exterior radiation problems using the wave...

Exterior sound radiation problems using the Wave BasedTechnique without spherical adaptation elements

P. Zsifkovits 1, T. Mócsai 21 AVL List GmbH, Graz, Austriae-mail: [email protected]

2 Laboratory of Acoustics and Studio Technology, Budapest University of Technology and Economics, Bu-dapest, Hungary

AbstractThe Wave Based Technique [1] is a method for airborne noise simulation, applicable in the low- and mid-frequency range. In the wave based model for exterior sound radiation problems, the radiating surface is enclosed by a bounding box, which lies in the interior of a truncation sphere. The domain between the box and the sphere consists of six spherical sections, the so-called spherical adaptation elements. The sound pres-sure is modeled using radiating solutions of the Helmholtz equation outside this sphere and wave function solutions inside. A large number of wave functions is needed in the spherical adaptation elements to model the sound pressure accurately. Consequently the computational effort is to a large part due to the spherical adaptation elements. This work investigates the possibility of using the bounding box itself as a truncation surface. As a result the system matrix will be notably smaller and faster in computation. An industry-like car combustion engine example is used to compare the results and performances of both methods.

1 The Wave Based Technique for unbounded acoustic problems

1.1 Mathematical formulation

This article considers a 3-dimensional exterior acoustic radiation problem (Figure 1) in a domain Ω =R3 − Ω+1, where Ω+ ⊂ R3 is a bounded domain, referred to as scattering obstacle or radiating object depending on the kind of boundary conditions, which are imposed.The propagation of time-harmonic acoustic waves in homogeneous, isotropic, and friction-free media having constant speed of sound c in absence of additional sound sources can be described as follows. Under the assumption of time-harmonicity the acoustic sound pressure p(r, t) takes the form p(r, t) = pω(r)eiωt, with the steady-state pressure pω(r) and circular frequency ω. The steady-state pressure pω(r) then satisfies

∆pω + k2pω = 0 (1)

This is Helmholtz’ equation with wave number k = ωc , cf. [2].

The particle velocity v(r, t) has the same time-harmonicity v(r, t) = vω(r)eiωt as the sound pressure. Thesteady-state particle velocity vω satisfies

vω(r) =i

ωρ∇pω(r) (2)

where ρ is the ambient fluid density.

1The notation Ω means the topological closure of Ω, i.e. Ω together with its boundary.

1333

n

x

y

z

r

Ω+

ΓpΓv

ΓZ

ΩU

ΩB

Figure 1: Helmholtz radiation problem

Different boundary condition types will be considered. It is assumed that the boundary of Ω is partitioned asΓ := ∂Ω = Γp ]Γv ]ΓZ as depicted in Figure 1, on which the following boundary conditions are imposed.

• On Γp a pressure boundary condition

pω(r) = pb(r) for each r ∈ Γp (3)

for a given boundary pressure function pb on Γp.

• On Γv a normal velocity boundary condition

nTvω(r) = vb(r) for each r ∈ Γv (4)

for a given boundary normal velocity function vb on Γv and the outward normal vector 2 n on ∂Ω.

• On ΓZ a normal impedance boundary condition

nTvω(r)−1

Zb(r)pω(r) = 0 for each r ∈ ΓZ (5)

for a given normal impedance function Zb on ΓZ .

• In addition the Sommerfeld radiation condition

∂

∂rpω(r) + ikpω(r) = o(r

−1) as r →∞ (6)

2Outward-pointing is meant as “out of Ω” here, but into the obstacle Ω+.

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is imposed uniformly in all directions3. Physically it means that the acoustic wave is not reflected atinfinity. It ensures that the boundary value problem is well-posed on unbounded domains. For detailsthe reader is referred to [3, 2].

1.2 Solution using the wave based technique

The idea of the Wave Based Technique (WBT) is to decompose the domain Ω into subdomains, to imposeadditional continuity boundary conditions on the subdomain interfaces, and to use a linear combination ofexact solutions of the Helmholtz equation (1) on each subdomain as an ansatz. Subsequently the boundaryconditions are enforced by an appropriate choice of parameters in this ansatz. For further information on theWave Based Technique the reader is referred to [1, 4].

The first step is to decompose Ω into a bounded and an unbounded part by introducing an artificial truncationboundary ΓT . The bounded and unbounded part are denoted by ΩB and ΩU respectively. The boundedpart ΩB is decomposed into subdomains Ω1, . . . , ΩN . To ensure convergence of the wave based method thegeometry of these subdomains has to be restricted4.

The pressure inside the domain Ωα is approximated using the functions

Φ(α)ij (x) :=

e−ik

(α)ijxx cos(k

(α)ijy y) cos(k

(α)ijz z) j = 1

cos(k(α)ijxx)e

−ik(α)ijy y cos(k(α)ijz z) j = 2

cos(k(α)ijxx) cos(k

(α)ijy y)e

−ik(α)ijz z j = 3

(7)

The Φ(α)ij are called wave functions. In order for Φ(α)ij to be an exact solution of the Helmholtz equation (1)

it is necessary and sufficient that the equation

(k(α)ijx )

2 + (k(α)ijy )

2 + (k(α)ijz )

2 = k2 (8)

be satisfied. In this paper the following choice is used.

(k(α)ijx , k

(α)ijy , k

(α)ijz ) :=

±√k2 − (n(α)ijy πL

(2)α

)2−

(n

(α)ijz π

L(3)α

)2,

n(α)ijy π

L(2)α

,n

(α)ijz π

L(3)α

j = 1n(α)ijxπL

(1)α

,±

√k2 −

(n

(α)ijxπ

L(1)α

)2−

(n

(α)ijz π

L(3)α

)2,

n(α)ijz π

L(3)α

j = 2n(α)ijxπL

(1)α

,n

(α)ijy π

L(2)α

,±

√k2 −

(n

(α)ijxπ

L(1)α

)2−

(n

(α)ijy π

L(2)α

)2 j = 3(9)

where L(j)α are the dimensions in x-,y- and z-direction of an axis-parallel rectangular box enclosing thedomain Ωα. The integers n

(α)ijx , n

(α)ijx , n

(α)ijx run from 0 to some upper bounds N

(j)α which are chosen depending

on the size of Ωα and on the frequency ω as will be described in Section 4.

For the pressure inside the domain Ωα the ansatz

pα(r) =3∑

j=1

N(j)α∑

i=0

p(α)ij Φ

(α)ij (r) (10)

is used. The parameters N (j)α is chosen in depending on Ωα and on the frequency ω as will be described inSection 4.

3Boldface letters are used to denote vectors. The plain letter r denotes the length of the vector r.4Convex domains are admissible, although more general domains could be allowed, as well, cf. [5].

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The pressure expansion in the domain ΩU reads

pU (r) =

NU∑l=0

l∑m=−l

p(U)lm Φ

(U)lm (r) (11)

The functions

Φ(U)lm (r, φ, θ) := hl(kr)Y

ml (φ, θ) (12)

are called radiation functions5. They satisfy the Helmholtz equation on ΩU and also the Sommerfeld radia-tion condition (6), cf. [2]. Here, again the parameter NU is chosen depending on the frequency ω.

For α = 1, . . . , N, U6 the notations Γαp := Γp ∩ Ωα, Γαv := Γv ∩ Ωα and ΓαZ := ΓZ ∩ Ωα are used.Furthermore the following residuals are defined to enforce the boundary conditions (3), (4) and (5).

R(α)p (r) := pα(r)− pb(r) for r ∈ Γαp (13)

R(α)v (r) :=i

ρωnT∇pα(r)− vb(r) for r ∈ Γαv (14)

R(α)Z (r) :=

i

ρωnT∇pα(r)−

1

Zb(r)pα(r) for r ∈ ΓαZ (15)

At the domain interfaces Γαβ := Ωα ∩ Ωβ the residuals

R(α,β)(r) :=i

ρωnT∇(pα(r) + pβ(r))

+1

Zc(pα(r)− pβ(r)) for r ∈ Γαβ

(16)

are introduced. The parameter Zc weights the relative importance of the pressure term against the normalvelocity term.

For brevity’s sake the notation

〈f, g〉Γ :=∫Γ

fg dS (17)

is employed.

Instead of requiring that the residuals vanish, it is merely imposed that they vanish in average. This isexpressed in the weighted residual formulation, which reads

〈nT∇q, R(α)p 〉Γαp + 〈q, R(α)v 〉Γαv + 〈q, R(α)Z 〉ΓαZ

+∑β 6=α

〈q, R(α,β)〉Γαβ = 0(18)

for α = 1, . . . , N, U and all q from an appropriate class of test functions7.5Here hl are spherical Hankel functions and Y ml are spherical harmonics.6To treat the domains Ω1, . . . , ΩN and ΩU in a coherent way, the index α runs through the numbers 1, . . . , N and U . Here U is

merely a symbol, not a number.7It will be assumed this class contains at least the wave functions (7) and the radiation functions (12).


By plugging the expansions (10) and (11) into equation (18) and using Φ(α)ij with α = 1, . . . , N, U as testfunctions, a linear system of equations is obtained. Which can be written in matrix form as

A1 C12 C1N C1U

C21 A2 C2N C2U

CN1 AN CNU

CU1 CUN AU

p1

p2

pN

pU

=b1

b2

bN

bU

(19)

2 Truncation surfaces

As described above, a Wave Based model for an unbounded problem partitions the exterior domain into abounded and an unbounded part, which are separated by a truncation surface. The bounded part contains thephysical model boundary and is further decomposed into geometrically restricted domains.

2.1 Spherical truncation surface

The Wave Based Method for unbounded problems as described in the literature (see e.g. [1], [4]) mostly usesa spherical truncation surface. From a mathematical point of view this seems to be the natural choice sincethe spherical harmonics in (12) form an orthonormal basis of the Hilbert space L2(S2) of square-integrablefunctions on the sphere [6, 7]. For the decomposition of the interior of the sphere the radiating object iscontained in a rectangular bounding box, which is completely contained in the sphere. The part between thebounding box and the truncation sphere is partitioned into six sphere caps (cf. Figure 2(b)), which are calledspherical adaptation elements. The interior of the bounding box is also partitioned further into convex parts.

2.2 Rectangular truncation surface

Due to the size of the spherical adaptation elements, a rather large number of wave functions is needed inthese domains. Therefore it might be possible to substantially reduce the degrees of freedom by using amodel without these elements. To achieve this the bounding box itself is used as truncation surface, whilethe subdomains inside the bounding box are kept unmodified.

3 Calculation of radiation functions and their normal derivatives

When using a rectangular truncation surface, unlike a spherical one the normal derivatives are not equal tothe radial derivatives, but have also parts in the azimuth and zenith directions. To calculate the entries ofWBT matrix system it is therefore necessary that all these derivatives of the radiation functions be known.This section describes the computation of these derivatives. The radiation functions are given by

Φ(r, φ, θ) = hl(kr)Yml (φ, θ) (20)

where r, φ, θ denote radius, azimuth and zenith respectively. The spherical harmonics Y ml (φ, θ) and thespherical Hankel functions hl are defined by

Y ml (φ, θ) :=

√(2l + 1)(l −m)!

4π(l + m)!Pml (cos θ)e

imφ and hl(z) :=

√π

2zH

(2)

l+ 12

(z) (21)

respectively, where Pml denote the associated Legendre functions and H(2)

l+ 12

the Hankel functions of the

second kind. For the radial derivative of Y ml only the derivative of the spherical Hankel functions is needed.


It can be obtained from the well-known formula (9.1.27. in [8])

d

dzhl(z) = hl−1(z)−

l + 1

zhl(z) (22)

The φ-derivative is elementarily determined.

∂

∂φΦ(r, φ, θ) = hl(kr)

√(2l + 1)(l −m)!

4π(l + m)!Pml (cos θ) im e

imφ (23)

Finally for the θ-derivative

∂

∂θΦ(r, φ, θ) = hl(kr)

√(2l + 1)(l −m)!

4π(l + m)!

d

dθPml (cos θ)e

imφ (24)

the derivative of the associated Legendre function Pml needs to be known ([8] formula 8.5.4.)

(µ2 − 1) ddµ

Pml (µ) = lµPml (µ)− (l + m)Pml−1(µ) (25)

From this we obtain with µ = cos θ and dµdθ = − sin θ = −√

1− µ2

∂

∂θPml (cos θ) =

dPmldµ

dµ

dθ=

lµPml (µ)− (l + m)Pml−1(µ)√1− µ2

(26)

for µ 6= 1. The case µ = 1 must be treated separately, but since it can be avoided by not evaluating thederivative on the z-axis, it will be left out here.

From the derivatives ∂Φ∂r ,∂Φ∂φ and

∂Φ∂θ the gradient ∇Φ can be obtained by coordinate transform

∇Φ =cos φ sin θ sin θ sin φ cos θ

1r

cos θ cos φ 1r

cos θ sin φ − 1r

sin θ

− 1r

csc θ sin φ 1r

cos φ csc θ 0

∂Φ∂r

∂Φ∂φ

∂Φ∂θ

(27)

4 Number of approximating functions

The following section describes how the number of approximating functions for a specific domain is de-termined (see also [1] and [4]). The parameters N (j)α and NU of equations (10) and (11) are defined asfollows.

N (j)α =TWF

λL(j)α , where j = 1, 2, 3 (28)

where λ = 2πk is the wavelength, TWF is a domain- and frequency-dependent truncation parameter and L(j)α

are the dimensions in x-,y- and z-direction of an axis-parallel rectangular box enclosing the correspondingdomain. Similarly for the unbounded domain

NU = 2rT TRF k + Noffset (29)

where TRF is again a frequency-dependent truncation parameter and rT is the radius of the truncation spherein case a spherical truncation surface is used and the radius of a maximally inscribed sphere in the case of arectangular truncation surface.


5 Validation

5.1 Description of the test models

Two different examples are used for the validation. First one with a very simple geometry is used. Afterwardsthe method is put to the test for a more complex car combustion engine model.

Radiating cube with cavity As first example a cube of side length 0.369 m with one face removed isconsidered. The inner face of the cube opposing the opening (shown in mangenta in Figure 2(a) and 2(b)) isexcited with a normal velocity of 1.0 m/s, while all the other inner and outer faces (shown in turquoise) arerigid. In the model with rectangular truncation surface only two domains are present: the radiation domainand the interior of the open cube. In the model with spherical truncation surface there are 7 bounded domains,namely the interior of the cube and the 6 spherical adaptation elements, and the radiation domain outside thetruncation sphere. The radius of the truncation sphere is 0.35 m.

(a) Rectangular truncation surface (b) Spherical truncation surface

Figure 2: Radiating cube with cavity

Car combustion engine model To test the method also with a more complicated model we use a simpli-fied car combustion engine model. The model with rectangular truncation surface consists of 47 boundeddomains and the radiation domain. Correspondingly the model with spherical truncation surface consists of53 bounded domains and the radiation domain, see Figure 3(b). The engine surface (Figure 3(a)) is uniformlyexcited with a normal velocity of 1.0 mm/s. The radius of the truncation sphere is 550 mm.

5.2 Description of the reference models

Two reference models are created and solved with the Boundary Element Method (BEM) using LMS Sys-noise 5.6. The details of the BEM models are described in the following two paragraphs.

Radiating cube with cavity The model is an indirect BEM model with 10673 nodes (dofs), the number ofelements is 10580. The mesh has a uniform mesh size of 8 mm and consists of linear quadrangle elements.With a rule of 10 elements per wavelength, the mesh is valid up to 4197 Hz.


(a) Physical boudary Γ (b) Bounded WBT domains Ωα and domain interfaces

Figure 3: Car combustion engine model

Car combustion engine model This model is also an Indirect BEM model. It consists of 17871 nodes(dofs) and 35738 elements. The mesh has an average mesh size of 10 mm with a maximum element lengthof 17 mm and consists of linear triangle elements. With a rule of 10 elements per wavelength, the mesh isvalid up to 1994 Hz (80% of the elements are valid up to 2760 Hz).

5.3 Description of the performed comparisons

For both cases the following comparisons between the models with and without spherical adaptation elementson one side and the BEM models on the other are performed.

1. Comparison of the pressure field along three circular surfaces in the xy-, yz-, and xz-planes.

2. Comparison of the pressure and velocity fields on a sphere enclosing the radiating surface.

The convergence behavior towards the BEM solution is investigated in terms of the truncation parameterTWF , the offset parameter Noffset and the number of Gauss integration points NGauss used in the approxima-tion of the integrals that occur in the entries of the matrix (19).

5.4 Results

Radiating cube with cavity The following calculations are performed on a Intel(R) Core(TM) i7-4900MQ2.80GHz CPU with 16.0 GB RAM.

In Figure 5(a) the relative L2 error of the solution for the WBT model with spherical truncation surfaceagainst the BEM solution on a sphere of radius 1.5 m is depicted for varying simulation parameters. It is


500 Hz

1500 Hz

2500 Hz

3500 Hz

Figure 4: Comparison of the sound pressure levels in dB (Ref.: 20 µPa) calculated with BEM (left), WBTwith rectangular truncation surface (middle, TWF = 3, TRF = 1, NGauss = 60, Noffset = 40) and WBT withspherical truncation surface (right, TWF = 3, TRF = 1, NGauss = 40)

observed that the number NGauss of integration points used almost no effect on the solution. However a smallimprovement can be observed if the value of TWF is increased from 3 to 5. Above 3200 Hz the calculationswith TWF = 5 run out of memory.


(a) Convergence behavior (b) Timing

Figure 5: Radiating cube with cavity (spherical truncation surface)

Figures 6(a) and 7(a) show the relative L2 error of the solution for the WBT model with rectangular truncationsurface against the BEM solution on a sphere of radius 1.5 m for varying simulation parameters. In Figure6(a) the values TWF = 3, TRF = 1 and Noffset = 40 are fixed and only the number of integration points isincreased. For NGauss = 40 the error starts to grow excessively at 800 Hz. If the number of integration pointsis increased to NGauss = 60 the error is stable until 3700 Hz, with NGauss = 80 to even higher frequencies.It is a common phenomenon that for fixed TWF , TRF and Noffset the solution accuracy increases with thenumber of integration points and then becomes stable.

(a) Convergence behavior as a function of NGauss (b) Timing

Figure 6: Radiating cube with cavity (rectangular truncation surface)

Figure 7(a) shows there is a slow error decrease of the solution with increasing Noffset = 10, 20, 40, but thedifference between the solutions with TWF = 3 and TWF = 5 are very small, except in the range above3700 Hz where it was already observed that error of the solution with NGauss = 60 grows.

The comparison of Figures 5(b) and 7(a) shows that in this simple example similar accuracy can be achievedwith spherical and rectangular truncation surfaces.

The differences in performance of both methods can be observed by comparing Figures 7(b) and 5(b). Whilethe model with spherical truncation surface performs generally better in the low frequency range, the CPUtime consumption curve has a steeper slope so, that in higher frequencies both models have similar perfor-mances. The weaker performance of the model with rectangular truncation surface in the low frequencyrange is due to the fact that a larger number of radiation functions is needed in that case. Compared to theWave Functions the Radiation functions need much more computational effort to calculate. But since themodel with spherical truncation surface has more domains which are also larger than those of the model withrectangular truncation surface, the number of Wave Functions grows faster in that model (cf. Equation (28)).


(a) Convergence behavior as a function of TWF and Noffset (b) Timing

Figure 7: Radiating cube with cavity (rectangular truncation surface)

Car combustion engine model Upon inspection of figure 8 it is observed that the use of a rectangulartruncation surface for the car combustion model causes clearly visible discontinuities along the truncationsurface.

1000 Hz 1800 Hz

Figure 8: Sound pressure level in dB (Ref.: 20 µPa) calculated with WBT with rectangular truncation surface,TWF = 3, TRF = 1, NGauss = 80, Noffset = 10

Similar to above the Figures 9(a) and 10(a) show the relative L2 errors of the solutions for the WBT modelsagainst the BEM solution on a sphere of radius 550 mm for varying simulation parameters. As opposed to thecube example while the model with spherical truncation surface still converges towards the BEM solution,the model with rectangular truncation surface fails to do so completely. Figure 10(a) shows that the error ofthe model with rectangular truncation surface is increasing with the frequency. Further it is not dependenton the number TWF of wave functions. That means that the model is apparently not converging when TWFis increasing.

6 Conclusion

For the relatively simple cube example the model with rectangular truncation surface still yields resultswhich, while not as accurate as those produced by the model with spherical truncation surface, are stillsensible. However for the more complex engine example the model with rectangular truncation surface fails.It seems that even for the small cube example it is not possible to reach a result with comparable accuracy


(a) Convergence behavior as a function of TWF (b) Timing

Figure 9: Car combustion engine (spherical truncation surface)

(a) Convergence behavior as a function of TWF (b) Timing

Figure 10: Car combustion engine (rectangular truncation surface)

and resource consumption to the model with spherical truncation surface. So in terms of accuracy the modelwith spherical truncation surface is the one to favor.

In terms of the performance it has been observed that even though the time consumption of the models withrectangular truncation surface does not grow as fast with frequency as in the model with spherical truncationsurface, it has a much higher time consumption already in the low frequency range. To take advantage of theslower growing time consumption one would have to perform computations at very high frequencies.

Therefore it must be concluded that in the current form the model with rectangular truncation surface are notsuitable for real world examples.

Acknowledgements

The first author would like to thank the Laboratory of Acoustics and Studio Technology at the BudapestUniversity of Technology and Economics for their hospitality. Part of this work was conducted duringhis visit there. This work is funded by the European Commission ITN Marie Curie proj. No. 605867“BATWOMAN” (Basic Acoustics Training- & Workprogram on Methodologies for Acoustics – Network).


References

[1] Bert Pluymers. Wave based modelling methods for steady-state vibro-acoustics. PhD thesis, KatholiekeUniversiteit Leuven, 2006.

[2] Frank Ihlenburg. Finite Element Analysis of Acoustic Scattering. Number 132 in Applied MathematicalSciences. Springer-Verlag New York, Inc., 1998.

[3] Rainer Colton, David Kress. Inverse Acoustic and Electromagnetic Scattering Theory. Springer-VerlagBerlin Heidelberg, 1992.

[4] Bart Bergen. Wave based modelling techniques for unbounded acoustic problems. PhD thesis,Katholieke Universiteit Leuven, 2011.

[5] Jevgenij Jegorovs. On the Convergence of the WBM Solution in Certain Non–Convex Domains. Proceed-ings of the International Conference on Noise and Vibration Engineering ISMA 2006, Leuven, Belgium,2006.

[6] Yuan Dai, Feng Xu. Approximation Theory and Harmonic Analysis on Spheres and Balls. SpringerMonographs in Mathematics. Springer-Verlag New York, 2013.

[7] Gerald B. Folland. Fourier Analysis and its Applications. Number 4 in Pure and Applied Undergraduatetexts. American Mathematical Society, 2009.

[8] Irene A. Abramowitz, Milton Stegun. Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables. New York: Dover Publications, 1972.


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