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Lightweight design of composite laminated structures with frequency constraint
Accepted Manuscript Lightweight design of composite laminated structures with frequency constraint Qimao Liu, Juha Paavola PII: DOI: Reference:
S0263-8223(15)00812-0 http://dx.doi.org/10.1016/j.compstruct.2015.08.116 COST 6821
To appear in:
Composite Structures
Please cite this article as: Liu, Q., Paavola, J., Lightweight design of composite laminated structures with frequency constraint, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.08.116
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Lightweight design of composite laminated structures with frequency constraint Qimao Liu ∗ Juha Paavola Department of Civil and Structural Engineering, Aalto University, Espoo, FI-00076 Aalto, Finland
Abstract This paper describes a method for lightweight design of composite laminated structures using optimization technique. The lightweight design optimization model aims to minimize the mass of the composite laminated structures with the fundamental frequency constraint. The design variables are the fibre volume fractions and fibre orientations of the layers. The first derivatives of the weight and frequencies with respect to design variables are computed. The lightweight design optimization model is converted into a series of linear constrain optimization problems using the interior point penalty function method. The sensitivity of the penalty function is calculated using the sensitivity information of the weight and frequencies. An optimization procedure based on the gradient projection algorithm is proposed. Finally, the lightweight designs of two composite laminated plates are performed by using the proposed optimization procedure. The merits of the proposed procedure are also discussed. Keywords: Composite structures; Laminated plate and shell; Lightweight design; Optimization; Vibration 1. Introduction Composite laminated structures are gaining more and more applications in aircraft, automobile, naval and defence industries because of their high performance characteristics, such as high strength-to-weight ratio, high stiffness-to-weight ratio, superior fatigue properties and high corrosion resistance [1, 2]. Therefore, the composite laminated structures are very attractive in the vehicle industries (aircraft, spacecraft, automobile, ship etc.) because the vehicles made of lightweight material can save energy (gas) and reduce the carbon emission. The composite laminated structures are believed to be promising in the ∗
Corresponding author. Tel.: +358 (0) 503502879; Fax: +358 (0)947023758. E-mail address: [email protected].
vibration environment because the lightweight material attracts smaller inertia forces than heavy material. Many researchers have employed optimization methods to achieve the optimal designs of the composite laminated structures in vibration environment. For example, Apalak et al. [3] used the Artificial Bee Colony algorithm to maximize the fundamental frequency of the symmetrical laminated composite plates by optimizing the fibre orientations. Honda et al. [4] employed the non-dominated sorting genetic algorithm to maximize the fundamental frequency and minimize the curvature of the curvilinear fibres for the laminated composite plates, and the design variables are the coefficients of shape of the curvilinear fibres. Bargh and Sadr [5] used the Particle Swarm Optimization Method to maximize the fundamental frequency of the composite finite strip by optimizing the fibre orientations. Bargh and Sadr [6]also employed the Elitist-Genetic Algorithm to maximize the fundamental frequency of the composite laminated plates by optimizing the fibre orientations. Topal [7, 8] performed the frequency optimization of composite laminated plates and skew sandwich plates using the modified feasible direction method to find the optimum fibre orientations.
Cho [9] implemented design optimization of the laminated
composite plates with static and dynamic considerations in the hygrothermal environments. The objective is the weight of the structures and the design variables are the thickness of the layers. A gradient-based modified feasible direction optimization algorithm is used to solve the optimization problem. Apalak et al. [10] demonstrated the layer optimization for maximum fundamental frequency of rigid point-supported laminated composite plates. The Genetic Algorithm combined with neural networks is used to carry out the optimization. Murugan et al. [11] employed response surface method to optimize the fibre orientations of the composite aerofoil cross-section so that the vibration of the helicopter is reduced. Narita [12, 13] conducted the layerwise optimization for the maximum fundamental frequency of laminated composite plates.
Many researchers also conducted research in static optimization [14, 15] and buckling
optimization [16, 17]. However, the paper is focus on the optimization design of composite laminated structures in vibration environment. Therefore, the static and buckling optimization designs of composite laminated structures are not reviewed in detail in this paper. From the above-mentioned we can see that the objectives are almost focused on the fundamental frequency and the design variables are almost the fibre orientations. The algorithms are almost focused on non gradient-based algorithms. Only the
literature [9] minimized the weight by adjusting the thickness of the layers. Cho [9] also used gradientbased optimization algorithm to search the optimum design. The sensitivity analysis results in the literature [18] show that the fibre volume fractions have much more influence on the frequencies of the composite laminated structures than the fibre orientations. The fibre orientations have no relationship with the weight of the structures. Therefore, the fibre volume fractions or thickness of the layers have to be treated as design variables in the lightweight design. The aim of the paper is to develop an optimization method for the lightweight design of composite laminated structures. First, the lightweight design optimization model is proposed to find the fibre volume fractions and fibre orientations of each layer to minimize the mass with the fundamental frequency constraint. The optimization model is also turned into dimensionless. Second, the first derivatives of the weight and frequencies with respect to the fibre volume fractions and fibre orientations are implemented based on author’s previous work [18]. Third, the lightweight design optimization model is converted into a series of linear constraint optimization problem by using the interior point penalty function method. The sensitivity of the interior point penalty function is obtained using the sensitivity information of the weight and frequencies. Fourth, an optimization procedure is proposed based on the gradient projection algorithm. Finally, the lightweight designs of the square and T-shape composite laminated plates are performed by using the proposed optimization procedure. The merits of the proposed procedure are also discussed. 2. Lightweight design optimization model Many researchers have conducted the layerwise optimization for the maximum fundamental frequency of laminated composite plates and shells. However, the goal of this paper is to find the fibre volume fractions and fibre orientations of the layers to minimize the mass of the composite laminated structures with the fundamental frequency constraint. The lightweight design optimization model is built as follows. I I Find d = rf( ) and θ ( )
( )) subject to ω ( r ( ) ,θ ( ) ) ≥ ω minimize W rf(
I
I
1
I
1
f
− 90o ≤ θ ( ) ≤ 90o I
0 ≤ rf( ) ≤ 1 I
( I = 1, 2,L , N )
(1)
where the design variable vector d includes the fibre volume fractions rf( I ) and fibre orientations θ ( I) of
( )
the layers. N is the total number of the layers of composite laminated structures. W rf( I )
(
ω1 rf( I ) ,θ ( I)
)
and
are the mass (objective function) and fundamental frequency (constraint function),
respectively. ω1 is the lower limit on the fundamental frequency. In order to make the objective function and constraint function at the same numerical value range, the optimization model is converted into dimensionless form as I I Find d = rf( ) and θ ( )
( ))
W rf(
I
( ) = W r ( ) = 1 (I )
minimize w rf
I
(
(I )
subject to g rf , θ
)=
( I)
f
(
(I )
ω1 rf , θ ( I) ω1
) −1 ≥ 0
(2)
− 90o ≤ θ ( ) ≤ 90o I
0 ≤ rf( I ) ≤ 1
( I = 1, 2,L , N )
I I where W rf( ) = 1 is the mass of the structure when the fibre volume fractions rf( ) = 1 ( I = 1, 2,L , N ).
3. Sensitivity analysis of objective function and constraint function 3.1 First derivatives of the objective function Differentiating the objective function in Eq. (2) with respect to the fibre volume fractions rf( I ) , we have the sensitivity of the objective function (the mass).
( )) =
∂w rf( ∂rf(
( ))
∂W rf(
I
I)
1
∂rf(
I W rf( ) = 1
I
(3)
I)
( ) and its derivatives with respect to fibre volume fraction r (I )
where the mass W rf
(I ) f
, i.e.,
( ) ) , can
∂W rf(
I
∂rf( I )
be computed as
( ) ) = ∑ ∑ ρ ( )Ω ( z
W rf(
N
I
k
e
k =1
N
= ∑Ω ×∑ ρ e
e
e
k =1
(k )
k +1
− zk )
( zk +1 − zk )
(4)
Here Ωe is the area of the e th composite laminated element. N is the total number of the layers of composite laminate. zk and zk +1 are the z coordinates of the top and bottom surfaces of the k th layer.
ρ ( k ) is the density of the k th layer.
(
)
ρ ( k ) = rf( k ) ρ f + 1 − rf( k ) ρ m
(5)
where ρ f and ρ m are the densities of the fibre and matrix, respectively. Differentiating Eq. (4) with respect to fibre volume fractions rf( I ) , we have
( )) =
∂W rf(
I
∂rf( I )
∑Ω
e
e
×
∂ρ ( I ) ( zI +1 − zI ) ∂rf( I )
(6)
where ∂ρ ( ) = ρ f − ρm I ∂rf( ) I
(7)
Since the fibre orientations have no influences on the mass, therefore we have
( )) = 0
∂w rf( ∂θ (
I
(8)
I)
3.2 First derivatives of the constraint function Differentiating the constraint function in Eq. (2) with respect to the fibre volume fractions rf( I ) and fibre orientations θ ( I ) , we obtain the sensitivity of the constraint function (fundamental frequency).
(
∂g rf( ) , θ ( ) I
I
∂rf(
(
I) I
∂θ
( I)
I
I
1
ω1
∂g rf( ) , θ ( ) I
) = 1 ∂ω ( r ( ) ,θ ( ) )
)
f
∂rf(
(
I)
( ) () 1 ∂ω1 rf , θ = I ω1 ∂θ ( ) I
I
(9)
)
(
)
In Eq. (9), the frequency ω1 rf( I ) ,θ ( I ) and its derivatives with respect to the fibre volume fractions rf( I )
and fibre orientations θ
( I)
(i.e.,
(
∂ω1 rf( ) ,θ ( ) I
∂rf( I )
I
)
and
(
∂ω1 rf( ) ,θ ( ) I
I
∂θ ( ) I
) ) have been achieved analytically in
author’s previous work [18]. The detailed formula derivations are demonstrated in the reference [18]. The computational procedure of the first derivatives of the frequencies is summarized as follows.
(1) The first derivatives of the engineering constants (i.e., the modulus of elasticity of the fibre E f , modulus of elasticity of the polymer-based matrix Em , Poisson's ratio of the fibre ν f , Poisson's ratio of the polymer-based matrix ν m , density of the fibre ρ f and density of the polymer-based matrix ρ m ), of the lamina, with respect to the fibre volume fraction and fibre orientation are computed based on the micromechanics theory. (2) The first derivatives of the stiffnesses (i.e., Aij , Dij , Bij and H ij ), of laminate, with respect to the fibre volume fraction and fibre orientation are calculated using the classical or first-order theories of composite plate and shell. (3) The first derivatives of the total stiffness matrix K and total mass matrix M with respect to fibre volume fraction and fibre orientations are computed based on the linear finite element methods of composite plates and shells of the classical or first-order theories. (4) Finally, the first derivatives of frequencies are calculated using the total stiffness matrix K and total mass matrix M and their first derivatives. 4. Penalty function and optimization algorithm 4.1 Penalty function and its sensitivity
It is convenient to calculate the gradient of the interior point penalty function when the direct differentiation method is used to obtain the first derivatives. For this reason, the interior point penalty function method is employed in this paper. There are various penalty terms in the interior point penalty functions. However, the popular forms are the inverse penalty term and the logarithm penalty term. Either of the terms can be used to construct the interior point penalty function. In this paper, the logarithm penalty term is used to construct the interior point penalty function. The interior point penalty function is defined in this paper as
(
) ( )
(
F rf( I ) , θ ( I ) , γ k = w rf( I ) − γ k ln g rf( I ) ,θ ( I )
)
(10)
where γ k is the penalty parameter. The penalty parameter is updated according to the following rule:
γ k +1 =
γk c
where the constant c = 5 to 50 ( c = 10 in this paper).
(11)
Therefore, the constraint optimization problem Eq. (2) has been converted into a series of linear constraint optimization problems.
(
Minimize F rf( ) ,θ ( ) , γ k I
I
Subject to − 90o ≤ θ
(I )
)
≤ 90o
(I )
(12)
( I = 1, 2,L , N )
0 ≤ rf ≤ 1
The first derivatives of the interior point penalty function with respect to the fibre volume fractions rf( I ) and fibre orientations θ ( I ) can be computed as follows.
(
∂F rf( I ) ,θ ( I ) , γ k ∂rf( I )
(
∂F rf( I ) ,θ ( I ) , γ k ∂θ
( I)
) = ∂w ( r ( ) ) − γ
(
∂g rf( I ) , θ ( I )
I
1
f
∂rf( I )
) = −γ
k
(
g rf( I ) ,θ ( I )
(
∂g rf( I ) , θ ( I )
1 k
(
(I )
g rf , θ
)
( I)
)
∂θ
)
∂rf( I )
)
(13)
( I)
The derivatives on the right hand side of Eq. (13) have been computed in Section 3. 4.2 Optimization algorithm of lightweight design
When the gradient of the objective function is known, the effective gradient-based optimization algorithms for a linear constraint optimization problem are the gradient projection algorithm and feasible direction algorithm. In this paper, the gradient projection algorithm is employed to solve a series of constraint optimization problem Eq. (12). The computation procedure of solving the mathematic model Eq.(1): Step 1. Choose the initial feasible design point d 0 , set γ 1 =10 (in this paper), define convergence criterion
ε1 =5 kg (in this paper) and let k =1. Step 2. Start from design point d k −1 , solve the linear constraint mathematic model Eq. (12) with the
gradient projection algorithm to obtain the optimum design d k . Step 3. If W ( d k −1 , rk ) − W ( d k , rk ) ≤ ε1 , d k is the best design. Print results and stop. Otherwise, update the
penalty parameter γ k +1 =
γk 10
, k = k + 1 and go to Step 2.
5. Examples
In this section, the optimization design of a T-shape composite laminated plate and square composite
laminated plate are implemented using the aforementioned optimization model and proposed lightweight design optimization procedure in this section. The two composite structures have the same material properties, i.e., the fibre material T800H: E f =294 GPa, ν f =0.2, ρ f =1.81 g/cm3 and the matrix material QY9511: Em =4.2 GPa, ν m =0.3, ρ m =1.24 g/cm3. Both composite structures have the same layered number N =8, thickness of each layer (5 mm) and thickness of laminate (40 mm). The laminate is symmetric and the fibre orientations are [0/90/45/-45]s. In the examples, only the fibre volume fractions are chosen to be design variables. The fibre volume fractions are rf( I ) =60% ( I = 1, 2,L , N ) in the initial designs of the square and T-shape plates. 5.1 Square composite laminated plate
The square composite laminated plate with 2 m × 2 m is shown in Fig. 1. One side of the square composite laminated plate is clamped. The lower limits on the fundamental frequency are ω1 =10 Hz (case 1) or ω1 =5 Hz (case 2), respectively, as shown in Table 1. The square plate is divided by 64 nonconforming rectangular elements [18].
2m
y 2m
x
Fig. 1 Square composite laminated plate Table 1 Initial design and optimum design of square plate Fibre volume fractions (%) Mass (kg) Frequency (Hz)
Initial design Optimum design 1 Optimum design 2
rf(1)
rf( 2)
rf( 3)
rf( 4)
W
ω1
60
60
60
60
253
10.34
0.1
56.3
0.1
0.1
211
10.22
0.3
10.6
0.8
0.1
201
5.33
Lower limit on frequency (Hz) CPU time (sec)
ω1
10 (case 1) 5 (case 2)
20 12
rf(1) =60, θ (1) =0
rf(1) =0.1, θ (1) =0
rf(1) =0.3, θ (1) =0
rf( 2) =60, θ ( 2 ) =90
rf( 2) =56.3, θ ( 2 ) =90
rf( 2) =10.6, θ ( 2 ) =90
rf( 3) =60, θ ( 3) =45
rf( 3) =0.1, θ ( 3) =45
rf( 3) =0.8, θ (3) =45
rf( 4) =60, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 5) =60, θ (5) =-45
rf( 5) =0.1, θ (5) =-45
rf( 5) =0.1, θ ( 5) =-45
rf( 6) =60, θ ( 6 ) =45
rf( 6) =0.1, θ ( 6 ) =45
rf( 6) =0.8, θ ( 6 ) =45
rf( 7 ) =60, θ ( 7 ) =90
rf( 7 ) =56.3, θ ( 7 ) =90
rf( 7 ) =10.6, θ ( 7 ) =90
rf(8) =60, θ (8) =0
rf(8) =0.1, θ (8) =0
rf(8) =0.3, θ (8) =0
(a) Initial design
(b) Optimum design ( ω1 =10 Hz)
(c) Optimum design ( ω1 =5 Hz)
Fig. 2 Layers of initial design and optimum design of square shape plate (view point from the positive x ) 5.2 T-shape composite laminated plate
The T-shape composite laminated plate and it dimensions are shown in Fig. 3. One side of the Tshape is clamped. The lower limit on the fundamental frequency are ω1 =1.5 Hz (case 3) or ω1 =1 Hz (case 4), respectively, as shown in Table 2. The fibre volume fractions are rf( I ) =70% ( I = 1, 2,L , N ) in both initial designs. The T-shape plate is divided by 256 nonconforming rectangular elements.
2m
2m
2m
y x
2m
2m
Fig. 3 T-shape composite laminated plate
Table 2 Initial design and optimum design of T-shape plate Fibre volume fractions (%) Mass (kg) Frequency (Hz)
Initial design Optimum design 1 Optimum design 2
rf(1)
rf( 2)
rf( 3)
rf( 4)
W
ω1
70
70
70
70
1049
1.60
0.1
91.6
0.1
0.1
877
1.53
2.1
24.6
13.5
0.1
830
1.0
Lower limit on frequency (Hz) CPU time (sec)
ω1
1.5 (case 3) 1.0 (case 4)
rf(1) =70, θ (1) =0
rf(1) =0.1, θ (1) =0
rf(1) =2.1, θ (1) =0
rf( 2) =70, θ ( 2 ) =90
rf( 2) =91.6, θ ( 2 ) =90
rf( 2) =24.6, θ ( 2 ) =90
rf( 3) =70, θ ( 3) =45
rf( 3) =0.1, θ ( 3) =45
rf( 3) =13.5, θ ( 3) =45
rf( 4) =70, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 5) =70, θ (5) =-45
rf( 5) =0.1, θ (5) =-45
rf( 5) =0.1, θ ( 5) =-45
rf( 6) =70, θ ( 6 ) =45
rf( 6) =0.1, θ ( 6 ) =45
rf( 6) =13.5, θ ( 6 ) =45
rf( 7 ) =70, θ ( 7 ) =90
rf( 7 ) =91.6, θ ( 7 ) =90
rf( 7 ) =24.6, θ ( 7 ) =90
rf(8) =70, θ (8) =0
rf(8) =0.1, θ (8) =0
rf(8) =2.1, θ (8) =0
(a) Initial design
131 47
(b) Optimum design ( ω1 =1.5 Hz) (c) Optimum design ( ω1 =1 Hz)
Fig. 4 Layers of initial design and optimum design of T-shape plate (view point from the positive x ) 5.3 Discussions
The results (Table 1 and Table 2) show that the fundamental frequencies of the optimum designs are almost full constrained (i.e., with a fundamental frequency value reaching or nearly reaching the allowable limit). Such results indicate that a composite structure can usually be made lighter/more economical (since the fibre is more expensive and has bigger density than the matrix) by allowing decreased fundamental frequency to occur and having the frequency constraint to attain a fully constrained condition. The case 1 (initial design: 253 kg and 10.34 Hz, optimum design: 211 kg and 10.22 Hz, lower limit: 10 Hz) and case 3 (initial design: 1049 kg and 1.6 Hz, optimum design: 877 kg and 1.53 Hz, lower limit: 1.5 Hz) show that even though the fundamental frequency values of the initial designs are very close to the allowable limit, the weight of composite laminated structures can be further reduced by adjusting the fibre volume fractions of the layers and the constraint is still satisfied. The
optimum designs (Fig. 2 (b) (c) and Fig. 4 (b) (c)) demonstrate that the proposed optimization model and computational procedure can effectively optimize the composite structure to obtain its most optimal distribution of the fibre volume fractions at the composite laminate. The maximum fibre volume fractions (Fig. 2 (b) and Fig. 4 (b)) are allocated at the 2nd and 7th layers in the optimum designs. Such results also indicate that the sandwich composite structures are reasonable composite type in the vibration environment. The computations are carried out using a desk computer (AMD Phenom(tm) II X2 550 Processor, and frequency of the CPU is 3.10 GHz). The computational time (CPU time in Table 1 and Table 2) is 10 sec and 20 sec for square plate, 47 sec and 131 sec for T-shape plate. It indicates the proposed optimization model and computational procedure are efficient to achieve the optimum designs. 6. Conclusions
The paper proposes a lightweight design method for composite laminated structures in vibration environment using the gradient algorithm based on the analytical sensitivity analysis method of frequencies which is developed in author’s previous work [18]. The numerical results indicate that the optimization model and computational procedure in this paper are effective and efficient to achieve the lightweight design of the composite laminated structures. In the situation, the fundamental frequency values of the initial designs are very close to the allowable limit, the composite laminated structures still can be lighter by adjusting the fibre volume fractions of the layers and the constraint can surely be satisfied. The optimum distributions of the fibre volume fraction illustrate that the sandwich composite structures are reasonable composite structure type in the vibration environment. In the future work, the lightweight design method proposed in this paper will be applied to the large scale engineering composite structures with complex boundary conditions and also subject to the static strength constraint. References
[1] Jones RM. Mechanics of Composite Materials. 2nd Ed. Philadelphia (PA): Taylor & Francis; 1999. [2] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. 2nd ed. Washington (DC): CRC Press; 2003.
[3] Apalak MK, Karaboga D, Akay B. The artificial bee colony algorithm in layer optimization for the maximum fundamental frequency of symmetrical laminated composite plates. Engineering optimization 2014; 46(3): 420-437. [4] Honda S, Igarashi T, Narita Y. Multi-objective optimization of curvilinear fiber shapes for laminated composite plates by using NSGA-II. Composites: Part B 2013; 45: 1071-1078. [5] Bargh HG, Sadr MH. Stacking sequences optimization of composite plates for maximum fundamental frequency using particle swarm optimization algorithm. Meccanica 2012; 47:719-730. [6] Sadr MH, Bargh HG. Optimization of laminated composite plates for maximum fundamental frequency using Elitist-Genetic algorithm and finite strip method. [7] Topal U. Frequency optimization of laminated composite plates with different intermediate line supports. Science and Engineering of Composite Materials 2012; 19(3): 295-306. [8] Topal U, Uzman U. Frequency optimization of laminated composite skew sandwich plates. Indian Journal of Engineering & Material Sciences 2013; 20:101-107. [9] Cho HK. Design optimization of laminated composite plates with static and dynamic considerations in Hygrothermal environments. International Journal of Precision Engineering and Manufacturing 2013; 14(8): 1387-1394. [10] Apalak ZG, Apalak MK, Ekici R, Yildirim M. Layer optimization for maximum fundamental frequency of rigid point-supported laminated composite plates. Polymer Composites 2011; 32(12):19882000. [11] Murugan MS, Ganguli R, Harursampath D. Surrogate based design optimization of composite aerofoil cross-section for helicopter vibration reduction. The Aeronautical Journal 2012; 116: 709-725. [12] Narita Y. Layerwise optimization for the maximum fundamental frequency of laminated composite plates. Journal of Sound and Vibration 2003; 263: 1005-1016. [13] Narita Y, Hodgkinson JM. Layerwise optimisation for maximising the fundamental frequencies of point-supported rectangular laminated composite plates. Composite Structures 2005; 69:127-135. [14] Manh TL, Lee J. Stacking sequence optimization for maximum strengths of laminated composite plates using genetic algorithm and isogeometric analysis. Composite Structures 2014; 116: 357-363.
[15] Hwang SF, Hsu YC, Chen Y. A genetic algorithm for the optimization of fiber angles in composite laminates. Journal of Mechanical Science and Technology 2014; 28 (8): 3163-3169. [16] Jing Z, Fan X, Sun Q. Stacking sequence optimization of composite laminates for maximum buckling load using permutation search algorithm. Composite Structures 2015; 121: 225-236. [17] Bohrer RZG, Almeida SFM, Donadon MV. Optimization of composite plates subjected to buckling and small mass impact using lamination parameters. Composite Structures 2015; 120: 141-152. [18] Liu Q. Analytical sensitivity analysis of frequencies and modes for composite laminated structures. International Journal of Mechanical Sciences 2015; 90: 258-277.
S0263-8223(15)00812-0 http://dx.doi.org/10.1016/j.compstruct.2015.08.116 COST 6821
To appear in:
Composite Structures
Please cite this article as: Liu, Q., Paavola, J., Lightweight design of composite laminated structures with frequency constraint, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.08.116
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Lightweight design of composite laminated structures with frequency constraint Qimao Liu ∗ Juha Paavola Department of Civil and Structural Engineering, Aalto University, Espoo, FI-00076 Aalto, Finland
Abstract This paper describes a method for lightweight design of composite laminated structures using optimization technique. The lightweight design optimization model aims to minimize the mass of the composite laminated structures with the fundamental frequency constraint. The design variables are the fibre volume fractions and fibre orientations of the layers. The first derivatives of the weight and frequencies with respect to design variables are computed. The lightweight design optimization model is converted into a series of linear constrain optimization problems using the interior point penalty function method. The sensitivity of the penalty function is calculated using the sensitivity information of the weight and frequencies. An optimization procedure based on the gradient projection algorithm is proposed. Finally, the lightweight designs of two composite laminated plates are performed by using the proposed optimization procedure. The merits of the proposed procedure are also discussed. Keywords: Composite structures; Laminated plate and shell; Lightweight design; Optimization; Vibration 1. Introduction Composite laminated structures are gaining more and more applications in aircraft, automobile, naval and defence industries because of their high performance characteristics, such as high strength-to-weight ratio, high stiffness-to-weight ratio, superior fatigue properties and high corrosion resistance [1, 2]. Therefore, the composite laminated structures are very attractive in the vehicle industries (aircraft, spacecraft, automobile, ship etc.) because the vehicles made of lightweight material can save energy (gas) and reduce the carbon emission. The composite laminated structures are believed to be promising in the ∗
Corresponding author. Tel.: +358 (0) 503502879; Fax: +358 (0)947023758. E-mail address: [email protected].
vibration environment because the lightweight material attracts smaller inertia forces than heavy material. Many researchers have employed optimization methods to achieve the optimal designs of the composite laminated structures in vibration environment. For example, Apalak et al. [3] used the Artificial Bee Colony algorithm to maximize the fundamental frequency of the symmetrical laminated composite plates by optimizing the fibre orientations. Honda et al. [4] employed the non-dominated sorting genetic algorithm to maximize the fundamental frequency and minimize the curvature of the curvilinear fibres for the laminated composite plates, and the design variables are the coefficients of shape of the curvilinear fibres. Bargh and Sadr [5] used the Particle Swarm Optimization Method to maximize the fundamental frequency of the composite finite strip by optimizing the fibre orientations. Bargh and Sadr [6]also employed the Elitist-Genetic Algorithm to maximize the fundamental frequency of the composite laminated plates by optimizing the fibre orientations. Topal [7, 8] performed the frequency optimization of composite laminated plates and skew sandwich plates using the modified feasible direction method to find the optimum fibre orientations.
Cho [9] implemented design optimization of the laminated
composite plates with static and dynamic considerations in the hygrothermal environments. The objective is the weight of the structures and the design variables are the thickness of the layers. A gradient-based modified feasible direction optimization algorithm is used to solve the optimization problem. Apalak et al. [10] demonstrated the layer optimization for maximum fundamental frequency of rigid point-supported laminated composite plates. The Genetic Algorithm combined with neural networks is used to carry out the optimization. Murugan et al. [11] employed response surface method to optimize the fibre orientations of the composite aerofoil cross-section so that the vibration of the helicopter is reduced. Narita [12, 13] conducted the layerwise optimization for the maximum fundamental frequency of laminated composite plates.
Many researchers also conducted research in static optimization [14, 15] and buckling
optimization [16, 17]. However, the paper is focus on the optimization design of composite laminated structures in vibration environment. Therefore, the static and buckling optimization designs of composite laminated structures are not reviewed in detail in this paper. From the above-mentioned we can see that the objectives are almost focused on the fundamental frequency and the design variables are almost the fibre orientations. The algorithms are almost focused on non gradient-based algorithms. Only the
literature [9] minimized the weight by adjusting the thickness of the layers. Cho [9] also used gradientbased optimization algorithm to search the optimum design. The sensitivity analysis results in the literature [18] show that the fibre volume fractions have much more influence on the frequencies of the composite laminated structures than the fibre orientations. The fibre orientations have no relationship with the weight of the structures. Therefore, the fibre volume fractions or thickness of the layers have to be treated as design variables in the lightweight design. The aim of the paper is to develop an optimization method for the lightweight design of composite laminated structures. First, the lightweight design optimization model is proposed to find the fibre volume fractions and fibre orientations of each layer to minimize the mass with the fundamental frequency constraint. The optimization model is also turned into dimensionless. Second, the first derivatives of the weight and frequencies with respect to the fibre volume fractions and fibre orientations are implemented based on author’s previous work [18]. Third, the lightweight design optimization model is converted into a series of linear constraint optimization problem by using the interior point penalty function method. The sensitivity of the interior point penalty function is obtained using the sensitivity information of the weight and frequencies. Fourth, an optimization procedure is proposed based on the gradient projection algorithm. Finally, the lightweight designs of the square and T-shape composite laminated plates are performed by using the proposed optimization procedure. The merits of the proposed procedure are also discussed. 2. Lightweight design optimization model Many researchers have conducted the layerwise optimization for the maximum fundamental frequency of laminated composite plates and shells. However, the goal of this paper is to find the fibre volume fractions and fibre orientations of the layers to minimize the mass of the composite laminated structures with the fundamental frequency constraint. The lightweight design optimization model is built as follows. I I Find d = rf( ) and θ ( )
( )) subject to ω ( r ( ) ,θ ( ) ) ≥ ω minimize W rf(
I
I
1
I
1
f
− 90o ≤ θ ( ) ≤ 90o I
0 ≤ rf( ) ≤ 1 I
( I = 1, 2,L , N )
(1)
where the design variable vector d includes the fibre volume fractions rf( I ) and fibre orientations θ ( I) of
( )
the layers. N is the total number of the layers of composite laminated structures. W rf( I )
(
ω1 rf( I ) ,θ ( I)
)
and
are the mass (objective function) and fundamental frequency (constraint function),
respectively. ω1 is the lower limit on the fundamental frequency. In order to make the objective function and constraint function at the same numerical value range, the optimization model is converted into dimensionless form as I I Find d = rf( ) and θ ( )
( ))
W rf(
I
( ) = W r ( ) = 1 (I )
minimize w rf
I
(
(I )
subject to g rf , θ
)=
( I)
f
(
(I )
ω1 rf , θ ( I) ω1
) −1 ≥ 0
(2)
− 90o ≤ θ ( ) ≤ 90o I
0 ≤ rf( I ) ≤ 1
( I = 1, 2,L , N )
I I where W rf( ) = 1 is the mass of the structure when the fibre volume fractions rf( ) = 1 ( I = 1, 2,L , N ).
3. Sensitivity analysis of objective function and constraint function 3.1 First derivatives of the objective function Differentiating the objective function in Eq. (2) with respect to the fibre volume fractions rf( I ) , we have the sensitivity of the objective function (the mass).
( )) =
∂w rf( ∂rf(
( ))
∂W rf(
I
I)
1
∂rf(
I W rf( ) = 1
I
(3)
I)
( ) and its derivatives with respect to fibre volume fraction r (I )
where the mass W rf
(I ) f
, i.e.,
( ) ) , can
∂W rf(
I
∂rf( I )
be computed as
( ) ) = ∑ ∑ ρ ( )Ω ( z
W rf(
N
I
k
e
k =1
N
= ∑Ω ×∑ ρ e
e
e
k =1
(k )
k +1
− zk )
( zk +1 − zk )
(4)
Here Ωe is the area of the e th composite laminated element. N is the total number of the layers of composite laminate. zk and zk +1 are the z coordinates of the top and bottom surfaces of the k th layer.
ρ ( k ) is the density of the k th layer.
(
)
ρ ( k ) = rf( k ) ρ f + 1 − rf( k ) ρ m
(5)
where ρ f and ρ m are the densities of the fibre and matrix, respectively. Differentiating Eq. (4) with respect to fibre volume fractions rf( I ) , we have
( )) =
∂W rf(
I
∂rf( I )
∑Ω
e
e
×
∂ρ ( I ) ( zI +1 − zI ) ∂rf( I )
(6)
where ∂ρ ( ) = ρ f − ρm I ∂rf( ) I
(7)
Since the fibre orientations have no influences on the mass, therefore we have
( )) = 0
∂w rf( ∂θ (
I
(8)
I)
3.2 First derivatives of the constraint function Differentiating the constraint function in Eq. (2) with respect to the fibre volume fractions rf( I ) and fibre orientations θ ( I ) , we obtain the sensitivity of the constraint function (fundamental frequency).
(
∂g rf( ) , θ ( ) I
I
∂rf(
(
I) I
∂θ
( I)
I
I
1
ω1
∂g rf( ) , θ ( ) I
) = 1 ∂ω ( r ( ) ,θ ( ) )
)
f
∂rf(
(
I)
( ) () 1 ∂ω1 rf , θ = I ω1 ∂θ ( ) I
I
(9)
)
(
)
In Eq. (9), the frequency ω1 rf( I ) ,θ ( I ) and its derivatives with respect to the fibre volume fractions rf( I )
and fibre orientations θ
( I)
(i.e.,
(
∂ω1 rf( ) ,θ ( ) I
∂rf( I )
I
)
and
(
∂ω1 rf( ) ,θ ( ) I
I
∂θ ( ) I
) ) have been achieved analytically in
author’s previous work [18]. The detailed formula derivations are demonstrated in the reference [18]. The computational procedure of the first derivatives of the frequencies is summarized as follows.
(1) The first derivatives of the engineering constants (i.e., the modulus of elasticity of the fibre E f , modulus of elasticity of the polymer-based matrix Em , Poisson's ratio of the fibre ν f , Poisson's ratio of the polymer-based matrix ν m , density of the fibre ρ f and density of the polymer-based matrix ρ m ), of the lamina, with respect to the fibre volume fraction and fibre orientation are computed based on the micromechanics theory. (2) The first derivatives of the stiffnesses (i.e., Aij , Dij , Bij and H ij ), of laminate, with respect to the fibre volume fraction and fibre orientation are calculated using the classical or first-order theories of composite plate and shell. (3) The first derivatives of the total stiffness matrix K and total mass matrix M with respect to fibre volume fraction and fibre orientations are computed based on the linear finite element methods of composite plates and shells of the classical or first-order theories. (4) Finally, the first derivatives of frequencies are calculated using the total stiffness matrix K and total mass matrix M and their first derivatives. 4. Penalty function and optimization algorithm 4.1 Penalty function and its sensitivity
It is convenient to calculate the gradient of the interior point penalty function when the direct differentiation method is used to obtain the first derivatives. For this reason, the interior point penalty function method is employed in this paper. There are various penalty terms in the interior point penalty functions. However, the popular forms are the inverse penalty term and the logarithm penalty term. Either of the terms can be used to construct the interior point penalty function. In this paper, the logarithm penalty term is used to construct the interior point penalty function. The interior point penalty function is defined in this paper as
(
) ( )
(
F rf( I ) , θ ( I ) , γ k = w rf( I ) − γ k ln g rf( I ) ,θ ( I )
)
(10)
where γ k is the penalty parameter. The penalty parameter is updated according to the following rule:
γ k +1 =
γk c
where the constant c = 5 to 50 ( c = 10 in this paper).
(11)
Therefore, the constraint optimization problem Eq. (2) has been converted into a series of linear constraint optimization problems.
(
Minimize F rf( ) ,θ ( ) , γ k I
I
Subject to − 90o ≤ θ
(I )
)
≤ 90o
(I )
(12)
( I = 1, 2,L , N )
0 ≤ rf ≤ 1
The first derivatives of the interior point penalty function with respect to the fibre volume fractions rf( I ) and fibre orientations θ ( I ) can be computed as follows.
(
∂F rf( I ) ,θ ( I ) , γ k ∂rf( I )
(
∂F rf( I ) ,θ ( I ) , γ k ∂θ
( I)
) = ∂w ( r ( ) ) − γ
(
∂g rf( I ) , θ ( I )
I
1
f
∂rf( I )
) = −γ
k
(
g rf( I ) ,θ ( I )
(
∂g rf( I ) , θ ( I )
1 k
(
(I )
g rf , θ
)
( I)
)
∂θ
)
∂rf( I )
)
(13)
( I)
The derivatives on the right hand side of Eq. (13) have been computed in Section 3. 4.2 Optimization algorithm of lightweight design
When the gradient of the objective function is known, the effective gradient-based optimization algorithms for a linear constraint optimization problem are the gradient projection algorithm and feasible direction algorithm. In this paper, the gradient projection algorithm is employed to solve a series of constraint optimization problem Eq. (12). The computation procedure of solving the mathematic model Eq.(1): Step 1. Choose the initial feasible design point d 0 , set γ 1 =10 (in this paper), define convergence criterion
ε1 =5 kg (in this paper) and let k =1. Step 2. Start from design point d k −1 , solve the linear constraint mathematic model Eq. (12) with the
gradient projection algorithm to obtain the optimum design d k . Step 3. If W ( d k −1 , rk ) − W ( d k , rk ) ≤ ε1 , d k is the best design. Print results and stop. Otherwise, update the
penalty parameter γ k +1 =
γk 10
, k = k + 1 and go to Step 2.
5. Examples
In this section, the optimization design of a T-shape composite laminated plate and square composite
laminated plate are implemented using the aforementioned optimization model and proposed lightweight design optimization procedure in this section. The two composite structures have the same material properties, i.e., the fibre material T800H: E f =294 GPa, ν f =0.2, ρ f =1.81 g/cm3 and the matrix material QY9511: Em =4.2 GPa, ν m =0.3, ρ m =1.24 g/cm3. Both composite structures have the same layered number N =8, thickness of each layer (5 mm) and thickness of laminate (40 mm). The laminate is symmetric and the fibre orientations are [0/90/45/-45]s. In the examples, only the fibre volume fractions are chosen to be design variables. The fibre volume fractions are rf( I ) =60% ( I = 1, 2,L , N ) in the initial designs of the square and T-shape plates. 5.1 Square composite laminated plate
The square composite laminated plate with 2 m × 2 m is shown in Fig. 1. One side of the square composite laminated plate is clamped. The lower limits on the fundamental frequency are ω1 =10 Hz (case 1) or ω1 =5 Hz (case 2), respectively, as shown in Table 1. The square plate is divided by 64 nonconforming rectangular elements [18].
2m
y 2m
x
Fig. 1 Square composite laminated plate Table 1 Initial design and optimum design of square plate Fibre volume fractions (%) Mass (kg) Frequency (Hz)
Initial design Optimum design 1 Optimum design 2
rf(1)
rf( 2)
rf( 3)
rf( 4)
W
ω1
60
60
60
60
253
10.34
0.1
56.3
0.1
0.1
211
10.22
0.3
10.6
0.8
0.1
201
5.33
Lower limit on frequency (Hz) CPU time (sec)
ω1
10 (case 1) 5 (case 2)
20 12
rf(1) =60, θ (1) =0
rf(1) =0.1, θ (1) =0
rf(1) =0.3, θ (1) =0
rf( 2) =60, θ ( 2 ) =90
rf( 2) =56.3, θ ( 2 ) =90
rf( 2) =10.6, θ ( 2 ) =90
rf( 3) =60, θ ( 3) =45
rf( 3) =0.1, θ ( 3) =45
rf( 3) =0.8, θ (3) =45
rf( 4) =60, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 5) =60, θ (5) =-45
rf( 5) =0.1, θ (5) =-45
rf( 5) =0.1, θ ( 5) =-45
rf( 6) =60, θ ( 6 ) =45
rf( 6) =0.1, θ ( 6 ) =45
rf( 6) =0.8, θ ( 6 ) =45
rf( 7 ) =60, θ ( 7 ) =90
rf( 7 ) =56.3, θ ( 7 ) =90
rf( 7 ) =10.6, θ ( 7 ) =90
rf(8) =60, θ (8) =0
rf(8) =0.1, θ (8) =0
rf(8) =0.3, θ (8) =0
(a) Initial design
(b) Optimum design ( ω1 =10 Hz)
(c) Optimum design ( ω1 =5 Hz)
Fig. 2 Layers of initial design and optimum design of square shape plate (view point from the positive x ) 5.2 T-shape composite laminated plate
The T-shape composite laminated plate and it dimensions are shown in Fig. 3. One side of the Tshape is clamped. The lower limit on the fundamental frequency are ω1 =1.5 Hz (case 3) or ω1 =1 Hz (case 4), respectively, as shown in Table 2. The fibre volume fractions are rf( I ) =70% ( I = 1, 2,L , N ) in both initial designs. The T-shape plate is divided by 256 nonconforming rectangular elements.
2m
2m
2m
y x
2m
2m
Fig. 3 T-shape composite laminated plate
Table 2 Initial design and optimum design of T-shape plate Fibre volume fractions (%) Mass (kg) Frequency (Hz)
Initial design Optimum design 1 Optimum design 2
rf(1)
rf( 2)
rf( 3)
rf( 4)
W
ω1
70
70
70
70
1049
1.60
0.1
91.6
0.1
0.1
877
1.53
2.1
24.6
13.5
0.1
830
1.0
Lower limit on frequency (Hz) CPU time (sec)
ω1
1.5 (case 3) 1.0 (case 4)
rf(1) =70, θ (1) =0
rf(1) =0.1, θ (1) =0
rf(1) =2.1, θ (1) =0
rf( 2) =70, θ ( 2 ) =90
rf( 2) =91.6, θ ( 2 ) =90
rf( 2) =24.6, θ ( 2 ) =90
rf( 3) =70, θ ( 3) =45
rf( 3) =0.1, θ ( 3) =45
rf( 3) =13.5, θ ( 3) =45
rf( 4) =70, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 4) =0.1, θ ( 4 ) =-45
rf( 5) =70, θ (5) =-45
rf( 5) =0.1, θ (5) =-45
rf( 5) =0.1, θ ( 5) =-45
rf( 6) =70, θ ( 6 ) =45
rf( 6) =0.1, θ ( 6 ) =45
rf( 6) =13.5, θ ( 6 ) =45
rf( 7 ) =70, θ ( 7 ) =90
rf( 7 ) =91.6, θ ( 7 ) =90
rf( 7 ) =24.6, θ ( 7 ) =90
rf(8) =70, θ (8) =0
rf(8) =0.1, θ (8) =0
rf(8) =2.1, θ (8) =0
(a) Initial design
131 47
(b) Optimum design ( ω1 =1.5 Hz) (c) Optimum design ( ω1 =1 Hz)
Fig. 4 Layers of initial design and optimum design of T-shape plate (view point from the positive x ) 5.3 Discussions
The results (Table 1 and Table 2) show that the fundamental frequencies of the optimum designs are almost full constrained (i.e., with a fundamental frequency value reaching or nearly reaching the allowable limit). Such results indicate that a composite structure can usually be made lighter/more economical (since the fibre is more expensive and has bigger density than the matrix) by allowing decreased fundamental frequency to occur and having the frequency constraint to attain a fully constrained condition. The case 1 (initial design: 253 kg and 10.34 Hz, optimum design: 211 kg and 10.22 Hz, lower limit: 10 Hz) and case 3 (initial design: 1049 kg and 1.6 Hz, optimum design: 877 kg and 1.53 Hz, lower limit: 1.5 Hz) show that even though the fundamental frequency values of the initial designs are very close to the allowable limit, the weight of composite laminated structures can be further reduced by adjusting the fibre volume fractions of the layers and the constraint is still satisfied. The
optimum designs (Fig. 2 (b) (c) and Fig. 4 (b) (c)) demonstrate that the proposed optimization model and computational procedure can effectively optimize the composite structure to obtain its most optimal distribution of the fibre volume fractions at the composite laminate. The maximum fibre volume fractions (Fig. 2 (b) and Fig. 4 (b)) are allocated at the 2nd and 7th layers in the optimum designs. Such results also indicate that the sandwich composite structures are reasonable composite type in the vibration environment. The computations are carried out using a desk computer (AMD Phenom(tm) II X2 550 Processor, and frequency of the CPU is 3.10 GHz). The computational time (CPU time in Table 1 and Table 2) is 10 sec and 20 sec for square plate, 47 sec and 131 sec for T-shape plate. It indicates the proposed optimization model and computational procedure are efficient to achieve the optimum designs. 6. Conclusions
The paper proposes a lightweight design method for composite laminated structures in vibration environment using the gradient algorithm based on the analytical sensitivity analysis method of frequencies which is developed in author’s previous work [18]. The numerical results indicate that the optimization model and computational procedure in this paper are effective and efficient to achieve the lightweight design of the composite laminated structures. In the situation, the fundamental frequency values of the initial designs are very close to the allowable limit, the composite laminated structures still can be lighter by adjusting the fibre volume fractions of the layers and the constraint can surely be satisfied. The optimum distributions of the fibre volume fraction illustrate that the sandwich composite structures are reasonable composite structure type in the vibration environment. In the future work, the lightweight design method proposed in this paper will be applied to the large scale engineering composite structures with complex boundary conditions and also subject to the static strength constraint. References
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