Inverse scheme for sound source identification in a vehicle trailer

Jonathan Nowak 1

TU Wien Getreidemarkt 9, 1060 Vienna, Austria

Reinhard Wehr 2

AIT Giefinggasse 4, 1210 Vienna, Austria

Manfred Haider 3

AIT Giefinggasse 4, 1210 Vienna, Austria

Manfred Kaltenbacher 4

TU Graz In ff eldgasse 18, 8010 Graz, Austria

ABSTRACT Tire / road noise is a highly relevant topic for improving the comfort and experience of drivers and residents living in high-tra ffi c areas. With growing numbers of electric cars, the relevance of tire noise even increases since it is the dominant sound source in the middle-speed range. We want to further investigate the acoustic sources relevant to tire / road interactions. Therefore, we apply di ff erent sound source localization algorithms to measurement data acquired inside a large measurement trailer equipped with microphone arrays. The methods for sound source identification used are well-known beamforming-based algorithms and a new inverse scheme using finite element simulations. The latter scheme requires the identification of the acoustic properties of the trailer in the stationary case. In this contribution, we present the characterization process and results of the sound source localization in this stationary case.

1. INTRODUCTION

Norms like CPX (Close Proximity Method) or OBSI (On-Board Sound Intensity Method), both used for the acoustic characterization of road surfaces, require sound pressure or intensity level measurements. However, no sound source localization at the tire is performed.

1 jonathan.nowak@tuwien.ac.at

2 reinhard.wehr@ait.ac.at

3 manfred.haider@ait.ac.at

4 manfred.kaltenbacher@tugraz.at

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

At AIT, the Austrian Institute of Technology, a new trailer for CPX measurements was built. This trailer shall be used for di ff erent sound localization algorithms at the rolling tire. One of these methods is a new approach that uses microphone array measurements and finite element (FE) simulations. With this inverse method, the boundary conditions of the given measurement setup can be taken into account. Thus, determining the correct acoustic properties of the trailer is crucial. In this contribution, the vehicle trailer’s characterization process and the FE model’s validation are presented, and sound source localization results are shown.

2. MEASUREMENTS SETUP

In this section, the measurement setup of the characterization process is presented. Figure 1 shows the inside of one compartment of the trailer. The trailer has two mirror-symmetric compartments with one tire in each compartment. They are closed on all sides and the top but completely open at the bottom. Inside the trailer, 33 microphones are mounted in a 3D array, such that the microphones surround the sound source. As in this preliminary study, the trailer is stationary, a loudspeaker is used as an acoustic source instead of a rolling tire. This loudspeaker is exited with sinusoidal frequencies between 200 ≤ f / Hz ≤ 1600. The electric current through the loudspeaker due to the excitation signal is measured parallel to the acoustic signals. With previous laser-scanning vibrometer measurements, the amplitude and phase of the membrane can be determined via the measured electric current, which is later used as excitation in the FE model. On the inside of the trailer compartment, porous absorbers are mounted for sound insulation and to absorb the radiated sound.

Figure 1: Used CPX trailer.

3. SOUND SOURCE LOCALIZATION – INVERSE SCHEME

The Inverse Scheme is a new approach for sound source localization. It identifies the acoustic sources located in a predefined source region Ω sc , see Figure 2, via solving the inverse problem of reconstructing the source distribution in the source region by minimizing the di ff erence between the measured and simulated acoustic pressure at given microphone positions x i . Sound sources are modeled as complex-valued monopole source strengths e σ at each node of the source region. Since the inverse problem is ill-posed, a Tikhonov regularization is used. The optimization is gradient-based, and for its calculation, the adjoint method is used. Details on the mathematical formulation can be found in [1,2].

Figure 2: Computational domain of the Inverse Scheme.

4. FORWARD MODEL

In contrast to known methods for sound source localization that use microphone arrays and analytic approaches as a source model, one advantage of the Inverse Scheme is that the boundary conditions, i. e. partially reflecting acoustic absorbers or scatterers can fully be taken into account. This, of course, requires knowledge of the material properties of the acoustic absorbers and obstacles. In other words, a good FE forward model is required to expect good results via the Inverse Scheme. In this section, the forward model is presented, and a comparison between measurements and simulations is performed.

4.1. Governing equations The forward problem in its strong form is defined in the frequency domain as

∇· ∇ p a + k 2 p a = σ in Ω , (1)

also known as the Helmholtz equation. In Equation (1) p a = p a ( x , ω ) denotes the acoustic pressure in frequency domain, ω the angular frequency, k = ω/ c 0 the wave number, σ the sound sources and Ω the computational domain. Since we solve the forward problem via the FE method, we derive the weak form of Equation (1) by multiplying with an arbitrary (complex-valued) test function ϕ and integrating by parts which leads to Z

 ∇ p a · ∇ ϕ − k 2 p a ϕ  d x = − Z

σϕ d x + Z

∂ Ω ∇ p a ϕ · d s ∀ ϕ ∈ V . (2)

Here, σ is a sound source distribution in the volume. Further, V denotes the function space of the test functions ϕ , which is a real Hilbert space [1]. With the linearized conservation of momentum, which reads in the frequency domain as

j ωρ 0 v a = −∇ p a , (3)

where v a denotes the acoustic particle velocity, we can rewrite the surface integral in Equation (2), assuming constant density as Z

Z

∂ Ω ∇ p a ϕ · n d s = − j ωρ 0

∂ Ω v n ϕ d s . (4)

The interface condition between the mechanic velocity v m and the acoustic particle velocity at an interface is

( v a − v m ) · n = 0 . (5)

With this, the movement of the loudspeaker’s membrane at Ω m , which was measured with a laser scanning vibrometer, can be used as an excitation in the FE model.

4.2. Computational mesh As computational domain, both trailer compartments were used with some simplifications. Sound hard obstacles inside the trailer with characteristic dimensions ≤ 40 mm, e. g. thin bars for mounting of the microphones were neglected in order to reduce the complexity of the mesh. The largest frequency of the measurements is f max = 1600 Hz, therefore the corresponding minimal wavelength is λ min = c 0 / f min = 340 m / s

1600 Hz ≈ 210 mm, with the speed of sound c 0 . Larger sound hard scatterers like the mounting of the tire were taken into account. The computational domain is discretized using linear nodal finite elements.The mesh size h is set according to the rule of thumb h ≲ λ min / 10, see [3].

4.3. Boundary conditions The trailer is – as mentioned – open at the bottom, and the lower edges of its lateral walls are ca. 8 cm above floor level. To model the sound propagation to the outside, an air volume around the gap between the trailer and the floor was added. This air volume is terminated with a region of perfectly matched layers (PML) to model free field radiation outside of the trailer. The remaining boundaries (without the microphone’s membrane) ∂ Ω \ Ω m of the domain Ω are modeled as sound hard. Regions with acoustic absorbers are modeled as complex fluid [4], i. e. a region with complex valued material properties compression modulus e K and density e ρ . The frequency dependent values of e K and e ρ were computed with the Johnson-Champoux-Allard-Lafarge (JCAL) model [5], where the parameters of the JCAL model are fitted via a genetic optimization such that the computed complex valued reflection coe ffi cient e r matches the measured values. Therefore, the cost function reads as

J r = || e r meas − e r JCAL || 2 . (6)

The measurements e r meas were done with the two-microphone method in an impedance tube [6]. This measurement method assumes plane waves. In a second step, the real and complex values of the compression modulus e K and of the density e ρ are the input parameters of a second genetic optimization, where the cost function changes to

J p = || e p meas − e p sim || 2 . (7)

In Equation (7) e p meas = e p meas ( x , ω ) denotes the vector of the measured sound pressures at the given microphone positions x and frequencies f = ω/ (2 π ). The simulated sound pressures e p sim = e p sim ( x , ω, e K , e ρ ) are calculated with the FE method solving (1) with respect to the boundary conditions determined by the material properties. With this second optimization, e ff ects that are not present in the impedance tube, e. g. sound waves impinging at an arbitrary non-normal angle at the absorbers, uneven thickness of the absorbers, etc. can be implicitly taken into account.

4.4. Results of forward computation In the following, a comparison between the measurements p ms a and forward simulations is presented. Since we measure the electric current through the loudspeaker synchronously with the acoustic pressure, we can compute the membrane’s velocity in amplitude and phase. Therefore, we can simulate the acoustic pressure without any scaling. Figures 3 and 4 show good agreement between measurements and simulations.

(a) Amplitudes, f = 300 Hz. (b) Phases, f = 300 Hz.

(c) Amplitudes, f = 500 Hz. (d) Phases, f = 500 Hz.

(e) Amplitudes, f = 1100 Hz. (f) Phases, f = 1100 Hz.

Figure 3: Comparison of sound pressures of measurements and forward simulation results.

» quieras

a simlation

neasurement

rs 0 i quieras

neasurement

(a) Amplitudes, f = 300 Hz.

(b) Phases, f = 300 Hz.

rs 0 i quieras

neasurement

1 5 0 quieras

(c) Amplitudes, f = 500 Hz.

(d) Phases, f = 500 Hz.

te sinmtation lh

1 5 0 i 2% quieras

(e) Amplitudes, f = 1100 Hz.

(f) Phases, f = 1100 Hz.

A simulation ys m

Figure 4: Comparison of sound pressures of measurements and forward simulation results.

neasurement paar

5. SOUND LOCALIZATION

Since the forward model is now validated, the inverse problem described earlier can be solved. As source region Ω sc in the inverse computation the unity of the loudspeaker’s membrane Ω m , the region Ω p around the membrane ( plate ) and the sidewalls Ω s of the loudspeaker enclosure is used. Further, for the inverse simulation a coarser mesh is used for lower frequencies to speed up the computation, see Figure 5.

(a) Forward simulation: Ω m (grey), Ω p (red), Ω s (yellow).

(b) Inverse simulation: Ω sc (grey).

Figure 5: Regions of the mesh for forward and inverse computation.

5.1. Results To compare the performance of the Inverse Scheme to known algorithms, the same input data, namely sound pressures p ms a , microphone positions, and source points, were processed with the advanced algorithm Functional Beamforming (funcBF) [7] and the deconvolution algorithm CLEAN-SC [8], which are both based on conventional beamforming. In Figure 6 sound source localization results are exemplarily shown at frequencies f 300 , 500 , 600 , 1100 Hz. A source strength level is depicted, which is computed analogously to the sound pressure level via

L σ = 20 log σ

σ ref , (8)

with the source strength σ and a reference source strength σ ref = 20 µ N / m. The dynamic range is chosen 20 dB, values L σ < L σ, max − 20 dB are omitted. The main sound sources are expected to be identified on the loudspeaker’s membrane, depicted as a circle on the top surface in Figure 6. While the distribution of the sources is reconstructed well also at lower frequencies with the Inverse Scheme, the results with Functional Beamforming are not satisfying at the lower frequencies. While the deconvolution algorithm CLEAN-SC identifies the main sound source in the upper part of the loudspeaker enclosure, for frequencies f < 1000 Hz they are not identified on its top plane. Please note that identified sources at surfaces facing backwards in Figures 6i to 6l are depicted circled and semitransparent. At higher frequencies f ≥ 1000 Hz both the Inverse Scheme and the beamforming based algorithms achieve good results. At those higher frequencies, the faster computation time of beamforming outweighs the benefits of the Inverse Scheme. This e ff ect is even amplified, as with increasing frequency the mesh size h has to decrease in order to fulfill h ≲ λ min / 10, which results in increasing computing time.

(a) f = 300 Hz .

(b) f = 500 Hz .

(c) f = 600 Hz .

(d) f = 1100 Hz .

(e) f = 300 Hz .

(f) f = 500 Hz .

(g) f = 600 Hz .

(h) f = 1100 Hz .

(i) f = 300 Hz .

(j) f = 500 Hz .

(k) f = 600 Hz .

(l) f = 1100 Hz .

Figure 6: Results of di ff erent sound source localization algorithms, circled sources are located at surfaces facing backwards.

6. OUTLOOK

As the sound localization results for a loudspeaker as a sound source inside the stationary trailer are promising, the following steps will be the application of the Inverse Scheme and other sound source localization algorithms to the rolling tire in the moving trailer.

REFERENCES

[1] Manfred Kaltenbacher, Barbara Kaltenbacher, and Stefan Gombots. Inverse scheme for acoustic source localization using microphone measurements and finite element simulations. Acta Acustica , 104:647–656, 2018. [2] Stefan Gombots. Acoustic source localization at low frequencies using microphone arrays . PhD thesis, TU Wien, Austria, 2020. [3] M. Kaltenbacher. Numerical Simulation of Mechatronic Sensors and Actuators: Finite Elements for Computational Multiphysics . Springer, 3 edition, 2015. [4] M.A. Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. i. low frequency range. Journal of the Acoustical Society of America , 28:168–178, 1956. [5] Jean Allard and Noureddine Atalla. Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials . John Wiley & Sons, 2009. [6] ASTM E2611-09. Standard Test Method for Measurement of Normal Incidence Sound Transmission of Acoustical Materials Based on the Transfer Matrix Method . West Conshohocken, PA, 2009. [7] Robert Dougherty. Functional beamforming for aeroacoustic source distributions. 06 2014. [8] Pieter Sijtsma. Clean based on spatial source coherence. International Journal of Aeroacoustics , 6:357–374, 05 2007.