Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp FEA-based methods for optimising structure-borne sound radiation Matthias Klaerner a,n, Mario Wuehrl a, Lothar Kroll b, Steffen Marburg c a Technische Universität Chemnitz, Institute of Lightweight Structures, 09107 Chemnitz, Germany Opole University of Technology, Institute of Mechanics, 45758 Opole, Poland Technische Universität München, Gerhard Zeidler Endowed Professorship for Vibroacoustics of Vehicles and Machines, 85748 Garching, Germany b c a r t i c l e i n f o abstract Article history: Received 16 April 2016 Received in revised form 24 June 2016 Accepted 11 July 2016 Lightweight components are typically stiff and thin-walled and thus tend to have significant sound radiation. Moreover using fibre reinforced plastics offers a wide range of adjusting the material properties such as stiffness and even damping by manipulating layup, fibre and matrix material or fibre volume content. With numerous free parameters within the composites, there is a need of efficient simulation methods in design and optimisation. In contrast, acoustic measures require complex multi-physical models with fluid–structure interaction and are commonly not implemented in standard FEA software. Different approaches based on the surface velocity of the component fill the gap. Namely, there is the equivalent radiated power, assuming a unit radiation efficiency all over the surface and neglecting local effects as an upper bound of structure-borne noise. In addition, the volume velocity provides good results for the lower frequency range with the frequency-dependent radiation efficiency as well as the lumped parameter model predictions being exact for dipole modes, too. Last, the kinetic energy is implicitly given in steady state FEA solutions and thus provides information about the dynamic behaviour without any additional efforts. Possibilities and limits of estimating the radiated sound power by these methods will be shown by numerical studies on a composite component. Moreover, the total power as an integral over frequency is used to demonstrate the feasibility and accuracy of such optimisation objectives. & 2016 Published by Elsevier Ltd. Keywords: Finite element analysis Sound radiation Approximation methods Optimisation MSC: 00-01 99-00 1. Introduction Lightweight structures are usually stiff and thin-walled and such tend to be sensitive for structure-borne sound. The sound radiation behaviour thus is a common optimisation criterion within lightweight design [1–3]. The optimisation procedure is related to the adjustment of the material properties of fibre reinforced plastics. There, stiffness and damping are contradictorily influenced by fibre volume content, fibre orientation as well as stacking sequence resulting in non-linear dependencies [4,5]. In addition, composites show significant uncertainties in fibre orientation [6]. Numerous finite-element simulations are required either for genetic or gradient based optimisations. Thus, efficient numerical measures of structureborne sound radiation are helpful as an effective objective function of such processes [7]. n Corresponding author. E-mail address: matthias.klaerner@mb.tu-chemnitz.de (M. Klaerner). http://dx.doi.org/10.1016/j.ymssp.2016.07.019 0888-3270/& 2016 Published by Elsevier Ltd. Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i 2 M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ In this study, the structural vibrations of continuous systems are used to determine the sound radiation behaviour in the surrounding fluid. The presented mathematical approaches are based on efficient numerical methods of vibration analysis such as mode-based steady state dynamic FE analysis. In contrast, the kinetic energy is implicitly given from the energy balance in steady state simulations and thus, is a very efficient but only qualitative measure. The treated methods and theories are validated by a thin-walled demonstrator part [8]. In detail, the radiated sound power is used to express the radiation of components and machines and is formulated as the integral of the intensity over the radiating surface. Analytical solutions of the sound power are limited to a few cases with regular geometries [9–11]. In addition, precise numerical approximation methods are used but are computationally expensive solving fluid–structure interaction in one or both directions. Another very popular approach for large-scale problems is the boundary element method (BEM) including fast-multipole techniques. This is limited for a large frequency range or modified structures within optimisation loops, e.g. [12]. Common simplification methods are based on some general assumptions. Stiff thin-walled structures with hard reflecting surfaces show identical particle velocity and structure normal velocity. There, the sound pressure is evaluated on the structure's surface. Basically, three different approaches of sound power estimation, the equivalent radiated sound power (ERP), the volume velocity (PVV) and the lumped parameter model (LPM), have been compared and opposed to the kinetic energy within the harmonic direct FEA analysis [13,8]. In addition to previous work, the total power over frequency has been determined and applied as objective for a parameter study. 2. Acoustic and mechanical energy and power estimations 2.1. Equivalent radiated sound power The radiation of vibrating parts is often estimated by the radiated sound power P representing the integral of sound intensity I in the normal direction over the closed surfaces Γ circumscribing the radiating object [13] P= ∫ →→ I · n dΓ → 1 ⎛ →⁎⎞ with I = R ⎜ pv ⎟ ⎠ 2 ⎝ (1) wherein n denotes a conjugate complex value as well as R the real part of a complex state variable. The velocity normal to →→ the surface vn = v n is imported from the dynamic FE-analysis and further on used to analyse acoustic fields by comparing different numerical estimates for the sound power. Therein, the equivalent radiated power includes a simple, popular and efficient approach for the sound pressure in the local relation p ≈ ϱf cf vn (2) with the fluid's density ϱf as well as its speed of sound cf. The relation between particle velocity and sound pressure is reduced to the fluid's characteristic impedance Z 0 = ϱf c f . (3) The approximation is typical in far fields and high frequencies and results in the sound power as a surface integral PERP = 1 ϱ cf 2 f ∫ vn 2 dΓ (4) or a discretised formulation for Ne constant elements with an area Sμ PERP = 1 ϱ cf 2 f Ne ∑ Sμ vnμ vn⁎μ . μ=1 (5) This simple formulation is based on the assumption of the same radiation efficiency σ = 1 for all elemental sources. It neglects effects such as interaction between local sources. Generally overestimating the radiation, it gives a good impression of an upper bound for convex rigid bodies and high frequencies. To reduce the overestimation at low frequencies, a Bessel-approach has been introduced [14]. There, the radiation efficiency depends on the frequency but still is constant over all elements: σ w (f ) = 1 − J1 (2kR eq ) kR eq (6) including the wavenumber k as well as the radius Req of a plane circular piston radiator with equal area as the radiation surface. The Bessel-function represents the dipole characteristics of the piston [10]. Thus, (5) is extended to the formulation Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3 of the weighted ERP (wERP) PwERP = ⎞ ⎛ J (2kR eq ) ⎟ 1 ϱf cf ⎜⎜ 1 − 1 kR eq ⎟⎠ 2 ⎝ Ne ∑ Sμ vnμ vn⁎μ . μ=1 (7) 2.2. Lumped parameter model Most accurate results can be achieved with the lumped parameter model (LPM) as well as the boundary element method (BEM) [13]. The LPM by Koopmann and Fahnline [15–17] is a discretisation of the Rayleigh-Integral and thus based on the assumptions for the validity of the integral. It includes a Taylor series of Green's function as a multipole expansion. This formulation is given for a source at xμ and a receiver at xν: PLPM = − 1 k ϱ cf 2 f with I { Gμν } = − Ne Ne ∑ ∑ Sμ Sν J { Gμν } R { vnμ vn⁎ν } μ=1 ν=1 sin (k|xμ − xν |) . 2π |xμ − xν | (8) (9) Therein, the imaginary part of Green's function weighs the interacting sources. The double summation is computationally more expensive than ERP but much more efficient than BEM. LPM predictions are exact for dipole modes. Besides, the accuracy depends on the mesh refinement as well as on the compliance with the assumptions of the Rayleigh-integral but it generally gives appropriate results in the low and mid frequency range. 2.3. Volume velocity The radiation estimation by the volume velocity u is an integral of the particle velocity on the surface [15] Ne u= ∫ v dΓ = ∑ vn Sμ. μ (10) ν=1 The derived radiated sound power PVV is understood as a reduction of the LPM: P VV = k2ϱf cf 4π ⁎ uu = k2ϱf cf 4π Ne Ne ∑ ∑ Sμ Sν R { vnμ vn⁎ν }. μ=1 ν=1 (11) It includes acoustic shortcuts such as dipole effects but only requires a single sum [13] compared to LPM. In this case all interactions have a constant but frequency dependent radiation efficiency. Thus, the PVV is suitable for the average power in the lower frequency range. It shows no convergence to mesh refinements. 2.4. Kinetic energy and input power Global quantities, e.g. potential and kinetic energy, are suitable to determine the acoustic behaviour [15]. In order to have thin-walled components, the sound radiation is dominated by out-of-plane waves. Thus, the displacement characteristics are mainly in surface normal direction. As kinetic energy and all previous sound power estimates depend on the squared velocity, this mechanical quantity is also assumed to be relevant for acoustic optimisation. Within the steady state finite element analysis the energy balance is estimated implicitly and consists of different elastic and dissipating components. Therein, the kinetic energy is directly accessible without any additional post-processing efforts. In detail, the kinetic energy of the whole part is given by the volume integral Wkin = ∫V 1 ϱ vvT dV 2 s (12) with the arbitrary oriented velocity v and the density of the solid surface ϱs. In this study we assume only harmonic vibrations and thus provide point-wise exact solutions of a steady state FEA. Therefore, the kinetic energy will be further on rated as the mechanical (input) power of the total vibrating system and is therefore transformed in every frequency step Pkin (f ) = Wkin (f )·f . (13) According to common standards in acoustics all power levels LW are referred to P0 = 10−12W . Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i 4 M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 2.5. Acoustic efficiency The mechanical power of the whole system is assumed to be much higher than the radiated power losses. Thus, the acoustic efficiency η= Pradiated Pkin (14) will be used to compare the given acoustic quantities. 3. FEA implementation Based on steady state FEA models with harmonic force excitation, all given sound power estimations have been determined in an external post-processing algorithm. All simulations have been done in ABAQUS 6.14 whereas the postprocessing is done by a python script [18] including NumPy and SciPy algorithms [19]. The implementation of all named sound power estimates (Section 2) is related to piecewise constant elements and follows [13] Ne P= Ne Ne − 1 μ=1 μ=1 ν=μ+1 Ne Ne ∑ ∑ Pμν = ∑ Pμμ + 2 ∑ ∑ μ=1 ν=1 Pμν. (15) The sound power portions Pμν are understood as partial monopole sound power contributions of all Ne constant elements. Therein, Pμμ considers the independent source distributions whereas Pμν (μ ≠ ν ) represents the interactions between these sources. This interaction matrix is symmetric and its elements can be determined by ⁎ 1 Pμν = Pνμ = 2 ϱf cf Sμ σμν R {vμ vν } (16) with a dimensionless radiation efficiency σμν . For the different sound power models, there is a general formulation of σμν including the distance Rμν between two elements μ and ν: σμν = δμν for ERP, (17) ⎡ J (2kR eq ) ⎤ ⎥ σμν (f ) = δμν ⎢ 1 − 1 kR eq ⎦ ⎣ σμν (f ) = k2 Sν 2π σμν (f ) = k2 Sν sin (k Rμν ) 2π k Rμν for wERP, for PVV, (18) (19) for LPM. (20) The algorithm is organised hierarchically starting with the common elements of (16) followed by the expressions (17)–(20) and thus can be limited to the fast approaches if necessary. 4. Sound radiation as objective function in optimisation processes In general, sound radiation is depending on frequency with dominant contributions at the resonance frequencies. Changing material or geometric parameters in optimisation processes may result in different number of modes within the considered frequency range. Next, scalar values are more likely to be implemented as an objective [7]. This total power value is estimated by frequency integral of Nf frequency steps: Ptot = 1 fu − fl Nf ⎛ ⎞ ⎝ ⎠ ∑ ⎜⎜ Δfn Pn ⎟⎟ n= 1 (21) with Δfn = fn + 1 − fn − 1 2 for 2 ≤ n ≤ Nf − 1 (22) Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 5 Fig. 1. Total power estimation by frequency integral. Δf1 = ΔfNf = f2 − f1 2 (23) fNf − fNf − 1 2 (24) referring to Fig. 1. The number and distribution of the frequency steps can be controlled by different parameters of the steady state FEA. For all further studies a mode-based approach with 100 frequency steps per mode within a spread window of 0.1·fn has been used. This leads to equidistant steps for each mode [20] but different step sizes between the modes. An optimisation loop has been implemented in Hyperstudy, combining standard FEA solutions of parametric models and user post-processing routines (Fig. 2). 5. Case study: oil pan vibrations Lightweight components are mainly thin-walled and stiff structures. For a proof of concept, an engine oil pan has been chosen exemplarily. Fig. 2. Optimisation loop with standard FEA solutions of parametric models and user-defined post-processing. Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i 6 M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Fig. 3. FEA model of an oil pan: shell elements, boundary conditions at red elements, normal forces (light blue) and arbitrary directed forces (yellow) at force locations 1 and 2. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.) The outer dimensions of the shape are approximately 250  210  100 mm and shown in Fig. 3. The mesh consists of 8500 quadratic shell elements with 2 mm edge length and 25,000 nodes. The frequency range was chosen according to common standards in acoustic analysis from 100 Hz up to 10 kHz. Thus, the model fulfils all recommendations with more than 15 elements per wavelength [21]. In detail, the oil pan dimensions keep the FE model relatively small compared to other radiating parts like trunk bottom or roof. In addition, the small geometry leads a smaller number of modes contributing within the hearing range. The oil pan implies about 50 modes starting at 1 kHz whereas larger parts have more low frequency contributions. Trunk bottom and roof show the first modes already at about 100 Hz with a high modal density in the high frequency range leading to more than 2500 modes altogether. This again increases the numerical efforts due to a mode based frequency step control. Originally, the part is of 1.7 mm thick steel blanks. For this study, the material has been replaced by a glass-fibre reinforced polypropylene (gfrp) with equal static bending stiffness for the local point force loadcases. The composite material consists of 12 layers of a 1:1 fabric layup with 0° and 90° plies oriented along the largest dimension of the pan. Each fabric layer is modelled with two separate 0.25 mm thick layers per orientation using the orthotropic elastic properties as given in Table 1. Moreover, the layup is symmetric avoiding any bending-tension coupling effects. The material substitution results in a mass reduction of 25% as well as a reduction of the total sound power of 75% due to the significant changes in damping (compare Table 2). Anisotropic damping referring to a damping model of Adams, Bacon and Maheri [22–24] has been implemented as modal damping including a weighting of strain energy components based on the mode shapes and the corresponding damping coefficients [25,26]. Related to Fig. 3, the part is fixed with displacement boundary conditions all along the red edge elements for static load case as well as the modal analysis. Further on, the oil pan has been excited normal to the mating surface with a 1g acceleration representing a base excitation. The acceleration amplitude has been kept constant within the whole frequency range. Dynamically and acoustically the model behaviour is dominated by about 50 modes in the given frequency range starting at 1001 Hz for the steel pan and 1109 Hz for the gfrp material. The mode distribution is concentrated within the third octave bands highest in frequency (Fig. 4). The model includes significant simplifications concerning boundary conditions. The engine block is supposed to be much stiffer and heavier than the thin-walled oil pan. Thus, the mount on the crankcase has been assumed to be rigid. Neglecting the compliance of the seal at the mounting as well as the influence of the oil filling are rather rough simplifications but sensible keeping tractable numeric efforts for a large number of simulations within an optimisation. The aim of choosing the oil pan demonstrator is rather showing the proof of concept than a very precise and expensive multiphysical simulation model. Considering damping, the influence of the oil filling is significant. Multi-physical numerical methods have been used to examine the modes of a filled oil pan using a Ritz method including residue iterations allowing the computation of damped structures coupled with fluids [27]. This method only includes the eigenvalues but no steady state solution. Further studies addressed the active damping control using piezo-ceramics [28] as well as the potential of material substitution [29] but do not take into account the oil within numerical simulations [30]. Last, the constant excitation amplitude has been used to demonstrate the oil pan behaviour and enable a scaling for frequency dependent excitations. Table 1 Material properties of unidirectional glass-fibre reinforced polypropylene. Elastic parameters Density Damping E1 (GPa) E2 (GPa) G12 (GPa) ν12 ϱ (g/cm3) η1 (%) η2 (%) η12 (%) 28.9 3.7 1.1 0.25 1.47 0.19 1.40 1.53 Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 7 Table 2 Material substitution – steel vs. gfrp: weight and total sound power estimates. Material Steel gfrp Weight (g) 786 514 Power/energy estimates (dB) ERP wERP LPM Pkin Wkin 53.6 47.2 53.9 47.5 51.6 56.1 99.3 91.6 65.8 57.8 Fig. 4. Modal density per third octave band of the oil pan model with gfrp material. 6. Numeric results 6.1. Power quantities in the frequency domain All comparative studies have been done with a 1g excitation in normal direction of the mating surface. Thus, all further interpretations are based on this loadcase. Fig. 5 illustrates equal frequency characteristics of all acoustical and mechanical power estimations. Reasonably, the mechanical power is significantly higher than all acoustic measures. In contrast to the modal density (Fig. 4), the sound radiation is dominated by the mid frequency range. In detail, the power estimates well follow the described frequency characteristics concerning validity in the frequency range, verified by the acoustic efficiency in Fig. 6. LPM is considered most accurate due to the refined characteristics throughout the whole frequency range. In comparison, the overestimation of ERP is represented in a higher acoustic efficiency. The Bessel-weighting of the ERP significantly changing sound radiation up to 2.5 kHz and is thus not more precise than the common ERP. Last, the PVV solution shows a feasible range of application up to 1.5 kHz but significant high deviations above. Additionally, the modal contribution of all power estimates has been analysed. Therein, the first mode contributes the Fig. 5. Mechanical and acoustic power estimates for the oil pan model. Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i 8 M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Fig. 6. Acoustic efficiency of the different sound power estimates for the oil pan model. most (Fig. 7). The correlation of the different approaches within the frequency range is similar to the previous interpretation of acoustic efficiency. The varying modal contributions cannot be clearly associated with the modal damping ratio due to the anisotropic damping approach. 6.2. Computational efforts The used numerical power estimations require different computational efforts. For the given model (Section 3) with about 5200 frequency steps, the total post-processing time is about two times more computationally expensive than the FEA Fig. 7. Mechanical and acoustic power contributions of the single modes and modal damping ratio of the anisotropic gfrp oilpan model. Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 9 Fig. 8. Calculation time of the post-processing steps related to the CPU time of the previous FEA solution. solution (Fig. 8). Therein, reading the model data and extracting the distance between each element is insignificant. Ac⁎ cessing the normal velocity fields and determining the matrix R { vμ vν } each take 11–13% of the original solution. According 2 to (17)–(20), ERP and PVV sums are of order Ne whereas LPM requires Ne steps. Thus, the ERP determination in total requires 38% of the FEA solution, PVV 81% and LPM 22% at least. In contrast to previous work, the time for reading the velocity fields and computing the ERP could be reduced by 50%. Nevertheless, within the complex numerics of the LPM, this is almost negligible. These high efforts are caused by a single-core solution of the matrix operations in contrast to a multi-core solution of the FEA problem. For a comparable number of CPU in both processes the acoustic post-processing might still be almost as expensive as the FEA solution. A major advantage of the kinetic energy is that no additional processing is required. 6.3. Evaluating the objectives As the PVV solution is inappropriate for the given frequency range, this estimate is not pursued any further. Next, a parametric study of the layup has been used to illustrate the objective evaluation. This common optimisation problem has been reduced to two independent layer orientations (φ1, φ3 ) being sequentially repeated and placed symmetrically regarding the mid surface. Both angles have been varied between 0° and 90° with a stepsize of 5°. The total power estimates P (φ1, φ3 ) of each layup have been determined and are compared in Fig. 9. On the one hand, the intensity plots show very similar dependencies of the power estimations subjectively with a variation of about 4 dB within one measure. On the other hand, the total values differ significantly in a range of 35 dB. For the given parameter field, three local minima are to be found in the corners differing less than 0.1 dB. The global minimum is achieved at the 0/90° solution in all cases. The periodically curved surfaces of the objectives are typically challenging for gradient based optimisation procedures. The total computational time for the fastest model – input power – is of about 3 h only for one single FEA solution and thus not appropriate for any genetic algorithm. To compare the different radiation measures objectively, a correlation coefficient CAB based on the norm of the sound power estimates depending on the two different angles is introduced: CAB = ∑i ∑j ⎡⎣ (Aij − A¯ )·(Bij − B¯ ) ⎤⎦ ⎡ ∑ ∑ (A − A¯ )2⎤ ⎡ ∑ ∑ (B − B¯ )2⎤ ⎣ i j ij ⎦ ⎣ i j ij ⎦ . (25) Therein, A and B are two different power level estimates P (φ1, φ3 ). This coefficient is a 2D-extension of the Pearson correlation [31–33] and very similar to standard MAC-analysis in vibro-acoustics [34]. The correlation of the radiation measures is objectively very good (Table 3) with a maximum deviation 1 − CAB of 1.12 %. Contrary to the expectations, the worst correlation is between ERP and wERP. In addition, the total wERP values are even smaller than the LPM solution. Thus, the downscaling in the low frequency domain seems to be too progressive. Astonishingly, the best correlation has been achieved for the most accurate and expensive LPM model and the fastest estimate Pkin. As a next step, the evaluated parameter fields (Fig. 9) can be used to determine valid starting points for deterministic optimisation processes finding the maximum precisely [35]. 7. Conclusions In summary, it is appropriate to use the structural vibrations of continuous systems to determine the sound radiation behaviour in the surrounding fluid for (almost)convex structures similar to piston or omnidirectional radiators without inclusions or resonators such as common thin-walled composite structures. The presented approaches are based on efficient numerical methods of vibration analysis such as the mode-based steady state dynamic FE analysis. The treated methods and theories are successfully validated by a thin-walled demonstrator part. All sound power estimates achieve qualitatively good results regarding similar frequency characteristics. In detail, the Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i 10 M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Fig. 9. Total sound power objective for varying laminate orientations: ERP, wERP, LPM and input power. Table 3 Correlation of the different power estimates as objective in a parametric layup study varying to layer angles. Sound power estimation ERP wERP LPM Pkin ERP wERP LPM Pkin 1.0000 1.0000 1.0000 0.9977 0.9975 1.0000 0.9978 0.9976 0.9998 1.0000 LPM is the most expensive but most accurate vibro-acoustic method. In contrast, PVV is an adequate solution up to 1 kHz whereas above the ERP achieves quite good results. Weighting the ERP by a Bessel-function according to a plane circular piston radiator seems to be too progressive. The overall frequency characteristics are very well represented by the mechanical power of the whole model, too. Nevertheless, the mechanical power is two orders of magnitude higher than the acoustic measures and can only be applied if the dominant deformations are out-of-plane. Although the computational time for the acoustic post-processing is significant, the correlation between the LPM and mechanical power proved to be very good. Nevertheless, the objective function determination for the full frequency range is still too long for expensive optimisation procedures like genetic algorithms. Deterministic approaches might fail due to the structure of the sound power objectives. Further investigations have to address a more precise validation by measurements, as fully coupled fluid–structure FEA or BEM simulations give a better impression of the accuracy and limits of the chosen measures. Please cite this article as: M. Klaerner, et al., FEA-based methods for optimising structure-borne sound radiation, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.07.019i M. Klaerner et al. / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 11 Moreover, other parts with different modal density will be additionally used to investigate especially the lower frequency range. Acknowledgements The paper arose in the context of the project DFG-KR 1713/18-1 Schallabstrahlung bei nichtlinearem und lokal variierendem Dämpfungsverhalten von Mehrlagenverbunden funded by the Deutsche Forschungsgemeinschaft (DFG) which is gratefully acknowledged. References [1] R. Ganguli, Optimum design of a helicopter rotor for low vibration using aeroelastic analysis and response surface methods, J. Sound Vib. 258 (2) (2002) 327–344. [2] Y. Narita, Layerwise optimization for the maximum fundamental frequency of laminated composite plates, J. Sound Vib. 263 (5) (2003) 1005–1016. [3] H. Denli, J. Sun, Structural-acoustic optimization of sandwich structures with cellular cores for minimum sound radiation, J. Sound Vib. 301 (1–2) (2007) 93–105. [4] I. Finegan, R. 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