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Optimization of multi-sandwich-panel composite structures for minimum weight with strength and buckling considerations

  • Hai-Tao Fan , Hai Wang EMAIL logo and Xiu-Hua Chen
Published/Copyright: September 22, 2016
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Science and Engineering of Composite Materials
From the journal Science and Engineering of Composite Materials Volume 25 Issue 2

Abstract

A genetic algorithm-based method is proposed to globally optimize the stacking sequence of multi-sandwich-panel composite structures for minimum weight with strength and buckling considerations. The prerequisites for the continuity between sandwich panels are first studied. To implement the summarized continuity rules in the evolutionary optimization, three newly constructed chromosomes are developed to encode the global stacking sequence with no additional repair. Genetic operators, including specialized mutation, swapping and crossover operators, are also developed to effectively explore the design space and keep the continuity rules followed. The Hashin criterion and maximum stress criterion are used to evaluate the strength of sandwich panels. A typical multi-sandwich-panel composite structure with identical and different core thicknesses is optimized to verify the validity and efficiency of the proposed method. It is found that much lighter solutions are obtained with an acceptable efficiency in all cases. It is also found that the weight of the multi-sandwich-panel composite structures can be further reduced when the core thicknesses are not identical.

Keywords: buckling; composite structure; genetic algorithm; optimization; sandwich panel

1 Introduction

A composite sandwich panel is formed by two laminated faces and a core, which is usually made of the honeycomb or foam material. As a commonly used structural construction, the composite sandwich panel is seen to have great bending stiffness because of its high-stiffness faces separated by the low-density core.

In engineering practices, the composite sandwich structure is known to have great potential in tailor-made design. The different regions in a multi-region sandwich structure are often subjected to different local loads and design constraints. In this paper, these regions are called sandwich panels. Usually, to get solutions with lower weights and fitter stiffness, these sandwich panels are designed to have different but continuous stacking sequences. The corresponding process is known as the design optimization of multi-sandwich-panel composite structures. The design issue involved is the ply drops between the faces of adjacent sandwich panels. As shown in Figures 1 and 2, the sandwich panel A and sandwich panel B are connected through common plies in their faces (different colors represent plies of different types) and the homogeneous core. In the transition region from A to B, most plies in the faces of A continue, while some plies are dropped.

Figure 1: Ply drops between adjacent sandwich panels with identical core thicknesses.
Figure 1:

Ply drops between adjacent sandwich panels with identical core thicknesses.

Figure 2: Ply drops between adjacent sandwich panels with different core thicknesses.
Figure 2:

Ply drops between adjacent sandwich panels with different core thicknesses.

Because of the discontinuities resulting from the restrictions to the definition of ply orientations and ply materials, the design optimization of the stacking sequences of a composite sandwich panel is always a process full of discreteness [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. In the design optimization of a multi-sandwich-panel structure, the directionality of fiber reinforced plies further raises the continuity requirement on the fiber placement between adjacent sandwich panels. The discreteness, together with the continuity requirement, has made the global stacking sequence optimization of multi-sandwich-panel structures hard to be performed.

Many methods have been developed to optimize the global stacking sequences of multi-panel composite structures [14], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. Kristinsdottir et al. [18] introduced the term blending and proposed the greater-than-or-equal-to blending rule, in which plies were dropped from the most critical panel, to get the blended solutions. Liu and Haftka [19] developed measures of continuity for two adjacent laminated panels and further used the inequality constraints to enforce the stacking sequence continuity. Then their method was demonstrated on a multi-laminated-panel composite wing. Soremekun et al. [20] proposed a sublaminate-based approach for optimizing the multi-panel composite structures with the genetic algorithm (GA). An 18-laminated-panel structure arranged in a horseshoe pattern was optimized and much smaller weight penalties were obtained. Adams et al. [21], [22], [23] introduced and implemented the concept of guide-based blending with GA to optimize multi-laminated-panel composite structures. In their method, the stacking sequences of all panels in the whole structure were obtained by deleting contiguous series of plies in a pre-defined guiding stack. By adding one variable per panel of the structure to represent the number of plies dropped from the guide, the continuity requirement was satisfied. Zehnder and Ermanni [24], [25] introduced a patch concept, in which the patch was a layer that covered the arbitrary part of the multi-laminated-panel structure and the stacking sequence of each panel was defined by the order and orientations of the patches. IJsselmuiden et al. [29] presented a multi-step framework for the design of multi-panel composite structures. In their work, the structures were first optimized using panel thicknesses and lamination parameters as continuous design variables. Then the stacking sequence of each panel was obtained using a guide-based GA. Fully blended designs were obtained at a small fraction of the computational cost of traditional blending algorithms. Seresta et al. [30], [31] proposed a multi-chromosomal GA to implement a novel parameterization of the stacking sequence and seamlessly blend stacking sequence across multiple composite panels. Irisarri et al. [32] introduced and implemented the concept of stacking sequence table (SST) with GA for the optimal design of multi-laminated-panel composite structures. The SST was used to describe the sequence of ply drops ensuring the continuity between adjacent panels. Each blended design was represented by an SST and a thickness distribution over the whole structure. Lots of design guidelines which had not been included in previous studies were also considered in their study. Lighter solutions were obtained with the SST method.

Researchers have successfully implemented their methods to optimize the multi-panel composite structures and obtained solutions with great improvements. However, the continuity requirements between laminates are mainly considered in these methods, while the optimizations of multi-sandwich-panel structures are rarely studied. In this paper, the prerequisites for the continuity between two adjacent composite sandwich panels are studied. Then a GA-based method, which follows the continuity rules, is proposed and implemented to globally optimize the stacking sequences of multi-sandwich-panel composite structures.

2 Problem description

In this paper, a ply type is represented by the combination of a ply material and a ply angle. So according to the candidate ply materials and ply angles, the candidate ply types for the optimization can be summarized. For example, the ply type whose material is A and ply angle is θ1 differs from the ply type whose material is A and ply angle is θ2(θ1≠θ2). The ply type whose material is B and ply angle is θ1 also differs from the ply type whose material is A and ply angle is θ1. It is assumed that there are npt different candidate ply types (PT1, PT2, …, PTnpt).

In the stacking sequence optimization of a multi-sandwich-panel structure, the first step is to make two sandwich panels continuous to each other. For two composite sandwich panels, their laminated faces should follow some rules to make them continuous. First considering the ply compositions of two adjacent sandwich panels, the faces of one sandwich panel should not have fewer plies of each type than another sandwich panel. For two adjacent sandwich panels, we can call the sandwich panel with not fewer plies “the parent sandwich panel” and another sandwich panel “the child sandwich panel”. For a parent sandwich panel and a child sandwich panel, their common plies in the faces are actually all plies in the faces of the child sandwich panel. Secondly, the common plies of two adjacent sandwich panels should be stacked in the same sequence. This continuity rule is realized by assigning a ranking factor to each ply. The ranking factor of a ply is defined to be a real number whose value determines the position of the ply. For example, there are sandwich panel A and sandwich panel B in a structure. A and B have 1 PT1-type ply (with a ranking factor of 0.2), 1 PT2-type ply (with a ranking factor of 0.4) and 1 PT3-type ply (with a ranking factor of 0.6) in common in their faces. In addition, A has another PT1-type ply (with a ranking factor of 0.1) and another PT2-type ply (with a ranking factor of 0.5). Then after ordering the plies by ranking these factors from small to large, we can get the layups of the face sheets of A and B. The layup of the face sheet of A is [PT1/PT1/PT2/PT2/PT3]. The layup of the face sheet of B is [PT1/PT2/PT3]. These ranking factors make the common plies of A and B stacked in the same sequence.

The two rules are the prerequisites for the continuity between the faces of sandwich panels. The quantities and relative positions of plies with various types should be constrained to follow these rules. Meanwhile the homogeneous core can be seen as a special ply type, which differs from those plies in the laminated faces due to its features as follows.

  1. For each sandwich panel, the thickness of the core is variable.

  2. The core should be positioned between the two faces of each sandwich panel.

  3. For two adjacent sandwich panels, their cores are naturally continuous to each other.

If the core thicknesses in a multi-sandwich-panel structure are identical as shown in Figure 1, the core can be seen as a ply with a variable thickness. If the core thicknesses in a multi-sandwich-panel structure are different as shown in Figure 2, then to realize the global optimization of the core thicknesses, the core can be seen as a group of homogeneous plies with small thicknesses. Here these homogeneous plies are called as core plies. No constraint should be subjected to the quantity and sequence of core plies.

To keep the continuity between adjacent sandwich panels, the continuity rules will be implemented in the GA-based optimization method.

3 Continuity rules-based optimization

3.1 Optimization objectives and constraints

The optimization objective in this paper is to get the lowest weight of a whole multi-sandwich-panel structure, which can be expressed as min(φ0). However, the discreteness in the optimizations usually leads to the existence of several equally optimal solutions. So on the premise that the lowest weight has been achieved, the largest minimum critical factor (λcr) across the whole structure is set as the second optimization objective. Here λcr<1 represents the failure of the whole structure and larger λcr represents the safer status. λcr is determined by the strength failure factor λs and the buckling failure factor λb. λs<1 represents the strength failure and λb<1 represents the buckling failure. So the critical factor of a sandwich panel can be expressed as

(1)λcr=min(λs, λb)

For two solutions that have equal weights, the one with the larger λcr is better.

In the optimization of a multi-sandwich-panel structure, the following constraints are considered.

  1. Continuity. The adjacent sandwich panels should be interconnected through the core and continuous plies in the faces. This is the key feature that distinguishes the optimization of a multi- sandwich-panel structure from that of a single sandwich panel.

  2. Symmetry. The stacking sequences of the laminated faces should be symmetric about the mid-plane of each panel.

  3. Balance. Each sandwich panel should have the same number of θ and −θ (0 and 90 excluded) plies in its faces.

  4. Strength and buckling. The strength and buckling failure should never occur across the whole structure.

The optimization problem of a multi-sandwich-panel structure can be formulated as follows:

(2)Minimize ϕ0=W

Subjected to,

  • Continuity constraint

  • Symmetry constraint

  • Balance constraint

  • Strength/buckling constraint: λcr>1

where W represents the total weight of the structure.

To incorporate the buckling and strength constraints, the critical factor λcr is added into the objective function through penalty strategies. So the objective function for the optimization of a multi-sandwich-panel structure can be reconstructed as

(3)λcr≥1, ϕ0=W+ε(1−λcr)λcr<1, ϕ0=Wλcrp

where ε is a small value for rewarding the strength/buckling constraint margin and p is a scaling parameter that controls the searching ability by cutting across the unfeasible design space. Kogiso et al. [4] studied the value of ε and concluded that “too large a value for ε would favor thick designs having a large feasibility margin over the optimum design”. It was also concluded that ε should be smaller than N(c−1)/(c3−1), where c>1 and N is the number of plies in a laminate. In this paper, ε=0.089 and p=20.

3.2 Multi-chromosomal GA

GA has long been used in the optimization of composite structures as an algorithm that can cope well with the discrete optimization problems. However, GA is an algorithm that needs to be designed, which is much more demanding in the optimization of a multi-sandwich-panel structure due to the more difficult handling of constraints. This is reflected in the construction of chromosomes and the design of genetic operators. Among those constraints mentioned above, the handling of the continuity constraint is usually thought to be the most difficult. In this paper, the continuity constraint, together with the symmetry and balance constraints, is incorporated into the chromosome construction.

3.2.1 Chromosomes

Three types of chromosomes, the ply-thickness chromosome, ply-composition chromosome and ply-ranking chromosome, are used to encode the global stacking sequence. The ply-thickness chromosome is constructed to optimize the thickness of thickness-variable plies. The ply-composition chromosome consists of the ply composition information, while the ply-ranking chromosome consists of the ply ranking information.

As shown in Figure 3, the ply-thickness chromosome is defined to be formed by npt real-coded genes. The ith gene Ti represents the thickness of the ith ply type. For the ply type that has a fixed thickness, the range of the corresponding gene would be restricted to a fixed value.

Figure 3: The ply-thickness chromosome.
Figure 3:

The ply-thickness chromosome.

The ply-composition chromosome helps implement the first continuity rule. As shown in Figure 4, a ply-composition chromosome consists of nss ply-composition genes. nss is the number of substructures across the whole structure. Each ply-composition gene, which represents the ply composition of a single sandwich panel, is composed of npt integer-coded child genes. npt is the number of ply types. Each integer gene represents the number or half number (for symmetry conditions) of plies of the corresponding type in the corresponding sandwich panel. For example, when subjected to the symmetry constraint, the value of Cj,i in Figure 4 represents the half number of plies of the ith type in the jth sandwich panel.

Figure 4: The ply-composition chromosome.
Figure 4:

The ply-composition chromosome.

The ply-ranking chromosome implements the second continuity rule. While the ply-composition chromosome determines the ply composition of all sandwich panels, the ply-ranking chromosome could further determine the sequence of these plies to get the full lay-up definition of each sandwich panel. As shown in Figure 5, a ply-ranking chromosome consists of Npt ply-ranking genes. The ith ply-ranking gene represents the ranking factors of PTi-type plies. Each ply-ranking gene is composed of several real-coded child genes, where the jth child gene in the ith ply-ranking gene represents the ranking factor of the jth PTi -type ply. The value of these ranking factors is in the range of (0, 1) for ply types that have no special position requirement. For ply types that have special position requirements, the value of the corresponding ranking factor can be set to beyond the range of (0, 1). For example, to keep the core always in the mid-plane of the panels, its corresponding ranking factor can be set as a specified value larger than 1.

Figure 5: The ply-ranking chromosome.
Figure 5:

The ply-ranking chromosome.

It is noted that to incorporate the balance constraint into these chromosomes, two plies with an angle of θ and −θ (θ≠0,90), respectively and the same material can be seen to belong to the same ply type. To incorporate the symmetry constraint, only half of each layup is encoded.

Figure 6 shows how to get the stacking sequence of a multi-sandwich-panel structure by decoding the ply-thickness, ply-composition and ply-ranking chromosomes.

Figure 6: Decoding of chromosomes.
Figure 6:

Decoding of chromosomes.

  1. We get the value of the ith child gene (integer coded) of the jth ply-composition gene in the ply-composition chromosome as c(j,i). c(j,i) represents the number of PTi-type plies in the jth sandwich panel.

  2. The first c(j,i) ranking factors can be found in the ith gene of the ply-ranking chromosome. So we get a group of plies whose ply types and ranking factors are determined. Thirdly, the ranking factors of all plies in the jth sandwich panel can be determined.

  3. By ordering the plies by these ranking factors, the stacking sequence of the jth sandwich panel can be obtained.

  4. The thickness of each ply type, which can be found in the ply-thickness chromosome, helps form the full definition of the jth sandwich panel.

An example for the decoding of a two-panel sandwich structure can be found in Figure 7. In this example, the ply types in the face sheets are restricted to 0, 45, –45 and 90 plies. The core is discreted to the core plies. The ply-composition chromosome has two genes, which contains the ply composition information of sandwich panel 1 and sandwich panel 2, respectively. For example, the first integer number in gene 1 is the number of 0° plies in sandwich panel 1, the second integer number in gene 2 is the number of ±45° plies in sandwich panel 2. The ply-ranking chromosome has four genes, which contains the ranking factors of different ply types (0 ply, ±45 ply, 90 ply and the core ply, respectively). The ply-thickness chromosome has four genes, which contains the thicknesses of different ply types (0 ply, ±45 ply, 90 ply and the core ply, respectively). Then we decode the chromosomes as in the figure.

Figure 7: An example for the decoding of chromosomes.
Figure 7:

An example for the decoding of chromosomes.

  1. From the ply-composition chromosome, we can get the number of 0° plies in sandwich panel 1. The number is 2. Then we get the first and second ranking factors in gene 1 of the ply-ranking chromosome for the first and second 0° plies.

  2. Like in step 1, we can get all plies whose ranking factors can be determined.

  3. Order these plies by ranking these ranking factors from small to large.

  4. Get the layups of sandwich panel 1 and sandwich panel 2.

  5. Get the thicknesses of all ply types from the ply-thickness chromosome.

  6. Get the core thickness as the sum of the thicknesses of all core plies.

  7. Then get the full definitions of all sandwich panels.

The global stacking sequence of the whole structure can be obtained by decoding the chromosomes in this way. The decoded stacking sequences of these sandwich panels are continuous to each other without any additional repair. This means the continuity constraint has been implemented through the constructed chromosomes.

3.2.2 Genetic operators

Three genetic operators, the mutation, swapping and crossover operators, are utilized to match the constructed chromosomes.

The general mutation is implemented on the ply-thickness chromosome and ply-ranking chromosome. This means that in each mutation operation the genes in the ply-thickness and ply-ranking chromosomes are mutated at a specified probability. The new value of each mutated gene is randomly generated in their ranges. In each mutation operation, the ply-composition genes are also mutated at a specified probability. However, due to the continuity constraint, the mutation inside each ply-composition gene differs from that of a ply-thickness gene. As shown in Figure 8, if the jth ply-composition gene is to be mutated, the first step is to mutate the total ply number Nj of the jth sandwich panel. An integer number ΔN is then randomly generated from the range of (ΔNmin, ΔNmax), where the values of ΔNmin and ΔNmax are defined to control the mutation range. In the cases of this paper, ΔNmin=−6, ΔNmax=2. The new total ply number is Nj+ΔN. Once the new total ply number is determined, the parent sandwich panels and child sandwich panels of sandwich panel j can be subsequently determined. According to the ply compositions of parent sandwich panels (or parent ply-composition genes), the upper limit of new Cj,i can be determined. Similarly, according to the ply composition of child sandwich panels (or child ply-composition genes), the lower limit of new Cj,i can also be determined. On the premise that the new total ply number is located between the lower and upper limits, the new value of Cj,i can be randomly determined. In this way, the corresponding sandwich panel of the mutated gene can keep its continuity to the adjacent sandwich panels.

Figure 8: Mutation of a ply-composition chromosome.
Figure 8:

Mutation of a ply-composition chromosome.

Swapping operations are implemented on both ply-composition and ply-ranking chromosomes. The swapping operation on a ply-composition chromosome is used to swap the ply compositions of the corresponding sandwich panel. The swapping operation is implemented on each ply-composition chromosome at a specified probability. As shown in Figure 9, in each swapping operation, the randomly selected i1th child genes of all ply-composition genes swap one or two plies with other randomly selected i2th child genes. This means for all sandwich panels some of the PTi2-type plies are changed into PTi1-type plies. Because every time the ply-composition swapping is implemented across the whole structure, the continuities between sandwich panels are preserved. The swapping operation on the ply-ranking chromosome is used to shuffle the location of plies without changing the composition of the sandwich panel. Once the locations of plies change, the flexural properties of the sandwich panels are modified while its in-plane properties are preserved. For each ply-ranking chromosome, the swapping operation is implemented at a specified probability. As shown in Figure 10, in each swapping operation the top k ranking factors of the randomly selected i1th ply-ranking gene swap with the top k ranking factors of the randomly selected i2th ply-ranking gene. This means some of PTi1-type plies swap their locations with some of PTi2-type plies in the corresponding sandwich panel.

Figure 9: Swapping operations in a ply-composition chromosome.
Figure 9:

Swapping operations in a ply-composition chromosome.

Figure 10: Swapping operations in a ply-ranking chromosome.
Figure 10:

Swapping operations in a ply-ranking chromosome.

The crossover operation is implemented on two individuals at a specified probability. Each individual, formed by the ply-thickness, ply-composition and ply-ranking chromosomes, is randomly selected to crossover with another randomly selected individual. In each crossover operation, the chromosomes of individuals are randomly selected to form the new-generation individuals.

The pseudocode for the whole optimization process is as follows:

  • initialize probabilities for swapping, mutation, crossover genetic operators

  • generation=0

  • initialize Population (generation)

  • decode and evaluate Population (generation)

  • while termination not detected do

  • apply crossover to Population (generation) giving Children (generation)

  • apply swapping to Children (generation)

  • apply mutation to Children (generation)

  • decode and evaluate Children (generation)

  • Population (generation+1)=select from [Population (generation)∪Children (generation)]

  • generation=generation+1

  • check for termination

  • end do

4 Results

Since there is no suitable existing case for the verification of the proposed optimization method, a seven-sandwich-panel configuration is constructed. The dimensions of the panels and the local loadings are shown in Figure 11. The loads are assumed to be fixed. All panels are assumed to be simply supported on their edges. Each panel has two laminated faces and a ROHACELL 51IG foam core. The material of the faces is T300/epoxy, whose properties are listed in Table 1. The properties of ROHACELL 51IG foam are listed in Table 2. The layup of each face is balanced. The layups of the upper face and the lower face are symmetric about the mid-plane of each panel. For the faces, the minimum number of plies is 14 and the maximum number of plies is 48. The ply orientations in the faces are restricted to 0, ±15, ±30, ±45, ±60, ±75 and 90. The optimization objective is to find a global stacking sequence that minimizes the mass (denoted by M) of the seven-sandwich-panel structure without any failure.

Figure 11: The seven-sandwich-panel structure.
Figure 11:

The seven-sandwich-panel structure.

Table 1:

Properties of T300/epoxy material.

Property
Density, ρ (kg/m3)1550
Thickness, t (mm)0.2
Longitudinal modulus, E11 (MPa)125,000
Transverse modulus, E22 (MPa)8000
Poisson’s ratio, µ120.3
In-plane shear modulus, G12 (MPa)5000
Longitudinal tensile strength, Xt (MPa)1600
Longitudinal compressive strength, Xc (MPa)1000
Transverse tensile strength, Yt (MPa)40
Transverse compressive strength, Yc (MPa)220
In-plane shear strength, S12 (MPa)80
Through-thickness shear strength, S23 (MPa)60
Table 2:

Properties of ROHACELL 51IG foam.

Property
Density, ρ (kg/m3)52
Elastic modulus, E (MPa)70
Poisson’s ratio, µ0.3
Shear modulus, G (MPa)19
Tensile strength, Xt (MPa)1.9
Compressive strength, Xc (MPa)0.9
Shear strength, S (MPa)0.8

The strength failure of the laminated faces is identified by the Hashin criterion [33], [34], in which the failures of the fibers and matrix are, respectively determined based on their different stress status and failure mechanism. The strength failure factor can be expressed with Hashin’s criterion as follows.

  1. Fiber mode (λsfiber is the strength failure factor of the fibers):

    For tensile conditions (σ11>0),

    (4)λsfiber=1(σ11Xt)2+(τ12S12)2

    where Xt and S12 are the longitudinal tensile strength and the in-plane shear strength, respectively.

    For compressive conditions (σ11<0),

    (5)λsfiber=−Xcσ11

    where Xc is the longitudinal compressive strength.

  2. Matrix mode (λsmatrix is the strength failure factor of the matrix):

    For tensile conditions (σ22>0),

    (6)λsmatrix=1(σ22Yt)2+(τ12S12)2

    where Yt is the transverse tensile strength.

    For compressive conditions (σ22<0),

    (7)λsmatrix=1(σ222S23)2+[(Yc2S23)2−1]σ22Yc+(τ12S12)2

    where Yc and S23 are the transverse compressive strength and the through-thickness shear strength, respectively. The strength failure of the homogeneous core is identified by the maximum stress criterion, with which the strength failure factor of the core can be expressed as

(8)λscore=min(|Xσ|, Sτ)

where X and S are the tensile/compressive strength and the shear strength, respectively.

Then the strength failure factor of a sandwich panel can be expressed as

(9)λs=min(λsfiber, λsmatrix, λscore)

The stress concentration in the transition areas between adjacent sandwich panels is not considered in this paper.

The buckling failure of a sandwich panel, which is subjected to in-plane loads and simply supported on four edges, is identified by the following equation:

(10)λb=π2D11(m/a)4+2(D12+2D66)(m/a)2(n/b)2+D22(n/b)4(m/a)2Nxx+(n/b)2Nyy

Where λb is the buckling factor and λb≤1 denotes the occurrence of the buckling failure. a and b are respectively the length and width of the panel. Nxx is the force per unit length in the longitudinal (x) direction and Nyy is the force per unit length in the transverse (y) direction. m and n are the number of half wavelengths along the x and y directions, respectively. D11, D12, D66 and D22 are bending stiffness terms of the sandwich panel. The minimum λb can be determined by different combinations of m and n. In this study, the minimization over m and n is performed by checking for all values of m between 1 and 10 and all values of n between 1 and 10.

The GA parameters are listed in Table 3. The optimization is implemented by programming in C#. The initial sandwich panels all have 24 ply faces and a 1-mm-thick foam core. The initial stacking sequences are randomly generated. The total mass of the initial seven-sandwich-panel structure is also 31.39 kg. The optimization terminates when the maximum number of generations is achieved.

4.1 Case 1: identical core thicknesses

In this case, the thicknesses of the foam cores in the sandwich panels are identical as shown in Figure 1. The core thickness is restricted to 1–3 mm. The foam core is taken as a foam ply with a variable thickness.

The typical evolutionary process is shown in Figure 12. The total mass and the critical factor of the best design solution reduce rapidly before the 20th generation. Then the total mass reduces slowly between the 20th and 100th generation, while the critical factor fluctuates between 1.000 and 1.128. The lowest total mass is mostly obtained before the 100th generation. The computation time for a run on a PC is about 20 min. So the lowest total mass is mostly obtained on a PC in about 10 min.

Since there are stochastic processes in GA, the convergence speeds in different runs of GA are usually different. So we executed the optimization program for 20 times to observe the convergence process. The lowest total mass obtained in each run is not larger than 10.63 kg. The best result obtained is 10.13 kg, while the corresponding critical factor is 1.000. The detailed optimization results are listed in Table 4. The cross-sections of the optimized solution in the views from the x and y directions, which further demonstrate the continuities in plies of the sandwich panels, can be found in Figure 13. Results show that the sandwich panels obtained in the optimization are continuous to each other. This means the multi-sandwich-panel structures obtained by the proposed optimization method are manufacturable. Meanwhile with identical core thicknesses, the lowest total mass of the seven-sandwich-panel structure obtained in the optimization is 68% lower than the total mass of the initial structure. It is verified that the proposed optimization method can cope well with the optimization of multi-sandwich-panel composite structures with identical core thicknesses.

4.2 Case 2: different core thicknesses

In this case, the thicknesses of the foam core in the sandwich panels are different as shown in Figure 2. The core thickness is restricted to 1–3 mm. The foam core is taken as a group of foam plies with a fixed thickness of 0.1 mm. So the number of foam plies is restricted to 10–30. It is noted that the continuity constraint is not implemented on the foam plies.

The typical evolutionary process is shown in Figure 14. The total mass and the critical factor of the best design solution reduce rapidly before the 20th generation. Then the total mass reduces slowly between the 20th and 30th generation, while the critical factor fluctuates between 1.016 and 1.280. The lowest total mass is mostly obtained before the 50th generation. The computation time for a run on a PC is about 20 min. So the lowest total mass is mostly obtained on a PC in about 5 min.

Similarly, since there are usually differences in the convergences of GA runs, the optimization program was executed for 20 times. The lowest total mass obtained in each run is not larger than 9.75 kg. The best result obtained is 9.29 kg, while the corresponding critical factor is 1.016. The detailed optimization results are listed in Table 5. Results show that the sandwich panels obtained in the optimization are continuous to each other. This means the obtained multi-sandwich-panel structure with different core thicknesses is manufacturable. Meanwhile the lowest total mass of the seven-sandwich-panel structure obtained in the optimization is 70% lower than the total mass of the initial structure. It is verified that the proposed optimization method can cope well with the optimization of multi-sandwich-panel composite structures with different core thicknesses. The results also show that the lowest total mass of the seven-sandwich-panel structure with different core thicknesses is 8% lower than that of the seven-sandwich-panel structure with identical core thicknesses.

5 Conclusions

In this paper, a GA-based method is proposed and implemented to optimize the global stacking sequence of multi-sandwich-panel composite structures. To explore the design space sufficiently in the optimizations, the prerequisites of the continuity between composite sandwich panels are studied. Then three chromosomes are constructed to include the continuity requirement in the evolutionary optimization of the global stacking sequence. Genetic operators are also designed to keep the diversity and follow the continuity rules. A seven-sandwich-panel composite structure with identical and different core thicknesses is optimized with strength and buckling considerations to verify the efficiency and validity of the proposed optimization method. Results show that the multi-sandwich-panel composite structures with identical and different core thicknesses can be well optimized with an acceptable efficiency. More important, solutions with much lower weights are stably obtained in the verifications. The results also reveal that the total mass of a multi-sandwich-panel composite structure can be further reduced when the core thicknesses are not identical.

Table 3:

GA parameters.

Population size100
Probability of mutation of the ply-thickness chromosome0.14
Probability of mutation of the ply-composition chromosome0.22
Probability of mutation of the ply-ranking chromosome0.01 for each gene
Probability of swapping of the ply-composition chromosome0.20
Probability of swapping of the ply-ranking chromosome0.20
Probability of crossover0.05
Termination criterion (generations)200
Figure 12: (A) The evolutionary process of the total mass. (B) The evolutionary process of the critical factor. The evolutionary process of design solutions with identical core thicknesses.
Figure 12:

(A) The evolutionary process of the total mass. (B) The evolutionary process of the critical factor. The evolutionary process of design solutions with identical core thicknesses.

Figure 13: (A) The cross-sections of face sheets in the view from the x direction. (B) The cross-sections of face sheets in the view from the y direction. The optimization results for the seven-sandwich-panel structure with identical core thicknesses.
Figure 13:

(A) The cross-sections of face sheets in the view from the x direction. (B) The cross-sections of face sheets in the view from the y direction. The optimization results for the seven-sandwich-panel structure with identical core thicknesses.

Figure 14: (A) The evolutionary process of the total mass. (B) The evolutionary process of the critical factor. The evolutionary process of design solutions with different core thicknesses.
Figure 14:

(A) The evolutionary process of the total mass. (B) The evolutionary process of the critical factor. The evolutionary process of design solutions with different core thicknesses.

Table 4:

Optimization results for the seven-sandwich-panel structure with identical core thicknesses.

PanelStacking sequences (C=core with the thickness of 1.4 mm)M (kg)λcr
1[45−3030−3030−450C0−4530−3030−3045]1.091.146
2[45−3030−3030−4500C00−4530−3030−3045]1.701.025
3[45−3030−3030−4500C00−4530−3030−3045]2.161.000
4[45−3030−3030−4500C00−4530−3030−3045]2.621.038
5[45−3030−3030−450C0−4530−3030−3045]1.011.001
6[45−3030−3030−4500C00−4530−3030−3045]1.051.197
7[4530−30−4500C00−45−303045]0.501.318
Total10.131.000
Table 5:

Optimization results for the seven-sandwich-panel structure with different core thicknesses (Tcore is the core thickness).

PanelStacking sequences(C=core)Tcore (mm)M (kg)λcr
1[−3045−4545−45300C030−4545−4545−30]1.31.091.016
2[−3045−4545−45300C030−4545−4545−30]1.91.501.049
3[−3045−4545−45300C030−4545−4545−30]1.91.901.066
4[−3045−4545−45300C030−4545−4545−30]1.72.301.016
5[−3045−4545−45300C030−4545−4545−30]1.51.001.025
6[−3045−4545−45300C030−4545−4545−30]1.50.921.047
7[−3045−4545−45300C030−4545−4545−30]0.90.581.488
Total9.291.016

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Received: 2015-4-16
Accepted: 2016-6-17
Published Online: 2016-9-22
Published in Print: 2018-3-28

©2018 Walter de Gruyter GmbH, Berlin/Boston

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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https://doi.org/10.1515/secm-2015-0171
Keywords for this article
buckling; composite structure; genetic algorithm; optimization; sandwich panel
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Frontmatter
  2. Review
  3. A review on the intensification of metal matrix composites and its nonconventional machining
  4. Original articles
  5. Optimization of multi-sandwich-panel composite structures for minimum weight with strength and buckling considerations
  6. An automated portable multiaxial pressure test rig for qualifications of glass/epoxy composite pipes
  7. Effects of nano-SiO2 on mechanical and hygric behaviors of glass fiber reinforced epoxy composites
  8. Comparison of the mechanical and wear behaviour of aluminium alloy with homogeneous and functionally graded silicon nitride composites
  9. Engineering behavior of clay soils stabilized with class C and class F fly ashes
  10. Preparation and erosion-corrosion behavior of polyetheretherketone (PEEK)/nickel foam co-continuous composites
  11. Optimization design, manufacturing and mechanical performance of box girder made by carbon fiber-reinforced epoxy composites
  12. Recent advances in the manufacturing processes of functionally graded materials: a review
  13. Numerical prediction of thermal conductivity in ZrB2-particulate-reinforced epoxy composites based on finite element models
  14. High-speed electrical sliding wear behaviors of Cu-WS2-graphite-WS2 nanotubes composite
  15. Adsorption removal of methylene blue from aqueous solution on carbon-coated Fe3O4 microspheres functionalized with chloroacetic acid
  16. Thermal degradation of coir fiber reinforced low-density polyethylene composites
  17. Preparation and analysis of polypropylene composites with maleated tea dust particles
  18. Predicting the thermal conductivity of polypropylene-multiwall carbon nanotubes using the Krenchel model
  19. Growth mechanism of 3D graphene-carbon nanotube hybrid structure
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  21. Sintered TiO2/recycled glass composites designed for the potential degradation of waterborne pollutants
  22. Evaluation of glass-fiber grafted by epoxide-terminated hyperbranched polymer on the effect of mechanical characterization of epoxy composites
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