Designing foam filled sandwich panels for blast mitigation using a hybrid evolutionary optimization algorithm

Composite Structures 158 (2016) 72–82

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

Designing foam filled sandwich panels for blast mitigation using a hybrid evolutionary optimization algorithm Idris Karen a,⇑, Murat Yazici b, Arun Shukla c a

Bursa Orhangazi University, Engineering Faculty, Mechanical Engineering Department, Yildirim Campus 16310, Bursa, Turkey Uludag University, Engineering Faculty, Automotive Engineering Department, 16059 Bursa, Turkey c The University of Rhode Island, Dynamic Photomechanics Laboratory, Department of Mechanical Industrial and Systems Engineering, 92 Upper College Road, Kingston 02881, RI, USA b

a r t i c l e

i n f o

Article history: Received 27 August 2015 Revised 24 June 2016 Accepted 29 July 2016 Available online 30 July 2016 Keywords: Sandwich panel Hybrid evolutionary algorithm Blast loading Corrugated steel core Polymer foam infill Shock tube

a b s t r a c t Developing sandwich structures with high energy absorption capability is important for shock loading applications. In the present study, a hybrid evolutionary optimization technique based on Multi-Island Genetic Algorithm and Hooke-Jeeves Algorithm is used in the design stage of the sandwich structures to obtain effective results. Optimum parameters of cell geometry were investigated using the hybrid optimization algorithm to design foam filled sandwich panels for three main boundary conditions. Shock tube experiments were conducted in order to simulate the shock load effects along with 3D and 2D finite element analysis. Using the experimental results, a simulation-based design optimization approach was prepared and used to develop the designs of new sandwich structures. Promising results were obtained for all three different boundary conditions. In the simply supported case, 21% improvement of shock absorption was achieved by using 57% less volume of foam with respect to the original fully foam filled sandwich panel. In the clamped-clamped case, 16% improvement of shock absorption with 52% less volume was obtained. In the rigid base case study, 6% improvement of shock absorption with 38% less volume usage was achieved. The structures developed in this study will be of use in the defense, automotive and other industries. Ó 2016 Published by Elsevier Ltd.

1. Introduction Recent studies relating to the protection of people from air shock loadings have emphasized the need for the development of novel sandwich structures. Sandwich structures that consist of different face sheets and cores have been developed to meet suitable shock absorption properties. Of particular interest are metallic sandwich structures composed of cellular material cores that combine high energy absorption capabilities with a lightweight design. The core material separates the face sheets and provides required stiffness of the sandwich structure. Furthermore, the metallic face-sheets provide the adequate strength for the structure [1–4]. The corrugated metallic core is one of the most popular core topologies in the sandwich panel constructions for blast loading. It provides high strength properties in both the normal and longitudinal directions. The corrugated cores also provide manufacturing advantages; they have an excellent flexibility feature in the ⇑ Corresponding author. E-mail address: [email protected] (I. Karen). http://dx.doi.org/10.1016/j.compstruct.2016.07.081 0263-8223/Ó 2016 Published by Elsevier Ltd.

design process and beyond, and their low-cost makes the sandwich structures attractive for mass production [5–11]. There is also a growing attention to using the polymeric foams as a filling material to improve the performance and energy absorption capabilities of traditional lightweight sandwich structures [12–17]. By combining a corrugated metallic cellular core with polymeric foams, it is possible to obtain acceptable shock absorption rates and diminish the transmitted shock load. Furthermore, filling the cores with foam supports the core cell walls against buckling and increases the strength of the structure. Filling the interstices of the metallic core sandwich structures with foam also provides some multifunctional advantages such as good acoustic and thermal insulation [12,14,18–21]. Yazici et al. [13] showed that adding foam to the interstices of metallic sandwich panels with corrugated cores doubles the performance of shock load mitigation compared to the unfilled panels. In their further study [14], various preferentially filled core configurations were examined for blast mitigation. Adding the foam to the backside cells of the sandwich structure and leaving the front side cells empty improves the shock absorption performance. Therefore, searching both the optimum core design and optimum foam filling distribution in structures exposed to air shock loading

I. Karen et al. / Composite Structures 158 (2016) 72–82

has practical significance in designing a blast-resistant structure that has not been previously investigated. In this study, the optimum distribution of foam material between corrugated core cells was examined to maximize the performance of the structure. The energy absorption capability of the structure is improved by using a hybrid optimization technique with the aim of increasing the resistance of sandwich panels exposed to blast loads. Also, from the experiments, it is observed that maximizing the energy absorption capability is equal to minimizing the back face deflection (BFD) of the sandwich structure exposed to the shock load. Applying the optimization methods in the development stage of the sandwich structures is increasing continuously for making brand new structure designs. Some design optimization studies have focused on finding the optimum thermo-mechanical and geometrical parameters of metallic corrugations in the sandwich panel [22,23]. Wadley et al. [1] investigated the optimum topology of metallic sandwich panels with periodic, open-cell cores. Wei et al. [24] searched the optimum design of prismatic core sandwich panels subject to bending loads in the in-plane directions. Rathbun et al. [25] tried to find the optimum core topology between pyramidal truss, tetrahedral truss, square honeycomb, and corrugated sheet. Kooistra et al. [26] applied a design optimization method to find the optimum geometric parameters of second-order corrugated cores. Liu et al. [27] used a quadratic optimizer to determine the optimal dimensions and weights of open-ended, internally pressurized sandwich cylinders. The aforementioned studies were done under static or quasi-static loading conditions. The studies carried out under dynamic loading conditions such as blast loads with high-strain rates are a bit more complex. Liang et al. [28] investigated the optimum geometric parameters of metallic corrugated core sandwich panels subjected to blast loads for the naval industry. Hou et al. [29] used multi-objective optimization techniques for the optimum values of structural parameters of corrugated sandwich panels under low-velocity local impact and planar impact with the crashworthiness criteria. Lim et al. [30] used hybrid sandwich plates for the optimal design of core structure in the sandwich panels against blast loads. Qi et al. [31] used a group of metallic aluminum foam-cored sandwich panels to find the optimum geometric and material design parameters of the structure for vehicle armors against blast loading. In this study, the optimum distribution of foam cores between the metallic corrugation sheets of the sandwich panel exposed to shock loads was investigated using an evolutionary optimization algorithm. In the literature, different optimization techniques have been developed for various types of problems [32–36]. In recent years, evolutionary algorithms are popular due to their practical, robust and heuristic properties. Because of the nonlinear and nondifferentiable properties of dynamical blast-exposed sandwich panel design optimization problems, classical mathematical optimization techniques cannot be used efficiently. These techniques primarily seek the optimum design solution around the starting point and easily get stuck in the local optimum design regions. However, evolutionary algorithms can search throughout the design space efficiently without struggling in local regions. Evolutionary algorithms mimic the laws of nature, especially biological evolutionary processes such as adaptation to rapid changes and intelligent social behaviors of species. These algorithms have been used largely after the 1960s. Among these; Genetic Algorithms [32,33], Differential Evolution [34], Ant Colony Optimization [35] and Particle Swarm Optimization [36] can be listed. In the present study, a hybrid algorithm based on the MultiIsland Genetic Algorithm and Hooke-Jeeves Algorithm was used as an optimization technique for the purpose of benefiting the powerful properties of two algorithms. Multi-Island Genetic Algorithm [37] is very efficient at global search, whereas the

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Hooke-Jeeves Algorithm [38] is good at local search. In the hybrid algorithm, these two important properties are combined in such a way that firstly, the Multi-Island Genetic Algorithm explores the design space globally to find a near optimal solution, and then the Hooke-Jeeves Algorithm continues searching the local design space deeply. By this way, the performance of the optimization process was increased significantly. The optimum distributions of foam filled cells in the sandwich panel core were investigated to develop a novel composite material that has high strength/weight and high rigidity/weight ratios in the design stage of the process. A shock tube apparatus was used in the experiments with high-speed cameras to examine the displacement and strain results of the sandwich structure. A simulation-based design approach integrated with the experimental results was prepared and used for supporting the designs of the new shock absorption sandwich structures. After validating the simulations, the hybrid evolutionary optimization approach was applied, and promising results were obtained for three different boundary conditions; simply supported, clamped-clamped and rigid base case. 2. Experimental study of the fully foam filled sandwich panel subjected to shock loading The sandwich panel used in the experimental study was prepared with corrugated steel plates containing fully filled foam cores (Fig. 1). Low-density polyurethane (PU) foam was used as a foam material. Each core surfaces were glued by a G/Flex epoxy adhesive (West System Inc., Bristol, RI). The dimensions of both front face and back face plates are 184.51
50.8
3.175 mm. A section part of the sinusoidal corrugated steel plates with dimensions in mm is given in Fig. 2. A shock tube that consists of a long (8 m) thick-walled hollow cylinder was used in the experimental study of the fully foam filled sandwich panel (Fig. 3a). It consists of a driver section with 0.15 m inner diameter, a driven section beginning with 0.15 m and ending with 0.07 m inner diameter, and a muzzle section with 0.05 m inner diameter. The lengths of the sections are 1.82 m, 3.65 m, and 2.53 m, respectively. The driver section is pressurized with helium gas because of its lightweight and non-explosive properties. The driven section has atmospheric air pressure in this case. There is a diaphragm separating the driver and driven sections. The driver section is pressurized until the diaphragm reaches the critical pressure where the rupture occurs. After a rapid release of pressurized gas, the shock wave is created, moving along the tube with a high pressure (incident pressure) and velocity. The shock wave contacts the specimen positioned at the end of the shock tube, reflecting with a higher pressure (reflected pressure) nearly five times greater than the incident pressure. Two pressure transducers are mounted at the end of the muzzle section for mea-

Fig. 1. The sandwich panel with fully foam filled corrugated cores used in the shock load experiments.

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Fig. 2. A section part of the sinusoidal corrugated steel plate (All dimensions are in mm).

Fig. 4. The 3D solid model of the sandwich panel simulation.

Fig. 5. The surface-to-surface explicit contacts between first and second corrugated plates.

Fig. 6. The 8-node linear brick, reduced integration with hourglass control element (C3D8R).

Fig. 3. a. The shock tube at the Dynamic Photomechanics Laboratory in the University of Rhode Island, b. The incident and reflected peak pressure profiles.

suring the pressure and velocity values (Fig. 3a). A typical pressure profile obtained from pressure transducers is plotted in Fig. 3b. The incident peak pressure of the shock wave was measured as 1.21 MPa, and the reflected peak pressure was measured as 5.57 MPa. Each experiment was repeated at least three times for robustness and repeatability. The pressure profile data given in Fig. 3b was later used in the finite element analyses (FEA) of the sandwich panels. The deformation of the sandwich composite panel specimen occurred after the shock load impingement and it was calculated photomechanically from the real-time side-view frames captured by the high-speed digital camera with 20,000 fps (Fig. 10).

3. Numerical study of the fully foam filled sandwich panel subjected to shock loading All numerical analyses were executed in the commercial finite element software, Abaqus/Explicit, to simulate the experimental study of the sandwich panels subjected to shock loading for 3 ms run time. The first three-dimensional (3D) simulation study was done including the simply supported with a roller boundary condition as in the experimental study. After the validation of the 3D simulation study with experimental results, further simulations were conducted with three different boundary condition cases; simply-supported, clamped-clamped, and rigid base. 3.1. Three-dimensional finite element simulations The fully foam filled corrugated sandwich panel simulation study illustrates an inexpensive approach to the prediction of a

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is the yield stress at nonzero strain rate, A, B, n, and m are where r metal plasticity constants measured at or below the transition temperature,
epl is the equivalent plastic strain, C and e_ 0 are material ^ is the parameters.
e_ pl is the equivalent plastic strain rate and h nondimensional temperature defined as;

^h 

Fig. 7. The 4-node bilinear plane strain quadrilateral with first-order reduced integration element (CPE4R).

Back Face Deflection (BFD) value that reveals the shock absorption energy. It is also observed from the experiments that maximizing the energy absorption capability is equal to minimizing the back face deflection (BFD) of the sandwich structure subjected to the shock load. The first 3D simulation case study was conducted to build a proper simulation model that was consistent with the shock tube experimental study. The 3D solid model of the simulation contains a front face plate, 33 foam cores, four corrugated plates, and a back face plate as used in the experimental study of the sandwich panel (Fig. 4). The material of the front and back face plates, and the corrugated plates was assigned as steel with the Young’s Modulus (E) of 205,000 MPa, the Poisson’s Ratio (m) of 0.29, and the density (q) of 7.8 g/cm3. A Johnson-Cook isotropic hardening model was used for simulating the plastic behavior of the material. In this model, the yield stress is provided as an analytical function of equivalent plastic strain, strain rate, and temperature [39]. The yield stress is calculated as;

"


e_ pl n r
¼ ½A þ Bð
epl Þ  1 þ Cln _ e0

!#

ð1  ^hm Þ

ð1Þ

8 > < > :

0 1

h < ht

for

ðh  ht Þ=ðhm  ht Þ for for

ht 6 h 6 hm

ð2Þ

h > hm

where h is the current temperature, ht is the transition temperature, and hm is the melting temperature [39]. The Johnson Cook parameter values were used as the same values in Schwer’s study [40] for the steel material. The foam cores’ material was defined as a hyperelastic material with uniaxial test data. The polyurethane (PU) foam material shows extremely compressible behavior and the porosity structure allows enormous volumetric change. The Ogden second order strain energy potential was selected as a material model due to its perfect match of the stress-strain curve to experimental data. The density was described as 0.0446 g/cm3. An explicit, dynamic analysis was used to solve the sandwich panel model with a very short dynamic response time (0.003 s). In this analysis, the total analysis time is divided by a large number of small time increments. They are used in the calculation of the accelerations and velocities for the dynamic equilibrium equations. Between the adjacent faces in the sandwich panel, surface-tosurface explicit contacts were identified using the penalty contact method as a mechanical constraint formulation with finite sliding formulation in Abaqus/Explicit (Fig. 5). The normal behavior of the contact was selected as a hard kinematic contact in which the master surface can penetrate the slave surface in a properly corrected configuration. Thus, the contact behavior has a flexible property for large deformations used in this explicit dynamic simulation. The tangential behavior was identified with a 35 MPa shear stress limit. The shock load was applied on the front face plate as a pressure profile with a tabular data taken by a pressure transducer in the shock tube experiment. As in the experiment, a simply supported boundary condition with a roller applied to the bottom edges was utilized. The Finite Element Model (FEM) of the dynamic expli-

Fig. 8. a. Pressure profile transferred from 3D FEM to 2D FEM, b. Shock pressure profile.

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Time (ms)

High Speed Camera Views

Simulation Results

0.0

0.5

1.0

1.5 Fig. 9. The rectangular and circular pressure forms.

cit problem was modeled with 8-node linear brick, reduced integration elements (C3D8R) with the hourglass control feature [39]. The C3D8R element is shown in Fig. 6. The second 3D simulation was conducted to predict the overall dynamic response of the sandwich structure under different boundary conditions using the same shock load pressures obtained from the experiment. In this simulation, simply supported, clamped-clamped, and rigid base cases can be compared for the same loading conditions. However, in the first 3D simulation, the clamped-clamped and the rigid base cases cannot be compared with the simply supported case because it is too hard to eliminate the overhanging and the edge effects of boundary condition variations at the supports that were used in the experimental study. For this reason, after the correlation of the first 3D simulation with shock load experiment, the second 3D simulation and 2D simulation were identified with the same settings of the first 3D simulation but under three different cases. The 2D simulation was used to reduce the high CPU time of the optimization loop involved with 3D simulations. By using 2D simulation, the CPU time is reduced by about 8000 times.

2.0

2.5

3.0

3.2. Two-dimensional finite element simulations In the 2D FEA, the explicit dynamic problem was modeled with a 4-node bilinear plane strain quadrilateral with first-order reduced integration elements (CPE4R). This element type has an hourglass control feature in the explicit linear dynamic analysis and was used specifically in the large strain analyses of sandwich structures subjected to shock loads. The CPE4R element is shown in Fig. 7. The experimentally obtained air shock pressure loads were imported into the Abaqus as tabular data. The shock pressure distribution on the sandwich specimen’s front face was arranged as a function of time and radius of the affected area as conical shapes. However, in 2D simulations this conically shaped pressure profile was reduced to 2D section. Both were arranged as a function of

Fig. 10. The comparison between the high-speed camera views and simulation results at first 3.00 ms [13].

time. In the previous studies, this shock pressure distribution on the sandwich panel front face was characterized as a combination of uniform and non-uniform sections [13,14]. The uniform pressure profile was applied to the front face directly from the muzzle section’s inner diameter. The non-uniform pressure profile was applied in such a way that it decays from inner muzzle pressure values at smaller circle to zero at the larger circle as shown in Fig. 8 [13]. The transformation of 3D shock pressure to 2D is shown in Fig. 8a. The 3D shock pressure has two parts; inside pressure

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x2 þ y 2 ¼ R 2

ð3Þ

The middle x-z plane of the sandwich structure is used for the 2D sandwich structure with symmetrical boundary conditions. However, if the same 3D pressure value is used in the 2D loading, the loading shape will be rectangular because of the symmetrical conditions (Fig. 9). The form of the shock pressure in the 2D model must be circular to apply the real shock pressure loading on the structure. To correct this issue, the 3D shock pressures (P3D, inside and P3D, outside) can be multiplied by correction values for handling the real 2D shock pressures (P2D, inside and P2D, outside), as follows;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2D;inside ¼ P3D r 2  x2 ;

P2D;outside ¼ P3D

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2  x2

Experiment

10

Simulaon

8 6 4 2 0 0

0.5

1

1.5 2 Time (ms)

2.5

3

Fig. 11. The correlation of simulation BFD results with experiment.

4.2. Correlation of 2D simulation results with 3D simulation results

ð4Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The correction values ( r2  x2 and R2  x2 ) were calculated from Eq. (3). So, the pressure varies circularly from the center (x ¼ 0) to the radius (x ¼ r or x ¼ R). When x ¼ 0, the 2D Pressure value will be equal to the 3D Pressure value. When x increases, the 2D Pressure value will decrease circularly same as in 3D loading. In Abaqus, the variation of this 2D pressure over x-axis was applied using the analytical expression fields in which special types of mathematical functions can be easily defined [39]. Selecting the optimum mesh size or element number is one of the important issues in a finite element analysis. If the selected element number is too low, then the accuracy of the simulation results can be compromised. On the contrary, if the element number is sufficient, the precision of the solution can remain stable, beyond which CPU runtime can increase dramatically. For this reason, mesh convergence has been performed to find the optimum element number for the finite element analysis of the sandwich panel. In the mesh convergence study, the number of the elements is increased in each simulation until a negligible change between results is achieved. The optimum number of elements found for the mesh convergence is 2131, the optimum number of nodes is 3152, and the DOF value is 6304 for this 2D FEM simulation.

The 2D FEM was verified using experimentally validated 3D FEM model. In these models, suitable boundary conditions were used to characterize the optimum behavior of the sandwich material. In Fig. 12, the 2D and 3D simulation results were given for the front and back face mid-point deflections (FFD and BFD). The results show excellent correlation values when compared to one another. The Pearson’s correlation coefficients, R and R2, were calculated using equations given in [41] and [42] as a means to evaluate the model’s accuracy. The correlation coefficient is a measure of the accuracy of the linear relationship between the 3D and 2D simulation results. The predictability of the finite element model over both the front and back faces of the simulations are shown in Table 1. The correlation coefficients R and R2 are 0.99 or higher indicating that the data of both the 3D and the 2D FEA models are in good agreement.

BFD (mm)

x2 þ y 2 ¼ r 2 ;

12

BFD (mm)

(P3D, inside) related to inner diameter r, and outside pressure (P3D, outside) related to outside diameter R. Because of the circular shape of the shock pressure, the equations can be defined, as follows;

4. The comparison studies between experiments, 3D simulation and 2D simulation Before the structural optimization study, the simulations were verified with the experimental study to perform the optimization with the correlated simulation model. Therefore, initially, the first 3D simulation results were compared with the experimental results. Then, the 2D simulation results were compared with the correlated 3D simulation results with the new boundary conditions.

In the experiments, a side view high-speed camera recording at 20,000 fps was used to record the Back Face Deflection (BFD). In the first 3D simulation, the BFD was calculated using a node set related to the center point of the FEM for 3.00 ms at which time the peak shock effect ended (Fig. 10). Fig. 11 shows the correlation of the BFD simulation results with the experimental results. The Pearson’s correlation coefficient R2 for the correlation is 0.95 indicating a favorable agreement between the experiments and simulation.

3D Simulaon 2D Simulaon

0.0

0.5

1.0

1.5

2.0

2.5

3.0

2.0

2.5

3.0

Time (ms)

a

FFD (mm)

4.1. Correlation of 3D simulation results with shock tube experimental results

35 30 25 20 15 10 5 0

35

3D Simulaon

30

2D Simulaon

25 20 15 10 5 0 0.0

0.5

1.0

1.5

Time (ms)

b Fig. 12. The comparison of BFD (a) and FFD (b) in the 3D and 2D simulations.

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According to these results, 3D to 2D simplifications concerned with the geometry and the applied shock loading are valid. In Table 2, the comparisons are given between 3D and 2D simulation performance for equivalent conditions showing the reduction in the CPU time in the simulation by more than 8000 times. 5. Design optimization of foam filled sandwich panels with a hybrid optimization algorithm based on the Multi-Island Genetic Algorithm with elitism and the Hooke-Jeeves Algorithm The optimum distribution of foam filled cells in the sandwich panel core were investigated to develop a novel composite material that has a high strength/weight and rigidity/weight ratios in the design stage of the process. In this optimization process, the main objective is to increase the performance of sandwich panels subjected to shock loads. For increasing the performance of a sandwich panel, the energy absorption capability of the structure must be maximized. In this study, the optimization process was executed for three different boundary conditions, such as simplysupported, clamped-clamped, and rigid base cases. It was observed from the experiments that maximizing the energy absorption capability is equal to minimizing the back face deflection (BFD) of the sandwich panel for the simply-supported (Fig. 13) and clamped-clamped cases. For the rigid base case, the more energy absorption capability means, the less produced reaction forces. For this reason, to maximize the energy absorption capability, the total reaction forces or the mean reaction force on the bottom surface of the sandwich structure must be minimized. The plastic deformations have an important role in the simulation. So, the stress and strain values do not need to be limited. For this reason, this optimization process has no constraints, in other words, the design optimization problem of the sandwich panel is an unconstrained optimization problem. There are 33 design variables of this optimization problem. Each core in the sandwich structure expresses one design variable. If a core is filled with foam then, the design variable of this core has the value of ‘‘1”, otherwise, (when the core is empty) it has the value of ‘‘0”. Consequently, the cores can be filled with 233 different possibilities. The main objective of this optimization problem is to

Table 2 The comparison between 2D and 3D simulation results.

CPU time (seconds) Number of Elements Number of Nodes Total Degrees of Freedom (DOF)

2D Simulation Model

3D Simulation Model

Difference (%)

105 2131 3152 6304

47,640 55,443 84,681 254,043

45,371 2602 2687 4030

Fig. 13. The BFD of foam filled corrugated sandwich panel (Image taken by Photron SA1.1 high-speed camera at 20,000 fps).

find the optimum distribution of these foam filled cells in the sandwich structure with maximizing the energy absorption capability for the boundary conditions studied. The objective values (BFD and the mean reaction forces) are calculated by the explicit dynamic FEA simulations explained in the numerical simulations section of this paper. 5.1. Unconstrained optimization of foam cores with the case study of simply-supported with a roller The objective of the optimization process in this case study is the minimization of the Back Face Deflection (BFD) calculated by the explicit dynamic FEA simulation. The design parameters of the optimization problem are the existence of foam filled cells which take ‘‘0” or ‘‘1” values as explained before. In this study, 33 cells are used as optimization parameters (Fig. 14). Objective (Minimization): The Back Face Deflection (BFD) calculated from the FEA

Table 1 The correlation ratios between 3D simulation and 2D simulation. Correlation Ratios

Time (ms)

0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.01 1.20 1.35 1.50 1.65 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 3.00 R R2

Back Face Mid-Point Deflections (BFD) (mm)

Front Face Mid-Point Deflections (FFD) (mm)

3D

2D

3D

2D

0.0000 0.0001 0.0030 0.2254 0.9680 2.0251 3.2497 4.8437 6.8422 9.1793 11.7848 14.5440 17.3154 20.0308 22.6660 25.1255 27.3051 29.1272 30.5606 31.6208 32.3604 0,9993 0,9986

0.0000 0.0002 0.0278 0.2936 0.9266 1.7037 2.7245 3.7336 6.1610 8.3526 10.7977 13.5559 16.4825 19.3829 22.1117 24.6143 26.8381 28.7652 30.4118 31.7967 32.9644

0.0000 0.1531 1.3501 3.1068 5.5083 8.6597 11.9539 14.9327 17.6589 20.1833 22.2714 23.9122 25.2172 26.2830 27.1657 27.9349 28.8278 29.8763 31.1433 32.6901 34.2479 0,9987 0,9974

0.0000 0.1287 1.2449 2.9906 5.5083 8.9131 12.3785 14.5919 18.4110 21.0554 23.1525 24.7292 25.9525 26.8464 27.4937 28.0710 28.7073 29.5123 30.5623 31.7371 32.9730

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The clamped-clamped boundary conditions of the simulation are shown in Fig. 16. All the translational displacements of x and y-directions of the bottom left, and bottom right corners are fixed. All the rotational displacements around z-directions of both corners are also fixed (Fig. 16). 5.3. Optimization of foam cores with the rigid base case study Fig. 14. Optimization parameters of simply-supported case study.

Fig. 15. The boundary conditions of simply-supported case study.

The design parameters of the rigid base optimization problem are the same. However, the objective is to minimize the mean reaction force on the bottom surface (Fig. 17). Objective (Minimization): The Mean Reaction Force (MRF) calculated from the FEA Design Variables: xi ¼ f0; 1g; ði ¼ 1; 2; . . . ; 33Þ After the sandwich structure is subjected to the shock load, the structure must absorb the shock energy as much as possible and transmit the residual energy to the back surface homogeneously. The boundary conditions of the rigid base case are shown in Fig. 17. All the translational and rotational displacements of the back surface are fixed. (Fig. 17). A hybrid optimization algorithm based on the Multi-Island Genetic Algorithm with elitism and the Hooke-Jeeves Algorithm was used in the design optimization of foam filled sandwich panels with the purpose of benefiting the powerful properties of two algorithms. Multi-Island Genetic Algorithm is very efficient at global search, whereas the Hooke-Jeeves Algorithm is good at local search. In the hybrid algorithm, these two important properties

Fig. 16. The boundary conditions of clamped-clamped case study.

Fig. 17. The objective and the boundary conditions of rigid base case study.

Design Variables: xi ¼ f0; 1g; ði ¼ 1; 2; . . . ; 33Þ The simply-supported boundary condition of the simulation is shown in Fig. 15. Both the translational displacements of x and y-directions of the bottom left corner are fixed, and only the translational displacement of y-direction of the bottom right corner is fixed (Fig. 15). 5.2. Optimization of foam cores with the clamped-clamped case study In this case study, the objective and the design parameters of the optimization process are the same as in the simply-supported case.Objective (Minimization): The Back Face Deflection (BFD) calculated from the FEADesign Variables: xi ¼ f0; 1g; ði ¼ 1; 2; . . . ; 33Þ

Fig. 18. The flowchart of the Hybrid Algorithm used in this study.

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Fig. 19. The optimization result (21% improvement) for the simply-supported with roller case study.

Fig. 20. The optimization result (16% improvement) for the clamped-clamped case study.

are combined in such a way that firstly, the Multi-Island Genetic Algorithm explores the design space globally to find a near optimal solution, and then the Hooke-Jeeves Algorithm continues searching the local design space deeply (Fig. 18). In this way, the performance of the optimization process was increased significantly. The hybrid algorithm starts the optimization process with creating the initial population. In the initial population, the chromosomes or individuals that represent the design variables are randomly generated. Each individual has a potential solution to the optimization problem and consists of genes in a binary system. Each has a fitness function that characterizes the structure of the optimization problem and evaluates how effective an individual will be compared with others. After creating the initial population, the population is divided into many sub-populations or islands (Fig. 18). Then the migration operation between the islands is performed randomly. In each island, the individuals are exposed to three fundamental operators, namely crossover, mutation, and selection after elitism. In this study, an elitism operator has been used for the purpose of protecting the fittest individual from being changed by crossover and mutation operations. In the crossover operation, two individuals are selected randomly, and some part of their genes exchange with each other. There are several types of crossover operations depending on how and from where they cut the chromosomes [43,44]. In this study, two-point crossover technique has been used for its efficiency and practicality [45]. The mutation operator has a very key role in many evolutionary algorithms in that it increases the diversity of the population. In mutation operation, one randomly selected gene of the individual or chromosome is changed. In the binary-coded system used in this research study, if the current value of the selected gene is ‘0’ then it is changed to the value of ‘1’. In the selection operation, the individuals that have better fitness functions have more chance to be

selected for the new generation. In this way, strong and healthy individuals give their genetic codes or genes to the new generation. The fitness function values are calculated from the results of explicit dynamic FEA simulations explained in the numerical simulations section of this paper (Fig. 18). The global search with Multi-Island Genetic Algorithm (MIGA) continues until reaching a specific generation number. In this study, 50 generations and 100 individuals was used. The HookeJeeves Algorithm starts from the best result of MIGA. It, firstly, evaluates the best result and applies an optimal line search by using exploratory moves. At each iteration, it explores each move with its own step size. If it cannot find a better point, then it presumes that the step size is very large, thus it decreases the step size to explore new point. By this way, the algorithm continues searching the local design space deeply until a specified step size value is reached (Fig. 18). 6. Results and discussion The deflection of the sandwich structure for simply-supported case after the hybrid evolutionary algorithm based structural optimization is shown in Fig. 19. The BFD of the fully foam filled corrugated sandwich structure is 32.96 mm after three milliseconds of shock load exposure. The BFD of the optimized foam filled corrugated sandwich structure decreased to 25.97 mm. The BFD decreased by 21% which indicates an improvement in the energy absorption for the novel sandwich structure. As shown in Fig. 19, the hybrid evolutionary based optimization process placed the foam cores near to the boundary condition regions in the structure. This placement of the foam core in the structure contributes to the increase in the energy absorption of the sandwich structure exposed to the shock load.

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Fig. 21. The optimization result (6% improvement) for the rigid-base case study.

Table 3 The volume differences between the fully filled and optimized structures.

Volume (mm3) Normalized Value Difference (%)

Fully Filled Foam Structure

Optimized Structure (Simply Supported)

Optimized Structure (ClampedClamped)

Optimized Structure (Rigid Base)

235,676 100

100,761 43

114,083 48

1,450,483 62

0

57

52

38

The deflection of the sandwich structure for the clampedclamped case after the hybrid evolutionary algorithm based structural optimization is given in Fig. 20. In clamped-clamped case, the BFD of the fully foam filled corrugated structure is 5.97 mm, however the BFD of the optimized core structure is 5.04 mm. The decrease in BFD is 16%, which indicates a good amount of energy was absorbed. The optimized placement of foam cores is symmetrical because the boundary conditions are the same at the two corners. The shape of the core structure is similar to a leaf spring, and it is behaving like a shock absorber during the shock loading process (Fig. 20). For the rigid base case study, the deflection of the sandwich structure is displayed in Fig. 21 where the Mean Reaction Force (MRF) is compared to the fully foam filled core structure. After three milliseconds of the shock load exposure to the front face of the sandwich panel, the MRF value is calculated. The MRF has been reduced by 6% from 1412 N in the case of the fully filled structure to 1328 N in the case of the optimized structure. The simulation results of the optimized structure of rigid base case showed that the minimum MRF values can be obtained by filling around half of the cores with foam (Fig. 21). The volume differences between the fully foam filled case and the other three study cases such as simply supported, clampedclamped and rigid base are given in Table 3. In the fully filled sandwich panel, there are 33 foam cores with a total volume of 235,676 mm3. In the optimized simply-supported case, the volume was decreased by 57%. In the optimized clamped-clamped case, the volume was decreased by 52%, and in the optimized rigid base case, the volume was decreased by 38% all with better energy absorption. As a result, using optimization techniques in the design stage of shock absorption materials such as sandwich structures, yields very efficient and practical results. 7. Conclusions The primary objective of this study was to develop a novel sandwich structure with improved performance under shock loading using structural optimization techniques to find the optimum

distribution of filled cores. The conclusions of the present study are summarized below; 1. The hybrid optimization algorithm based on the Multi-Island Genetic Algorithm and the Hooke-Jeeves Algorithm can be used effectively to optimize the blast response of cellular structures. 2. Cellular structures can be optimized for properties such as energy absorption, weight, and cost by adjusting the number and location of foam filled cells. 3. The best improvements in back face deflections were achieved in the simply-supported case. 4. Symmetric deflection and distribution of core structure was obtained in both the clamped-clamped and simply-supported models. 5. In the rigid base case study, the deflection of core structure was not symmetric, and this is an artifact of the optimization technique used. The simply supported case was found to be more sensitive than the fixed supported case to foam filled core cells 6. This technique allows for a reduction of the filling material by obtaining the optimum core cell infill map.

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