Aerospace Science and Technology 30 (2013) 183–191

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Aerospace Science and Technology

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Multi-discipline optimization of sandwich cylinders under a pointforce excitation

Chongxin Yuan ∗, N. Bert Roozen, Otto Bergsma, Adriaan Beukers

Delft University of Technology, Faculty of Aerospace Engineering, Structural Integrity & Composites, Delft, The Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 December 2012Received in revised form 23 May 2013Accepted 7 August 2013Available online 20 August 2013

Keywords:OptimizationSandwich cylinderAcousticMechanicalFuselage

The aircraft fuselage was idealized as a sandwich cylinder and the minimization of the inner soundpressure of the cylinder was studied. Subjected to a point force excitation, the inner sound pressure ofthe sandwich composite cylinder was predicted using the FEM/BEM. The acoustic transfer vector methodwas adopted in the numerical model. The predicted results were validated with the experimental results.Using the verified numerical model, the structural parameters including core properties, core thickness,sandwich layup and fiber orientation, were studied for their influences on the inner sound pressure.Finally, an optimization method integrating the genetic algorithm and acoustic transform vector methodwere developed to minimize the inner pressure of a fuselage section. It has been demonstrated that theoptimization method can improve the computation efficiency and give a good compromise between theweight, the mechanical performance and the acoustic properties of the sandwich fuselage.

© 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

Due to the high structural efficiency, the large crash absorbingproperties [22], as well as the good thermal and acoustic insulationproperties, composite sandwich structures have increasingly beenapplied on the aircraft fuselage. For example, the Extra 500, a six-seat business aircraft, has a sandwich fuselage with CFRP facingsand an Aramid honeycomb core. The sandwich skin is integrallystiffened with frames in the vicinity of cut-outs and at the wingsupport [27]. Current jet transport aircrafts require noise controltreatments with minimal weight penalties to achieve comfortableinterior noise levels in the cabin. The maximum A-weighted soundlevel anywhere in the fuselage interior is set as 80 dBA [32]. Thetypical exterior noise levels are 125 dB and a reduction of the or-der of 30 to 50 dB is needed below 1000 Hz [31]. Therefore manyefforts have been done to improve the sound insulation of sand-wich structures [10–12,16,20].

The sound transmission is related to the dynamic behavior ofthe structure and a review of the dynamic analysis of compositeshells is given in [23]. The sound transmission of sandwich fuse-lages has been analytically studied in [5,26,33]. In these analyticalmethods, the fuselage is usually infinite in length, its equation ofmotion of the cylindrical shell is derived from the Love’s theory[25], and the frequency response of the cylinder is a superposi-tion of the natural modes. Li [21] has developed a mathematicalmodel that can be used to study the sound transmission into a fi-

* Corresponding author.E-mail address: c.yuan@tudelft.nl (C. Yuan).

1270-9638/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.ast.2013.08.002

nite sandwich cylindrical structure. However it was assumed thatthe end capes of the cylinder were rigid and only the cylindri-cal shell radiated the sound into the cavity, as a simple case ofthe finite enclosed cylinder. Cheng [3] investigated the sound ra-diation of a cylinder into its enclosed cavity. The author used anartificial spring system to simulate the shell–plate joints. Gener-ally speaking, analytical methods are only suitable for analyzingvibroacoustic properties of structures with simple geometries andboundary conditions.

As the progress of computer technology, numerical methodssuch as the finite element method (FEM) and the boundary el-ement method (BEM) are frequently used to analyze the vibroa-coustic problems. Assaf [1] predicted the sound transmission lossthrough sandwich plates using FEM/BEM. The plate formulationwas derived from Kirchhoff’s theory and Mindlin’s theory for thecore. Fernholz and Robinson [14] studied the coupled fluid/struc-ture vibration of a finite composite cylinder with finite element(FE) method. They also studied the influence of lamination angleson the internal noise level of a composite aircraft in [15]. For com-plex structures or the case where repeated computation is needed,the acoustic transfer vector (ATV) algorithm is often adopted forimproving the computation efficiency. The ATV links the input ofthe structural velocity of the radiating surfaces and the sound pres-sure levels at the desired output field points. It depends on thefollowing system parameters: geometry of the vibrating surfaces,acoustic treatment of the surfaces (acoustic boundary conditions,in terms of impedance or admittance), microphone (field point)location, frequency, physical properties of the acoustic medium(speed of sound & density). However, it is independent of the op-erational structural response and the loadings [17]. This represents

184 C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191

Table 1The property of the sandwich cylinder and its materials.

Material property

E1 [Pa] E2 [Pa] v12 G [Pa] ρ [kg/m3]

UD-Glass/epoxy 3.66e+10 5.4e+09 0.3 4.2e+09 1800Tycor Foam 1.51e+08 1.51e+08 0.1 6.87e+07 70Wood for end caps 6e+09 6e+09 0.25 2.4e+09 700

Geometry and weight

Radius (R) 0.25 Length 1 m Total weight 10.34 KgFace thickness 3.18 mm Core thickness 15 mm End caps thickness 45 mm

an enormous advantage in that loading and design parameters canbe varied without having to run original solvers again, as long asthe acoustic model is not changed. Some investigations on the vi-broacoustic behaviors of structures using the ATV method wereintroduced in [4,35].

The optimization of the structural properties to achieve the bestnoise insulation has been discussed by some researchers. The pa-rameters for the optimization include the material properties suchas the stiffness, damping, Poisson ratio; the sandwich layup suchas the layer thickness, the reinforcement orientation and the stack-ing sequence; the geometry and topology such as the microstruc-ture of the core and the shape [7]. Denli and Sun [8] introduceda coupled FEM/BEM to minimize the interior sound in cylindershells under the exterior acoustic excitations. They found that thelamination angles were effective to minimize sound transmissioninto the interior of a cylindrical shell. A similar work by Denli andSun was discussed in [19]. Wennhage [28,29] conducted a weightoptimization for a sandwich railway car body with acoustic andmechanical constraints. The computation model for the acousticresponse was not discussed in details in the papers.

The prediction the inner pressure of composite sandwich cylin-ders using FEM/BEM has not been validated by the experimen-tal tests to date. In addition, the multi-objective optimization ofsandwich cylinders in multi-discipline fields has not been exten-sively studied. Therefore, the focus and the main contribution ofthe present work are to validate the developed FEM/BEM modelusing the experimental results, and to develop an efficient multi-discipline optimization system of sandwich cylinders, including themechanical requirements, the total weight and the acoustic re-sponse, by adopting the generic algorithm and the ATV method.

2. Experimental measurement of the inner pressure

A sandwich cylinder was produced by means of the filamentwinding method. The skins of the sandwich cylinder were com-posed of fiberglass/epoxy and the fibers were placed in the hoopdirection. Tycor® foam was used as core material because it can fitthe curved surface of the cylindrical model. The materials proper-ties and geometry of the sandwich cylinder are shown in Table 1.

The sandwich cylinder was enclosed by two wooden panelswith a thickness of 40 mm. In order to avoid flanking transmis-sions, the junctions between the cylinder and wooden ends weresealed with AirTech® sealant tape. A shaker (B&K 4809) controlledby a white noise signal was used to excite the cylinder. The in-ner pressure was recorded with a microphone (B&K 4196), and theoutside noise was recorded with another microphone (B&K 4196).The structural response of the cylinder was recorded using an ac-celerometer (B&K 4344). All the signals were collected by an inputmodule (Lan-Xi 3050-B-040), and were post-processed with Mat-lab.

As shown in Fig. 1, a point force excitation was applied at thecenter of the cylinder surface. A shaker is connected to the point ofexcitation by means of a rod in order to avoid forces and momentsother than the desired force in normal direction. Two microphones

Fig. 1. Setup for the inner pressure measurement of cylinders subjected to a forceexcitation.

were placed inside the cylinder to measure the sound pressure in-side cavity. The two microphones were fixed on a support at aradius equal to 2/5 and 4/5 part of the inner radius of the cylin-drical shell. The axial position of the microphones was changed to1/7, 3/7, 5/7 part of the half axis, and at 0◦ , 90◦ , 180◦ , 270◦ inthe tangential direction respectively (as illustrated in Fig. 1). Therewere totally 24 measurement points. Finally the transferred pres-sure levels were spatially averaged by:

pav = 1

n

√p2

1 + p22 + · · · + p2

n (1)

where pav denotes the average sound pressure (Pa) and p1, p2, pn

denote the sound pressure at the specific point respectively, n =24.

3. Numerical model description

A numerical model was developed to predict the inner pres-sure of a sandwich cylinder under a force excitation. It should benoted here that although the only FEM can be used to obtain theinner pressure of the sandwich cylinder under a force excitation, itwas not adopted. One reason is that three-dimensional meshes areneeded to model for the fluid cavity if only the FEM is used, andthis would increase lots of computation efforts in the frequencyrange of interest (400–2000 Hz). In addition, the coupling of theFEM structure/FEM fluid is needed to compute in each iteration ofthe optimization. To improve the computation efficiency, the innersound pressure of the cylinder was calculated using the coupledFEM/BEM. The structural response of the cylinder was obtained bythe FEM and the pressure at the field points inside the cylinderwas calculated using the acoustic transfer vector (ATV) method.

3.1. FE model for calculation of the surface velocity

The cylinder in the numerical model had the same geometryas in the experimental test. The inner radius equals 0.25 m andthe length equals 1 m. The geometry, mesh, properties and load

C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191 185

Fig. 2. Numerical models of the FEM structure and the BEM fluid, the largest element length is 0.028 m.

cases of the cylinder were built in MD Patran. As shown in Fig. 2,since the cylinder was excited by a point source at the symmetryplane in the axial direction, only a quarter of its geometry wasmodeled. The simply supported boundary condition (Tx, T y, T z =0) was applied at the cylinder edge. A unit point force for all thefrequencies was applied on the center of the cylinder surface.

Since the stress distribution in the thickness direction is notof interest in vibroacoustic analysis, shell elements with laminatedproperties represented the sandwich cylindrical structure and shellelements with isotropic properties represented the wooden endcaps. The cylindrical surface was meshed with rectangle QUAD4elements, and the two end caps were meshed with triangle TRIA3elements. The material properties refer to Table 1 and the dampingloss factor of the sandwich cylinder was set as 0.02. A point forcewas applied on the cylinder outer surface at [−0.25,0,0] in thenormal direction. A modal frequency response of the model wascomputed using MD Nastran, and the calculated structural velocitywas used to obtain the pressure at the field points.

3.2. The BE model for calculation of pressure at field points

The calculation of the pressure at the field points using the ATVmethod is introduced below briefly. The BE formulation can be de-veloped by discretizing the continuous Helmholtz integral equationto a discrete system. The Helmholtz integral equation is as fol-lows [18]:

c · p(R) =∫S

(p(R0)

∂ g

∂n0− g

(|R − R0|) ∂ p

∂n0

)dS (2)

g(|R − R0|

) = e−ik|R−R0|

4π |R − R0| (3)

v = i

ρ0ω

∂ p

∂n0(4)

where g is the Green’s function, S the boundary surface, and kthe wave number. R0 represents the points located at the bound-ary surface. R denotes the points set inside the boundary domain,p the sound pressure at the points, v the normal velocity of pointson the boundary surface. c is a factor dependent on the position ofR: c = 1 when R is inside the domain, c = 1/2 when R is on thesmooth boundary of fluid domain, c = Ω/4 when R is at the non-smooth edge, Ω is the edge angle. Considering a constant elementon a smooth surface, whose p and v are assumed to be constant,c = 1/2, and Eq. (2) can be discretized as:

1

2pi −

N∑j=1

(∫S j

g dS

)p j = iρ0ω

N∑j=1

(∫S j

g dS

)v j (5)

where j refers to elements on the boundary surface, and i refersto receiver nodes (field points), N is the total number of the el-

ements. Since it is assumed that p and v are constant over eachelement, they are labeled as p j and v j for element j. It should benoted that non-unique solutions problems would occur in solvingEq. (5). The Burton–Miller formulation [2], which is a linear combi-nation of the Helmholtz integral equation and its normal derivativeequation, is usually used to avoid the non-uniqueness of the math-ematical problem.

For the computation convenience, two matrices are introduced:

Hi j =∫S j

g dS, Gi j =∫S j

g dS (6)

By introducing

Hi j = 1

2δi j − Hi j (7)

where δi j is Kronecker delta, δi j = 0 when i �= j, and δi j = 1 wheni = j. Thus Hi j = Hi j when i �= j, Hi j = 1/2 when i = j [18]. Therelationship between the surface velocity and the surface pressurecan be rewritten in the matrix form as:

N∑j=1

Hi j p j = iρ0ω

N∑j=1

Gi j v j (8)

According to Eq. (5), the pressure at the interior field points canbe obtained by:

pi =N∑

j=1

Hi j p j + iρ0ω

N∑j=1

Gi j v j (9)

Combining Eq. (8) and Eq. (9), pi can be expressed in terms ofv j as the following:

pi(ω) = iρ0ω[Hi jH

−1i j (ω)Gi j(ω) + Gi j(ω)

]v j(ω) (10)

Let ATV = iρ0ω[Hi jH−1i j (ω)Gi j(ω) + Gi j(ω)]T , the pressure at

field points p f (matrix composed of pi) can be determined interms of the surface velocities v (matrix composed of vi ) as:

p f (ω) = ATVT (ω) · v(ω) (11)

As the surface velocity can be obtained from the FE analysis, thepressure at the field points can be calculated according to Eq. (11).In this model, the mesh of the BE model was identical to that ofthe FE model, and the mesh of the field points were the same withthe measurement position of the microphones and there were 18field points (see Fig. 2).

186 C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191

Fig. 3. Experimental and predicted spatially averaged inner pressure of the sandwichcylinder in a narrow-band frequency range (glass/epoxy facing, 90/90/90/Tycor®

/90/90/90, tf = 3.18 mm, tc = 15 mm).

Fig. 4. Experimental and predicted structural response of the sandwich cylinder ina narrow-band frequency range (glass/epoxy facing, 90/90/90/Tycor®/90/90/90, tf =3.18 mm, tc = 15 mm).

4. Comparison between the experimental and the numericalresults

The sound pressure inside the sandwich cylinder was predictedusing the model introduced in Section 3. The predicted and theexperimental results are compared in Fig. 3. It can be found thatthe numerical model gives a satisfactory prediction on the innersound pressure of the sandwich cylinder at most frequencies. Thereis a discrepancy of the first peak at low frequencies (at approx-imately 200 Hz). As shown in Fig. 4, the measured accelerance(acceleration/force) has a peak at 200 Hz while the predicted ac-celerance does not show this peak. Besides this area, there existssome discrepancy between the predicted accelerance level and theexperimentally measured accelerance level. The difference may bedue to the mounting conditions in the experimental test, i.e. theend caps were not perfectly contacted or connected with the cylin-der, while the end caps and the cylinder share the same nodesat the interface in the FE model. Another possible reason is thatthe properties used in the model refer to the material datasheetprovided by the material manufacturers and there could be somedifference between the property values in the datasheet and thepractical properties in the experimental test.

In general, as the cavity resonances also play an important roleon the sound transmission, the numerical model proved to be ableto predict the spatially average sound pressure inside the sandwichcylinder, despite the fact that there exists a bit larger difference of

Fig. 5. Spatially and frequency averaged inner pressure for various core types in1/6th octave bands (five frequencies averaged in each band), the facings are allglass/epoxy with layup 90/90/90/core/90/90/90, facing thickness = 3.18 mm, corethickness = 15 mm.

structural response of the cylinder between the prediction and themeasurement.

5. Numerically parametric study on the inner sound pressure ofsandwich cylinders

A parametric study on the sound insulation of the sandwichcylindrical shell was conducted using the numerical FEM/BEMmodel. The parameters include the core type, core thickness, sand-wich layup, and fiber orientations in the facings. The interestedfrequency range was from 400 to 2000 Hz in 1/6 octave steps.

5.1. The effect of core types

When sandwich structures are subjected to a bending momentor an out-of-plane force, one skin is subjected to tension and theother is subjected to compression, while the core is subjected toshear. Thus the core shear stiffness Gc plays a vital role withrespect to the dynamic response of sandwich structures. With re-spect to the propagation of sound waves in the sandwich struc-tures, at low frequencies, the wavelength is large compared to thepanel thickness and the sandwich panel behave like a uniformpanel with an equivalent stiffness in bending; at middle frequen-cies, the stiffness is controlled by the shear modulus of the core;at high frequencies, the bending stiffness of each skin dominatesthe sound transmission.

Three different kinds of core were chosen to investigate theinfluence of mechanical properties of the core on the sound insu-lation. As shown in Table 2, the mass densities of the three kindsof cores are nearly equal so that the influence of the mass onthe sound transmission can be neglected. Among the three kindsof cores, the aluminum honeycomb has the largest Gc and theRohacell® foam has the smallest Gc. The results in Fig. 5 show thatthe sandwich structure with the Tycor® foam core has the bestsound insulation at frequencies below 1000 Hz (around the coin-cidence frequency), and the sandwich with Aluminum honeycombhas the best sound insulation at 1000–2000 Hz. This phenomenoncan be explained in terms of the radiation efficiency and the struc-tural velocity. The radiation efficiency, which represents the abilityof the panel to radiate sound power into the fluid cavity per unitsquare of structural velocity, increases as the Gc increases belowthe coincidence frequency, while decreases as the Gc increases[13]. On the other hand, the structural surface vibration veloc-ity always decreases as the Gc increases. Therefore, the Tycor®

sandwich with a medium Gc has an optimal balance between the

C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191 187

Table 2Properties of three different kinds of cores (for the honeycomb, Ec denotes the compressive modulus, GL and GW denotes the transverse shear stiffness in the lengthand width direction, here is the length direction is the axial of the cylinder, G12 of the two honeycombs are neglected in most cases, here G12 = 10 MPa is used for thecomputations.)

Name Type Density [kg m−3] Ec [MPa] GL [MPa] GW [MPa] G12 [MPa]

Aluminum honeycomb 3/16-5052-.0015 70 999.7 468.8 206.8 10Tycor® foam glassfibers/PVC foam 70 151 68.7 68.7 68.7Rohacell® foam 71A/ PMI 75 92 29 29 29

Table 3Mechanical properties of the different sandwich structures.

A [MPa m] B [Pa m2] D [Pa m3]

tc = 5 mm 20.4 6.09 0 232 69 06.09 133 0 69 1560 00 0 14.9 0 0 172

tc = 15 mm(symmetrical)

20 5.9 0 1250 373 05.9 120 0 373 8190 00 0 14 0 0 916

tc = 25 mm 19 5.7 0 2880 859 05.7 108 0 859 18 300 00 0 13 0 0 2080

tc = 15 mm(unsymmetrical)

20 5.9 0 −8.4E4 −2.5E4 0 1160 348 05.9 120 0 −2.5E4 −5.8E5 0 348 7570 00 0 14 0 0 −6.3E4 0 0 848

Fig. 6. Spatially and frequency averaged inner pressure for various core thicknessin 1/6th octave bands (five frequencies averaged in each band, the facings are allglass/epoxy with layup 90/90/90/Tycor®/90/90/90).

structural response and the radiation efficiency, and thus it has thebest insulation below 1000 Hz. The aluminum honeycomb sand-wich with the highest Gc has the best insulation above 1000 Hz.

5.2. The effect of core thickness

Three sandwich cylinders with different core thickness (tc =5 mm,15 mm and 25 mm) were compared with respect to thesound insulation. Note that the skin thickness was varied accord-ingly to keep the total sandwich weight constant so that influencesof the mass law on the sound transmission differences can be ex-cluded. The skin thickness and stiffness matrices of the three sand-wiches are shown in Table 3. It can be seen that the membranestiffness A has a little change and bending stiffness D increasesas the core thickness increases. The spatially averaged sound pres-sure inside the three sandwich cylinders are shown in Fig. 6. Itcan be seen that the inner sound pressure of the cylinder withtc = 15 mm is the lowest below 1200 Hz and the cylinder withtc = 25 mm has the lowest inner sound pressure above 1200 Hz.

Fig. 7. Radiation efficiencies of sandwich cylinders with different core thickness.

The cylinder with tc = 5 mm has the worst sound insulation atmost frequencies. The radiation efficiencies of the three sandwichcylinders were calculated using the statistic energy method, asshown in Fig. 7.

Below 1200 Hz, as tc increases from 5 to 25 mm the radi-ation efficiency increases. On the other hand, the structural ve-locity decreases because that the bending stiffness D increases.The radiation efficiency and the structural velocity counteractthe sound insulation. In this situation, the sandwich with tc =15 mm achieved the best compromise between the radiation ef-ficiency and the structural velocity. Therefore the sandwich withtc = 15 mm presents the best insulation at low fluencies. Above1200 Hz, the radiation efficiency and of the cylinder with tc =25 mm is the lowest, and its surface velocity is the lowest be-cause of the highest bending stiffness. Therefore the cylinder withtc = 25 mm has the lowest sound pressure in this frequency range.

In addition, it can be found that the coincidence frequency de-creases as the core thickness increases, i.e., it is about 6000 Hz fortc = 5 mm, and about 2000 Hz for tc = 25 mm. The reason is thatthe coincidence frequency is proportional to the mass-to-stiffnessratio, and an increase of tc leads to a decrease of the mass-to-stiffness ratio.

188 C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191

Fig. 8. Spatially averaged inner pressure for different sandwich layups in 1/6th oc-tave bands (five frequencies averaged in each band, the facings are all glass/epoxy,t f = 3.18 mm, tc = 15 mm).

Fig. 9. Spatially averaged inner pressure for different fiber orientation in 1/6th oc-tave bands (five frequencies averaged in each band, the facings are all glass/epoxy,tf = 3.18 mm, tc = 15 mm).

5.3. The effect of sandwich layup

An unsymmetrical sandwich layup was compared with a sym-metrical layup, and their mechanical properties are shown in Ta-ble 3. By comparing the unsymmetrical layup to the symmet-rical layup, it can be found that A is not changed and D de-creases slightly. The most significant change is that the bending-extensional coupling matrix B is not zero anymore. This leads to asignificant decrease of the sound insulation, as shown in Fig. 8. Inmost cases, if B is non-zero, an in-plane wave causes an out-planewave as well, which is not beneficial for vibration and noise con-trol. Whitney [30] studied the effect of bending-extensional cou-pling of laminated plates and concluded that the coupling couldincrease the maximum deflection by as much as 300%. In addition,the bending-extensional coupling may also cause some unexpecteddamage to the structure. Thus the unsymmetrical layup is usuallyavoided in the engineering design.

5.4. The effect of fiber orientation in the facings

The fiber orientations in the facings were varied to investi-gate its influence on the sound insulation of sandwich cylinders.The inner sound pressures of the three sandwich cylinders withdifferent facing layups are presented in Fig. 9. The fiber orien-tation value of 0 denotes that the fibers are along the hoop

direction and 90 denotes along the axial direction of the cylin-der. Results showed that the curve representing the layup of0/10/20/core/70/80/90 presented more fluctuations than the othertwo curves. This could attribute to the bending-extensional cou-pling of the unsymmetrical laminated layup. With respect to thecomparison of fiber orientation, the 0 degree showed a similartrend with the 90 degrees below 1024 Hz. However, it is diffi-cult to determine which orientation shows better sound insulationthan the other in a broad frequency range (400–2000 Hz). In spiteof this phenomenon, the fiber orientation has a large influenceon the sound insulation in a narrow-band frequency. For example,the transferred pressure difference between the 0/0/0/core/0/0/0and the 0/10/20/core/70/80/90 is about 15 dB at 1093 Hz. Thusthe fiber orientation could be a useful parameter for sound opti-mization in a narrow frequency range, while it is not effective insituations with broad band excitations.

6. Optimization of the sandwich cylinder: a case study

The Genetic Algorithm (GA) is one of the most powerful op-timization tools when the objective function and constraints arehighly nonlinear, and contain many local optima and discontinu-ities in the design domain. In such situations, gradient based tech-niques can fall into a local optima near the starting configuration,while GA can avoid it and reach a more global optimum [9]. There-fore an optimization tool which integrates the FEM/BEM and GAmethod is used to make a multi-discipline and multi-objective op-timization. To demonstrate the ability of the codes, a simple casestudy is shown in this section.

6.1. Introduction of the optimization algorithms

In most real-life optimization situations, there is no unique op-timal solution (Pareto solution) for the multiple-criteria or multi-objective problems. The function gamultiobj, which is an optimiza-tion tool based on the Non-dominated Sorting Genetic Algorithm II(NSGA-II) in Matlab, was used for the multi-objective optimization.The gamultiobj solver attempts to create a set of Pareto front for amulti-objective minimization [6]. The Pareto front, which denotesthe set of all Pareto solutions, gives a trade-off limit as an aid ofdecision making for the complex optimization. The flow chart ofthe optimization system is shown in Fig. 10. The global bucklingand the strain were computed using MD Nastran during the op-timization. The input data and output data for MD Nastran wereread by Matlab. The inner pressure was calculated in Matlab us-ing the ATV and the structural velocity from Nastran, according toEq. (10). In this way, the gamultiobj in Matlab could control theoptimization loop for both the mechanical analysis and the acous-tic analysis. Some parameters of the NSGA-II optimizations are:crossover fraction = 0.8, migration/mutation fraction = 0.2, Paretofraction = 0.5.

As shown in Fig. 11, a fuselage section composed of sandwichcomposites was studied for the optimization. The facing mate-rial was carbon/epoxy and its mechanical properties are: E1 =1.81e11 Pa, E2 = 1.03e10 Pa, G12 = 7.17e9 Pa, ρf = 1600 kg/m3,υ12 = 0.28. The core material was rohacell200 and its mechanicalproperties are: Ec = 3.5e8 Pa, Gc = 1.5e8 Pa, ρc = 205 kg/m3. Itshould be noted that Gc was set as one of the design variables,and Ec was also changed according to Ec = 2Gc(1 + υc), where υcwas kept as 0.3. The sandwich layup was [90/0/45/core/45/0/90].The layer thickness in the top facing tf1_layer, the core thickness tcand the layer thickness in the bottom facing tf2_layer were designvariables.

For the mechanical analysis, the fuselage was subjected to acombined loading case including a bending moment, a torsion mo-ment, a shear force and an inner pressure. A simply supported

C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191 189

Fig. 10. The flowchart of the multi-objective and multi-discipline optimization in Matlab.

boundary condition was applied at one end of the cylinder, and theother end was free. The combined loading was applied at the freeend of the cylinder. The global buckling and the material failureswere set as the design constraints. For the computations of thebuckling and the materials failures of composite sandwich cylin-ders, reference was made to [34].

For the acoustic analysis, the cylinder was subjected to twosymmetric point force excitations at 60 Hz, because the mechani-cal excitation from the shaft engine is usually tonal and it is 60 Hzin a turbo-prop aircraft according to [24]. The boundary conditionwas the same as the mechanical analysis. There were six seats inthe fuselage section, and only three of them were chosen to ob-serve the noise level because the cylinder was symmetric in the xdirection. Three points were used to simulate the passengers’ ears,pressures p1, p2 and p3, their positions are shown in Fig. 11. Thespatial average noise level at 60 Hz was set as one of the objec-tives. Finally, a summary of the optimization is given as follows:

Objectives: Min(Ntf1_layerρf + tcρc + Ntf2_layerρf),

Min(

√13 (p2

1 + p22 + p2

3))

where N denotes the layer number for the facings, here N = 3.

Design variables: 0.1 mm < tf1_layer < 1 mm, 20 mm < tc <

100 mm, 0.1 mm < tf2_layer < 1 mm,

0.38 MPa < Gc < 384 MPa

Constraints: Global buckling (|λ| > 1)Strain criterion of the facing (|ε11|, |ε22|, |ε12| < 0.4%)Strain criterion of the core (|εc| < 3.5%)

where |λ | denotes the absolute value of the eigenvalue of thebuckling analysis.

Fig. 11. The fuselage section for the multi-objective case-study optimization.

6.2. Results and discussion

Four optimizations with different population sizes and gener-ations were studied and their optimization results are shown inFig. 12. The Pareto front results imply that the total weight iscompeting with the internal noise pressure level. In addition, theinfluences of the population sizes and generations on the optimiza-tion results can be found by comparing the Pareto fronts of thefour optimizations. The third optimization (opti 3) has the worstoptimization results in that the sound pressure is higher at otheroptimizations with a comparable mass. The reason should be thegeneration number of the third optimization is too low, which isonly two. The fourth optimization (opti 4) has the largest diversityin the Pareto fronts and this is a characteristic indicating a bet-ter optimization. In addition, the second optimization (opti 2) also

190 C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191

Table 4Design variables and the corresponding objectives of the four points in Fig. 12.

Point tf1_layer (mm) tc (mm) tf2_layer (mm) Gc (MPa) Weight (Kg/m2) Noise level (dB)

1 0.37 26.6 0.16 230 7.99 55.552 0.34 30.5 0.22 306 8.94 50.443 0.31 36.8 0.14 355 9.76 44.554 0.85 32.7 0.32 175 12.30 40.97

Fig. 12. The Pareto front and the Score diversity of the multi-objective optimization.

shows a better optimization results because its Pareto front is thelowest when the total weight is at the range of 9–12 kg/m2. It canbe learned from the Pareto front analysis that a better NSGA-II op-timization would be achieved if a larger population size and moregenerations are set for the optimization. Moreover, it is good torun different optimizations with varied parameters so that bettercompromises between the sound pressure and the total weight canbe obtained.

In order to give more detailed information of the optimiza-tion, four points representing the best optimization results wereselected, see Fig. 12. The design variables of the four points areshown in Table 4. Although the parameter Gc has a wide range tovary in the design (0.38–384 MPa), its final value is located at arelatively narrow range (175–355 MPa) after the optimization Thisindicates that the optimization algorithm makes a compromise be-tween the acoustic property and the mechanical requirement. Sim-ilarly, the core thickness is also confined to a narrow range afterthe optimization, which is in the range of 26.6 to 36.8 mm. In ad-dition, the layer thickness of the outer facing is larger than thatof the inner facing and the scale factor is 1.5–2.6. Eventually it isdemonstrated that a multi-discipline and multi-objective optimiza-tion can be done using the developed codes in Matlab. Based onthe obtained Pareto front, a further compromise can be made be-tween the weight and the acoustic performance according to theother design requirements of the aircraft, such as the operationalcost and the thermal insulation.

Finally, it should be noted that a large amount of computa-tion time can be saved by adopting the described optimizationmethod, because the ATV matrix is solved only once in the opti-mization iterations. In this optimization example, the computationfor the iteration took about 1.5 min using the Intel i7-2637M dual1.7 Hz CPU. When ATV concept was not used, the computationof the full FEM/BEM model took about 40 min for each iteration.The mentioned computation efficiency difference is only for a sim-ple cylinder model and much more time would be saved for morecomplex models in reality.

7. Conclusions

An FE/BE model was built to predict the inner pressure of thecylinder under a mechanical excitation. Shell elements were usedto build the FE model and the laminate theory was used to com-pute the stiffness matrix. The acoustic transfer vector (ATV) wasused to improve the computation efficiency. A parametric studywas conducted on the sound insulation of sandwich cylinders anda multi-objective optimization was carried out to improve thesound insulation of sandwich fuselage considering the mechanicalconstraints.

The FE/BE models for predicting the sound transmission ofsandwich cylinders at low frequencies have been successfully vali-dated by experimental measurements.

The influence parameters on the sound insulation of the sand-wich cylinder were studied numerically and it was found that:The coincidence frequency, fc, is an important characteristic forthe sound transmission of sandwich structures. A higher bendingstiffness leads to higher radiation efficiency below the fc while itleads to lower radiation efficiency above the fc. Therefore a higherbending stiffness is beneficial for the noise control at high fre-quencies (> fc). At low frequencies (< fc), there exists an optimalbending stiffness which can make a good compromise between thestructural response and the radiation efficiency. In other words, in-creasing the core thickness to a certain extent (there is an optimalcore thickness for the largest bending stiffness in case of constantweight) or the core shear stiffness can improve the TL at high fre-quencies while right values of the core thickness or core shearstiffness exist for the best sound insulation at low frequencies. Theunsymmetrical layup causes the extensional-bending coupling, in-creasing the vibration of the structure and thus is not beneficialfor sound insulation. The sound insulation of composite cylindersin a narrow frequency range can be improved by changing the fiberorientation. However, it is difficult to find an optimal fiber orienta-tion layup to have the best sound insulation in a broad frequencyrange.

The optimization system which integrates the FEM/BEM, ATVand GA methods has been proven to be efficient and robust forsolving the multi-variable nonlinear problems. The codes writtenin Matlab are able to make a multi-discipline (the mechanical andacoustic analysis) and multi-objective optimization (minimizationof weight and minimization of inner noise). It is found that theATV method can improve the computation efficiency significantly,because that the ATV matrix builds the relationship between thesurface velocity and the pressure at field points and it is onlyneeded to compute once during the optimization loop. A seriesof possible configurations representing compromises among theweight, the mechanical performance and the acoustic propertieswere achieved by using the developed optimization system. Al-though the optimization system is only demonstrated with a sim-ple fuselage section structure, the essential algorithm of the systemcan be extended for the optimization work of more complex struc-tures, and the parameters in the gamultiobj function can be variedto find more reasonable solutions of the practical problems.

C. Yuan et al. / Aerospace Science and Technology 30 (2013) 183–191 191

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