BEM Simulation of tube acoustics using thin elemens
Wolfgang KREUZER(1) , Veronika WEBER(2)(1) ÖAW, Acoustics Research Institute, Austria, wolfgang.kreuzer@oeaw.ac.at(2) ÖAW, Acoustics Research Institute, Austria, vero_weber@gmx.at

Abstract
The acoustic of hollow tubes and pipes is of interest in many research areas, e.g. noise control, simulation of musicalinstruments or when modelling the human vocal tract. As an alternative to the “classical” tube models that assume
plane wave propagation inside the tube, the boundary element method offers a flexible alternative, where differentmaterials and different shapes of the tubes (e.g. curved tubes) can be easily simulated. In this paper we present aBEM model of a hollow tube using thin elements. We use this model to calculate the resonance frequencies and thesound radiation of a so called Klangröhre or Boomwhacker, which is a musical toy for children tuned to the note C4.
The simulation results are then compared with measured resonance frequencies from the same Boomwhacker.
Keywords: Sound, Tube, BEM

1INTRODUCTION

The simulation of sound wave propagation and acoustic radiation in connection with (half) open tubes andpipes has a long tradition in acoustic engineering. Among the many numerical methods used in acoustics
the boundary element method (BEM) plays an important role because its flexibility with respect to geometry,material parameters and radiation condition, especially at infinity. In our talk we present a BEM model for a
tube where the wall thickness is assumed to be infinitely small. This has the advantage that no coupling between
interior and exterior domain of the tube and no special boundary conditions at the tube ends are necessary. Thetube is assumed to be made of hard plastic and vibrations of the material are neglected. This plastic tube is63 cm long tube with radius r = 1.75 cm and when hit against an object it produces the musical note C. The
BEM is used to calculate the sound radiation for frequencies between 100 Hz and 1500 Hz. To verify the BEM
calculations the resonance frequencies are compared with measurements. In the conclusion we briefly discusspossible numerical problems in connection with the model.

2SIMULATION OF TUBES

2.1 Classical tube models
In music theory, the most common used approach for simulating tubes and pipes (e.g. for flutes [1] or tosimulate the human vocal tract [2]), is to assume that the sound wave inside the tube can be represented as
the sum of two traveling (plane) waves, one from right to left, and the other from left to right, reducing the3D wave propagation problem to a one dimensional system. The assumption of a traveling plane wave isvalid as long as the cross section of the tube is small enough compared to the wavelength of the travelingwave. Additionally, most of these models assume that the tube walls are rigid and that there are no losses
do viscosity or heat conduction (cf. [2]). The models are mostly used in the frequency domain to calculate
resonance frequencies.
A second drawback of the 1D tube approach is modeling the sound radiation at the tube ends. The soundradiation is sometimes considered by including an additional tube segment with larger cross sections at theends, by considering a radiation model cf. [3, 4]), or by introducing correction factors dependent on the radiusof the tube to correct the fundamental frequency of the tube (cf. [5]).

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Compared to the 1D-tube models full 3D-BEM simulations have the advantage, that different materials can beused, the shape does not need to be a straight tube, and modeling radiation at the tube ends can be easily done.
But like in the case of a finite element analysis standard BEM simulations suffer from stability problems if thewall width is very small. One way to model a tube in 3D could be to close both ends of the tube, and thena)b)c)Figure 1. Different possibilities to model a tube with BEM
couple an interior and exterior problem using appropriate boundary conditions (see Figure 1a). In the different
approach one could model the tube with standard 3D elements like in Figure 1b). But in that case stability
problems are expected and probably very small elements will have to be used. A third alternative is to use
thin elements comparable to shell or plate elements for FEM (see Figure 1c). In such a formulation the wholeproblem can be seen as an exterior problem, and as in case b) the radiation at the ends poses no problem atall.
2.3 Thin BEM elements
In the literature thin BEM elements have already been used for simulating e.g. cracks, loundspeakers, or glass
noise barriers. In this work we use the BEM formulation from [6]. Starting from the direct BIE formulationc(x)u(x) −ZZ

H(x, y)u(y)dΓy +ΓG(x, y)v(y)dΓy = 0(1)ΓFigure 2. Scheme for a thin BEM element. For the final element ε → 0.

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where c(x) = 0, 21 , 1 if x is inside the scatter Ω, on a smooth part of the boundary Γ = δ Ω of the structure, andat a point outside the structure (x ∈ Ωe ), respectively, the formulation for the thin elements are given by:u(x) −ZZG(x, y)∆v(y)dΓy = 0, x ∈ Ωe ,H(x, y)∆u(y)dΓy +Γ(2)Γu+ (x) + u− (x)− H(x, y)∆u(y)dΓy + G(x, y)∆v(y)dΓy = 0, x ∈ Γ ,2Γ
ΓZ
Zv+ (x) + v− (x)− E(x, y)∆u(y)dΓy + H 0 (x, y)∆v(y)dΓy = 0, x ∈ Γ .
2
ΓΓZZ(3)(4)

In Eqs. (2) to (4) u(x), u+ (x), u− (x), and ∆u(x) denote the velocity potential at an exterior point x, the velocity
potental at one side of the infinitely thin element, the velocity potential on the second side, and the difference ofthe potentials ∆u = u+ + u− , respectively. v(x) is the particle velocity, G(x, y) is the free field Green’s function
for the Helmholtz equation, and H, H 0 , and E denote the double layer, the adjungated double layer and thehypersingular operators:
eikr, r = ||x − y||,4πr∂H(x, y) =G(x, y),∂ ny∂
G(x, y),H 0 (x, y) =∂ nx
∂2E(x, y) =G(x, y),∂ nx ∂ ny
G(x, y) =(5)
(6)
(7)
(8)

where ∂∂nx denotes the derivative with respect to x in the direction of the normal vector at x, k = 2πc f denotes
the wavenumber, and i2 = −1.
The ’positive’ side is defined by the direction of the normal vector (see also Figure 2).

3

NUMERICAL EXPERIMENTS

To test the BEM formulation the sound radiation and resonance frequencies of a straight plastic tube (Klangröhre, Boomwhacker) with radius 1.75 cm and a length of 63 cm was investigated. The dimensions of thetube were chosen to match a tube producing the note C4 with a fundamental frequency of about 262 Hz (anadditional measurement of the real tube resulted in a fundamental frequency of 264 Hz).
3.1 BEM simulation
For the BEM simulation we used collocation with constant elements. The cylindrical surface of the tube wassubdivided into 16 · 63 elements quadrilateral elements, thus along the length of the tube the element length
was 1 cm. The tube was assumed to be sound hard, and as excitation force a velocity boundary condition attwo elements 3 cm from one tube ending was chosen (see the bright elements in Figure 3). Additional to thesurface mesh evaluation grids where placed along the tube ends and parallel to the tube ends in 3 cm distancefor determining the resonance frequencies. All simulations were done between 100 and 1500 Hz in steps of 1
Hz. As the stiffness matrix was relatively small a direct solver from the LAPACK library was used.
3.2 Simulation results
In Figure 4 the radiation pattern at the first resonance frequency and the sound pressure inside the tube at 1296
Hz are given. In Figure 4b the typical nodes and antinodes for tubes can be observed quite clearly.

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Figure 3. BEM mesh and the absolute value of the calculated sound pressure at a frequency of 100Hz. The
grid around the tube is used for the evaluation of the sound pressure and its radiation at the tube ends. The
two bright elements on the tube mark the elements where the velocity boundary condition was applied.

In Figure 5 the measured resonance frequencies can be compared with the calculated results. As can be seenby the data, the measured and calculated frequencies agree up to some Hertz. One possible reason for the small
difference may lie in the assumption of the tube to be sound hard. If one looks for example at Figure 5b) the
typical theoretical harmonic structure of the resonances for a rigid tube is slightly more pronounced.

4CONCLUSIONS

Using thin elements in combination with the boundary element method offers an attractive alternative for solving
wave propagation problems in connection with thin walled tubes. Compared to some other methods, the BEM
is relatively flexible with respect to shape and materials, and it has the advantage that no special consideration
has to be made to model the sound radiation at the tube ends.
However, it has to be mentioned that the method has some possible drawbacks. In general, it is recommended touse at least 6 elements per wavelength (cf. [7]). In case of the current tube simulation the size of the meshes
used had to be smaller than suggested by the above rule of thumb to achieve robust resonance frequencies.
Heuristically this can be motivated by the fact that the position of the antinodes (i.e. the low amplitude regionsin Figure 4b)), are very important for the radiation and the resonance frequencies, so the meshsize should
be chosen in a way, that these zero crossings can be resolved accurately, especially in the case of constantelements.
A second drawback is the BIE-part including the hypersingular operator. Especially at low frequencies and withsmall grid the condition number of the system becomes quite large, which is a problem, even if the linearsystem is solved by a direct solver, because the resonance frequencies can be shifted a few Hertz which is
unacceptable at low frequencies. To that respect the tube would be a nice benchmark for algortithms that adapt
the representation of the solution to the frequency at hand.

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a)

b)

Figure 4. a) Absolute sound pressure at the evaluation grid at the first resonance frequency 259 Hz. b) Absolute
sound pressure inside the tube at a frequency of 1296 Hz.

60
f0 = 262

f2 = 789

110

f3 = 1055
f4= 1317

403020a)f1 = 518

f2 = 777

f3 = 1036

f4 = 1296

9080
7060

100f0 = 259

100
Aver. Pressure (dB)

Sound pressure (db)50

f1 = 521

0200400600
800
Frequency

1000

1200501400b)

200400600
800 1000
Frequency

1200

1400

Figure 5. Averaged a) measured and b) calculated sound pressures at the evaluation grid.

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[4] Dalmont, J. -P.; Nederveen, C. J.; Jolly, N. Radiation Impedance of Tubes with different Flanges: Numerical and experimental investigations. J. Sound Vibr., Vol 224(3), 2001, pp 505–534.
[5] Ruiz, M. J. Boomwhackers and End-Pipe Corrections. Pys. Teach., Vol. 52, 2014, pp 73–75.
[6] Chen, Z.-S.; Hofstetter, G.; Mang H. A. A Symmetric Galerkin Formulation of the Boundary Element
Method for Acoustic Radiation and Scattering. J Comput. Acoust, Vol 5(2), 1997, pp 219–241.
[7] Marburg, S. Six boundary elements per wavelength. Is that enough? J. Comput. Acoust. Vol. 10, 2002, pp25–51.

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