PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Application of the Energy based Finite-Element-Method to determine
the sound emission of vibrating ship structures in the HighFrequency Domain
Henning LOHMANN1; Boris DILBA1; Olgierd ZALESKI1; Otto VON ESTORFF1
1

Novicos GmbH, Veritaskai 8, 21079 Hamburg, Germany

ABSTRACTFor large scale structures the applicability of classical numerical methods such as Finite-Element-Method(FEM) or Boundary Element Method (BEM) is restricted to the low and mid frequency range. The fine meshresolution for higher frequencies rapidly leads to problem sizes which exceed given computational resources.
Here, the energy based Finite Element Method (EFEM) provides an efficient solution.
The EFEM is based on balancing input, output and dissipated powers over a specific control element and isapplicable to fluid and structure problems. Exchange of energy between elements is described bytransmission coefficients. The unknowns of the EFEM are energy densities which aren’t oscillatory in nature.
Thus, the required discretization is independent of the frequency and results in a smaller system of equations
compared to classical methods.
In this contribution the application of the EFEM to a ship is outlined. After a brief introduction to theoreticalaspects the focus will be on the coupling of fluid and structure problems based on the radiation efficiency. In
addition to determining the radiation efficiency the influence of the coupling parameters on the resulting
system of equations is shown and utilized to define a lower frequency bound for the application of the EFEM
to vibrating ship structures.

1. INTRODUCTION
Especially in the design and development phase of ships the prediction of underwater soundcharacteristics is of great interest. In the lower frequency range classical numerical methods like theFinite Element Method (FEM) are a suitable tool to predict the required physical quantities . In thehigher frequency range the computational costs for classical numerical methods exceed the commonly
available computer capacities. Energy based methods provide an efficient possibility to calculate
acoustic quantities in the higher frequency range. A widely used energy based method is the Statistical
Energy Analysis (SEA). To apply this method the whole model is divided into larger substructures
before the calculation. An energy density is than assigned to each substructure as unknown value. For
the application of the energy based Finite Element Method (EFEM) the substructures are further
divided into finite elements, which yields the possibility to determine the local distribution of energy
density on substructures. The power flow in coupling regions is therefore described more accurately
due to this additional discretization.
The appropriate frequency range of applicability of the EFEM is often desired. Several approaches
have been made regarding this topic (1, 2). In this contribution the influence of fluid-structure
coupling on the system of equations is investigated in detail to give a lowest bounding frequency for
EFEM calculations considering the solvability of the system matrix.
The correct determination of the input power is a crucial point for numerical calculations withenergy based methods. Besides the vibrating systems on board of the ship, propeller noise has a large
influence on the resulting sound pressure in the fluid since there is a direct transfer path into the water.
The rotating propeller leads to a sound excitation with different excitation mechanisms that are
explained in context of energy based calculations in the fourth chapter. A sensitivity analysis of thedifferent excitation transfer paths is presented for the investigated model. This contribution has been
produced in collaboration with the WTD 71.

1

gf@aaa.com

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2. THEORETICAL BASICS
2.1 Basic equation of energy transport
The EFEM is based on an equation that is similar to the heat transfer equation, since the energy flow
between different areas or volumes is described proportional to the gradient of energy density between
these two domains. The gradient ο݁ is scaled depending on the group velocity of the specific wave
type ܿ௚ , the structural damping constant ߟ and the angular velocity ߱ of the calculated frequency.
The transportation of energy is described by the first term on the right hand side of equation 1.
Furthermore the dissipated energy in the area or volume is taken into account. This amount ofdissipated energy flow is described by the second term on the right hand side of equation 1. Since theamount of energy is not changing in a stationary state of the system, the dissipated and transported
energy sums up to the incoming energy ȫ௜௡ for a small domain. This leads to the basic equation of the
EFEM and reads
ȫ௜௡ ൌ

௖೒మ
ఎఠ

ο݁ ൅ ߟ߱݁.

(1)

The unknowns of the system of equations are the energy densities of different wave types on each nodeof the discretized system (3). Basic elements in an EFEM model are beams, plates and acoustic cavities.
For each basic element the energy densities of each wave type are distinguished. In the case of plates
longitudinal, shear and bending waves are assumed.
2.2 Structural Coupling
At joint connections, a distinction is made between dissipated and reflected power flows of thecoupled energy densities, which also depend on the wave type. These power flows are described bytransmission coefficients. The analytical calculation of the transmission coefficients depends on
geometrical and material properties and is done before the system matrix is set up. The transmission
coefficients are collected in the transmission matrix ࣎ො, which is further used to calculate the joint
matrix ࡶ෠. The joint matrix ࡶ෠ finally describes the transmitted power flows depending on the level ofenergy densities in the connected domains and reads(2)ࡶ෠ ൌ ሺࡵ െ ࣎ොሻሺࡵ ൅ ࣎ොሻି૚ ࡯,with the identity matrix ࡵ and a diagonal matrix ࡯, containing the group velocities of the coupledwave types. The joint matrices are added to the system of equations after setting up the element
matrices. The solution is further obtained using an LU decomposition.
2.3 Fluid-Structure-Coupling
The coupling between fluid and structure domains is realized in a similar way. In this case thetransmission coefficients are used to describe the power flows between the fluid and the structural
domain. The transmission coefficientߚܿ଴ଶ ߪ
(3)߬௙௟௨௜ௗǡ௦௧௥௨௖௧௨௥௘ ൌǡܿ஻ ݂݄
describes the amount of energy that is transmitted from the fluid into the structure and depends on thequotient of the characteristic impedances ߚ, the speed of sound in fluid ܿ଴ , the group velocity of thebending wave ܿ஻ , the specific calculated frequency ݂, the thickness of the plate ݄ǡ and the radiation
efficiency ߪ. The amount of energy that is transmitted from the structure into the fluid is described by
the transmission coefficientʹߚߪ߬௦௧௥௨௖௧௨௥௘ǡ௙௟௨௜ௗ ൌǤ(4)ʹ ൅ ߚߪ
The transmission coefficients in equations 3 and 4 both depend on the radiation efficiency (4). The
determination of the radiation efficiency is a crucial point and its calculation includes several
properties of the coupled domain. A direct determination of the radiation efficiency is only possible
with a large numerical effort. A direct calculation of the radiation efficiency using the FEM is only
suitable in the lower frequency range. The determination in the high frequency range for a complete

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ship structure is not possible due to insufficient numerical capacities. Fortunately, analytical
approaches exist that can be used to determine the desired value based on the equations of motion ofbasic fluid-structure coupled configurations. In this contribution the analytical approach of Rumerman
is used (5). Here, the equations of motion of a simply supported, fluid loaded plate are used to give an
estimation of the radiation efficiency. The radiation efficiency depends on the edge width of thecoupled structure. Therefore, an algorithm has been implemented to detect the needed geometricalproperties of the coupled areas. One specific radiation efficiency is prescribed on each structural panelof the investigated ship.

3. COUPLING PARAMETERS
A comparison between the analytical approach of Rumerman and an FEM determined radiation
efficiency is shown in Figure 1. The depicted radiation efficiency is obtained for a rectangular simply
supported steel plate that is coupled to a water domain. A strong frequency dependency is observed forthe radiation efficiency that is calculated with the FEM. This is due to the modal effects that influence
the radiating behavior of the plate. These modal effects are neglected in the approach of Rumerman.
Nevertheless there is a good averaged agreement of the analytical approach especially for higher
frequencies. In the lower frequency range the radiation efficiency calculated with the analytical
approach of Rumerman approaches zero, whereas the FEM calculated radiation efficiency is still in arange between ͳͲିସ and ͵ ‫ିͲͳ ڄ‬ଷ.

Figure 1 – Radiation efficiency determined with the approach of Rumerman and the FEM
The influence of the radiation efficiency on the transmission coefficient ߬௦௧௥௨௖௧௨௥௘ǡ௙௟௨௜ௗ is shownfor the frequency range below 1 kHz in Figure 2. All values are in a physically reasonable range. The
qualitative deviations are similar to the ones detected in the direct comparison of the radiation
coefficients.

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Figure 2 – Transmission coefficient for energy flow from structure to fluid
A comparison of the transmission coefficients ߬௙௟௨௜ௗǡ௦௧௥௨௖௧௨௥௘ with the prescribed Rumermanradiation efficiency and the FEM calculated radiation efficiency is given in Figure 3. Here, thetransmission coefficient that is calculated with the FEM radiation efficien cy leads to high values in arange of up to ͳͲଷ for low frequencies below 20 Hz.

Figure 3 – Transmission coefficient for energy flow from fluid to structure
To assess the solvability of the system of equations a consideration of the largest and smallest
eigenvalues of the system matrix is made. The frequency dependent behavior of the largest eigenvalue
is shown in Figure 4 for system matrices that are set up for systems in different coupling conditions
and with different prescribed radiation efficiencies. Besides the eigenvalues of the coupled system, theeigenvalues of the particular structural and fluid systems are shown. The largest eigenvalue of eachsystem is independent of the prescribed radiation efficiency value. This value is determined by the
structural component of the model since the largest eigenvalue of the system matrix of the particular
fluid domain is clearly smaller in the whole considered frequency range.

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Figure 4 – Largest eigenvalue of EFEM system matrix
The smallest eigenvalue for different coupling conditions is depicted in Figure 5. All eigenvalues
tend to zero for very small frequencies. This leads to a poor condition of the system of equations. The
unsteady behavior of the eigenvalue of the coupled system with a prescribed FEM calculated radiation
efficiency is noticeable. This behavior is not observed if the transmission coefficient ߬௙௟௨௜ௗǡ௦௧௥௖௧௨௥௘ is
set to zero, which is depicted in the FEM – one way – curve. Therefore this drop of the lowest
eigenvalue is due to the very high values of the corresponding transmission coefficient.

Figure 5 – Smallest eigenvalue of system matrix
This drop leads to eigenvalues that are nearly zero and therefore influences the solvability of thesystem of equations. For the investigated system a frequency of 100 Hz can be regarded as a lower
bound for the solvability of the numerical fluid-structure coupled system of equations.

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4. PREDICTION OF SHIP NOISE
The sound emission of a ship mainly results from the structure-borne sound that is transferred over
the wetted surface to the surrounding fluid as well as the propeller induced pressure fluctuations. For
the research ship Planet of the German Navy the structure-borne sound radiated into the water ismainly generated by the dynamic behavior of the propulsion system which comprises the drive shaft
and the electric main and auxiliary engines. Due to the SWATH design of the ship the generators can bepositioned on deck between the twin-hull and thus they contribute less to the generated underwater
noise than the propulsion. The propeller induced sound emission results from different mechanisms.
Besides the turbulence noise and the sound radiated at the blade passing frequencies, the occurringpressure fluctuations on the hull lead to vibrations of the submerged structure and thus to sou nd
emissions induced by the propeller. Additional noise is generated if propeller cavitation occurs. The
equivalent sound source exhibits a broadband spectrum which varies with the rotational speed of thepropeller.
In the following example the previously outlined vibro-acoustic formulation of the EFEM is applied for asensitivity analysis of the sound emission of the ship in the mid and high frequency range with respect tothe above mentioned main contributors. According to measurements the sound emission of the ship isdefined in terms of the maximum sound pressure level while crossing a hydrophone, as illustrated in Figure
6.

Figure 6 – Vibroacoustic EFEM model of the research ship Planet

Provided that the hydrophone is passed with a constant speed, it can be assumed that the vibrations of theship are at a steady state. Furthermore, it is supposed that the transfer functions from the sound generating
components to the hydrophone are independent of the excitation. Thus, the sound emission of the ship fordifferent passing speeds is determined by scaling the transfer functions with the according input power
spectra of the different wave types. The required input powers of the structural components can either bemeasured or calculated. In contrast, the determination of the sound power generated by the propeller is aneven more challenging task since it is, if even possible, hard to measure. Furthermore, appropriate
simulation methods, especially for the high frequency range, are entailed with very high computationalcosts. The development of more efficient numerical methods is the subject of current research (6, 7).
Figure 7 shows the corresponding transfer functions for a hydrophone located midships at a depth of about
25m below the water surface as depicted in Figure 6. By comparing the transfer functions of thestructure-borne sound sources it is obvious that the hydrophone is least sensitive with respect to thegenerators. This is due to the fact that the transfer path is longer than for the other structural components,
thus more energy is dissipated before it propagates through the hull into the water. The engine units and thethrust bearing of the shaft are mounted on the submerged part of the ship which results in a shorter transfer
path to the hydrophone and hence yields a higher noise sensitivity. In order to point out the effect of thereinforced propulsion foundation with respect to the sound emission an additional transfer function for a
direct excitation on the hull is determined. The according transfer function is denoted as shaker and shows asensitivity regarding the hydrophone that is up to 100 times higher than for the propulsion components.
Furthermore, by comparing the transfer functions of the structure-borne sound sources it turns out that thestructural damping leads to a decreasing acoustic sensitivity especially at higher frequencies. The longer thetransfer path, the faster the acoustic sensitivity decreases.

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Figure 7 – EFEM transfer functions with respect to hydrophone no. 7 for different excitation components
However, the highest acoustic sensitivity is observed for the propellers which are modelled as acoustic
sources including the interactions with the ships structure. The direct transfer path to the hydrophone
results in a sensitivity, which is up to ͳͲ଼ times higher than for the less sensitive generators.
In addition to the transfer functions the energy distribution in the entire acoustic domain provides further
information about the sound emission properties of the ship. Among other things directivity characteristics
as well as the sound power can be determined. Figure 8 shows the resulting energy density distribution in
the acoustic domain due to a unit load at the mounting points of the backboard engine at 1.6kHz.

Figure 8 – Energy density distribution in the fluid domain at 1.6kHz due to a unitload at the mounting points of the backboard engine

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5. CONCLUSIONSIn this contribution a limiting lowest frequency is shown for a simplified model of a single platethat is coupled to a fluid volume. In the frequency range below this limiting frequency an influence ofthe radiation efficiency on the numerical solvability of the system of equations is observed. For highradiation efficiency values unphysical transmission coefficients in a range greater than one aredetected in this lower frequency range. In this case the solution of the system of equations is possible
but should not be interpreted as physically correct.
An application of the coupling parameters is studied for the calculation of sound radiation of theresearch ship Planet. The sensitivities of the sound pressure level to the different excitation
components are shown via the frequency dependent transfer functions and qualitative comparisons
between the vibration exciting systems are presented.

REFERENCES
1. Moens I, Vandepitte D, Sas P. A wavelength criterion for the validity of the Energy Finite Element
Method for plates. Proceedings of ISMA25; Belgium 2001. p. 599-607.
2. Kong X, Chen H, Zhu D, Zhang W. Study on the validity region of Energy Finite Element Analysis.
Journal of Sound and Vibration. 2014; 333: p. 2601-2616.
3. Bernhard R, Wang S. Energy Finite Element Method. Computational Acoustics of Noise Propagation in
Fluids - Finite and Boundary Element Methods; Berlin 2008. p. 287-306.
4. Bitsie F, Bernhard FJ. Structural acoustic energy finite element method. Journal of the AcousticalSociety of America. 1996; 99(4): p. 2585-2603.
5. Rumerman ML. The effect of fluid loading on radiation efficiency. Journal of the Acoustical Society ofAmerica. 2002; 111: p. 75-79.
6. Testa C, Ianiello S, Salvatore F, Gennaretti M. Numerical Approaches for Hydroacoustic Analysis of
Marine Propellers. Journal of Ship Research. 2018; 52: p. 57-70.
7. Young-Zehr K, Kao JH. Underwater acoustic field and pressure fluctuation on ship hull due to unsteadypropeller sheet cavitation. Journal of Marine Science Technology. 2011; 16: p. 241-253.

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