Evaluating acoustic properties based on sound energy for interior problems
Caglar GUERBUEZ(1) , Steffen MARBURG(1)(1) Chair of Vibroacoustics of Vehicles and Machines, Technical University of Munich, Germany, caglar.guerbuez@tum.deAbstract
The analysis of sound radiation of vibrating structures is of great importance in order to develop low-noise
emitting structures. For exterior problems, it was shown that the non-negative intensities shall be considered
in order to evaluate the acoustic properties of a structure, since using this method accounts for the acoustic
short circuit. This acoustic short circuit is a phenomenon which is present when evaluating the sound intensityof a structure. Regarding interior problems, the sound pressure level is commonly analyzed to characterize
the acoustic behavior. However, the sound pressure level is strongly dependent on the position of the field
point of interest. Thus, the analysis of the sound pressure level is highly sensitive to deviations of the field
point position. This paper introduces the sound energy as an appropriate quality criteria for interior problems
due to the global characteristic. The sound energies are compared with the sound pressure level for numericalbenchmark examples and real world applications. Based on the time harmonic problem formulation, the acoustic
interior problem is solved with the boundary element method (BEM).
Keywords: Acoustic Radiation, Interior Problem, Boundary Element Method, Sound Energy

1INTRODUCTION

For exterior problems, the non-negative intensity (NNI) is an appropriate quality criterion since it circumvents
the effect of the acoustic short circuit [11]. The formulation of the NNI is derived by the acoustic radiation
modes which are obtained by the eigenvalue analysis of the impendance matrix [13]. Based on the NNI, thesurfaces which are contributing to the sound power can be identified [3, 11]. Unfortunately, the NNI is notsuitable for interior problems because of the near-field and reflection effects which are present in the interior
problem [2]. Nevertheless, this motivates an energy-based approach which is suitable for interior problems.
Usually, interior problems are analyzed by evaluating the sound pressure level at several field points. This
makes the sound field evaluation strongly sensitive to deviations of the field point position. In order to avoid
the problem of the locality, Koopmann and Fahnline [2] introduce sound energies as an appropriate quality
criterion. As a global acoustic quantity, the sound energies are unaffected by field point deviations, whilst theypreserve the acoustic behavior of an enclosure.
Based on a BEM discretization, this contribution deals with the evaluation of the sound energy as an appropriatequality criterion. It will be compared with the sound field evaluation considering a numerical benchmark
example and an application of the automotive industry, e.g. the acoustic behavior of a sedan cabin compartment.

2BOUNDARY ELEMENT METHOD

In this contribution, the derivation of the boundary element method (BEM) is presented only in a condensedform. The presentation of the BEM is reduced to the main equations and assumptions which are relevant forthis paper. For a comprehensive and detailed description, the interested reader is referred to the lecture notes[12]
2.1 Helmholtz equation and boundary integral equation
Considering the time-harmonic scalar wave equation for the sound pressure p in a bounded domain Ω ⊂ R3with the boundary Γ = ∂ Ω and the unit normal vector n outward to Ω, the Helmholtz equation for acoustic7447

Figure 1. Definition of the interior problem with the domain Ω and its complementary Ωc . The unit normal non the boundary Γ points outward to Ω.
radiation problems can be expressed as∆p(xx) + k2 p(xx) = 0

withx ∈ Ω ⊂ R3(1)with k = ω/c as the wavenumber. The wavenumber k is evaluated by the angular frequency ω and the speedof sound c. In Figure 1, the acoustic interior problem is depicted.
The Helmholtz equation satisfies the boundary conditions on Γ as well as the Sommerfeld radiation condition
at infinity. The boundary condition on Γ is equivalent to a Robin boundary condition which results into aboundary condition of Neumann type for zero admittance. The Robin boundary condition is defined asv f (xx) − vs (xx) = Y (xx)p(xx)with x ∈ Γ ⊂ R2(2)with Y representing the boundary admittance. The boundary admittance establishes the relation between thesound pressure p and the difference of the fluid particle velocity v f and the structural particle velocity vs in
normal direction. By means of the linearized Euler equation, the fluid particle velocity in normal direction canbe expressed by the derivative of the sound pressure in normal directionv f (xx) =1 ∂ p(xx)iρω ∂ n(xx)(3)

with ρ as the ambient density of the fluid and i the imaginary unit. The method of weighted residuals regarding
the Green’s function for the test function results into the Kirchhoff-Helmholtz integral equation. Incorporating
the admittance boundary condition, the associated boundary integral equation readsZ c(yy)p(yy) +ΓZ
∂ G(xx, y )− iωρG(xx, y )Y (xx) p(xx)dΓ(xx) = iρω G(xx, y )vs (xx)dΓ(xx)∂ n (xx)Γ(4)with the Green’s function asG(xx, y ) =7448

1 eikr(xx,yy)
4π r(xx, y )(5)where r is the Euclidean distance between the field point x and the source point y. The coefficient c represents
the integral free term. The integral free term is c = 0.5 for y on a smooth surface which will be the case in thefollowing numerical examples.
2.2 Collocation method
The boundary integral equation in Eq.(4) is discretized by testing it with delta functions. Moreover, the acoustic
field quantities are approximated by the same Lagrangian interpolation polynomials ϕ . The resulting linear
system of equations can be then expressed byH − GY )pp = G v s(H(6)

where the system matrices G and H are dense, complex valued and non-hermitian. In the numerical experiments,
discontinuous Lagrangian elements with linear interpolation functions are applied. The system of equations inEq.(6) is solved iteratively using a GMRes algorithm.
2.3 Field point evaluation and sound energies
Regarding the boundary data for the sound pressure and the particle velocity, one can obtain the sound pressure
at an arbitrary field point. For this purpose, the boundary integral equation in Eq.(4) is rearranged for the fieldpoint y ∈ ΩZp(yy) =iωρG(xx, y )v f (xx)dΓ(xx) −Γ∂ G(xx, y )p(xx)dΓ(xx).
x)Γ ∂ n(xZ(7)

The discretized version of Eq.(7) can be expressed asp(yy) = g T (yy)vv f − h T (yy)pp(8)

with p and v f as the nodal data for the sound pressure and the particle velocity, respectively. The data in thecolumn matrices g and h is evaluated as rows of the system matrices G and H with the field point replacing
the collocation point
Zg j (yy) = iωρG(xx, y )ϕ j (xx)dΓ(xx),(9)∂ G(xx, y )ϕ j (xx)dΓ(xx).
∂ n(xx)(10)

Γand
Zh j (yy) =ΓThe acoustic field quantities, e.g. sound pressure and fluid particle velocity, are strongly dependent on theposition of the field point. Thus, the analysis of the sound pressure level will be highly sensitive to deviationsof the field point position. Koopmann and Fahnline introduce sound energies as an appropriate quality criterion
for interior problems [2]. The total sound energy includes the potential energy and the kinetic energy. The
potential energy describes the compression of the fluid particles and is evaluated by
Ep =

1
4ρc2

Z|p(xx)|2 dΩ(xx).

(11)ΩThe kinetic energy, which comprises the motion of the fluid particles, can be expressed by
Ek =ρ4Zv(xx) · v(xx)∗ dΩ(xx).

(12)ΩThe total energy Etot can be then evaluated by7449

Etot = E p + Ek .

3(13)NUMERICAL EXAMPLES

Based on a BEM discretization, two numerical examples are considered in this contribution. Firstly, travelingand standing waves for long ducts are investigated. Secondly, an industrial application, e.g. sedan cabincompartment, is analyzed. In both examples, the sound pressure level at specific field points are comparedwith the sound energies.
3.1 Traveling and standing waves in long ducts
The first example is an air-filled duct as introduced in [1]. The duct has a length L = 3.4 m and a square crosssection A = 0.04 m2 . Moreover, the speed of sound and the ambient density are assumed to be c = 343 m/s andρ = 1.21 kg/m3 , respectively. Regarding the boundary conditions, a particle velocity is applied to one end, e.g.
vs (x = 0) = v0 with v0 = 1.0 m/s, and an admittance boundary condition to the other end. In the first case, thestanding waves are analyzed. Hence, the admittance boundary condition will be assumed to Y1 (x = L) = 0. In
order to observe traveling waves, the admittance will be Y2 (x = L) = 1/ρc. The duct is discretized with 140linear, discontinuous elements where the nodes are positioned at the zeros of the Legendre polynomials. A
schematic description of the first numerical example is depicted in Figure 2.

Figure 2. Schematic description of the duct example. The duct will be excited by a vibration at x = 0. The
resulting waves are then fully reflected (first case, Y1 ) or fully absorbed (second case, Y2 ) at x = L.
In the following analyses, the potential energy E p is firstly approximated by the mean sound pressure derived
from a series of sound field evaluations
Ep ≈

1 1 N
∑ |p(xx)|2 .
4ρc N k=1(14)The results of the sound pressure level at the field points and the potential energy are shown in Figure 3. In
the analysis, the sound pressure level is evaluated at ten field points along the x-axis.

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Figure 3. Sound radiation for the interior duct problem. In the upper row, the sound pressure level at several
field points and potential energy are shown for zero boundary admittance, Y1 = 0. The lower row depicts theresults regarding a fully absorbing admittance boundary condition, Y2 (x = L) = 1/ρc.
For standing waves (upper row), the results show that different field point position lead to different soundpressure levels. The potential energy however, represents the global characteristics of the standing waves without
the deviations of the field points. In the lower row, the results for the traveling waves are depicted. Since thewaves are fully absorbed at x = L, the deviations of the field points are nearly irrelevant. As in the case for thestanding waves, the potential energy gathers the acoustic behavior in global sense.
3.2 Sedan cabin compartment
In addition to the duct problem, an industrial application, e.g. a sedan cabin compartment is investigated.
Regarding the boundary conditions, a fictitious excitation excitation is realized by applying a particle velocity
vs = 1.0 m/s at the lower left front area of the cavity. Moreover, a uniform boundary admittance of Y = f / f0 ρcwith the reference frequency f0 = 2800 Hz is assumed at the surfaces of the cavity [9]. The mesh of the sedancabin compartment is visualized in Figure 4.

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Figure 4. Mesh of the sedan cabin compartment.

4CONCLUSIONSThis contribution introduces the sound energies as an appropriate quality criterion for interior acoustic problems.
The sound pressure level, which is usually analyzed for interior problems, is highly sensitive to deviations of
the field point position. On the contrary, the sound energy is unaffected by the deviations of the position, since
it represents a global acoustic quantity. In this work, a numerical benchmark example, e.g. the duct example,
and an application from the automotive industry were investigated. In these analyses, the sound pressure level
was compared to the potential energy, which was capable of representing the global behavior of the underlying
models without the dependencies on the field point positions. Future work will include the analysis of kinetic
sound energy and the total sound energy. Moreover, the sound energy, which was approximately evaluated by
the mean sound pressure of a series of sound field evaluations, needs to be computed for the entire domain.
Since the analyses are performed with the BEM, the transformation of the volume integral into a surface
integral, e.g. by the Gaussian divergence theorem, seems a promising approach.

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