Use of boundary conditions of impedance, continuity ... · 22nd International Congress on...

Buenos Aires – 5 to 9 September, 2016 Acoustics for the 21

st Century…

PROCEEDINGS of the 22nd International Congress on Acoustics

NT - Numerical Techniques (others): Paper ICA2016-118

Use of boundary conditions of impedance, continuity, Sommerfeld, for evaluate sound attenuation of mixed

passive silencers with FEM-2D.

Alexander Garcia Luque (a), Mauricio Ruiz (b). (a)

Universidad Politécnica de Madrid / FiberGlass-Isover SG, Colombia, [email protected]

(b) Universidad Nacional de Colombia, Colombia, [email protected]

Abstract

In this paper we study the influence and importance of boundary conditions (BC) in the formulation, implementation, and solution for acoustic problem silencers in mixed passive silencers. The analyzed model corresponds to a muffler chamber expansion, using FEM-2D (two-dimensional Finite Element Method). As a first approximation to describe the dissipative condition of chamber walls, we define an impedance BC; but it forces a certain idealized acoustic behavior of the muffler. In order to achieve more reliable and accurate predictions, we model the impedance walls as an absorbent sub-domain, which is coupled with the rest part of the muffler. Due to the characteristics of the problem (knowledge of the physical phenomena) was necessary to fix conditions like Impedance, Continuity and Sommerfeld. Delany-Bazley approximation is used to describe the property of porosity of the absorbent material and medium. The introduction of the absorbent sub-domain makes the sound pressure at each point of the domain more sensitive (“real”) in this kind of muffler and yields to better results. The calculated transmission loss values show good agreement with the experimental measurements.

Keywords: Silencer, Boundary conditions, FEM, Noise Control.

22

nd International Congress on Acoustics, ICA 2016

Buenos Aires – 5 to 9 September, 2016

Acoustics for the 21st

Century…

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Use of boundary conditions of impedance, continuity, Sommerfeld, for evaluate sound attenuation of mixed

passive silencers with FEM-2D.

1 Introduction Our interest is focused on studying from the perspective of acoustics engineering the noise as

pollution and public health problem; particularly using the numerical Finite Element Method

(FEM) for analysis of a bidimensional model corresponding to phenomenon of sound

propagation within a duct. The chosen typology is a mixed passive muffler and the

implementation was in Comsol Multiphysics® and Matlab-MathWorks ®.

2 Problem formulation 2.1 Formulation, constitutive equation The sound propagation within the duct is described by the wave equation,

∇2𝑃 −

1

𝑐2𝜕2𝑃

𝜕𝑡2= 0. (1)

P, is sound pressure [Pa], c, is speed of sound in the medium [m/s], and, t, is time [s].

For harmonic solution (Plan wave) the result is Helmholtz equation, and 𝑘, is the wave number.

∇2𝑃 + 𝑘2𝑃 = 0. (2)

The acoustic problem is in Figure 1, 2D sketch for symmetry expansion chamber silencer, Fig. 1a) the boundaries are denotes. Fig. 1b) shows the model with absorbent subdomain.

Source: (Author, 2016) Figure 1: Bidimensional model for problem

http://www.mathworks.com/products/matlab/

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2.2 Boundary conditions For the analysed model we have three different types of boundary conditions.

Sommerfeld Radiation Condition: As in input and output sides the silencer is open the boundary condition must be sound radiation where the wavefront radiation is divergent.

Rigid Wall Condition: Corresponds to the condition of sound reflection (insulation), where the acoustic wave find a wall hit and is reflected. Specifically it is a Neumann boundary condition, where due to the rigidity of the wall there is no pressure fluctuation at the edge.

Impedance Condition: Is an edge with absorbent inner lining, where exist a change of propagation medium (different material), which means that will happen sound refraction and the edge will have a damping effect. Is a Robin boundary condition, where energy loss according to the characteristic impedance of the coating.

2.3 Study Cases. Then the mathematical formulation of the boundary conditions is presented and described what was presented in Figure 1.

A. Impedance boundary condition in chamber. (Absorption edges). This case are coated with absorbent material the inner walls of the expansion chamber, the corresponding math formulation is,

∇2𝑝 + 𝑘2𝑝 = 0, in Ω.

−�⃗� (−1

𝜌0∇𝑝 + (𝑗𝑘 + 𝜅(𝑟))

𝑝

𝜌0) = (𝑖𝑘 + 𝜅(𝑟) − 𝑗(�⃗� �⃗� ))

𝑝0

𝜌0𝑒−𝑗�⃗� �⃗� , in Γ1, and Γ7. (3)

𝜕𝑝

𝜕�⃗� = 0 , in Γ2 and Γ6 . (4)

.−1 𝜌0⁄ (∇𝑝 ∙ 𝑛) − 𝑗𝜔

𝑍𝑝 = 0 , in Γ3, Γ4 and Γ5 . (5)

In (3), p0 is 1Pa for Γ1, source in input edge; while for Γ7 output from silencer p0 = 20𝜇Pa (Air

pressure). The boundary conditions in Γ1 y Γ7 are Sommerfeld, boundaries in (4) correspond to Newmann condition (Rigid walls), finally Γ3, Γ4 and Γ5 are impedance BC (5). The impedance of

the absorber edge is define by Rf, material flow resistivity [1], function of density of and fibre diameter according to

𝑅𝑓 =3,18 ∙ 10−9𝜌𝑎𝑏𝑠

1,53

𝑑𝑎𝑏𝑠2 (6)

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𝑍 is the material impedance. When inner walls of the chamber are coating is important

to know the absorbent properties of the material using Delany-Bazley approximation [2]

Z = (ρ0c)

{

1 + 0,0485Rf

−0,754 − j0,087Rf−0,73 ; Rf > 60

1

2πRf+ j1,4

(−1,466 + j o,212

Rf)12⁄; ; Rf < 60

(7)

kc = (ρ0c) {

1 − j0,189Rf−0,6185 + 0,0978Rf

−0,6929 ; Rf > 60

−(1,466 − j o, 212

Rf)

12⁄

; Rf < 60 (8)

B. Internal subdomain absorbent in silencer.

It shows more real situation than case B., it is a refined model of the problem which absorption is included as subdomain based on the thickness of coating, likewise the exterior edge from the chamber is hard and rigid BC. The Fig.1b) describes the model, the formulation is,

∇2𝑃 + 𝑘12𝑃 = 0, in Ω1 (9)

∇2𝑃 + 𝑘22𝑃 = 0, in Ω2 (10)

−�⃗� ∙ (−1

𝜌0𝛻𝑝 + (𝑗𝑘1)

𝑝

𝜌0) = (𝑖𝑘1 − 𝑗(�⃗� 1�⃗� ))

𝑝0

𝜌0𝑒−𝑗𝑘1⃗⃗ ⃗⃗ �⃗� , in Γ1 y Γ8

𝜕𝑝

𝜕�⃗� = 0 , in Γ2, Γ3, Γ5, Γ6 y Γ7

𝜕𝑝

𝜕�⃗� Ω1=

𝜕𝑝

𝜕�⃗� Ω2 , in Γ4 (11)

where, Ω1, is air, k1, is the wave number in air.Ω2, is absorption material, k2 = kc, , is the

wave number in the absorbent medium.

kc is related with sound speed, density and impedance of coating. 𝑘𝑐 = 𝜌𝑐𝜔

𝑍𝑐, 𝑘𝑐 y Zc is

calculate using Delany-Bazley approximation with (7) and (8).

Γ4, is an interior edge, corresponding to continuity or interface condition, that coupled wave

equations in the air and in the absorbent material.

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2.4 FEM – Finite Elements Method This simulation method is used to solve the acoustic problem defined by the Helmholtz equation

and the boundary conditions of model for the case. First is transformed to variational

formulation, then is discretized the domain and borders, after the problem is reduced to linear

system of equations (mass and stiffness matrices), and finally this is solved with the boundary

conditions. The variational form for the problem is: Find p ∈ H1(Ω) such that,

∫𝜕𝑝

𝜕𝑛𝜙

𝛤1

𝑑𝑥 + ∫𝜕𝑝

𝜕𝑛𝜙

𝛤7

𝑑𝑥 + ∫ 𝜕𝑝

𝜕𝑛𝜙

∪𝑖=26 𝛤𝑖

𝑑𝑥 − ∫ 𝛻𝑝 ∙ 𝛻𝜙𝑑𝑥 +𝛺

∫ 𝑘2𝑝𝜙𝑑𝑥 = 0𝛺

(12)

∀ ϕ ∈ H1(Ω). For domain discretization is used a triangular mesh 𝒯 = {T1, T2… , Tr} ,

formed by r closed triangles contents in Ω so,

Ω̅ =∪i=1r Ti. ℎ𝑖 = 𝑑𝑖𝑎𝑚(𝑇𝑖). ℎ = ℎ𝑖𝑖=1…𝑝

𝑚á𝑥 .

It is denoted by {x1, x2… , xm} vertices of the triangulation 𝒯, with this is associated a space

of linear functions continuous piecewise,

Vh = {𝓋 ∈ C ( Ω̅ ) ∶ lineal in T for all T ∈ 𝒯}. In addition to each j ∈ {1,… ,m} we introduce the functions ϕj ∈ Vh such that 𝜙𝑗(𝑥𝑗) = 1

and 𝜙𝑗(𝑥𝑖) = 0 for all i ≠ j , which form a basis for Vh, therefore,

𝑃(𝑥) = ∑ 𝑃𝑖𝜙𝑖(𝑥) ∀ 𝑃 ∈ 𝑉ℎ .𝑚𝑖=1

, where Pj = P(xj) are P values in nodes. Then considering that the internal domain points

begin from 𝑖 = 1 to 𝑚, the boundary is listed from 𝑚+ 1 to 𝑛, and have the following linear

system of equations:

.∑ ∫ 𝑃𝑖𝛻𝜙𝑖𝜙𝑗𝜕𝛺𝑛𝑖=𝑚+1 − ∑ ∫ 𝑃𝑖𝛻𝜙𝑖𝛻𝜙𝑗𝛺

𝑚𝑖=1 + 𝑘2∑ ∫ 𝑃𝑖𝜙𝑖𝜙𝑗𝛺

= 0𝑚𝑖=1 .

(13)

And results, stiffness �̂� and mass matrix �̂�,

.�̂�𝑖𝑗 = 𝑘2 ∫ 𝛻𝜙𝑖𝛻𝜙𝑗Τi

𝑑𝑥𝑑𝑦. . . �̂�𝑖𝑗 = ∫ 𝜙𝑖𝜙𝑗Τi𝑑𝑥𝑑𝑦. (14)

The general form for linear system resulting is,

.−�̂�𝜙𝑖+ �̂�𝜙

𝑖= �̂� (15)

,where F̂ depends form boundary conditions.

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For Impedance boundary condition in chamber, (Absorption edges).

�̂� is for impedance boundary in walls of chamber is,

�̂� = ∫𝜕𝑝

𝜕𝑛𝜙

𝛤1

𝑑𝑥 + ∫𝜕𝑝

𝜕𝑛𝜙

𝛤7

𝑑𝑥 + ∫𝑗𝜔𝜌0𝑍

𝑝 𝜙∪𝑖=35 𝛤𝑖

𝑑𝑥. (16)

And finally,

�̂� =∑𝑗𝜔𝜌0𝑍

∫ 𝑝𝑖𝜙𝑖 𝜙𝑗Ω

𝑞

𝑙+1

+∑ ∫ −�⃗⃗� ∙ (𝑗𝑘)𝑝𝑖 + (𝑗𝑘 − 𝑗(�⃗� �⃗� ))Ω

(1)𝑒−𝑗�⃗� �⃗� 𝜙𝑖𝑔

𝑞+1𝜙𝑗 +

∑ ∫ −�⃗⃗� ∙ (𝑗𝑘)𝑝𝑖 + (𝑗𝑘 − 𝑗(�⃗� �⃗� ))Ω

(20𝜇)𝑒−𝑗�⃗� �⃗� 𝜙𝑖𝑛

𝑔+1𝜙𝑗 . (17)

For internal subdomain absorbent in silencer. Impedance coating material (absorbent medium)

is solved the Helmholtz equation Ω1 in (air) and Ω2 (material).

3 Results The acoustic parameter used for comparisons and analysis is transmission loss TL ,

TL = 10log (Wo

Wi) (18)

, where Wo = ∫|p|2

2ρcsΓOutput , i.e. Γ7.

3.1 Validation Experimental data are used as a benchmark to validate simulations. The assembly is measured according to ISO 11820[3], experimental data obey test described by Tao (2003)[4]. The study

from Rivas (2010)[5] is taken as reference. The silencer parameters and geometry are:Rf: 14000Rayls; material thickness: 0,15m (6inch)., total duct length1,828m (72inch). The simulations are performed for two cases of analysis: a half geometry model, and another with full geometry muffler. The results presented are: TL graphics, table describing the discretization, graphic with sound propagation inside the duct for the bands from 31Hz to 4kHz, and table with error calculated vs measurement.

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3.2 Half geometry model An axis of symmetry is assumed (for a mirror for the model behavior). TL for different polynomial orders: linear, quadratic and cubic. The resulting TL values are in Figure 2.

Source: (Author, 2016)

Figure 2: TL FEM half geometry model The bases and elements numbers associated can be seen in the following table of Figure 2. Also is used mesh refinement discretization looking better result with respect to the experimental data. For complementarily analysis are calculated absolute and relative errors (Table 1) quantifying differences simulation methods regarding experimental values.

Table 1: Error for half geometry model

FEM

Linear Linear

Refined FEM

Quadratic Quadratic Refined

FEM Cubic

Cubic Refined

Relative Error 14,25% 7,81% 6,92% 4,67% 4,93% 4,03%

Absolute Error (Δ dB) 51,9 33,4 33,2 25,8 26,4 24,2

The internal modes, resonance and pressure field in silencer from 31Hz to 4kHz can see in Figure 3.

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Figure 3: SPL Surface and contour for half geometry model

3.3 Full geometry model According to the results shows that the refinement of the mesh discretized domain for half geometric model does not yield accurate respect to the values of measurements. For this reason, it is decided detailing best in performing a simulation model for a complete geometry. So, a geometrically refined model of silencer is used. The resulting TL values for full geometry model are in Figure 4, and a table shows data discretization for full model.


Figure 4: TL FEM full geometry model

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Figure 5 shows the internal sound pressure in full duct from 31Hz to 4kHz.


Figure 5: SPL Surface and contour for half geometry model According simulated for full geometry model results muffler best values are observed with respect to the experimental data, except for the linear model unrefined. The error calculation is observed in Table 2.

Table 2: Error for full geometry model

FEM

Linear Linear

Refined FEM

Quadratic Quadratic Refined

FEM Cubic Cubic Refined

Relative Error 13,69% 1,98% 2,42% 1,01% 1,44% 0,62%

Absolute Error (Δ dB) 39,0 6,9 8,9 4,6 5,7 3,7

4 Conclusions Although the half geometric model is symmetrical (Identical BC and sizes), the graphs of Figures 3 and 5 show that do not have a complete symmetry in the acoustical behavior in spectrum, which is cause of diffractions and reflections producing resonances (modes) dependent wavelength within the domain . The mathematic description of the absorption condition involves the Delaney-Bazley approach, which determines the impedance and vector wavenumber in the porous medium. So, in the case of this heuristic expression the model has an important consideration for refine that is

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linked to BC and/or subdomain loss. Then the coating being a fabric is essentially random surface and can be described as an anisotropic material. In detailing the model analyzing muffler complete shape the results are able to improve their correspondence with measured values and more expected in terms of accuracy than the half geometry model. This comparison is validated with errors resulting relative and absolute. The Figure 6 shows the general comparison, green curve are experimental values, in red color is the results for simulation of half geometric model, and finally the blue curve is for TL values obtained by FEM method for full geometry model and refined cubic lagrangian polynomials with better results than half shape duct.


Figure 6: General Comparison TL Given the results obtained is understood to be more efficient considering the full geometry of domain and refine the discretization. The most detailed and comprehensive model for the numerical solution allows a more accurate approximation of the acoustic phenomenon given the type of muffler and mixed characteristic, i.e., reactive - dissipative. With the above it is understood that to simulate the confluence of diffraction , refraction and sound reflection refinement is needed, describe best the boundary conditions especially impedance ( absorbing medium ) and numerically more of elements , degrees of freedom , that is , the more points calculation of sound pressure.

References [1] D. A. Bies and C. H. Hansen, “Acoustical properties of fibrous absorbent materials,” Appl. Acoust.,

Vol.14, 1980, pp. 357–391.

[2] M. Delany, E. Bazley. Acoustic properties of fibrous absorbent materials. Applied Acoustics, Vol.3, 1970, pp.105-116.

[3] ISO 11820:1996 Acoustics — Measurements on silencers in situ

[4] TAO Z. and SEYBERT A. F., A Review of Current Techniques for Measuring Muffler Transmission Loss. University of Kentucky. (2003).

[5] RIVAS Cristian., Pérdida de Carga en Silenciadores Pasivos Disipativos tipo Splitter. Universidad Austral de Chile. Tesis (2010).

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