Pyramid Tracing vs. Ray Tracing For The Simulation of Sound Propagation in Large Rooms
Pyramid Tracing vs. Ray Tracing For The Simulation of Sound Propagation in Large Rooms
Pyramid Tracing vs. Ray Tracing For The Simulation of Sound Propagation in Large Rooms
Abstract
The aim of this paper is to introduce a new computational model (RAMSETE)
for the simulation of sound propagation in large rooms; the model can easily be
adapted to work outdoor, and can consider diffraction effects around screen
edges and sound paths passing through (light) panels.
However, this paper focuses on room acoustics, and particularly on rooms
with non-Sabinian behaviour. In fact, the Pyramid Tracing algorithm does not
involve an hybrid computation scheme, with a reverberant tail superposed to
the deterministic early reflections estimate, as it is common with other
diverging beam tracers (cone tracers, gaussian beam tracers, etc.). This make it
possible to study also sound fields characterised by double-slope sound decays,
inside spaces with not comparable dimensions and inhomogeneous sound
absorption.
It is well known that the same capabilities were already present in the
(original) Ray Tracing scheme, but requiring much longer computation time. In
fact, a correct Ray Tracing implementation can be considered as the reference
standard for any (faster) numerical code based on the Geometrical Acoustics
assumptions.
After a brief introduction to some important details of the two algorithms,
the results obtained in three cases are presented. The first is a typical Sabinian
room (a reverberating chamber), the second is the coupling of two rooms with
different average absorption (a theatre with its stage), the third is a typical
industrial building (having an height very little compared to other dimensions)
with non-uniform sound absorption (baffles under the ceiling).
The results show how the Pyramid Tracing can give results very similar to
the original Ray Tracing, provided that a proper adjustment of the parameters is
performed. On the other hand, the magnitude of the errors that can be done
with improper parameter settings is delimited and discussed.
1. Introduction to the two algorithms
Before we can present the results of the comparison tests, it is better to explain
briefly the working principles of the two codes used here. Both of them run on
a standard PC, under MS-Windows, and share the same input and output file
formats, so the comparison is easy.
All the files are plain-ASCII, with auxiliary strings embedded to make easy
to understand the meaning of each row of data. The input data file is produced
by a dedicated 3D CAD program, and the output files are processed through a
set of graphical utilities capable of reconstructing, from the “raw” impulse
response data, the usual descriptors used in room acoustics: Levels, Early-to-
Late ratios, Lateral Efficiency, Center Time, STI, etc. . The only difference
during the post processing is that the impulse responses produced by the
pyramid tracing must be corrected prior of calculating such parameters, as
explained in another paper (Farina [1]).
S
L
The contribution W’ to the total energy density W that each ray leaves
inside the receiving sphere is proportional to the length of the intersection L
and to the initial energy reduced for multiple absorption on the boundary
surfaces (with absorption coefficients αi) and for the air absorption (with
coefficient γ multiplied for the path length x):
W '=
Pwr ⋅ Q ϑ
N rays ⋅ c ⋅ Vsphere
⋅ L⋅ [∏ i (1− α i )] ⋅ e− γ ⋅x (1)
This formulation avoids the common inconsistencies present in other
detections schemes (as surface intensity over the sphere surface or over a
circular disk), that are not physically compatible both with free field and
reverberant spaces.
Another remarkable point is the ray generation at the source. Although the
Ray Tracing scheme requires a random generation, it must be ensured that the
generation is almost uniform on the surface of a spherical source (the source
directivity Qθ is managed along with the energy assigned to each ray, as shown
in eqn 1). The simple assumption of three random generators for the three
versor components of the ray is not correct, as this produce a “cube of rays”
instead of a sphere; it is possible to “cut away” the corners of the cube
(discarding each vector having modulus greater than 1), but it was preferred to
employ a semi-probabilistic generator, in which the sphere surface is
mathematically divided in a large number of equal areas (actually 400=20x20),
each of them being “brushed” with the random generators.
This task was accomplished employing just two random generators (RND1
and RND2), and projecting their values over the sphere to obtain the versor
components of the ray:
2
i + RND1 i + RND1 j + RND2
vx = 2⋅ − ⋅ cos 2 ⋅ π ⋅
20 20 20
2 (2)
i + RND1 i + RND1 j + RND2
v y = 2⋅ − ⋅ sin 2 ⋅ π ⋅
20 20 20
i + RND1
v z = 1− 2⋅ i = 0..19 j = 0..19
20
In this case the absorption coefficients are the same everywhere, so the
acoustic field is surely Sabinian, and just one receiver need to be considered.
The comparison is made plotting on the same graph the Backward
Integrated Impulse Response in dB for the octave band of 1 kHz, computed
with the Ray Tracing (128000 Rays) and with the Pyramid Tracing (the latter
with various number of pyramids). In figure 5 the comparison is made twice:
on the left the Ramsete’s responses are reported without tail correction, on the
right the same are corrected with the theoretical values of α=2 and β=0.3. It can
be shown that these values make the Pyramid Tracing nearly coincident with
the Ray Tracing, even for a very little number of pyramids.
100 100
90 90
80 80
Sound Level (dB)
70 128000 Rays 70
1024 Pyramids
60 60
256 Pyramids
50 50
64 Pyramids
40 40
30 30
20 20
0 0.5 1 1.5 2 0 0.5 1 1.5 2
Time (s) Time (s)
The accuracy of the results can be checked comparing the numerical values
of the reverberation time T30 with that obtained by the Ray Tracing (2.768 s):
Table 1 - Values of T30 computed with Ramsete
Number of Pyramids T30 w/out correction T30 with correction
1024 2.368 2.614
256 2.136 2.828
64 1.784 2.691
2.2 Coupled Volumes with different absorption
In figure 6 both the geometry and the results are reported for this case: it is the
Theatre Buero Vallejo recently built in Spain, at Alcorcon (near Madrid), with
architectural project of Isicio Ruiz. The simulation is representing the hall
completely furnished, while the stage is empty (and reverberating) at all.
The graph in fig. 6 shows the Impulse Response (not integrated) in the
octave band of 1 kHz obtained in receiver # 19 with the Ray Tracing program
and with Ramsete at various number of pyramids, the latter being corrected
with α=5.78 and β=0.0153. The double slope of the decay is quite evident.
80
60
50
40
0 0.5 1 1.5 2
Time (s)
64 Pyramids 256 Pyramids 1024 Pyramids
4096 pyramids 100000 Rays
In this case the results show very large discrepancies with 64 Pyramids,
and also with 256 pyramids the results are quite poor. Nevertheless, increasing
the number of pyramids to 1024 or more, the responses become practically
indistinguishable from the Ray Tracing results, while the computation times are
still reasonable (7min+43s for 1024 pyramids on a 486 DX-2 66 MHz) .
90
85
80
0 1 2 3 4
Number of distance doublings
Ray Tracing: 3.55 dB/doubling Ramsete1: 3.65 dB/doubling
Free Field: 6.0 dB/doubling Ramsete2: 2.91 dB/doubling
3. Conclusions
The pyramid tracing algorithm has the main advantage of being very fast, but
the tail correction required is quite delicate. As it was shown here, a proper
adjustment of the post-processing parameters α and β is required to obtain
results comparable with a “reference” (and very slow) Ray Tracing program.
The values of the parameters that produce good results can be obtained
with the simple rule used for the above cases: α and β were chosen as the
values that minimise the sum of squared differences between the results
obtained with two different pyramid generations (i.e. 256 and 4096 pyramids).
An automatic adjusting utility is actually being implemented to make this
“self-calibration” easy for everyone.
References
1. Farina, A. RAMSETE - a new Pyramid Tracer for medium and large scale
acoustic problems, Proc. of Euro-Noise 95, Lyon, France 21-23 march 1995.
2. Farina A., Cocchi A., Garai M., Semprini G., Old churches as concert halls: a
non-sabinian approach to optimum design of acoustic correction, F5-7, Proc. of
14th ICA, Beijing , China, 1992.
3. Farina, A. & Maffei, L. Sound Propagation Outdoor: Comparison between
Numerical Previsions and Experimental Results, in COMACO95, Proc. of Int.
Conf. on Comput. Acoustics and its Environmental Applications, Southampton,
England, 1995, Computational Mechanics Publications, Southampton 1995.
3. Verbandt F.J.R., Jonckheere R.E., Bench-mark of acoustical ray-tracing
computer programs, Proc. of INTERNOISE 93, Leuven, Belgium, 1993.
4. Lewers T. A combined Beam Tracing and Radiant Exchange computer model of
Room Acoustics, Applied Acoustics, 1993,Vol. 38, no.s 2-4, pag. 161-176.