its contents, its computing power, its versatility and its sophistication, being still and always the object of a continuous improvement program (refinement of modeling steps, extension of validation programs) for the provision of users, such as those usually working in design offices or R&D structures specializing in acoustics, of a tool for which performance (reliability, precision, speed for obtaining calculation results) is inseparable with ease of use (which is not nothing in the matter).
If necessary, the ITS human resource can  of course  also perform calculations with Module 1 of the SILDIS^{®} software  Prediction of the acoustic and aeraulic (aerodynamic) performance of silencers  or, in some specific cases, with other tools (FEM, BEM) as part of engineering missions at all levels, in all contexts.
^{[1]} Finite Element Method
^{[2]} Boundary Element Method
^{[3]} NF EN ISO 7235 Acoustics — Laboratory measurement procedures for ducted silencers and airterminal units — Insertion loss, flow noise and total pressure loss
^{[4]} simulations of plasterboard, wooden facings (also possible with Module 1 of the software) are generally not involved in silencer calculations, being otherwise however useful for other applications in relation to the limitation of reverberation in rooms
^{[5]} the imperfection of some standardized test facilities with respect to the generation of plane waves can be reflected (in order to make comparable what can or must be compared), for what concerns the evaluation of reflection loss with the software, by considering the presence of higher order modes such as sometimes observed in practice
^{[6]} cf. R&D reports list
Computer Aided Design (CAD): SILDIS^{®} calculation software for acoustics and aeraulics in the construction sector
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