Room acoustic simulations using the spectral element methodF. Pind, A.P. Engsig-Karup, C.-H. Jeong, M.S. Mejling & J. Strømann-Andersen

Introduction & motivationThis poster describes the work being carried out in a cross-disciplinary research project, where the aim is to develop a newway of simulating sound in spaces (“room acoustics”), withunprecedented accuracy and efficiency. The areas of applica-tion are many-fold, including architecture, virtual reality, com-puter game audio, music, hearing research and entertainment.The team consists of researchers from the Acoustic TechnologyGroup and the Scientific Computing Section at DTU and fromthe Danish architectural firm Henning Larsen.

Historically, room acoustic simulations have been carried outby means of so-called “geometrical acoustics” (GA) methods,such as ray-tracing. In GA, several simplifying approximationsregarding sound propagation are made, which make the compu-tational task more manageable, but at the cost of significantlyreduced accuracy. It is desirable to use numerical methods in-stead of GA methods, due to their inherent accuracy. However,numerical simulation of room acoustics is a particularly chal-lenging problem, due to the large 3D domains, complex geome-tries and a wide frequency range of interest. Thus, there is a needfor a highly efficient numerical method that simultaneously canhandle complex geometries.

Geometrical acoustics Numerical acoustics

Here, a numerical scheme based on a high-order spatial dis-cretization using a spectral element method (SEM), coupledwith an explicit high-order Runge-Kutta temporal integrationmethod, is presented. The scheme supports the use of adaptiveunstructured meshes, high-order polynomial basis functions andcurvilinear mesh elements, for accurate representation of acous-tic wave propagation in complex geometries. Various numericalexperiments reveal the accuracy and efficiency of the proposednumerical scheme and highlight the need for high-order methodswith high geometric flexibility for accurate and cost-effectivesimulations of room acoustics.

Numerical discretizationThe sound field in a room can be described by the linearizedEuler equations with zero mean flow

vt = −1

ρ∇p,

in Ω,pt = −ρc2∇ · v,

(1)

where p is the acoustic pressure, v is the particle velocity, ρ isthe density of the medium and c is the speed of sound. This canbe discretized in the standard spectral element method approach,yielding the following semi-discrete system

Mu′ = −1

ρSxp, Mv′ = −1

ρSyp,

Mw′ = −1

ρSzp, Mp′ = ρc2

(STx u + STy v + STz w

),

(2)

where u, v, w represent the x, y, z components of the particle ve-locity, M is the global mass matrix and S are the global stiffnessmatrices and. These matrix operators are constructed through

the assembly of local nodal element basis functions, defined onstandard elements and transformed to represent arbitrary ele-ments on the mesh. In the derivation of the local element ba-sis functions, the orthonormality of modal basis functions is ex-ploited to avoid the use of numerical quadrature rules.

Properties of the numerical scheme

ConvergenceA convergence test is presented using a periodic 3D domain.The figure reveals the favorable numerical error properties whenusing high-order polynomial basis functions.

100

101

102

h−1

10-8

10-6

10-4

10-2

100

ǫ

P=1P=2P=3P=4P=5P=6O(h2)O(h4)O(h6)

Dispersion analysisThe objective of the analysis is to understand the relationshipbetween the exact physical wave speed c and the discrete wavespeed cd, which will vary with frequency, causing dispersion er-rors. A general, multi-modal method for analyzing these proper-ties has been developed in this project. This method takes boththe spatial and the temporal discretization into account. An ex-ample of a dispersion relationship, for a given spatio-temporalresolution, is shown below. It is clear that the high-order dis-cretization results in a wider frequency range where the numer-ical wave speed accurately captures the physical wave speed.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

ω∆t

0.95

1

1.05

c d/c

P = 1

P = 2

P = 4

P = 6

Computational work effortDue to the increased accuracy of the high-order discretization, asshown above, a coarser spatial discretization can be used whenusing the high-order basis functions. This can improve the com-putational efficiency of simulation drastically. The figure belowshows the estimated computational work effort needed to prop-agate a 3D wave, as a function of simulation duration, measuredin wave periods.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Nw

100

105

1010

1015

WP

P = 1

P = 2

P = 4

P = 6

P = 10

P = 14

P = 18

P = 20

Simulation results

2D discThis test case is chosen to illustrate the geometric flexibility ofthe SEM. When curved boundaries occur, using straight-sidedmesh elements will introduce errors the response, unless an ex-tremely fine mesh is used to capture the geometry, which de-stroys performance. This can be mitigated by using curvilinearmesh elements.

-0.5 0 0.5

x [m]

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

y[m

]

The acoustic response of the disc is simulated with and withoutthe use of curvilinear boundary elements, in both cases usingP = 4 basis functions. The resulting responses are shown be-low.

100 200 300 400 500 600 700 800 900 1000

-10

-5

0

5

10

15

20

25

30

Clearly, the use of curvilinear boundary elements mitigates theapparent mistuning of the modal frequencies of the disc, whicharise when the straight sided elements are used. The error isfurther analyzed in the figure below.

1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

3D room

The acoustic response of a 3D cube shaped room is shown be-low, simulated using different basis function orders. In all casesthe same “fineness” in spatial discretization is used, i.e. the samenumber of DOF’s on the mesh. When high-order basis func-tions are used, the frequency range of the simulation effectivelyextends.

100 200 300 400 500 600 700 800 900 1000

-120

-100

-80

-60

-40

-20

0

20

Conclusions

The SEM appears to be a particularly suitable candidate fornumerical room acoustic simulations, due to its accuracy, effi-ciency and flexibility. The usage of these methods, once fullymatured, will allow for much more realistic virtual acoustics,and thereby improved immersion into the virtual space, than isknown today.

Interested? Want to collaborate?

We are always looking for talented students to collaborate with,e.g. by creating special courses or doing master thesis’. Thereare many interesting challenges that remain unsolved, such as

• Implementation of a matrix-free solver

• Implementation on modern many-core hardware (GPU’s)

• Improved meshing techniques

• Reduced order techniques

•Adding viscothermal losses to the medium

•Modelling sound sources (loudspeakers and the human voice)

• Comparison of different algorithms

• etc, etc, etc

Please contact us if you are interested!