Multi-objective structural optimization of honeycomb cells

Xu Zhang1 , Zhaoming Su2 , Wei Li3 , Xuejuan Niu4 , Zituo Wang5

1, 2, 4, 5Tianjin Key Laboratory of Advanced Mechatronics Equipment Technology, Tiangong University, Tianjin, 300387, China

3School of Energy and Safety Engineering, Tianjin Chengjian University, Tianjin, 300384, China

3Corresponding author

Journal of Vibroengineering, Vol. 24, Issue 4, 2022, p. 714-725. https://doi.org/10.21595/jve.2022.22210
Received 15 September 2021; received in revised form 20 December 2021; accepted 17 January 2022; published 30 June 2022

Copyright © 2022 Xu Zhang, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Creative Commons License
Abstract.

This paper develops an optimum cell structure design method considering the in-plane tensile/compression and shear properties to improve the stiffness and strength of the honeycomb core. The equivalent elastic modulus in the X or Y direction and shear modulus in the XY plane are derived using Energy Method for hexagonal, quadrilateral and concave hexagonal cells, and are compared with the results in the related literatures. The multi-objective optimization model in which the vertical wall length, wall thickness and inner angle of the cell are taken as design variables is solved by Genetic Algorithm to maximize the equivalent elastic moduli. The static and dynamic characteristics of the honeycomb cores with original and optimized cells are studied using Finite Element Method. The results show that after the cell optimization, the maximum displacement, stress and strain obviously decrease, thus improving the structural performance of the honeycomb core. The research provides significant guidance for the design of the cell structure.

Multi-objective structural optimization of honeycomb cells

Highlights
  • A cell considering in-plane tensile/compression and shear properties is optimized.
  • The results confirm structural performance improvements.
  • The research provides significant guidance for the design of cell structure.

Keywords: honeycomb cell, structural optimization, elastic modulus.

1. Introduction

The metallic and polymeric honeycomb cores widely used in panel structures, energy absorbers and insulation layers are formed by periodic permutation and combination of prismatic cellular microstructures [1-3]. Their tensile/compression and shear properties can be improved effectively by the cell structure optimization [4, 5]. Hence, much effort is usually spent on the optimization of geometric parameters of the cell.

Qin et al. [6] optimized the cell wall thickness and its variation gradient index using Non-dominated Sorting Genetic Algorithm (NSGA) to maximize the energy absorption ratio and minimize the collision force peak. Zhang et al. [7] optimized the cell size of a honeycomb structure by NSGA to maximize the energy absorption efficiency. Qiu et al. [8] optimized the size and key point coordinates of a hexagonal cell using Genetic Algorithm (GA) to minimize the flexibility of a chiral honeycomb panel. Wang et al. [9] optimized the relative density and size of the cell to improve the heat dissipation capacity of a composite sandwich honeycomb panel. Zhang et al. [10] optimized the cell wall thickness of an aluminum honeycomb using software Hyperstudy and LS-DYNA integrated optimization technique. Sorohan et al. [11] adopted the optimization module of ANSYS software to optimize the cell geometric parameters of a hexagonal honeycomb core with out-of-plane isotropic properties. Cinar et al. [12] optimized the thickness, diameter and inner angle of a three-dimensional cell using GA combined with a finite element solver to minimize the natural frequency difference square sum of the honeycomb panel. Gholami et al. [13] and Namvar et al. [14] optimized the cell structure parameters by Particle Swarm Optimization and GA to minimize the deflection of a honeycomb core under uniformly distributed normal load and static load, respectively.

To date, it has not been investigated that on the premise of ensuring the honeycomb core light weighting, the cell structure is optimized to improve the tensile/compression and shear moduli in in-plane. The present study firstly derives the formulas of the equivalent elastic moduli of three types of cells by Energy Method, and analyzes the calculation accuracy by comparison with the results of Refs. [15] and [16]. Then, the multi-objective optimization model is established to maximize the equivalent elastic moduli, and the size and shape of the cell are optimized using GA. Finally, the structural performance of original and optimized honeycomb cores are investigated using statics and modal analysis.

2. Equivalent elastic modulus of the cell

The inclined wall length l, vertical wall length αl, wall thickness βl and inner angle θ of the basic cell, and the honeycomb core thickness ηl are shown in Fig. 1. The non-dimensional treatment for the inclined wall length is produced, that is, [l]=1, and the dimensionless quantities of the vertical wall length, wall thickness and honeycomb core thickness are α, β, η, respectively.

According to θ>0°, θ=0° and θ<0°, the cell is divided into three types: hexagon cell, quadrilateral cell and concave hexagon cell, and the quarter structure and size parameters are shown in Fig. 2. The tensile and compressive stress in the X or Y direction and shear stress in the XY plane generate when the force acts in the honeycomb core plane. The equivalent elastic and shear moduli are derived, and the processes are the same for hexagon, quadrilateral and concave hexagon cells. The concave hexagonal cell is chosen as a testbed to illustrate the calculation of material constants.

Fig. 1. Honeycomb core

Honeycomb core

a) Honeycomb core

Honeycomb core

b) Basic cell

Fig. 2. Quarter structure and size parameters of hexagonal, quadrilateral and concave hexagonal cells

Quarter structure and size parameters of hexagonal, quadrilateral and concave hexagonal cells

a)θ>0°

Quarter structure and size parameters of hexagonal, quadrilateral and concave hexagonal cells

b)θ=0°

Quarter structure and size parameters of hexagonal, quadrilateral and concave hexagonal cells

c)θ<0°

2.1. Equivalent elastic modulus in the X direction

The basic cell is subjected to a normal stress along the X direction, σX, when the tensile deformation of the honeycomb core occurs, as shown in Fig. 3(a). The displacement of a quarter of the cell is calculated due to the symmetry, and is the same as that of the inclined wall under the combined action of stretching and bending. The vertical wall is regarded as a fixed-end constraint in view of the interaction between cells, and then the inclined wall is simplified to be a cantilever beam model, as shown in Fig. 3(b). The tensile load FX and bending load MX are expressed by:

(1)
FX=12σXαηl2,MX=14σXαηl3sinθ.

Fig. 3. Concave hexagonal cell subjected to σX and its equivalent model.

Concave hexagonal cell subjected to σX and its equivalent model.

a) Concave hexagonal cell subjected to σX

Concave hexagonal cell subjected to σX and its equivalent model.

b) Equivalent cantilever beam model

The axial force FXNx and bending moment MXZx of the inclined wall OA are as follows:

(2)
FXNx=12σXαηl2cosθ,MXZx=14l-x2σXαηl2sinθ.

The total deformation energy under the coupling action of FXNx and MXZx, UX, is as follows:

(3)
UX=UM XZ+UFN=0lMXZ2(x)2EsIZdx+0lFXN2(x)2EsAdx=σX2η2l7sin2θcos4θ32EsIZ+σX2η2l5sin4θcos2θ2EsA,

where UMXZand UFXN are the deformation energies produced by MXZx and FXNx, respectively; Es is the elastic modulus of matrix material; IZ and A are the moment of inertia and the area of the section.

The displacement of the cell along the X direction, δX, is obtained by Castigliano’s Theorem:

(4)
δX=UXFX=σXηl5sin2θcosθ12EsIZ+σXηl3cos3θEsA.

The strain along the X direction, εX, is as follows:

(5)
εX=δXαl=2σXcosθ(cos2θ+β2sin2θ)αβ3Es.

The equivalent elastic modulus along the X direction, EX*, is expressed according to Hooke’s Law:

(6)
EX*=σXεX=Esαβ32cosθ(cos2θ+β2sin2θ),     θ<0°.

For hexagonal and quadrilateral cells, the equivalent elastic modulus along the X direction is written as:

(7)
EX*=Es(α+2cosθ)β32cosθ(cos2θ+β2sin2θ),     θ0°.

2.2. Equivalent elastic modulus in the Y direction

The equivalent elastic modulus in the Y direction is calculated based on the cantilever beam model when the cell is subjected to a normal stress along the Y direction, σY, as shown in Fig. 4. The tensile load FY and bending load MZ are expressed by:

(8)
FY=σYηl2cosθ,MZ=12σYαηl3cosθ.

Fig. 4. Concave hexagonal cell subjected to σY and its equivalent model

Concave hexagonal cell subjected to σY and its equivalent model

a) Concave hexagonal cell subjected to σY

Concave hexagonal cell subjected to σY and its equivalent model

b) Equivalent cantilever beam model

The axial force FYN(x) and bending moment MYZ of the inclined wall OA are as follows:

(9)
FYNx=σYηl2sinθcosθ,MYZx=12l-xσYηl2cos2θ.

The total deformation energy under the coupling action of FYN(x) and MYZ, UY, is as follows:

(10)
UY=UMYZ+UFN=0lMYZ2(x)2EsIZdx+0lFYN2(x)2EsAdx=σY2η2l7cos4θ24EsIZ+σY2η2l5sin2θcos2θ2EsA.

The displacement of the cell along the Y direction, δY, is obtained by Castigliano’s Theorem:

(11)
δY=UYFY=σYηl5cos3θ12EsIZ+σYηl3cosθsin2θEsA.

The strain along the Y direction, εY, is as follows:

(12)
εY=δYαl=σYηl2cosθ(cos2θ+β2sin2θ)Esβ3ηlα.

The equivalent elastic modulus along the Y direction, EY*, is expressed according to Hooke’s law:

(13)
EY*=Esαβ3cosθ+βcos2θ+β2sin2θ,     θ<0°.

For hexagonal and quadrilateral cells, the equivalent elastic modulus along the Y direction is written as:

(14)
EY*=Esβ3α+sinθcosθ+βcos2θ+β2sin2θ,     θ0°.

2.3. Equivalent shear modulus in the XY plane

The displacement of the cell subjected to the shear stress τ in Fig. 5(a) is obtained by calculating that of point B. Similarly, the vertical wall is regarded as a fixed-end constraint, and the cell is simplified to a symmetric beam with antisymmetric loads, as shown in Fig. 5(b). The load on point B, FXY, is expressed by:

(15)
FXY=τηl12αl+lsinθ.

The displacement caused by the shear deformation, δXY, is obtained by Castigliano’s Theorem:

(16)
δXY=8cos2θ(α3+3α2+6αsinθ+7sin2θ)FXYl33(α3+12α2+24αsinθ+16sin2θ)EsIZ.

Fig. 5. Concave hexagonal cell subjected to τ and its equivalent model

Concave hexagonal cell subjected to τ and its equivalent model

a) Concave hexagonal cell subjected to τ

Concave hexagonal cell subjected to τ and its equivalent model

b) Equivalent symmetric beam model

The in-plane shear strain γXY is as follows:

(17)
γXY=δXY2lcosθ.

The equivalent shear modulus GXY* is expressed according to Hooke’s law:

(18)
GXY*=Esβ3α3+12α2+24αsinθ+16sin2θ8αcosθα3+3α2+6αsinθ+7sin2θ,      θ<0°.

For hexagonal and quadrilateral cells, the equivalent shear modulus is written as:

(19)
GXY*=Esβ3α3+12α2+24αsinθ+16sin2θ8cosθα3+3α2+6αsinθ+7sin2θα+2sinθ,      θ0°.

2.4. Validation of equivalent elastic and shear moduli

Table 2 gives the values of EX*, EY* and GXY* from the presented method, method of Ref. [15] and experiment of Ref. [16] for the cells with the parameters in Table 1. The calculated results from the methods of this paper and Ref. [15] are within the range of experimental values for Sample 1, while EY* from the method of Ref. [15] is out of the range of experimental values for Sample 2. This is because Ref. 15 does not consider the tensile deformation of the cell wall. The displacement caused by the tensile deformation can be ignored due to the thinner wall of Sample 1, and need be taken into account as the wall thickness of Sample 2 is closer to the inclined wall length. Hence, the presented method has good accuracy.

Table 1. Size and performance parameters of Sample cells in Ref. 16

Size and performance parameters
Sample 1
Sample 2
l / mm
6
2.5
αl / mm
6
2.5
βl / mm
0.04
0.03
θ / (°)
30
30
Es / GPa
72
72
ρs / (kg.m-3)
2700
2700
υs
0.34
0.34

Table 2. Calculated and experimental values of EX*, EY* and GXY*

Analysis method
Sample 1
Sample 2
EX* / MPa
EY* / MPa
GXY* / MPa
EX* / MPa
EY* / MPa
GXY* / MPa
Presented method
0.045
0.049
0.005
0.262
0.283
0.030
Method of Ref. 15
0.049
0.047
0.012
0.287
0.216
0.072
Experiment of Ref. 16
0.045-0.055
0.043-0.053
0.260-0.300
0.270-0.310

3. Optimization method of cell structure

The vertical wall length, wall thickness and inner angle of the basic cell are chosen as the design variables:

(20)
H=α,β,θ,

where H is the vector of design variables.

The elastic modulus is an important index to measure the strength and stiffness of the honeycomb core. Hence, the design objective is to maximize the equivalent elastic and shear moduli:

(21)
FH=maxλ1EX*+λ2EY*+λ3GXY*,

where λ1, λ2, λ3 are the weight coefficients, and λ1+λ2+λ3=1.

Too thin or too thick cell wall increases the processing difficulty and results into the solid cell, respectively, and the cell will disappear if the inner angle is too great. Besides, the relative density ρs* is used to characterize the mass in design space. Hence, α, β, θ and ρs* are bounded by:

(22)
0.5α1,0.004β0.008,-30°<θ<30°,ρs*ρori*,

where ρori* is the relative density of the original cell.

The flow chart of the cell structure optimization is shown in Fig. 6. The multi-objective GA starts the search from a string set of solutions, generating a large number of non-inferior solutions in one optimization thus avoiding the disadvantage that the integrated algorithm is easy to fall into a local optimum. For GA parameters, the crossover probability is 0.7, the mutation probability is 0.02, and the maximum iteration number is 100. To ensure the calculation accuracy, the design variables and optimized results keep 4 decimal places using binary encoding:

(23)
2t-1<Hmax-Hmin×10n2t,

where Hmax and Hmin are the upper and lower limits of design variables, respectively; n is accurate to the nth decimal place; t is the length of the encoded gene, and is 33.

Fig. 6. Flow chart of the cell structure optimization

Flow chart of the cell structure optimization

The optimizer performance is illustrated in Fig. 6(a), and the good convergence is achieved after 19 iterations. The structural parameters are shown in Table 3, and Fig. 7 gives the original and optimized cells.

Fig. 7. Optimizer performance and cells

Optimizer performance and cells

a) Optimizer performance

Optimizer performance and cells

b) Original and optimized cells

Table 3. Structural parameters of optimized honeycomb cell

Optimized parameters
Initial values
Optimized values
αl / mm
6.0000
5.5230
βl / mm
0.0400
0.0384
θ / °
30.0000
16.6920
ρs*
0.0153
0.0133

4. Structure performance analysis of original and optimized honeycomb cores

The original and optimized cells are respectively tiled to form the honeycomb cores, and then the finite element models are established with APDL language in ANSYS environment and are meshed by shell93 element, as shown in Fig. 8. The statics and modal analysis are performed with an aim to study the effects of the cell optimization on the displacement, stress, strain, nature frequency and mode shape.

Fig. 8. Finite element models of original and optimized honeycomb cores.

Finite element models of original and optimized honeycomb cores.

a) Before the cell optimization

Finite element models of original and optimized honeycomb cores.

b) After the cell optimization

Fig. 9. Displacement, stress and strain distributions of original and optimized honeycomb cores

Displacement, stress and strain distributions of original and optimized honeycomb cores

a) Displacement before the optimization

Displacement, stress and strain distributions of original and optimized honeycomb cores

b) Displacement after the optimization

Displacement, stress and strain distributions of original and optimized honeycomb cores

c) Stress before the optimization

Displacement, stress and strain distributions of original and optimized honeycomb cores

d) Stress after the optimization

Displacement, stress and strain distributions of original and optimized honeycomb cores

e) Strain before the optimization

Displacement, stress and strain distributions of original and optimized honeycomb cores

f) Strain after the optimization

4.1. Statics analysis

The planes A1, A2 and A3 are fully constrained, and the uniform load of 1000 N along the positive direction of Y is applied on the planes A4, A5 and A6. Fig. 9 gives the distributions of the displacement, stress and strain for original and optimized honeycomb cores. The nodal displacement is corrugated from the left to the right, and decreases. The maximum displacement occurs at the cell located in the bottom-left of the honeycomb core, and the maximum stress and strain are at the bottom-left cell connections. The maximum displacement, stress and strain decrease by 50.70 %, 51.69 % and 47.45 % after the cell optimization, respectively.

4.2. Modal analysis

The modal analysis is carried out by Lanczos algorithm, and the frequencies and mode shapes of original and optimized honeycomb cores are obtained. Table 4, Fig. 10 and Fig. 11 give only the first six frequencies and mode shapes due to the greater influence of the lower frequencies on the stability. The first six mode shapes of optimized honeycomb are the same as those of original honeycomb. The second-, third-, fourth- and sixth- order frequencies increase by 2.95 %, 13.38 %, 24.29 % and 2.95 % after the optimization, while the first- and fifth- order frequencies decrease by 2.28 % and 8.33 %.

Table 4. Frequencies and mode shapes of original and optimized honeycomb cores

Frequency order
Before the optimization
After the optimization
Frequency / (Hz)
Mode shape
Frequency / (Hz)
Mode shape
1st
167.39
First-order shimmy vibration
163.57
First-order shimmy vibration
2nd
357.19
First-order torsional vibration
367.73
First-order torsional vibration
3rd
832.18
First-order bending vibration
943.55
First-order bending vibration
4th
1109.41
Second-order bending vibration
1378.84
Second-order bending vibration
5th
1252.38
Second-order shimmy vibration
1148.04
Second-order shimmy vibration
6th
1339.38
Second-order torsional vibration
1378.84
Second-order torsional vibration

Fig. 10. Mode shapes of the original honeycomb core

Mode shapes of the original honeycomb core

c) First-order

Mode shapes of the original honeycomb core

b) Second-order

Mode shapes of the original honeycomb core

c) Third-order

Mode shapes of the original honeycomb core

d) Fourth-order

Mode shapes of the original honeycomb core

e) Fifth-order

Mode shapes of the original honeycomb core

f) Sixth-order

Fig. 11. Mode shapes of the optimized honeycomb core

Mode shapes of the optimized honeycomb core

c) First-order

Mode shapes of the optimized honeycomb core

b) Second-order

Mode shapes of the optimized honeycomb core

c) Third-order

Mode shapes of the optimized honeycomb core

d) Fourth-order

Mode shapes of the optimized honeycomb core

e) Fifth-order

Mode shapes of the optimized honeycomb core

f) Sixth-order

5. Conclusions

Based on the cantilever and symmetric beam models, the equivalent elastic and shear moduli of the basic cell are derived by Energy Method, respectively. Compared with the method of Ref. [15], the accuracies of the formulas are higher since the influences of tensile and bending deformations of the cell wall are considered. The cell structure parameters are optimized with the consideration of in-plane tensile/compression and shear mechanical properties, and the displacement, stress, strain, frequency and vibration mode of original and optimized honeycomb cores are analyzed. The maximum displacement, stress and strain of the honeycomb core decrease by 50.70 %, 51.69 % and 47.45 % after the cell optimization, thus weakening the stress concentration and deformation and increasing the strength and stiffness. The natural frequencies of optimized honeycomb core mostly increase and are all beyond the excitation frequency range, indicating that the optimized structure is less prone to the resonance.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 51805369), Science and Technology Planning Project of Tianjin (Grant No. 20YDTPJC00820) and Natural Science Foundation of Tianjin (Grant No. 18JCYBJC89000).

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