On the improvement of the solution accuracy for exterior acoustic problems with BEM and FMBEM

Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

On the improvement of the solution accuracy for exterior acoustic problems with BEM and FMBEM R. D’Amico a,n, J. Neher b, B. Wender b, M. Pierini a a b

Dipartimento di Meccanica e Tecnologie Industriali, Universita degli Studi di Firenze, Via di Santa Marta 3, 50139 Firenze, Italy Department of Structural Mechanics and Acoustics, University of Applied Sciences, Prittwitzstr. 10, 89075 Ulm, Germany

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 May 2011 Accepted 3 February 2012 Available online 3 March 2012

The boundary element formulations suffer from the non-uniqueness of the solution. Consequently, fictitious resonances degrade the exterior field prediction. This paper investigates the benefits of using different approaches to mitigate fictitious resonances to improve the solution accuracy in real industrial cases. For direct BEM simulations, over-determination points are added to the interior of the cavity as suggested by the CHIEF method. For indirect BEM and Fast Multipole BEM simulations, the impedance condition is put over the interior boundary and two different values of absorption are applied to observe the effects on the response. The different BEM methods are applied to simulate the exterior sound radiation of three different gearbox housings. The numerical results are compared with high quality measurements enabling the benefits and the improvements on the solution accuracy of each method to be evaluated. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Non-uniqueness problem BEM Fast Multipole BEM Internal impedance method CHIEF Fictitious resonances

1. Introduction Achieving accurate predictions in numerical vibro-acoustics is a challenging task. Reliable Computer Aided Engineering (CAE) tools are essential to improve the design process and, for the acoustic optimization of vehicles, to reduce the radiated noise. Gearbox housings are an excellent example. Due to their thinwalled and light-weight design, they radiate 30% of the sound emission of the assembly; this is primarily caused by the engine [1]. The Boundary Element Method (BEM) [2] is one of the most powerful and suitable techniques for predicting the noise around a vibrating system. Compared to the Finite Element Method (FEM) [3], the BEM reduces by one the mathematical dimension of the problem. Consequently, the BEM is perfectly suitable for handling exterior unbounded domains. As with all the element-based techniques, the accuracy of the BEM solution is strictly connected to the mesh refinement. At least six elements per wavelength are required to obtain an accurate solution, excluding the simulation of large models and mid-frequency problems. However, over the last few years, the Fast Multipole BEM (FMBEM) [4,5] has enabled these limitations to be dealt with. Instead of solving the model in one go, the FMBEM splits it up into subdomains which are successively split into smaller domains and treated as classical

n

Corresponding author. E-mail address: Roberto.damico@unifi.it (R. D’Amico).

0955-7997/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2012.02.009

BEM models [6]. As a result, solution time is shorter, the memory usage lower and the frequency range of analysis can be extended for large acoustic problems [7,8]. Nevertheless, the boundary element (BE) approaches are not without shortcomings. Their formulation is affected by the non-uniqueness of the solution, and the exterior response can be seriously degraded by fictitious resonances, especially at high frequencies where the modal density increases. With the exception of simple academic cases, it is not possible to know how and where the solution will be compromised and the correct result can only be obtained by using proper approaches. In the past, several methods have been proposed for overcoming this problem. Brunduit [9] used an indirect integral formulation with layer potentials and proved that this approach does not suffer from non-uniqueness. Another branch of methods suggests simultaneously assembling the interior and exterior surface integrals or adding integral equations referring to internal points. A method using the aforementioned approach was developed by Kupradze [10], for the theory of elasticity, and Copley [11], for the acoustic problem. Schenck [12] successively improved the method as the Combined Helmholtz Integral Equation Formulation (CHIEF), which is widely used in many acoustic software packages. Implementation methodologies are developed for simple and complex cases [13] and applied to three-dimensional industrial problems by Bartolozzi [14]. Other improved formulations are based on an integral equation which combines the original surface integral and its normal derivative. Panich [15] originally applied this technique to the

R. D’Amico et al. / Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

wave problem and the Maxwell’s equation, resulting in a very computationally expensive approach. Further developments have been made by Kussmaul [16], Burton and Miller [17], and later Zaman [18] in order to obtain an easily applicable integral equation with unique solution over the whole frequency range. Clouteau [19] later proposed the modified Burton and Miller method to reduce the computational effort in elasto-dynamic applications. Ursell [20,21] used a modified Green’s function built as the sum of the original Green’s function and an analytic wave function on the exterior domain and over the boundary. Some practical methods exist to address the problem involving the indirect BEM (IBEM) but their applicability is limited to exterior problems since they distort the internal response. Dummy elements with impedance can be put on the interior of the model to damp internal resonances and reduce the pollution on the exterior [22], but this leads to larger models whose solution accuracy is often unreliable. Wu [23] proposed overcoming the non-uniqueness of the Neumann problem by simultaneously employing the velocity and unequal impedance boundary conditions. An alternative BE formulation proposed by Ambarisha [24] applies discontinuous boundary conditions. This allows the pollution to be alleviated while not to eliminate fictitious eigenfrequencies. Finally, the impedance boundary condition can be applied to the inner side of the cavity obtaining a formulation which is mathematically equivalent to the Burton and Miller method [25,26]. Usually a positive þ rc is used as the impedance value. D’Amico [27] proposed the use of a negative rc to improve the accuracy. Numerical results, obtained using this negative value, have not been compared with experimental ones yet. The aim of this paper is to highlight and demonstrate the benefits of applying different approaches to avoid the nonuniqueness of the solution within the BE techniques. The direct BEM (DBEM) is enhanced by applying the CHIEF method, while the internal impedance method is applied for the IBEM and the FMBEM. Since the results yielded by imposing positive and negative impedance have not been compared in previous works, in the following article they are investigated. To show the influence of fictitious resonances and their mitigation in real applications, the exterior noise radiation from three different gearbox housings is numerically predicted and compared with experimental data. The high accuracy of the sound pressure measurements and the velocities applied as boundary conditions represent a reliable basis to show the effects of the fictitious resonances and the benefits yielded by the aforementioned approaches. However, examples from the literature on fictitious resonances are often focused on problems with academic boundary conditions. In this paper the structure-borne noise from three real models is investigated. The fluid is mainly excited by the resonant behaviour of the structure which does not necessarily coincide with the resonant behaviour of the acoustic domain. This does not assure that fictitious resonances will not degrade the solution, since the consequent poor conditioning could degrade its accuracy. In some cases, the pollution is negligible and applying those approaches is not necessary. On the other hand, they guarantee an improvement of the solution accuracy. Moreover, using three different models provides consistent information about the sensitivity of each method to the non-uniqueness. Such an accurate investigation and comparison is very rare in BEM literature and represents the main contribution of this research. The theoretical introduction to the BE techniques and the nonuniqueness problem are given in Sections 2 and 3. Section 4 deals with the application cases. Firstly, the measurement campaign is presented. Secondly, the results for each BE technique and each treatment are compared.

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2. The boundary element method and Fast Multipole BEM 2.1. The Helmholtz equation and boundary conditions Many problems related to steady-state oscillations lead to the Helmholtz equation,

r2 pðxÞ þk2 pðxÞ ¼ 0 on D

ð1Þ

where p is the acoustic pressure at x, k is the wavenumber o=c, o is the circular frequency and c is the speed of sound. With respect to the model, the acoustic domain D is partitioned in the exterior D þ and interior D domain of boundaries @D þ and @D , as described by Fig. 1. Solving the acoustic problem (1) requires the application of known conditions to the boundary. The most general type of boundary condition is called Robin or impedance boundary condition, @p ro ðxÞ þ i pðxÞ ¼ g @n Z

on @DZ

ð2Þ

where r is the density of the medium and the function g is nonzero only for radiation problems. By means of the prescribed impedance Z, this condition directly connects the pressure and its normal derivative on the boundary @DZ . The rate of increase of the pressure on the boundary @Dv in the n direction corresponds to the Neumann boundary condition and is expressed as a function of the prescribed normal velocity vn. Finally, the Dirichlet boundary condition defines the prescribed pressure on the boundary @Dp . 2.2. The boundary element method Two different boundary integral approaches exist to solve the problem (1): the DBEM and the IBEM. In order to obtain the former, Eq. (1) is integrated twice via Green’s theorem over one side of the domain (i.e. D þ ), using the free space Green’s functions gðx,yÞ, where y is placed on the boundary and serves as integration variable [2]. This leads to the direct Helmholtz Integral Equation (dHIE) Z Z @gðx,yÞ @pðyÞ dsðyÞ gðx,yÞdsðyÞ ð3Þ cs ðxÞpðxÞ ¼ pðyÞ @n @D @D @n where cs ðxÞ is a coefficient dependent on the position of the point x. If x is located inside the domain, cs is equal to 1; if it is outside the domain, cs is equal to 0; and if it is on the boundary, cs is equal to 12 for smooth surfaces. Taking the derivative of Eq. (3) with respect to the normal direction at x leads to the direct Normal Derivative Helmholtz Integral Equation (dNDHIE). dHIE and dNDHIE allow the problem to be solved either in the exterior D þ or in the interior D domain. To extend their validity to the whole D, the solutions from both subdomains can be superposed, obtaining an equivalent formulation [28]; taking into account the normal to the surface leads to the indirect Helmholtz Integral

Fig. 1. Domain definition.

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Equation (iHIE) Z Z @gðy,xÞ dsðyÞ cs ðxÞpðxÞ ¼ mðyÞ sðyÞgðy,xÞdsðyÞ @n @D @D

3. The non-uniqueness of the solution ð4Þ

where xA D þ [ D . Eq. (4) expresses the acoustic pressure as the field in free-space due to monopole and dipole distributions over the boundary known as the double layer potential, with density m, and the single layer potential, with density s,

mðyÞ ¼ p þ ðyÞp ðyÞ sðyÞ ¼

@p þ ðyÞ @p ðyÞ  @n @n

ð5Þ ð6Þ

with p þ ðyÞ (p ðyÞ) and @p þ ðyÞ=@n ð@p ðyÞ=@nÞ the pressure and its derivative on the positive (negative) side of the boundary @D þ (@D ). Computing the derivative of Eq. (4) with respect to the normal direction at x leads to the indirect Normal Derivative Helmholtz Integral Equation (iNDHIE). The collocational approach is normally used to solve a BEM problem with direct formulation, leading to complex and nonsymmetric matrices. On the other hand, a variational approach is used for the IBEM [29,30]. This allows hypersingular integrals to be reduced to a weak singular form. Moreover, minimizing the functional leads to a symmetric and complex system of equations " #" # " # fs r A BT ¼ ð7Þ fm l B C where A, B and C are n  n matrices, with n the number of degrees of freedom (dofs), and f s and f m represent the terms containing the boundary conditions of the problem. Since the system of Eq. (7) contains 2n dofs, n conditions have to be applied and can be written as a function of the densities using Eqs. (5) and (6). If the same type of condition is applied on both the exterior and the interior boundary a reduced system of equations is obtained. Supposing that the pressure on both @D þ and @D is known, the density of the double layer potential m is consequently known (Eq. (5)) and the only unknown is the single layer potential density on the boundary. Consequently the system dimension can be reduced to n. Similar results can be obtained by solving the Neumann problem. Applying different Robin boundary conditions to each side prohibits these kinds of simplifications and both densities remain unknown. However, they are connected by the boundary condition itself. The application of an impedance boundary condition results in an increase in the system dimensions. In fact, by applying it over the whole boundary the number of equations becomes 2n. 2.3. The Fast Multipole BEM The FMBEM represents an enhancement of the standard BEM, improving its computational limits and accelerating the solving process. The algorithm divides the domain into near and far fields. While the former is solved using the classical BEM, in the latter, a clustering of boundary elements is formed and the solution is evaluated through a multipole expansion. In the standard BEM, the interaction between nodes needs to be computed, while in the FMBEM, the contribution of nodes is centralised and the interaction between distant nodes is treated only once. This results in a matrix reorganization and consequently the iterative solver does not require the full matrix assembly to compute the necessary matrix–vector products. The computational cost is reduced from OðN 3 Þ for a standard BEM analysis to OðN logðNÞÞ with the FMBEM. For the sake of briefness, mathematical operators have not been reported explicitly, however, they can be found in literature [31,6].

Both the direct and the indirect approaches fail in providing a correct solution to exterior acoustic problems [32]. Indeed, once the integral formulation has been chosen, the rank-deficiency generates fictitious resonances that pollute the solution and its accuracy due to the non-uniqueness [33–35]. Supposing to solve an exterior Neumann problem by means of the dHIE, the prediction will present an infinite set of values for which the equation has multiple solutions. These values correspond to the eigenfrequencies of the dHIE associated with the internal Dirichlet problem. Things are slightly different for the IBEM because the Neumann problem is polluted by the corresponding internal Neumann problem eigenfrequencies. This is simply due to the fact that only one indirect formulation describes both the exterior and the interior domains. Consequently, eigenfrequencies result in resonances on both solutions. It is important to note that, the non-uniqueness for the IBEM occurs only when the model is a closed surface. If the surface has an opening, the distinction between external and internal domains is no longer consistent. Thus, the peaks on the solution cannot be properly defined as fictitious resonances. The most popular approaches for overcoming the fictitious resonances for the DBEM are the Burton and Miller [17] and the CHIEF [12] methods. However, apart from some improved indirect formulations, some practical approaches are also used for the IBEM. The basic idea behind them consists of damping internal resonances in order to avoid the eigenfrequencies polluting the external solution. Finally, it is worth noting that, since the FMBEM is an iterative solver, fictitious resonances also degrade its computational performance. In the proximity of an eigenfrequency, the matrices suffer from poor conditioning and a larger number of iterations are required to achieve an accurate solution. As a consequence, eliminating fictitious resonances enables faster convergence as well as better accuracy to be achieved. 3.1. The Burton and Miller method Burton and Miller [17] overcome the non-uniqueness problem with a linear combination of the dHIE and dNDHIE. Both the integral equations suffer from non-uniqueness and two infinite sets of non-unique wavenumbers exist. If these equations are linearly combined, a third set of eigenfrequencies is obtained whose values are correspondent to the common values of the first two sets. If the coefficient of the linear combination a has a nonzero imaginary part, the resulting formulation does not suffer from non-uniqueness. Hence the Burton and Miller method provides an unpolluted solution over the whole frequency range. a is usually chosen equal to i/k where i is the imaginary unit. Even if this approach is very efficient, the connection between the dHIE and dNDHIE increases the computational load to solve the problem. 3.2. The CHIEF method The CHIEF method, proposed by Schenck [12], suggests to use some additional Helmholtz integral relations evaluated at points x interior to @D. For those points, Eq. (3) is evaluated by using a coefficient cs equal to zero. The discretization process leads to an over-determined system of equations which can be solved in a least-squares sense. The system of equations consists of n@D þ nI equations and n@D unknowns, where n@D are the dofs of the problem and nI the number of CHIEF points. To successfully apply the CHIEF method, determining the position of the internal points to be used is necessary. Choosing

R. D’Amico et al. / Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

these points is not straightforward because certain conditions have to be satisfied. Many authors have dealt with optimizing the choice, e.g. [36]. This methodology is easily implementable and computationally cheap, however, it assures good accuracy only at low frequencies. 3.3. The internal impedance method The internal impedance method consists of applying the impedance boundary condition to the interior of an IBEM model. Internal resonances are damped and do not pollute the external solution. Moreover, it has been proved that applying the internal impedance method leads to an indirect formulation mathematically equivalent to the Burton and Miller method for DBEM [26]. In fact, writing the Burton and Miller linear combination for the exterior problem on D þ , and properly substituting the potential jumps of the indirect approach leads to an indirect form in which the coefficient of the linear combination corresponds to the impedance value. The impedance of a perfect absorber is rc. Some authors suggest imposing an impedance equal to þ rc over the inner surface of models with outward normals. D’Amico proposes using a negative value of impedance rc, since this value corresponds to the coefficient i/k, usually used for the Burton and Miller approach [25]. The equivalence between these methods guarantees the mitigation of fictitious resonances over the whole frequency range. Nevertheless, as explained previously, imposing the impedance condition doubles the number of unknowns and, consequently, greatly increases the solving time. To decrease this effort, the impedance must be applied over a proper percentage of the elements [27].

high quality [37]. This constitutes a solid basis for investigating the benefits that can be obtained applying the aforementioned approaches. Moreover, it is possible to evaluate the dependency of the sound predictions on the complexity of the radiating surfaces and on the numerical methods by means of the comparisons with measurements. The context of the investigation and simulation of different methods of the sound radiation calculation is given in [38]. The use of individual components means the quality of the simulated structure-borne noise for the sound radiation simulation is higher than for assemblies, because the uncertainties caused by the modelling of joints are not included [39,40]. The structure-borne noise input values for the sound radiation simulation are updated with measurements in several steps. Thus, the difference between measured and simulated airborne-noise is due to the numerical prediction itself and the deviations from previous simulations steps are hardly accumulated. The results for the simulated airborne-noise are compared with measurements and the deviation indicates the capacity of the methods to mitigate the non-uniqueness problem. The measurement data are relevant at the eigenfrequencies of the structures; for slightly damped structures, the radiated noise is dominated by the resonant behaviour, while for non-resonant frequencies the contribution to the overall noise is negligible. A detailed description of the measurements is given in [37]; the following results from the measurements are used from this reference for the three different gearboxes.

 Results from the Experimental Modal Analyses (EMA) with



4. Application cases This section investigates the solution accuracy obtained with different methods to mitigate the non-uniqueness by means of a comparison between numerical results and measurements. 4.1. Models and measurements Three gearbox housings with varying complexity (Table 1) and extensive experimental data from [37] are used to investigate and compare different methods of mitigating the non-uniqueness problem for industrial sized models. Due to the high accuracy of the experimental campaign, the boundary conditions for the numerical problems and the reference measurements are of very

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hammer excitation and freely supported boundary conditions: eigenfrequencies, mode shapes and modal viscous damping ratios. Structure velocities from laser vibrometer measurements with hammer excitations at a measuring point and measurements at all measuring points. The velocities at the surface of the structure are normalised by the exciting force and selected at the eigenfrequencies. Sound pressure on the exterior surfaces of the housings at the eigenfrequencies, normalised by the exciting force. Measurements are made with condenser microphones at a distance of about 3 mm from the radiating surface. Excitation and measuring conditions according to the measurements of the sound velocities. Only the measurements at the closed surface of the gearbox housings are used for the investigations. Thus, the sound radiation at the openings of the components does not influence the results.

By using detailed test models with about 300 measuring points per gearbox, mode shapes at higher eigenfrequencies can still be

Table 1 Gearboxes geometrical features. Gearbox model

Model Surface Dimensions (mm) (height/width/depth) Mass (kg) Material Modulus of elasticity (GPa)

Oval principal gearbox

Daimler Chrysler axis housing

ZF 6S850 gearbox ecolite

OPG Simple 380/390/270 4.64 AlSi7Mg 74–75

DC-HAG Ribs, recesses 287/237/230 11.6 GGG40 169

ZF-G Many ribs, complex 425/410/249 13.6 GD-AlSi9Cu3 75–76

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described accurately. The evaluated frequency range includes 40 eigenfrequencies per housing. Thereby, deflections of mode shapes, structural velocities and sound pressures are available at these frequencies for all three gearbox housings at all measuring points. Additionally, all modal damping ratios from the experimental modal analyses are provided. 4.2. Structure-borne noise input data update The structure-borne noise input for the sound radiation simulation is calculated with the FEM. In order to do so, finite element (FE) models (Table 2) are derived from the structures and updated with the results of the modal analysis. Using digitised models from [37], considerable improvements are made compared with CAD-based models, because the influence of the production tolerances are minimised. FE models are obtained from the envelope of the surfaces. In the following, the generation of their structure velocities is described. In the first simulation step, a modal analysis is carried out with the updated models to determine 40 mode shapes and eigenfrequencies of the FE models. The mode shapes from measurements and simulations are compared for all housings by using the MAC values (Modal Assurance Criterion) [41]. Table 2 shows that these values are high when compared to the documented values in the literature [42–45], which proves the exceptional quality of the models [37]. The mode shapes and eigenfrequencies are used for the frequency response analysis together with the measured modal damping ratios in the second calculation. Certain points are excited separately with a unit force at the eigenfrequencies. The velocities are calculated at the FE nodes close to the measuring points first. The comparison with the prepared results of the measured frequency response functions shows the mean deviation of the velocities for all measuring points. In the next step, the exciting unit forces are modified by correction parameters for all eigenfrequencies in a way that the mean velocities of the simulation and the measurements comply with each other. The scaling equalises the uncertainties of the measured damping ratios, which are mainly responsible for the deviations despite Table 2 FE models of the gearbox housings. Model

OPG

DC-HAG

ZF-G

Mean element size (mm) Degrees of freedom Averaged MAC

11.4 77,226 89.9%

3.0 2,477,727 98.0%

4.1 3,739,533 97.6%

the accurate FE models. The mean correction parameters are only 10% away from the ideal value of one. Most of the values vary between 0.5 and 1.5. For some frequencies with small amplitudes the correction parameters are more distant. The variance strongly depends on the amplitudes. The correction parameters for strong vibrations are smaller due to the high accuracy of the measurements. These parameters, obtained by the measured velocities, are used for the calculation of the velocities for the detailed surface model in the last step. The distributions of the velocities are not influenced by the executed correction and the strong agreement between the modes shows that they are very close to reality. The evaluation of the deviations between measured and simulated velocities is performed by v-MAC values. In this case, velocity vectors vTest and vSim are used for the correlation of simulation and measurement at the eigenfrequencies, as indicated by the following equation: 2

v-MACðvTest ,vSim Þ ¼

9vTTest vSim 9 ðvTTest vTest ÞðvTSim vSim Þ

ð8Þ

A value of one indicates coincident vectors and zero stands for completely different vectors. The evaluation of the v-MAC values shows that the measured distributions are presented accurately by the simulated velocities. A mean value of 9571% is reached for all gearbox housings and excitation points. For all measurements and components, the values strongly depend on the mean vibration amplitudes, which are shown e.g. for the ZF-G in Fig. 2. Very good v-MAC values are found for all housings at strongly vibrating eigenfrequencies. Thus, complex velocities at all nodes of the surface models are available for 40 structural eigenfrequencies per component. They are useful for the acoustic simulation in the following. Remaining deviations between the simulated and measured pressures are mainly caused by the sound radiation calculation and the uncertainties on the sound pressure measurements. 4.3. Sound radiation simulations and comparison with measurements The BE meshes used for different sound radiation simulation methods are identical and the results can be compared directly. All meshes are closed, because the DBEM solver requires closed surfaces. No structure velocities are available at the few openings of the housings. Thus, their radiation of sound is excluded for the simulation and the surfaces are modelled as reflective. As previously described, the sound pressure is evaluated close to the structure surface. Therefore, the evaluation is not influenced by the sound radiation at the openings. This is also verified by specific simulations.

1 0.9

v−MAC

0.8 0.7 0.6

excit. point 114 excit. point 141 excit. point 145 excit. point 169

0.5 0.4 10−4

10−3

10−2

Mean Measured Velocities [m/s] Fig. 2. v-MAC values for ZF-G model, four excitation points.

10−1

R. D’Amico et al. / Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

The six-element-per-wavelength rule defines the limits of the frequency range of analysis. Coarse meshes lead to inaccurate results and fine meshes increase the solving times drastically. Nevertheless, when the number of dofs is low due to the clustering operations, the FMBEM becomes computationally less efficient than the classical BEM [6]. For FMBEM simulations this represents a lower boundary for the frequency range of analysis, although its violation does not entail a loss of accuracy. For the classical BEM simulations this lower boundary does not exist. These limits are automatically computed by the software LMS Virtual.Lab Rev.9 [46]. They are reported in Table 3 as suggested frequency ranges and can be compared with the effectively simulated ones. Some eigenfrequencies lie below the lower threshold and the negative influence on the FMBEM solving time is acceptable in the following analysis. The upper limit is not reached by the highest frequencies of the DC-HAG (Table 3). Graf [37] proved that the model leads to very good results anyway, and this is borne out by Harlacher [47] who demonstrates that a smaller element size has not a significant influence on the results in this case. Thus, the mentioned frequency criterion must only be seen as a reference value. Geometry and boundary conditions influence the accuracy of the results but the numerical integration and the applied solver are also important for the choice of the element size [22]. The DBEM, IBEM and FMBEM are implemented in the software [46]. For DBEM the over-determination point (ODP) positions are automatically chosen in the interior of the model. For IBEM and FMBEM, different values of impedance can be applied on either side of the boundary. This allows the internal impedance method to be used. It is worth noting that the FMBEM is based on an

Table 3 Mesh composition of the BE models and frequency ranges. Model

OPG

DC-HAG

ZF-G

Elements Nodes Mean el. size (mm) El./wavelength for fmax Suggested freq. range (Hz) Simulation freq. range (Hz) Eigenfreq. lower than freq. range Eigenfreq. higher than freq. range

11,048 5526 12 11 871–5833 393–2682 6 –

5125 2564 10 5 1194–6364 738–6945 1 10

28,697 14,351 8 11 807–8670 315–3944 3 –

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indirect formulation and its solution is dependent on the iterative solver parameters. They are set as the default values: 200 is the maximum number of iteration and 0.01 is the maximum tolerance. All numerical computation is done on a workstation Intel Core 2 Duo E8500, 3.16 GHz, 4 GB RAM. The velocities at the radiating surface of the FE models are interpolated automatically to the coarse BE models. Field point meshes are created by an offset function based on the BE meshes. For the DBEM the problem of the non-uniqueness is prevented by 100 ODPs within the BE model. For IBEM and FMBEM classical impedance boundary conditions are applied to the interior element side of the BE models regardless of the solving time. Two different values are applied to see the effects on the solution accuracy: positive and negative rc, as mentioned in Paragraph 3.3. After the acoustic simulation, results of the airborne-noise simulation are available for IBEM, FMBEM and DBEM at 40 eigenfrequencies per gearbox housing. They are subsequently compared with measured sound pressures. Table 4 shows the sound pressures on the field point mesh of the second and the seventh eigenfrequencies of the OPG, placed at 419 Hz and 890 Hz respectively. DBEM, IBEM and FMBEM predictions, treated with the aforementioned methods, are compared with the FEM results which are not dealt within this paper, although they are reported in [38]. The sound pressure distribution looks similar but the values are different. The plot in the third column represents the IBEM prediction without the application of the impedance method. The pressure distribution at the second eigenfrequency is not polluted significantly. On the contrary, at the seventh eigenvalue the solution is heavily distorted and the highest pressure value occurs at the flange region. This clearly shows the effects of the non-uniqueness on the mode shape and on the pressure amplitude. Because the results are extensive, they are reduced before the evaluation. In Table 5 the comparison between simulations and measurements is shown for each investigated case. DBEM, IBEM and FMBEM refers to the solver used. ODP means the application of the CHIEF method. Impedance and its sign is indicated with the symbols þ rc or rc. Finally, if no abbreviation follows the type of solver, no method has been applied to mitigate fictitious resonances. For each eigenfrequency, the mean and maximum sound pressures of simulation and measurement are compared and combined with their mean values (MV), spreads (Max-Min) and

Table 4 Sound pressure distribution for the OPG gearbox at 419 Hz (top) and 890 Hz (bottom). DBEM (ODP)

IBEM ( þ rc)

IBEM (no imp.)

FMBEM ( þ rc)

FEM

max. 0.45 Pa

max. 0.44 Pa

max. 0.40 Pa

max. 0.57 Pa

max. 0.56 Pa

max. 1.71 Pa

max. 1.5 Pa

max. 10.40 Pa

max. 2.12 Pa

max. 2.13 Pa

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Table 5 Comparison between numerical results and measurements [38]. Mean values (dB)

OPG OPG OPG OPG OPG OPG OPG OPG OPG

DBEM DBEM-ODP IBEM IBEM ð þ rcÞ IBEM ðrcÞ FMBEM FMBEM ð þ rcÞ FMBEM ðrcÞ

DC-HAG DC-HAG DC-HAG DC-HAG DC-HAG DC-HAG DC-HAG DC-HAG DC-HAG ZF-G ZF-G ZF-G ZF-G ZF-G ZF-G ZF-G ZF-G ZF-G

DBEM DBEM-ODP IBEM IBEM ð þ rcÞ IBEM ðrcÞ FMBEM FMBEM ðþ rcÞ FMBEM ðrcÞ

DBEM DBEM-ODP IBEM IBEM ð þ rcÞ IBEM ðrcÞ FMBEM FMBEM ð þ rcÞ FMBEM ðrcÞ

Maximum values (dB)

MV

SD

Max–Min

MV

SD

Max–Min

 2.23  2.24 1.89  2.92  2.13  0.09  0.19  0.19

0.51 0.50 4.31 0.54 0.57 0.71 0.53 0.53

2.46 2.43 18.86 2.83 2.60 4.56 2.59 2.58

 2.17  2.16 1.56  2.88  2.10 0.00  0.25  0.25

1.12 1.14 5.31 1.15 1.25 2.11 1.19 1.19

5.32 5.58 19.30 5.68 5.70 14.84 5.90 5.90

0.90 0.90 0.59 0.89 0.90 0.89 0.90 0.90

 2.96  2.94 4.03  3.75  2.27  0.53  1.02  1.05

0.70 0.70 3.82 0.64 0.62 0.85 0.70 0.71

3.35 3.19 22.14 2.85 2.68 3.26 3.32 3.36

 3.29  3.28 1.34  4.04  2.40  1.09  1.14  2.10

1.27 1.32 5.20 1.61 1.30 1.36 1.33 1.25

6.01 5.74 23.81 7.94 6.60 6.17 5.53 5.70

0.84 0.84 0.51 0.83 0.85 0.83 0.85 0.85

 6.43  6.42 0.16  6.87  6.00  4.46  4.73  4.74

2.35 2.35 3.33 2.27 2.17 2.51 2.30 2.30

9.08 9.05 14.26 9.00 8.45 8.93 8.95 8.96

 5.58  5.56  0.89  5.87  5.07  3.77  3.89  3.90

3.79 3.77 4.94 3.69 3.59 3.72 3.65 3.67

16.48 15.94 19.89 14.86 14.40 13.76 13.97 14.14

0.76 0.76 0.54 0.76 0.76 0.75 0.76 0.76

standard deviations (SD). The deviations of the sound pressures are also evaluated. Measured and assigned simulated sound pressures are written in vectors for each evaluated frequency. The mean value of the sound pressure vectors from the simulation is adjusted to the level of the vectors from the measurements, otherwise the deviation of their mean values would influence the characteristic value for the sound pressure deviation. This problem does not occur for the deviations of the structure velocities because the mean values are equal due to the correction parameters of the excitation forces. Following this preparation, the p-MAC values are determined by means of the relation 2

p-MACðpTest ,pSim Þ ¼

p-MAC

9pTTest pSim 9 ðpTTest pTest ÞðpTSim pSim Þ

ð9Þ

If pressure measurements pTest and simulations pSim are in good agreement, p-MAC is close to one; otherwise the value can decrease towards zero. The simulated sound pressure is in general smaller than the measurements: this can also be seen in the negative mean values of Table 5. Detailed results of the mean pressure values at the eigenfrequencies are reported in the following: Figs. 3(a), 4(a), 5(a) 6(a), 7(a) and 8(a) show, for example, the mean sound pressures and the p-MAC values of the OPG as function of frequency. The curves obtained with the different methods display a similar behaviour. For the classic BEM simulations a continuous offset is confirmed. For the other models, the behaviour of the mean values is comparable. In general, the highest accuracy is achieved using the FMBEM, whose results can be compared with the FEM ones, e.g. for two eigenfrequencies in Table 4. Detailed explanations of the deviation behaviour are not included here, for the sake of briefness. In all the cases deviations

are very small ð o1 dBÞ especially at high pressures, which confirms the accuracy of the results [38]. 4.4. Effect of the CHIEF points on the solution In Fig. 3 the measurement results are compared with the responses obtained from both the classical BEM and the BEM enhanced with the CHIEF method. For the latter, 100 ODPs are placed in the interior of the cavity. Comparing the curves shows that the simulation results are in good agreement with the measurements for both the mean responses and the p-MAC values, shown in Fig. 4. The effects of fictitious resonances are negligible for the OPG (Figs. 3(a) and 4(a)) and the ZF-G (Figs. 3(c) and 4(c)) cases, where the ODPs do not consistently improve the solution accuracy. Little benefit can be observed on the DC-HAG curves (Figs. 3(b) and 4(b)), where the CHIEF points assure higher level of accuracy. Even if the effect of fictitious resonances is not predictable, an improvement can be made using this approach. It is clear that in each case, none of the analysed frequencies exactly correspond to a fictitious resonance, and the poor conditioning around fictitious eigenvalues does not influence the response. This result is consistent for all the gearbox housings and it seems to suggest a dependency on the DBEM approach. In terms of solving time, the CHIEF offers a computationally very cheap approach. Since the system dimension is only increased by 100 equations, the computational time does not increase drastically. In the case of the OPG, the solving time increases of about 1 min and the total simulation time is about 58 min. For the DC-HAG model, the solution time of the standard BEM calculation is 12 min; the enhancement with CHIEF increases the solving time by less than 1 min. Finally, for the ZF-G the solution time of the ODP simulation is about 583 min, 6 min longer than the classic DBEM prediction.

R. D’Amico et al. / Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

1

Measurements OPG DBEM OPG DBEM CHIEF

7 6 5 4 3

OPG DBEM OPG DBEM CHIEF

0.95 p−MAC

Mean Sound Pressure [Pa]

8

2

0.9 0.85 0.8

1 0.75

0 500

1000

1500

2000

500

2500

1000

1500

2000

2500

Frequency [Hz]

Frequency [Hz] 7

1

Measurements DC−HAG DBEM DC−HAG DBEM CHIEF

6 5

0.9

4

0.85

3

DC−HAG DBEM DC−HAG DBEM CHIEF

0.95

p−MAC

Mean Sound Pressure [Pa]

1111

0.8 0.75

2

0.7

1

0.65

0 1000

2000

3000

4000

5000

6000

1000

7000

2000

3000

6000

7000

6 4

ZF−G DBEM ZF−G DBEM CHIEF

0.9 p−MAC

Mean Sound Pressure [Pa]

8

5000

1

Measurements ZF−G DBEM ZF−G DBEM CHIEF

10

4000

Frequency [Hz]

Frequency [Hz]

0.8 0.7 0.6

2

0.5

0 500

1000

1500

2000

2500

3000

3500

Frequency [Hz]

500

1000

1500

2000

2500

3000

3500

Frequency [Hz]

Fig. 3. Mean sound pressure at 40 eigenfrequencies of OPG (a), DC-HAG (b) and ZF-G (c) evaluated by measurement, DBEM and DBEM enhanced with CHIEF.

Fig. 4. p-MAC at 40 eigenfrequencies of OPG (a), DC-HAG (b) and ZF-G (c) evaluated by DBEM and DBEM enhanced with CHIEF.

4.5. Effect of the internal impedance method

Introducing the positive value of impedance þ rc leads to a mitigation of the fictitious resonances and to a more accurate solution. Results in Fig. 5 show that imposing the negative impedance rc leads to further improvements in the solution accuracy. p-MAC also presents higher values, as illustrated in Fig. 6. In each case improvements are confirmed, proving their consistency. It is worth noting the mathematical meaning behind the choice of the impedance. As previously mentioned, solving an indirect problem with impedance applied to the interior boundary is mathematically equivalent to the Burton and Miller method for DBEM. To guarantee an unpolluted solution over the whole frequency range, the latter requires the use of an imaginary coefficient. This corresponds to a real value of impedance in the Robin boundary condition Eq. (2). Consequently, using either positive or negative impedance will reduce the pollution in all cases. As previously mentioned, introducing the impedance boundary condition doubles the number of equations and the solving time is more than doubled when compared to the case without impedance. In fact, applying the impedance to the OPG model requires a solving time of about 141 min instead of 54 min. For the DC-HAG case, it increases from 12 to 26 min. Finally, the solution time of the ZF-G with impedance is about 1.600 min and about 502 min without impedance.

The effects of fictitious resonances are worse when the problem is solved with the IBEM. In Fig. 5 measurements and IBEM results are compared. If no impedance is applied, the solution accuracy is strongly degraded and the sound pressure is overestimated. Fig. 6 shows that the non-uniqueness has drastic consequences on the p-MAC curve as well. The results are consistent in each case and in general the results are much more polluted when compared with the DBEM ones. This does not mean that the DBEM is exempt from non-uniqueness. In fact, at a fictitious eigenfrequency, the matrices suffer from poor conditioning and it is difficult to determine a priori the magnitude of the pollution. Again, accuracy can be increased by using the impedance. In Table 4 it is possible to observe the effects of the non-uniqueness on the pressure distribution of the OPG for the IBEM case. The pollution at the second eigenfrequency is not strong even if it is visible. At the seventh eigenfrequency the mode of vibration is completely different, indeed the p-MAC value is very low at that frequency (4%), as shown in Fig. 6(a). The shape of the variable distribution is strictly related to the matrices, thus the solver used and its algorithm play a crucial role, as confirmed by the results in [38].

R. D’Amico et al. / Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

12

1

Measurements OPG IBEM no imp. OPG IBEM imp. +ρc OPG IBEM imp. −ρc

10 8

0.8 p−MAC

Mean Sound Pressure [Pa]

1112

6

0.6 OPG IBEM no imp. OPG IBEM imp. +ρc OPG IBEM imp. −ρc

0.4

4 0.2

2

0

0 500

1000

1500

2000

500

2500

1000

1500

2500

1

8

Measurements DC−HAG IBEM no imp. DC−HAG IBEM imp. +ρc DC−HAG IBEM imp. +ρc

7 6 5

DC−HAG IBEM no imp. Mean pr. 19.6 Pa, 2623 Hz Mean pr. 9.4 Pa, 2984 Hz

4 3

0.8 p−MAC

Mean Sound Pressure [Pa]

2000

Frequency [Hz]

Frequency [Hz]

2

0.6 0.4 DC−HAG IBEM no imp. DC−HAG IBEM imp. +ρc DC−HAG IBEM imp. −ρc

0.2

1 0

0 1000

2000

3000

4000

5000

6000

1000

7000

2000

3000

4000

5000

6000

7000

Frequency [Hz]

Frequency [Hz]

8

0.9 0.8 0.7 p−MAC

Mean Sound Pressure [Pa]

1

Measurements ZF−G IBEM no imp. ZF−G IBEM imp. +ρc ZF−G IBEM imp. −ρc

10

6 4

0.6 0.5 0.4 0.3

2

ZF−G IBEM no imp. ZF−G IBEM imp. +ρc ZF−G IBEM imp. −ρc

0.2

0

0.1

500

1000

1500

2000

2500

3000

3500

Frequency [Hz]

500

1000

1500

2000

2500

3000

3500

Frequency [Hz]

Fig. 5. Mean sound pressure at 40 eigenfrequencies of OPG (a), DC-HAG (b) and ZF-G (c) evaluated by measurement, IBEM and IBEM enhanced with the internal impedance method.

Fig. 6. p-MAC at 40 eigenfrequencies of OPG (a), DC-HAG (b) and ZF-G (c) evaluated by IBEM and IBEM enhanced with the internal impedance method.

With the FMBEM, the effects of fictitious resonances do not degrade the simulation accuracy as much as it does with the IBEM, as shown by Figs. 7 and 8. The results confirm their consistency for each geometry and the errors are only assignable to the numerical method. As explained in Section 2.3 the FMBEM applies a matrix reorganization. This probably has benefits on the solution and leads to a less polluted prediction. However, when the non-uniqueness does not lead to a very poor conditioning number, the effects of fictitious resonances become negligible. As a consequence, either using the positive þ rc or a negative rc impedance values guarantee accurate results which are closely superposed over the measured ones, as shown in Fig. 7. Again, the effects of the non-uniqueness are not easily predictable even though the solution is anyway polluted. Fig. 9(a) shows the iterative solver information for the OPG case. The plot in Fig. 9(b) illustrates the time used by the solver to compute the solution of the system of equations at each frequency. The first graphic illustrates the number of iterations needed to reach the convergence. The total solving time is strictly connected to the number of iterations and to the system dimensions. If no impedance boundary condition is applied, due to

fictitious resonances, the matrices suffer from ill-conditioning and a larger number of iterations may be required to reach the convergence. On the other hand, the number of equations is minimal and solving time is small. If we apply the internal impedance boundary condition, the system dimensions are doubled. Nevertheless, the matrix conditioning is improved and convergence is reached with a smaller number of iterations. As a result, applying the impedance requires a larger amount of computational resources, both in terms of solving time and memory usage. Positive impedance displays faster convergence when compared with the result obtained with the negative one. It is interesting to note that the computational time is higher for low frequencies than for high frequencies. Indeed, the FMBEM only becomes more advantageous with large models or high frequencies. Since the FM cells are about a quarter of a wavelength when frequency is low, the subdomains are large. Due to the clustering procedure, solving the problem is even more expensive than with a standard BEM. When the wavelength becomes smaller the FM algorithm results in a faster procedure even if the clustering operations and the multipole expansions have to be computed.

R. D’Amico et al. / Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

1

Measurements OPG FMBEM no imp. OPG FMBEM imp. +ρc OPG FMBEM imp. −ρc

8 7 6

0.9 0.8 p−MAC

Mean Sound Pressure [Pa]

9

5 4

0.7

3

0.6

2

0.5

OPG FMBEM no imp. OPG FMBEM imp. +ρc OPG FMBEM imp. −ρc

1 0.4

0 500

1000

1500

2000

500

2500

1000

1500

7

2500

1

Measurements DC−HAG FMBEM no imp. DC−HAG FMBEM imp. +ρc DC−HAG FMBEM imp. −ρc

6 5

0.95 0.9 0.85 p−MAC

Mean Sound Pressure [Pa]

2000

Frequency [Hz]

Frequency [Hz]

4 3

0.8 0.75 0.7

2

0.65

1

DC−HAG FMBEM no imp. DC−HAG FMBEM imp. +ρc DC−HAG FMBEM imp. −ρc

0.6 0.55

0 1000

2000

3000

4000

5000

6000

7000

1000

2000

3000

Frequency [Hz] 12

8

5000

6000

7000

1

Measurements ZF−G FMBEM no imp. ZF−G FMBEM imp. +ρc ZF−G FMBEM imp. −ρc

10

4000

Frequency [Hz]

0.9 0.8 p−MAC

Mean Sound Pressure [Pa]

1113

6 4

0.7 0.6

2

ZF−G FMBEM no imp. ZF−G FMBEM imp. +ρc ZF−G FMBEM imp. −ρc

0.5

0 500

1000

1500

2000 2500 Frequency [Hz]

3000

3500

Fig. 7. Mean sound pressure at 40 eigenfrequencies of OPG (a), DC-HAG (b) and ZF-G (c) evaluated by measurement, FMBEM and FMBEM enhanced with the internal impedance method.

5. Conclusions This paper investigates the prediction accuracy obtainable using the BE techniques and the effects of two different approaches for mitigating fictitious resonances. For the DBEM, the CHIEF method has been used, adding a set of ODPs to the interior of the model. For the IBEM and the FMBEM, an impedance boundary condition has been applied to the interior boundary. Two different values of absorption have been imposed to observe the consequences on the solution accuracy and the results have been compared. The theoretical approaches presented are applied to real industrial cases: the exterior acoustic radiations of three different gearboxes are calculated as test cases (Table 1). As a starting point, the measurement campaign carried out by Graf [37] is presented. The exterior sound pressures and updated velocities over the model surfaces are evaluated at 40 structural eigenvalues mainly responsible for contributing to the overall sound pressure. Successively these results are used as input data for the numerical simulations. The high accuracy level of the experimental data is proven by the v-MAC coefficient (Fig. 2) and constitutes a reliable basis for investigating the effects of the fictitious resonances on the numerical simulations. However, the non-uniqueness occurs

0.4 500

1000

1500

2000

2500

3000

3500

Frequency [Hz] Fig. 8. p-MAC at 40 eigenfrequencies of OPG (a), DC-HAG (b) and ZF-G (c) evaluated by FMBEM and FMBEM enhanced with the internal impedance method.

at resonant frequencies of the acoustic cavity which are not necessarily the same resonances of the structure. The proposed comparison enables the sensitivity of each method to fictitious resonances to be shown. These kinds of comparisons and real application examples are very rare in the literature discussing fictitious resonances. The classic DBEM predictions show a negligible pollution at the 40 eigenfrequencies and a good prediction accuracy, also proved by high MAC values (Figs. 3 and 4). When enhanced with 100 CHIEF points, benefits are obtained where the solution accuracy has been degraded. From a computational point of view, the CHIEF method does not consistently increase the system dimensions. In contrast, the IBEM solutions are strongly degraded by fictitious resonances (Fig. 6). Applying a positive internal impedance þ rc allows the pollution to be reduced. The numerical results are closer to measurements when a negative impedance rc is imposed (Fig. 5). The improvement of the solution accuracy is consistent, since all the cases show similar behaviour. These results confirm that using the negative value of the impedance improves the solution accuracy, as suggested in [25]. In general, the treatment requires high computational effort. Finally, the FMBEM provides less polluted predictions than the IBEM for all cases (Fig. 8). Applying the internal impedance

1114

R. D’Amico et al. / Engineering Analysis with Boundary Elements 36 (2012) 1104–1115

Iterations

200

OPG FMBEM OPG FMBEM (+ρ c) OPG FMBEM (−ρ c)

150 100 50 0 5

10

15

20 25 Frequency ID

30

35

40

5

10

15

20 25 Frequency ID

30

35

40

Solving Time [s]

700 600 500 400 300 200 100

Fig. 9. Iterations to convergence (a) and solving time (b) for the OPG model and FMBEM solver.

boundary condition leads to unpolluted results and differences between positive and negative values of impedance are negligible (Fig. 7). In terms of computational efforts, damping fictitious resonances reduces the number of iterations necessary to reach the convergence at each frequency but the system dimensions are doubled. In this case, solving time increases and positive impedance assures a faster convergence than negative impedance.

Acknowledgments This research has been carried out in collaboration with the University of Florence and the University of Applied Sciences of Ulm. The contents are discussed from a theoretical point of view by Dipartimento di Meccanica e Tecnologie Industriali at the University of Florence. Numerical applications, measurements and their comparisons have been provided by the Laboratory of Structural Mechanics and Acoustics of the University of Applied Sciences of Ulm, which has been researching the sound radiation of gearboxes for many years [37,48,49]. References ¨ ¨ ¨ [1] Steffens CS. Anregungsvorgange und Korperschallfl usse im Gesamtsystem Motor-Getriebe. PhD thesis. RWTH Aachen, Verlag Mainz; 2000. [2] Brebbia CA, Dominguez J. Boundary elements, an introductory course. US: McGraw Hill Inc.; 1989. [3] Zienkiewicz OC, Taylor RL. The finite element method, vols. 1, 2.4th ed. London: McGraw-Hill; 1991. [4] Greengard L, Rokhiln V. A fast algorithm for particle simulations. J Comput Phys 1987;73:325–48. [5] Greengard L, Rokhiln V. On the evaluation of electromagnetic interactions in molecular modelling. Chem Scr 1989;29A:139–44. [6] Gumerov NA, Duraiswami R. Fast multipole methods for the Helmholtz equation in three dimensions. Oxford: Elsevier Ltd.; 2004. [7] Hallez R, De Langhe K. Solving large industrial models with the Fast Multipole Method. In: The sixteenth international congress on sound and vibration (ICSV16), Krakow, Poland; 2009. [8] Chen JT, Chen KH. Applications of the dual integral formulation in conjunction with fast multipole method in large-scale problems for 2-D exterior acoustics. Eng Anal Boundary Elem 2004;28(6):685–709.

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