Sensitivity analysis for hybrid room acoustic simulation regarding spatial dataof receiverPhilipp SCHÄFER, Michael VORLÄNDER
Institute of Technical Acoustics, RWTH Aachen, Germany, philipp.schaefer@akustik.rwth-aachen.de

Abstract
To overcome the limitations of methods using geometrical acoustics, Ray Tracing and Image Source algorithms
can be combined with well-known numerical approaches - such as Finite Element Method (FEM) - which
consider wave-based effects and therefore produce more sophisticated results in the lower frequency range.
For this purpose, a room is modeled for both methods and then simulated separately. In a second step, theoverall result can be received by uniting the distinct results in the frequency domain. A challenging aspect ofroom acoustic simulations is the acquisition of input data such as the receiver positions. The question arises
how accurate this data has to be measured for the purpose of room acoustic simulations. Thus in the present
work, a sensitivity analysis is conducted to determine the inﬂuence of deviations in the receiver position on thesimulation result.
Keywords: Room Acoustics, Simulation, FEM, Ray Tracing, Image Sources

1

INTRODUCTION

The question of how the sound ﬁeld in a room changes over space has been tackled in prior studies. Usually,
the spatial ﬂuctuation is investigated by comparing change of room acoustic single number quantities with the
distance between two receiver points. Studies, such as by Witew et al. [6], are based on simulation methods using approaches from geometrical acoustics. Thus, in the present work a hybrid approach combining the strengthof geometric and wave-based simulations is made. This paper focuses on the low to medium frequency range
where wave-based effects like room modes are considered. For this purpose, investigations of the spatial deviation of reverberation time T30 in a reverberation chamber are carried out. In this context, it is also investigatedwhether these deviations have a directional component.

2

HYBRID WAVE-BASED AND GEOMETRICAL ACOUSTIC SIMULATION TOOL

When combining wave-based and geometric acoustic (GA) methods for room acoustic simulations, two models
have to be designed. The models should be equivalent in their geometric and acoustic properties. Furthermore,
the simulation settings have to be adjusted to the considered problem. Afterwards, both simulations can be
carried out independently. Both results are then combined using a cross-fade ﬁlter in the frequency domain [3].
Since the wave-based result is usually band-limited, the crossover frequency should be in an adequate distance
below the upper cut-off frequency. The overall procedure of this simulation approach is illustrated in ﬁgure 1.
More details on the distinct simulation methods are given in the following sections.
2.1 FE simulation via COMSOLFor the calculation of the low frequency range room transfer function, the well-known ﬁnite element method(FEM) is utilized. This method allows to numerically solve the Helmholtz wave equation giving a volume meshof the considered room. As additional input, the boundary and source conditions have to be speciﬁed using
impedances and velocities. This method allows to evaluate the sound pressure at each point within the room
using a single simulation. In this work, all simulations are carried out using COMSOL Multiphysics 5.4 [1]including the Acoustics Module.

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Hybrid room
acoustic simulationgeometric
properties

S
R

Wave-based
methodS

acoustic
properties

R

simulation
settingsGA
methodband-limited
RTF
f < fmaxcrossover filter
RIR
RTF

Figure 1. Block diagram of hybrid wave-based and geometrical acoustic simulation.
2.2 RAVEN
The geometrical acoustic simulation tool RAVEN [5] uses a hybrid approach of image sources and stochastic
ray tracing. Giving an simple surface model of the considered room, it allows to predict the room impulse
response (RIR) of speciﬁed source-receiver pairs. The boundary conditions are modeled using the frequencydependent absorption and scattering coefﬁcients α and s. Although the sources are considered to be points,
arbitrary sources can be modeled using a directivity and free ﬁeld sensitivity. On the other hand, the receiver ismodeled as point for the image source method but as sphere for the ray tracing method.

3

SIMULATION SCENARIOSThe scenarios considered in this research take place in the reverberation chamber at ITA of RWTH Aachen.
They are based on a former investigation of Pelzer et al. [4]. Of the three suggested boundary condition
scenarios, only the ﬁrst two are considered here - the empty reverberation chamber and a scenario with mineralwool at two of the walls (see [4], section 3, scenario a) and b)). Both scenarios are shown in ﬁgure 2. A single
source is used to excite the room. Since only mono RIRs are considered in this work, all receivers are modeled
as omnidirectional. Thus, no further geometries (such as a dummy head) are added to the FE room model. In
the following, the two considered scenarios are referred to as• Empty room
• Absorber wall
3.1 Boundary conditions
The boundary conditions are the only parameters that are varied between the simulation scenarios. There are
two materials considered here: The reverberant wall of the room and a mineral wool that acts as absorber. Their
acoustic parameters - speciﬁc impedance, absorption coefﬁcient and scattering coefﬁcient - are shown in ﬁgure
3. The data is taken from [4]. While in scenario "Empty room" only the ﬁrst material is used, in scenario"Absorber wall", the mineral wool is applied to two walls (see ﬁg. 2).

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(-8.11, 5.574, 2.688)m(0, 0, 2.688)m(0, 0, 0)m(-8.591, 0, 0)m

Figure 2. Room model of the reverberation chamber at ITA of RWTH Aachen. The source position is marked
as blue, the center position of the receiver grid as red circles. For the scenario "Absorber wall", the mineralwool is applied to the two walls highlighted in light blue. The room has a rectangular shape except for the left
wall which is placed in an angle greater than 90◦ and two beams at the ceiling.
3.2 Source model
A single source is used to excite the room. Although compared to [4] the source position is unchanged at(−6.902, 1.714, 1.543) m, it is modeled as perfect monopole with a ﬂat frequency response for both simulation
methods. Thus, no directivity is applied to the GA source model.
3.3 Receiver positions
In context to the research question, multiple receivers are placed in form of a grid for the simulation. The gridhas a resolution of 1 cm in x-, y- and z-direction respectively and is centered at (−1.817, 3.591, 1.696) m. It
spans from −5 cm to +5 cm in each direction around the center leading to a total number of receivers of
Ngrid = 113 = 1331.

(1)

3.4 Simulation settings
The simulation settings are summarized in table 1. It is important to mention that the crossover frequency
roughly corresponds to the Schroeder frequency of the considered room. Furthermore, a highpass ﬁlter (butterworth, order 8) is applied to the FE results to remove potential zero order modes.
Table 1. Simulation settings
COMSOL
Simulation frequencies
Highpass ﬁlter
Mesh frequency(5 elements per wavelength)
Combination ﬁlter
Crossover frequency
Filter type
Filter order5984

0.1 Hz : 0.1 Hz : 400 Hz
fcutoff = 10 Hz
340 Hz340 HzButterworth
16

RAVEN
Image source orderRadius of receiver sphere
Number of rays
(per frequency band)Ambient settings
Static pressure
Temperature
Relative humidity

101000 Pa
291.35 K (18.2◦ C)
43 %

4
0.2 m
100,000

80
Reverberant wall
Mineral wool
60

400

40

200

20

0
20

40

60

100

200

1

0
1k

400

45

[°]

Reverberant wall Mineral wool Both materials - s

0.8

/s

| |

600

Coefficients

800

0

0.6
0.4
0.2

-45
-90
20

40

60

100

200

400

0
20

1k

40 60 100

200

400

f [Hz]

1k

2k

4k 6k 10k

20k

f [Hz](a) Speciﬁc impedance(b) Absorption and scattering coefﬁcient

Figure 3. Acoustic boundary conditions of the two utilized materials according to [4]. Note that the speciﬁc
impedance magnitude of the mineral wool is plotted using a scale of factor 10 (right y-axis). Furthermore,
the speciﬁc impedance for the reverberant wall is real-valued and calculated from the corresponding absorption
coefﬁcient. The scattering coefﬁcient is assumed to be 5 % for both materials at all frequencies.

4

RESULTSFor each scenario and each receiver position, a combined broadband room impulse response is simulated. Exemplarily, the room transfer functions at the grid’s center position are shown in ﬁgure 4. To compare the results
60

Scenario: Empty room
Scenario: Absorber wall

50
40

Modulus in dB

30
20
10
0
-10
-20
-30
-40
20

40

60

100

200

400

1k

2k

4k

6k

10k

20k

Frequency in Hz

Figure 4. Room transfer functions for both scenarios evaluated at the center position of the receiver grid.
between different receiver positions, single value room acoustic parameters can be utilized [6]. The reverberation time is one of the most frequently used room acoustic parameters. Especially, in the context of absorption
coefﬁcient measurements in reverberation chambers it is of special interest. Thus, the reverberation time T30 is
calculated for each RIR according to [2] for third-octave bands. No noise compensation is used, since the noise
ﬂoor of the simulation results is expected to be very low (numerical noise only).
4.1 Statistical spreading over receiver grid
In order to get an impression of how the reverberation time changes over the receiver grid volume, the relative
deviation is calculated for each point. As a reference, the mean value over all grid points r i
Ngrid

Tmean =∑i

T (rr i )

Ngrid

(2)

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is used. Then, the relative deviation at observation point r becomesηT (rr ) =|(T (rr ) − Tmean )|
.
Tmean

(3)

Relative deviation T30 [%]The results are shown in the boxplots in ﬁgure 5. Note that the minimum frequency is limited to 80 Hz due

15

15

10

10

5

5

0

0

-5

-5

-10

-10

-15

-15
80

125 200 315 500 800 1.25k 2k 3.15k 5k

f [Hz](a) Scenario: Empty room

8k 12.5k 20k80

125 200 315 500 800 1.25k 2k 3.15k 5k

8k 12.5k 20k

f [Hz](b) Scenario: Absorber wall

Figure 5. Boxplots of relative deviation of reverberation time over the receiver grid for both scenarios and thirdoctave bands. The maximum whisker size is set to four times the interquartile range.
to the maximum length of the RIRs (tmax = Δ1f = 10 s) and the high reverberation time of the room at low
frequencies (T  10 s). As expected, ηT is very small at frequencies below 125 Hz where the wavelength ismuch larger than the receiver grid. Furthermore, it can be observed that the statistical spreading is higher forthe scenario "Absorber wall". For both scenarios, ηT has the highest variance in the lower frequency bands
around approximately 200 Hz where the result is based on the FE simulation. Above the crossfade frequencyof 340 Hz the deviation is much smaller. Only in scenario "Empty room" for f ≥ 10 kHz an increase of ηT isobserved again which is most likely caused by the effect of air absorption. A visual inspection of the EDCs of
the lower frequency bands shows that the energy decay is not strictly exponential but shows ripples (not shownfor brevity). Thus, the observed deviations might be caused by a non smooth decay process. Also for scenario"Absorber wall", there is a mismatch in the variance at the crossover between the FE and GA results. Thissuggests that in this frequency range, wave-based effects are not negligible and therefore the GA approach is
not sufﬁcient. In the following section, further investigations are conducted for the FE frequency bands.
4.2 Distance between receiver points
In order to set the deviation of the reverberation time in relation to the distance between observation points,
additional parameters are introduced. Assuming two receiver positions r n and r m and the vector(4)Δrr = r n −rr m = Δrx Δry Δrz
connecting them, then the spatial three-dimensional distance between both receiver points can be expressedtaking the absolute value of each component:⎞⎛ ⎞ ⎛|Δrx |dx(5)
d = ⎝dy ⎠ = ⎝|Δry |⎠dz|Δrz |

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On the other hand, the absolute inter-receiver distance is calculated by applying the l 2 -norm:d = dd  = Δrr  = Δrx2 + Δry2 + Δrz2

(6)Regarding the simulated reverberation times, the absolute deviation between these two observation points is
ΔT (rr n ,rr m ) = |T (rr n ) − T (rr m )| .

(7)

The relative deviation is derived normalizing eq. 7 to the mean reverberation time of the receiver grid (eq. 2):
ηT (rr n ,rr m ) =ΔT (rr n ,rr m )
.
Tmean

(8)

For the investigation in the following two sections, eq. 8, 5 and 6 are evaluated for all receiver pairs along thereceiver grid resulting in
Ngrid − 1
= 885115(9)
Ndata = Ngrid ·2
data pairs of d and ηT or d and ηT respectively. It is important to mention, that Ndata exceeds the number ofpossible inter-receiver distances. Thus, values of ηT referring to the same distance d or d are summarized usinga respective mean operation:(10)
ηT (dn ) = ηT (di = dn )ηT (dd n ) = ηT (dd i = d n )(11)4.3 Evaluation of absolute distanceIn this section, the inﬂuence of the absolute distance between two observation points on the reverberation time isinvestigated. For this purpose, the relative deviation of the reverberation time is evaluated according to section
4.2 and processed according to eq. 10. Note that at high distances (d > 0.14), the number √of data samples
used for the average is very small (e.g. there are only four receiver pairs with d = dmax = 3 · 0.1 m). The
results for both scenarios are shown in ﬁgure 6. As expected, ηT grows with increasing inter-receiver distance.
18

3.5
f = 125Hz
f = 160Hz
f = 200Hz
f = 250Hz
f = 315Hz

[%]

14

2

1.5

1

T30

2.5

12

Relative deviation

Relative deviation T30 [%]3

f = 125Hz
f = 160Hz
f = 200Hz
f = 250Hz
f = 315Hz

16

10
8
6
4

0.5

0

2

0

0.02

0.04

0.06

0.08

0.1

0.12

Inter-receiver distance d [m](a) Scenario: Empty room

0.14

0.16

0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Inter-receiver distance d [m](b) Scenario: Absorber wall

Figure 6. For both scenarios, the relative deviation in the reverberation time ηT is shown as function of theabsolute inter-receiver distance d. Values that refer to the same distance are summarized using eq. 10.

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Furthermore, there are large differences between the considered frequencies. This is likely to be caused bymodal effects. For most frequencies, the scenario "Absorber wall" has a higher deviation in reverberation timethan the scenario "Empty room". At 250 Hz and 315 Hz, it is even above 15 %. Due to the absorber material on
two of the walls, the sound ﬁeld is less diffuse. Thus, a higher ﬂuctuation of the reverberation time over spaceis expected. ηT might also be increased since the receiver grid is placed very close to the absorber walls.
4.4 Evaluation of 3D-distance
Now, an investigation similar to the previous section is done regarding the three-dimensional distance d . Therefore, the reverberation time data ηT (dd ) is processed according to eq. 11. The results are shown in multiple(a) Scenario: Empty room@ 200 Hz(b) Scenario: Absorber wall@ 200 Hz(c) Scenario: Absorber wall@ 250 Hz(d) Scenario: Absorber wall@ 315 Hz

Figure 7. Scatter plot of the relative deviation of reverberation time ηT as function of the inter-receiver distancevector dx , dy and dz . Values that refer to the same distance vector are summarized using eq. 11.
scatter plots in ﬁgure 7. For the considered frequencies, it can be seen that ηT not only depends on the interreceiver distance, but also on its directional components dx , dy and dz . The differences between the directions

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can be large. For example at 250 Hz in the "Absorber wall" scenario, the deviation at dx = 0.1 m can be up to
16 % whereas at dx = 0 m it is below 1 %. In this case, dx seems to be the "dominant" direction for the changeof the reverberation time. Comparing the three results for scenario "Absorber wall", this direction also seems
to change with frequency. Furthermore, it seems to be inﬂuenced by the boundary conditions as can be seen
comparing the results at 200 Hz. Additional investigations are necessary to check whether this phenomenon canbe reproduced in further simulation scenarios.

5

CONCLUSIONSIn the course of this work, it could be shown that for the considered scenarios the reverberation time T30
calculated from hybrid room acoustic simulations signiﬁcantly deviates over a small receiver grid. The highﬂuctuation can be observed in the frequency bands just below the Schroeder frequency where the results are
based on the FE tool.
In the scenario with absorber around the crossfade frequency of 340 Hz, a mismatch in the variance of the relative deviation of T30 between the FE and GA results could be observed. Here, the approximation of geometricacoustics might not be valid due to the strong difference of the absorption coefﬁcients of the walls. Thus, theFE simulation should be extended to higher in future studies in order to validate this.
It could be shown that in average, the relative deviation of T30 increases with the distance between two observation points. Also for the frequency bands around 200 Hz, this spatial deviation seems to be directional.
Depending on the considered frequency and the boundary conditions, the deviation towards one direction (x, yor z) is greater than towards the others. As expected, this phenomenon is more prominent in the scenario withabsorber.
The ﬁndings of this study suggest that - depending on the considered frequency range, the room geometry
and boundary conditions - the FE results of a hybrid room acoustic simulation might vary signiﬁcantly evenfor small changes in the receiver position due to modal effects. Additional investigations have to be done toevaluate this. In this context, a variation of the receiver grid position and size is of interest. Most importantly,
the spatial deviation of additional room acoustic parameters should be investigated.

REFERENCES
R 5.4, COMSOL AB, Stockholm, Sweden. www.comsol.com.
[1] COMSOL Multiphysics

[2] ISO EN DIN 3382-1, Measurement of room acoustic parameters - Part 1: Performance spaces. International
Organization for Standardization, Brussels, Belgium, 2009.
[3] M. Aretz. Combined wave and ray based room acoustic simulations of small rooms. Doctoral dissertation,
RWTH Aachen University, Germany, 2012.
[4] S. Pelzer, M. Aretz, and M. Vorländer. Quality assessment of room acoustic simulation tools by comparing
binaural measurements and simulations in an optimized test scenario. Proceedings of Forum Acusticum, 2011.
[5] D. Schröder. Physically based real-time auralization of interactive virtual environments. Doctoral dissertation,
RWTH Aachen University, Germany, 2011.
[6] I. Witew, P. Dietrich, S. Pelzer, and M. Vorländer. Comparison of different modelling strategies to predict
the spatial ﬂuctuation of room acoustic single number quantities. In 40th Italian (AIA) Annual Conference on
Acoustics and the 39th German Annual Conference on Acoustics (DAGA), 2013.

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