PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Single versus multi-domain analysis in diffusion equation modelingfor sound field analysis of distinct room shapes
Zühre SÜ GÜL1; Erinç ODABAŞ2; Mehmet ÇALIŞKAN3
1

Bilkent University, Turkey

2,3

Mezzo Studyo Ltd, Turkey

ABSTRACT
Diffusion equation modelling (DEM) has gained popularity recently in the application of room acoustics
analysis mostly due to its computational efficiency. The method has been used in proportionate rooms
confidently, while some research has highlighted its applicability in disproportionate room typologies withcertain assumptions. In this study specifically, coupled rooms and long enclosures that cannot be very welldefined by statistical theory, or under-estimated by ray tracing are studied by DEM analysis. The diffusion
coefficient, dependent upon the volume and surface absorption area of individual domains, is one major
indicator in this analysis. Defining domains in coupled volumes or in multiple long spaces coupled each otheris an important problem. The limits where the domains can be considered as a single space or as amulti-domain system are quite vague and necessitate a systematic investigation. Some real structures
including multi-volume monuments and two-tracked subway stations are examined in that respect, byapplying DEM in a finite element medium. The results of this investigation are assistive in regards to thecoupling factor limits when different volumetric and aperture size relations are established. The
pre-knowledge of applying either a single domain or a multi-domain solution in a specific combination of
volumes will augment the speed of analysis.
Keywords: Diffusion equation model, room acoustics simulations, disproportionate rooms

1. INTRODUCTION
Sound energy decay analysis is essential in room acoustics predictions for predicting key
characteristics of an enclosure. Alternative modeling and analysis methods are developed initially toestimate acoustical indicators for proportionate rooms with diffuse sound field. However, in the caseof disproportionate room ratios or when multiple sub-volumes are coupled to each other, the interiorsound field is much more complex and statistical theory is not applicable. Such distinct room shapes
and configurations including coupled spaces and long enclosures have been investigated for several
decades and a number of theoretical formulae have been established. Among those diffusion equation
model has found to be a reliable (1-3) and practical method which is further discussed within thisstudy.
One advantage of diffusion equation model is that, meshing size takes its value from the mean freepath (MFP), rather than the wavelength, thus reducing the size of the model and thereby increasing thecomputational speed. While, diffusion coefficient takes into account the room morphology via its
mean free path (4). In that sense diffusion coefficient becomes an important variable within a DEM
solution. The critical issue is that when it comes to coupled volumes the dec ision on defining one or
multiple diffusion coefficients may affect the results, which depends on the strength of coupling in
between different spaces through apertures. In order to understand the conditions, when the coupledvolumes behave as distinct spaces or a single volume, and that multiple diffusion coefficients should
be defined, different cases are investigated in this research. Among those cases multi -domed
1

zuhre@bilkent.edu.tr
erinc@mezzostudyo.com
3
prof.mehmetcaliskan@gmail.com
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superstructures and a two-track subway station are discussed over their single and multi-domain DEM
solutions.

2. METHODOLOGYThis study applies one of the most recent methods in room acoustics that is the diffusion
approximation. Diffusion equation model (DEM) can be solved analytically for simple geometries,
and numerically for complex enclosures. DEM is based on the propagation of sound particles with thesame constant energy, propagating along straight lines and striking walls or scattering objects. In thissection, the governing interior and boundary equations applied to case structures are presented.
2.1 Interior Diffusion Equation
In a room region or domain (V) with time-dependent energy density (w) at a position (r) and as afunction of time (t) the sound energy flow vector ( J ) caused by the gradient of the sound energy
density can be expressed by Fick’s law;J  r , t    Dw  r , t (1)where D is the diffusion coefficient, which takes into account the room morphology via its mean freepath (  ) given by (4);
D

c
3



4Vc
3S

(2)

where V is the volume of the room, S is the total interior surface area of the room and c is speed ofsound. In the presence of an omni-directional sound source q(r,t) within a domain (V), the soundenergy density w changes per unit time as;w  r , t  D 2 w  r , t   q  r , t  ,  Vt(3)

Lastly, in a time dependent solution of DEM the energy flow levels (5) can be obtained by;12  w  r , t   2  w  r , t   2  w  r , t   2 
J L  r , t   10 log10  
 
 
 
  x   y   z  

(4)

2.2 Boundary Conditions
The effects of enclosing room surfaces can be analytically expressed by boundary equations
defined on the boundary surfaces (S). The boundary condition established to include energy exchanges
on enclosing surfaces is (4-6);

J  r , t   n   Dw  r , t   n  AX cw  r , t  , on S

(5)

where A x is an exchange coefficient, or the modified absorption factor, expressed as follows (34);AX 
4(1   / 2)(6)where α is the absorption coefficient of the specific surface or boundary. DEM with modified
absorption factor (6) is capable of modeling rooms with low absorption, as well as for mixed boundary
conditions associated with high absorption for specific room surfaces. This study utilizes Eq. ( 5) in Eq.
(6) and the resulting system boundary equation is as follows;D w(r , t )cw(r , t ), on S
n4(1   / 2)

(7)

For application of DEM in a multi-domain solution, there exists another boundary condition
applied for coupling apertures that is the continuous boundary condition (7). The aperture boundary
surface at intersection of two domains can be defined by the following;6021

nˆ   D1w  rb , t   D2w  rb , t    0(8)which represents a continuity boundary condition on interior boundaries at the aperture position,
where D1 is the diffusion coefﬁcient in the primary room, and D2 is the diffusion coefﬁcient for thesecondary room.

3. MATERIALS
3.1 Multi-domed Superstructures
Multi-domed monuments are interesting venues for acoustical studies as they have many subspacesthat are coupled to each other through arches, augmenting the potential for non-exponential energy
decay formation. Two sacred spaces, namely Süleymaniye Mosque and Hagia Sophia in İstanbul, areselected in that respect in search of application of proper diffusion coefficients within DEM. In orderto implement the DEM numerically in a finite element medium, initially the acoustical models of eachstructure are built. The effect of coupling of different sub-volumes are searched in single andmulti-domain solutions. Thus, first, results are obtained for the single domain, mea ning a single
diffusion coefficient, Eq. (2), assigned for the whole structure. Second, specific diffusion coefficientsin relation to their mean free paths (MFP) are defined for sub -volumes or sub-domains. Coupling
apertures of both structures are the arches that connect sub-spaces to each other, sheltered mostly withdomes or vaults of different sizes. Meshed models are fine-tuned with field test results taking thereverberation time into account (3,8).

Figure 1 - Süleymaniye Mosque mesh model (on the left); Hagia Sophia mesh model (on the right);
different domains indicated with individual colors
The geometric model of Süleymaniye Mosque has 164,468 linear Lagrange -type mesh elements,
while Hagia Sophia model has 691,865 linear Lagrange-type mesh elements (Fig. 1). Minimum meshsize is 0.62 m for Süleymaniye mosque and 0.39 m for Hagia Sophia. Maximum mesh size is 4.96 m
for Süleymaniye Mosque and 5.31 m for Hagia Sophia. As long as the maximum mesh size is smaller
than the MFP of the room, the DEM is applicable. In this case the range of mesh sizes are in between
1/4 to 1/11 of MFP for Süleymaniye Mosque and 1/4 to 1/14 of MFP for Hagia Sophia. Thus, maximum
mesh sizes of both models satisfy the MFP criteria for the DEM. The time-dependent simulation takes
approximately 13 mins on a computer with Intel(R) Xeon(R) E5-1650 CPU, @ 3,60 GHz processor for
Süleymaniye Mosque and 1h/4min for Hagia Sophia models. Table I lists the volume and the total
surface area of individual domains. Accordingly, the mean free paths (MFP) and diffusion coefficients(D) are calculated for different domains of case structures (3).

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Table I. Volume (V), surface area (S), mean free path (MFP) and diffusion coefficient (D) information for
single and multi-domain scenarios of Süleymaniye Mosque and Hagia Sophia models
Süleymaniye

V (m3)

S (m2)

Single Dom.

73,84818,293D055,525

D1

3.2

MFP

MFP

D

Hagia Sophia

V (m3)

S (m2)

16.2

1,846

Single Dom.

145,020

38,579

15.0

1,720

10,873

20.4

2,336

D0

95,96017,647

21.8

2,487

2,240

1,300

6.9

788

D1

2,575

1,241

8.3

949

D2

2,279

1,115

8.2

934

D2

625

468

5.3

611

D3

1,892

1,129

6.7

766

D3

4,430

2,158

8.2

939

D4

3,063

1,557

7.9

900

D4

6,771

3,434

7.9

902

D5

2,820

1,289

8.8

1,000

D5

2,395

1,728

5.6

635

D6

4,254

2,328

7.3

836

D7

2,499

1,190

8.4

960

D8

782

584

5.4

613

D9

3,625

1,938

7.5

855

(m)

(m)

D

Subway Stations

Subway stations are long enclosures with different acoustical properties from normal rooms, due to theirdisproportionate room ratios. The specific acoustic properties in long enclosures have been investigatedfor several decades and a number of theoretical formulae have been established (9-11). Sound
attenuation in real subways is even more complicated. In search of reliable methods of sound field
estimations in long and disproportionate spaces, this study applies DEM on two case subway stations
from Üsküdar-Ümraniye metro line in İstanbul.

Figure 2 – Station BAG mesh model (below); station USK mesh model (above); different domains
indicated with individual colors
Out of two stations, station BAG has a rectangular cross section, packs system, two tracks andconnecting corridors, and station USK has a circular cross section and two tracks with a connecting

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hall. For computation of DEM in a finite element medium the meshed models of both structures are
generated. The meshed model of station BAG has total of 282,701 linear, Lagrange -type meshelements with maximum mesh size of 4.1 m. While meshed model of s tation USK has total of 94,046
linear, Lagrange-type mesh elements with maximum mesh size of 4.9 m (Fig. 2). In this case themaximum mesh sizes of both BAG and USK are smaller than their estimated MFPs, so the DEM isapplicable (Table II).
Table II. Volume (V), surface area (S), mean free path (MFP) and diffusion coefficient (D) information for
single and multi-domain scenarios of (D1-D5) of station BAG and station USK
MFP
BAG

V (m3)

S (m2)

(m)

MFP
D

USK

SingleV (m3)

S (m2)(m)

D

SingleDomain12,988

11,143

4.7

533

Domain14,187

11,250

5.0

577

D1

3,575

2,995

4.8

545

D1

4,274

4,674

3.7

418

D2

2,084

2,810

3.0

340

D2

5,419

3,875

5.6

640

D3

368

393

3.8

429

D3

4,459

3,548

5.0

575

D4

504

521

3.9

442

D5

691

694

4.0

455

4. RESULTS
4.1 Sound Energy Flow Decays and Coupling Factor
In a previous study the occurrence of multi-slope decay formation in multi-domed superstructures
are presented (7). It is known that energy flow decays figuring energy flow dips are correlated withthe turning points in a non-exponential energy decay (5). The effects of architectural variables
including, coupling apertures, volume and geometry of domains can better be analyzed through energy
flow distribution in a DEM solution. In the modeling and computation phase, it is critical to correctly
define the individual domains with different diffusion coefficients for reliability of DEM solution.
In Fig. 3 single versus multi-domain DEM energy flow decays of Süleymaniye Mosque and Hagia
Sophia are presented for sample source-receiver configurations, where double-slope energy decays arepreviously detected through Bayesian decay parameter estimations (7). The convex forms of energy
flow decays indicate an energy return, either a full return or partial. Deeper the energy flow dip, theenergy flow pattern approaches to a complete return (3), indicating a higher difference between decay
levels and decay times of different slopes. A sample source-receiver position is S 1 R 4 for Hagia Sophia(Fig. 3), where the difference between energy flow dips in a single domain solution versus
multi-domain solution of DEM is greater.
In an earlier study Billon et al. (12) applied mean coupling factor (k) for quantifying the degree of
acoustical coupling. Mean coupling factor is defined as follows;



Sc2

 Sc  AR  Sc  AS 

(9)where S c is the coupling area, AR is the equivalent absorption area of the receiving room and AS is theabsorption area of the source room. According to that k ≈ 1 denotes a stro ng coupling, while k ≈ 0
indicates a weak coupling. A strong coupling means that the aperture sizes are big enough, so the twovolumes should not be considered as separate domains, but instead should be considered as a single
volume. Conversely, a weak coupling indicates that the apertures are small enough to consider the twovolumes as part of a multi-domain system.

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Figure 3 - DEM solutions for single (‘s’) versus multi-domain (‘m’) energy flow decay results; for
1 kHz, computed at different source (S) and receiver (R) configurations, Süleymaniye Mosque (on theleft), Hagia Sophia (on the right)
A weak coupling necessitates the assignment of individual diffusion coefficients to each volume asin the case of Süleymaniye Mosque and Hagia Sophia. Table III lists k values obtained by Eq. (9) for
specific source-receiver configurations tested in case structures. As can be observed in Table III,considering sub-volume to main volume coupling (Table I and Table II), the coupling factors are
smaller than 0.30 for Süleymaniye Mosque and much smaller than 0.10 for Hagia Sophia. Although thelimits of weak to strong coupling are not well defined, in a strong coupling indication by mean
coupling factor of 1, the values lower than 0.30 can securely be assumed to b e a weak coupling. Thus,
the individual volumes should be treated as individual domains with specific diffusion coefficients in
DEM simulation of both cases. There is still a considerable difference in the mean coupling factors
between Süleymaniye Mosque and Hagia Sophia. This is mainly due to the larger coupling apertures(arches) of Süleymaniye Mosque that divides main volume from smaller sub -volumes, in comparison
to those of Hagia Sophia. For that reason, the energy flow dips or flow returns of Süleyman iye Mosque,are not as sharp as those of Hagia Sophia (Fig.3).
Table III. Mean coupling factors (k) for specific source (S) receiver (R) configurations in relation to coupling
aperture area and absorption areas of individual domains (D) coupled to each other by arches for
Süleymaniye Mosque and Hagia Sophia
Süleymaniye (k) S1R4

S3R7

S4R5

S2R1

Hagia Sophia (k)

S1R4

S1R5

S2R1

S2R2

D0-D1

0,25

-

-

-

D0-D1

0,05

-

-

-

D0-D2

-

0,27

-

-

D0-D3

-

0,07

-

-

D0-D4

-

-

0,28

-

D1-D0

-

-

0,05

0,05

D2-D0

-

-

-

0,27

4.2 Decay Rate Estimates in Long Enclosures
In this section single and multi-domain DEM solutions of case subway stations are comparatively
evaluated over their T30 results for different receiver positions. The reason is basically forunderstanding if coupling of volumes, in form of tunnels to sub tunnels and tracks to halls havesignificant effect on the outcomes of different DEM solutions. The model of station BAG is tuned by
its field test results and known sound absorption coefficient data of applied materials within the station.
On the contrary to their geometrical features, interior finish materials are identical in both station BAG
and station USK that are ceramic tiles and painted concrete on walls and track tunnels, stone tile
flooring and mineral wool backed perforated metal ceilings (13).
Fig 8. compares field test, single and multi-domain DEM and ray-tracing results of T30 in station

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BAG for different receiver positions. For both field test results and simulated decay rates thereverberation increases as the distance from the source increase. According to that decay rates ofmulti-domain DEM solution are 0,20 s to 0,40 s higher than single domain DEM solution. On the other
hand, single domain DEM results are much closer to field test results when they are compared tomulti-domain DEM results and ray-tracing. This means the station BAG space does not act as acoupled-volume system and the apertures connecting different volumes (halls and two tunnels) to eachother are not small enough to cause non-exponential sound energy decay as previously observed indifferent structures. Ray-tracing over-estimates the decay rates and there are abrupt jumps inreverberation times on different receiver positions. In overall, single domain DEM results among
others, are much reliable and highly correlated with field test results.
Fig. 5 compares single and multi-domain DEM and ray-tracing results of T30 in station USK for
different receiver positions. Multi domain DEM decay rate results in overall parameters are 0.1 s to 0.2
s higher than single-domain DEM solution. This slight variation is even smaller than the deviations
observed in station BAG. This is due to the fact that USK station is even much less fragmented thanstation BAG (Fig. 2) and the station space together with its two track volumes and central platform
area behaves as a single volume. Thus, multi-domain investigation is proved not to be necessary.

Figure 4 - Comparison of field test, single and multi-domain DEM and ray-tracing results of T30 instation BAG for different receiver (R1 to R14) positions for 1 kHz

Figure 5 - Comparison of single and multi-domain DEM, and ray-tracing results of T30 in station
USK for different receiver (R1 to R12) positions for 1 kHz

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5. CONCLUSION
In this study diffusion equation model is discussed over rooms with particular room shapes including
coupled spaces and disproportionate long rooms. Main concern is the comparison of single-domain to
multi-domain DEM solutions in complex structures, for predicting the reliable modeling technique indifferent cases. Results of multi-domed superstructures highlight that the multi-domain DEM analysis
results are much indicative in multi-rate decay analysis in comparison to the single-domain solutions. In
single domain solutions, at some particular positions the energy dips are not observed, though the same
locations indicate an obvious dip over their multi-domain energy flow decays. There is always a difference
between single versus multi-domain DEM solutions. As the coupling factor gets weaker, for instance bysmaller apertures, the difference is much obvious. Thus, for both case structures with coupling factors
below 0,30 the reliable method is found to be the multi-domain solution.
The optimization of materials in acoustical design of subway stations starts with the decay rateestimations and in that sense diffusion equation model is a practical method. In case subway stations single
versus multi-domain DEM solutions are compared to field test and ray-tracing results of T30, in order tounderstand the effect of coupling and for defining the proper diffusion coefficients. Single-domain DEM
results are found much compatible to field test results considering decay rates. The results highlight that for
both case stations, the coupling of volumes in form of tunnels to inner corridors or central halls is not weakenough, meaning the aperture sizes are big and the equivalent absorption area is small. Thus, single domain
DEM solution is found to be suitable for the case subway stations.

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