Behrooz Keshtegr .Trung Nguyen-Thoi .Tm T.Truong .Shun-Peng Zhu

a School of Mechanical and Electrical Engineering.University of Electronic Science and Technology of China.Chengdu.611731.China

b Department of Civil Engineering.Faculty of Engineering.University of Zabol.P.B.9861335856.Zabol.Iran

c Division of Computational Mathematics and Engineering.Institute for Computational Science.Ton Duc Thang University.Ho Chi Minh City.Viet Nam

d Faculty of Civil Engineering.Ton Duc Thang University.Ho Chi Minh City.Viet Nam

Keywords: Adaptive kriging Laminated composite plates Buckling optimization Smooth finite element methods Cell-based smoothed discrete shear gap method (CS-DSG3) mproved PSO

ABSTRACT An effective hybrid optimization method is proposed by integrating an adaptive Kriging(A-Kriging)into an improved partial swarm optimization algorithm (IPSO) to give a so-called A-Kriging-IPSO for maximizing the buckling load of laminated composite plates(LCPs) under uniaxial and biaxial compressions.In this method.a novel iterative adaptive Kriging model.which is structured using two training sample sets as active and adaptive points.is utilized to directly predict the buckling load of the LCPs and to improve the efficiency of the optimization process.The active points are selected from the initial data set while the adaptive points are generated using the radial random-based convex samples.The cell-based smoothed discrete shear gap method(CS-DSG3)is employed to analyze the buckling behavior of the LCPs to provide the response of adaptive and input data sets.The buckling load of the LCPs is maximized by utilizing the IPSO algorithm.To demonstrate the efficiency and accuracy of the proposed methodology,the LCPs with different layers(2,3,4,and 10 layers),boundary conditions,aspect ratios and load patterns(biaxial and uniaxial loads) are investigated.The results obtained by proposed method are in good agreement with the literature results.but with less computational burden.By applying adaptive radial Kriging model.the accurate optimal results-based predictions of the buckling load are obtained for the studied LCPs.

1.Introduction

Structural optimization aims to design the best structures with better mechanical behaviors and an appropriate manufacturing cost.Therefore,researchers have been considerably investigated for establishing the novel optimization methods to search the optimal condition with accurate results.The optimization algorithms-based population strategy as genetic algorithm (GA) [1].differential evolutions(DE)[2],particle swarm optimization(PSO)[3],artificial bee colony algorithm (ABC) [4].cuckoo search (CS) [5].Ant colony optimization algorithm (ACO) [6]and harmony search algorithm(HSA)[7]can be used as suitable optimizers for global conditions of complex engineering problems.In recent years.these techniques have been widely used in optimization of laminated composite structures.For example.the GA algorithm was applied to find the optimal solutions of composite sandwich plates[8]and laminated composite plates (LCPs) [9]while the ACO was utilized by Sebaey et al.[10]to optimize laminated composite panels under biaxial loading.Omkar et al.[11]utilized ABC for multi-objective design optimization of composite structures.Almeida [12]used HSA to determine the maximum buckling load of composite plates.Ho-Huu et al.[13]proposed the improved DE to maximize the buckling load of LCPs.Vo-Duy et al.[14]presented the lightweight design optimization of LCPs subjected to frequency constraints via an adaptive elitist differential evolution (aeDE) algorithm.Barroso et al.[15]carried out a hybrid PSO-GA algorithm for optimization of laminated composites structures.Kamarian et al.[16]exhibited thermal buckling optimization of composite plates using firefly algorithm.By applying the FE-GAs-PSO method,Vosoughi et al.[17]presented optimum stacking sequences of thick LCPs formaximizing the buckling load.Topal [18]employed Teaching-learning-based optimization to maximize the fundamental frequency of simply supported antisymmetric LCPs.The optimization of the dynamic response for rectangular composite plates under electromagnetic field and thermal constraint was investigated using black-box optimization solver [19].Kaveh [20]displayed an application of the biogeography-based optimization(BBO) for buckling load maximization of the LCPs.An optimal design of anti-buckling behavior of graphite/epoxy laminated composites achieved by differential evolution and simulated annealing method was given by Ak?air [21].The main efforts in optimization solver are to investigate the ability of optimizer for searching the optimal conditions.In general.optimization algorithms are commonly integrated with numerical approaches as finite element models.In these approaches,the calculation domain is divided into many elements and each element is represented by a set of element equations.In addition.optimization process always requires a vast number of function assessment to reach the optimal result.Obviously.these procedures are very time consuming.For this reason.researchers have proposed an effective approach by combining the optimization algorithm with surrogate models to reduce the computational cost in optimization process.Recently,various surrogate models have been developed and applied in engineering field such as artificial neural network (ANN) [22].deep neural network (DNN) [23].deep convolution neural network(DCNN) [24].extended fourier amplitude sensitivity test (EFAST)[25].deep collocation method (DCM) [26].adaptive neuro-fuzzy inference system (ANFIS) [27]and Kriging [28].Among these surrogate models.the Kriging model with its ability to approximate accurately complicated systems has been popularly applied.The efficiency of this approach has been demonstrated through many studies.For instance.Jie et al.[29]proposed a Kriging assisted multi-objective PSO method to solve multi-objective optimization problems.To validate the effectiveness of the proposed method,twelve benchmark functions with different degrees of difficulty and an engineering design problem were investigated.Results revealed that the computational cost by the new introduced approach is significantly reduced in comparison with the conventional method.The Kriging as efficient metamodeling approach was used to optimize the structural mechanical responses under crashworthiness criteria [30].Qin et al.[31]used the Kriging model ensemble with genetic algorithm for model updating in complex bridge structures.In which.the Kriging model was constructed to approximate implicit relationship between structural parameters and responses while the GA was utilized to search the best solution on the entire design space.To validate the proposed method,Caiyuanba Yangtze River Bridge.a double decked of roadway and light railway bridge with a main span of 420 m were examined.The results of the study indicated that the Kriging model has high accuracy in predicting response of structures and can be used as a surrogate in GA to reduce the computational cost.With the purpose of saving the computational cost.an adaptive Kriging meta-model integrated with the GA for variable stiffness composite cylinder design under combined loadings was proposed by Zhong [32].In which.the Kriging model was utilized to approximate the structural behavior of the cylinder and the GA was employed to search optimum solution.Qian et al.[33]proposed general sequential constraints updating approach based on the confidence intervals from the Kriging surrogate model for solving engineering optimization design problem.A Kriging surrogate model-assisted PSO algorithm was introduced to solve the problems with high computational cost[34].Saad et al.[35]presented a new Kriging-Bat algorithm for solving global optimization problems.Relied on the above literature review.it can be found that a combination of the Kriging surrogate model and IPSO algorithm for maximizing the buckling load of LCPs under uniaxial and biaxial compression loads has not yet been studied.The modelling approaches can help improve the computational burden of the optimization process.However.the accuracy of the optimal results is strongly depended on the input data set and modelling structure.Thus.the adaptive-based active data points can be used as an accurate prediction method to improve the efficiency and accuracy in this complex problem.

Therefore.this study aims to introduce a novel hybrid methodology for maximizing the buckling load of LCPs subjected to uniaxial and biaxial compression loads.In which.an adaptive model-based active Kriging model is proposed to improve the accuracy and efficiency of optimization process in LCPs.An improved partial swarm optimization algorithm is used to search the optimum solution of the LCPs.Various numerical examples are presented to inspect the optimum solution of LCPs with distinct layers,boundary conditions and load patterns.The obtained optimum solutions are compared with those available in the literature to illustrate the accuracy and efficiency of the introduced method.The outline of the paper is presented as follows:the governing equation and formulation of the cell-based smoothed discrete shear gap method (CS-DSG3) for buckling analysis of LCPs are expressed in Section 2.Section 3 states the optimization approach using adaptively active Kriging model and the detailed description of IPSO algorithm.Several numerical examples are given in Section 4.Finally.some remarkable conclusions are drawn in Section 5.

2.A brief of finite element model of laminated composite plates

2.1.Governing equation

Consider a LCP with the length a.the width b and the total thickness h as shown in Fig.1.In the first order shear deformation theory(FSDT)[36],the displacement field of LCPs at a point is given by

in which u0,v0and w0are the displacements of a point on the plane z = 0; φxand φydefine the rotations to the normal mid-plane around y -axis and x -axis respectively.

The strain field of the LCP is computed as

where εm,κband γsare the membrane,bending and shear strains,respectively.The Galerkin weak form of the LCP under the in-plane pre-buckling stresses is stated as

Fig.1.Geometry of a laminated composite plate.

where,Dm.Dmb.Dband Dsdefine the material matrices regarding the membrane.the coupling of membrane and bending.the bending and shear deformations;εgis the geometric strain denoted as

and

2.2.Formulation of the CS-DSG3 for buckling analysis

The cell-based smoothed discrete shear gap method(CS-DSG3)was firstly developed by Nguyen-Thoi et al.[37]for analyzing the mechanical characteristic of the Reissner-Mindlin plates.The main idea of this methodology is to combine the cell-based strain smoothing technique in CS-FEM [38]with the discrete shear gap(DSG3)method[39]using three-node triangular elements to avoid the transverse shear locking phenomenon and improve the accuracy of the DSG3 method.In the CS-DSG3,each triangular element Ωeas illustrated in Fig.2 is first separated into three sub-triangle elements Δj(j=1，2，3) by simply associating the central point O of the element to three nodes.Next.the strain fields in each subtriangle are computed by means of using the DSG3 method.The smooth strain field is finally obtained by applying the cell basedstrain smoothing operation in the CS-FEM method as

Fig.2.A typical triangular element Ωe.

where Aeis the area of the element Ωeand AΔjis the area of the subtriangle Δj.

The element stiffness and geometric matrices of the CS-DSG3 are computed as

The finite element model of buckling problem is expressed in the form of the following linear algebraic equations

where λcris the critical buckling load and u is the nodal displacement vector.

The numerical results indicated that the CS-DSG3 is free of shear locking and achieves high accuracy compared to the exact solutions and other existing elements in the literature.Due to these outstanding properties.the CS-DSG3 then has been extended to analyze various kinds of structures such as flat shells[40],stiffened plates[41],FGM plates[42],piezoelectricity plates[43],composite and sandwich plates[44],plates resting on viscoelastic foundation subjected to a moving mass [45,46].cracked Mindlin plates [47],limit analysis of plates [48].optimization of folded laminated composite plates [49].analysis of stiffened folded plates [50].free vibration analysis of cracked Reissner-Mindlin shells [51].etc.

3.Hybrid optimization and modelling approach

3.1.Optimization approach

3.1.1.Optimization formulation

The optimization formulation for searching the maximization of the buckling force of LCPs is given as follows [13]:

where.λ(θ，t) is the buckling load factor obtained by the CS-DSG3 and θ are the integer design variables of fiber orientation angles(which are varied from -90°to 90°) and t are continuous design variables of layer thicknesses (which are varied from 0.005 to 0.095).The objective function of this optimization problem is to obtain the maximum buckling load factor under NL composite laminated layers.In order to handle the constraint function of the thickness in the optimization process.the penalty method is used and the constraint optimization problem is transformed into the following unconstraint optimization problem:

where P is the penalty factor given by P=105.The penalty function i.e.is defined using tolerance term of ε=10-6and total thickness of the plate h.The function in Eq.(13)is implemented in optimization process that various optimization methods-based meta-heuristic algorithms can be generally used to find the optimal design condition of LCPs.In this current study.a random improved partial swarm optimization is applied for optimization of the above function.

3.1.2.Improved partial swarm optimization (IPSO)

The partial swarm optimization(PSO)is improved by a random adjusting process for searching the maximization of the buckling force of LCPs.This random improvisation is involved two main adjustments for providing new positions of each partial.In the first stage.the following relations are formulated as:

in which.gbestand pbestare respectively the best population in all position and best population in current positions.The gbestand pbestare applied for adjusting the new velocity(Vk+1)and new position(Xk+1) of the particles;ωkis the inertia weight computed by:

In the second stage,the best population of partials is randomly adjusted using normal standard distribution as follows:

where k and NI are the current and total number iterations in optimization process.respectively.Based on Eq.(16).the best particle gbestis adjusted by using a normal random value generated by a Gaussian distribution function with mean of 0 and standard deviation of 1(by operator of Normrand(0，1)).As seen from Eq.(16),the best population is adjusted by a dynamical factor which is given a large value in the initial iterations whenis large and a smaller value in the final iterations whenis small.Eq.(15)and Eq.(16)are combined for optimization process by the following relation:

where r is a random number r∈(0,1).It can be seen that the new particles are adjusted based on the best population with the probability of 0.2+k/NI.In the improved PSO,the initial velocity of each particle is randomly computed in the domain of vmaxand vminin which maximum (vmax) and minimum (vmin) velocities for simulating randomly initial velocities are respectively given as=-vmax(xUand xLare the upper and lower bounds for x); c1and c2are acceleration coefficients which are set with the value of 2.The parameters of IPSO are given as ωmin=0.4 and ωmax=0.9 and r1and r2are two random uniform numbers in the range from 0 to 1 i.e.r1.r2∈[0,1][52,53].

3.2.Modelling process

In this optimization problem in Eq.(13).λ(θ，t) should be computed based on an analytical method or modelling approach by the limited data set.The main effort of this paper is to propose an active Kriging model for predicting λ(θ，t) of the LCPs.The input data set is very important for accurate prediction of λ(θ，t) in optimization process.

In the current work.the active data set is selected to build the Kriging model and the several adaptive points are provided byiterative process to increase the active data set.The active data points from initial random data and the selected adaptive data are used to improve the accurate prediction of the optimal solution by using the Kriging-based modelling approach.The schematic view of this method for optimization is presented in Fig.3.

According to Fig.3.the active points are selected using convex data set from simulated points in the first stage while the adaptive points are generated to increase the effective optimization region for modelling process of the Kriging in the second stage.

3.2.1.Kriging model

The Kriging model is a well-known modelling tool for estimating the engineering problems by the following predicted function:

where β is the vector of unknown coefficients;U(X)represents the random part of the models which is generally considered based on Gaussian process and G(X)is polynomial basis functions which are considered as second-order functions in the current study.The stationary Gaussian process between U(Xi)and U(Xj)is computed by the covariance function as follows:

where σ2denotes the variance and R (Xi.Xj,θ) represents the correlation function for U(X); θ is unknown correlation parameters.The correlation function using Gaussian covariance basis function between samples Xiand Xjis given by

where,rijis the distance byand θ ＞ 0.The predictive model of Kriging can be obtained by computing the unknown correlation parameter as follows:

where,

The function in Eq.(21) can be used to simulate the buckling force of the LCPs.As can be seen,the input data points Y and X are the major parameters to build this model.

3.2.2.Active learning process

An iterative modelling process-based nonparametric regression is introduced for optimization problems in this section.This iterative modelling approach-based Kriging is structured by the active sample set which is given from the initial simulating data.The proposed optimization method involves two major sequentially loops.in which one loop is operated using the improved PSO and one modelling loop is structured by the active Kriging as shown in Fig.3.

As seen from Fig.3,the Kriging model is calibrated based on two sample sets in which the active data set is given from initial data points in the first stage and the adaptive data set is provided using the CS-DSG3 in the second stage.

In the first stage.the active points are given based on the optimal design point (X*) as shown in the following steps:

i Normalize the input data by:

where x is the variables from the initial random input set with NV total input variables;are respectively the upper and lower bounds of the ithdesign variable;

Fig.3.Schematic framework of the proposed optimization -based active modelling method.

In the second stage,the adaptive points are randomly computed by using the normalized optimal design point (Z*) as follows:

in which C and Rm are controlling factors to simulate the adaptive points as 0.1 ≤ Rm ≤ 1 and 0.5 ≤ C ≤ 2;t is the iteration number of the Kriging model for approximating the optimization function;rand() is the random number as rand()∈[0,1]; exp is the exponential operator and RDiis the random input data for the ithactive point.By applying the adaptive data points which are simulated by Eq.(28).the data points in the training phase of Kriging model at the optimal domain are increased.Consequently,the Kriging model is trained based on the active process of input data for optimization problem.The steps to simulate the adaptive points are presented as follows:

i Set the number of adaptive points (NA).factors C and Rm;

ii Generate random data as RD = rand(NA,RV).where NA and RV are respectively number of adaptive points and number of variables;

iii Compute the normalized adaptive points based on Eq.(28);

iv Transfer normalized adaptive data set to original space using Eq.(27);

v Compute the buckling loads of plates based on data obtained by step iv using CS-DSG3;

By using two sets of data as active and adaptive points.the Kriging model is calibrated and this model is used to search the optimal design point in optimization process.These two processes of the input data may provide the accurate results for optimization of complex engineering problems with more efficient computational cost.

3.3.Optimization-based active modelling algorithm

The proposed improved PSO and the modelling-based active Kriging are sequentially applied to search the optimal design point of the LCPs.The proposed formwork in Fig.3 using hybrid intelligent method for optimization is presented by the following steps:

Step 1 Set parameters of the optimization and modelling process;

Step 2 Generate the random data point as the initial dataset for training of model;

Step 3 Compute the buckling loads of the LCPs using the CSDSG3;

Step 4 Build the Kriging model using input data set for prediction of the buckling load factor;

Step 5 Apply the optimization process using the IPSO to search the optimal design point;

Step 6 Check the convergence asif converge then stop or else go to Step 7;

Step 7 Select the active data set using the optimum conditions of Step 5 and random generated input data in Step 2;

Step 8 Compute the adaptive data set using random process and optimal design point in Step 5;Step 9 Set the active-based adaptive input data set using Step 7 and Step 8 then t = t+1 and go to Step 3.

In order to reduce computational burden and to improve the accurate prediction of the optimal design point.an iterative procedure is proposed in this current optimization-based modelling scheme.Fig.4 presents the flowchart of proposed optimization approach for searching the optimal solution of the LCPs.Four main subroutines are used to find the optimal result.in which the first subroutine uses the CS-DSG3 as an analytical analyzer to compute the buckling loads,and,in the second subsection,the Kriging model is called to build the buckling load models.In the third subroutine,the optimization method is then applied to search the optimal design point using the Kriging model,and in the last subroutine the adaptive and active data sets are determined.

Applying the adaptive and active data sets to build the Kriging model of the buckling load factor is the major difference of the proposed framework with other optimization methods combined by modelling approaches.

4.Numerical results

The effectiveness and reliability of the proposed A-Kriging-IPSO method for maximizing the buckling force of the LCPs are demonstrated through several numerical examples.In the first investigation.the optimum results of the LCPs with 2-layer under biaxial and uniaxial loads are presented.Next,the effect of different number of layers such as 2,3,4,and 10-layer of the LCPs subjected to uniaxial load is further inspected in the second stage.Finally,the influence of different boundary conditions including simply supported.clamped and free support is investigated for the 2-layer composite plates under biaxial loads.The parameters of the active Kriging are given by εa= 1.25.Rm=0.5 and C =1 for all problems.

4.1.2-Layer composite plates

This part investigates two distinct cases of optimization problems of the 2-layer composite plate with different kinds of design variables.For the Case 1,only the fiber orientation angles(θ1，θ2)are considered as integer design variables.Meanwhile both the fiber orientation angles(θ1，θ2)and the layer thicknesses(t1，t2)are used as the integer and continuous design variables in the Case 2,respectively.The optimum results attained by the present study are compared with those of the published one in order to validate the accuracy and efficiency of the proposed methodology.

The optimal results obtained by the proposed iterative modelling approach for the LCPs under biaxial loads in Case 1 are presented in Table 1.The obtained results give the values of the optimum objective function of 12.158.optimal design fiber orientations of[-45°/45°]with number of total call function of 260.The A-Kriging-IPSO provides the accurate optimal results in comparison with those extracted from reference [13]while the proposed method entails much less the number of total call function than the traditional optimization methods combined with numerical method for this problem.Additionally.the optimum fitness values gained by the Kriging and the CS-DSG3 exactly converge to each other after four iterations.The results reveal that the proposed method is very efficient and robust in optimizing the buckling force of the LCPs.especially in saving the computational cost.

The iterative process to obtain the optimal design solution is plotted in Fig.5 for the Case 1-biaxial LCPs.It can be observed that the prediction results of the Kriging model can be improved by applying the adaptive points to find the optimum conditions.andthe almost additive points are generated in a hyper-sphere domain with center of the optimal design point computed by the A-Kriging-IPSO method.

Fig.4.Schematic flowchart of the active Kriging combined with the IPSO for optimization of the LCPs.

Table 1 Optimization process of the hybrid adaptive Kriging and IPSO for the LCPs under biaxial loads in Case 1.

The results for each iteration of the A-Kriging-IPSO are presented in Table 2 for the LCPs under uniaxial load.Fig.6 illustrates the four last iterations of data samples for training the Kriging model.It can be seen from the results of Table 2 and Fig.6 that the A-Kriging-IPSO exactly provides the final optimum results for the LCPs in comparison with the FEM-IPSO at Iteration 5 and Iteration 6.In addition.the optimal results obtained by the A-Kriging-IPSO agree well with those of Ho-Huu et al.[13]while it is more computationally efficient compared to the traditional optimization approaches.Hybrid intelligent method-based optimization algorithm and meta-models can provide the accurate results with acceptable computational burden compared to the traditional optimization approaches combined with the FE model.

The optimal results for the LCPs under biaxial and uniaxial loads in Case 2 are tabulated in Tables 3 and 4.respectively.It can be found that the optimum results gained by the active Kriging model in both two cases agree well with those of the FEM model and those of Ho-Huu et al.[13].Meanwhile the Kriging model requires much less the number of total call function than the model using the differential evolution combined with FEM in Ref.[13].The obtained results again show the effectiveness and reliability of the proposed method in reducing the computational burden of the optimization problem.

Fig.7 depicts the buckling load factor for different fiber orientation angles of layers 1 and 2 for Cases 1 and 2 under a) uniaxial and b)biaxial loads.As seen,the symmetric results are obtained for both uniaxial and biaxial loads for Case 1.This means that the 2-layer LCPs with same thicknesses have two optimal design points while the 2-layer LCPs with different thickness show individual optimal points.The optimum thicknesses of layers improve the capacity of the LCPs.In Case 1,fiber orientation angle between two layers should be selected about 90°to obtain the acceptable buckling factor while one of the layers is located with the fiber orientation angle from 40°to 50°.In Case 2,the capacity of the LCPs is strongly depended on the thicker fiber layer and it may provide the less load capacity in terms of fiber orientation angle more than 70°for thick layer.

Fig.5.Iterative process of data samples as active points and adaptive points of Kriging model and optimization results of LCPs under biaxial load in Case 1.

Table 2 Optimization process of the hybrid adaptive Kriging and IPSO for the LCPs under uniaxial load in Case 1.

4.2.Effects of layers

Three conditions of layers as 3.4.and 10 with same total thickness of laminated composites plates as 0.1 h are investigated in this section.The integer random variables as fiber orientation angle of layers are considered in this parametric study.For the plate with 10-layer.five integer design variables are considered as the symmetric pattern for layers as [θ1.θ2.θ3.θ4.θ5]s.We have three composite laminated plates with 3.4 and 5 design variables that their iterative results using the A-Kriging-IPSO method are presented in Tables 5-7,respectively.It can be seen that the optimum results gained by the A-Kriging-IPSO match well with those of the FEM-IPSO and Ref.[13].However.the computational effort to find the optimal design point by the A-Kriging-IPSO is less than the obtained results by Ho-Huu et al.[13]about 840 iterations for 3-layer.400 iterations for 4-layer and 3625 iterations for 10-layer.

By comparing the results from Table 6 for the 4-layer laminated composite plate.the optimal design point obtained by the present study is different with the results extracted from Ho-Huu et al.[13].This means that the applied optimization method by Ho-Huu et al.[13]may provide the local optimum while the proposed method can search the global optimal results(the maximum bucking factor is improved from 30.7700 to 35.4351).The computational burden of optimization method as well as differential evolution may increase by increasing the design variables as 10-layer laminated composite plate.The proposed method can help reduce the computational cost of this complex problem about eight times faster than the differential evolution.

Influences of fiber orientation angles on the buckling load factor of the LCPs with different layers are shown in Fig.8.As observed,the layers in the LCPs with 2-layer have the same effect and theinner layers have the weaker effect on the buckling force of the LCPs when the number of layers increases.The inner layers 5 and 4 in 10-layer LCPs do not significantly improve the buckling force compared to the outer layers 1 and 2.For 2-layer and 4-layer LCPs,there are two optimum points found for maximizing the buckling force,while for 3-layer and 5-layer LCPs there is only one optimum point.

Fig.6.Iterative process of data samples using active points and adaptive points of Kriging model and optimization results of the LCPs under uniaxial load in Case 1.

Table 3 Optimization process of the hybrid adaptive Kriging and IPSO for the LCPs under biaxial loads in Case 2.

4.3.Effects of boundary conditions

The effects of mixed boundary conditions as clamped.simply supported and free support are investigated for the 2-layer LCPs under biaxial loads.The optimum results of three mixed boundary conditions as SSCC,SSSC and SSFC are evaluated using the proposed A-Kriging-IPSO.The obtained results of the proposed optimizationmethod are compared with the extracted optimum results obtained by Ho-Huu et al.[13].These results for different methods of the AKriging-IPSO.DE and modified DE algorithms are tabulated in Table 8.It is found that the A-Kriging-IPSO can help reduce strongly the computational burden to determine the buckling load factor compared to the optimization approaches of the DE and the modified DE.On the other hand.the accuracy of the proposed method at the final iteration of modelling to search optimal design point is varied from 0 to 0.008%.This means that this active modelling approach can provide the highly accurate results for this engineering problem.For all comparative results presented in Table 8.the A-Kriging-IPSO shows the larger load factor than the optimization method of the DE and the modified DE (for example,the load factor increased from 6.33676 to 6.4228 for LCPs with SSFC boundary condition).The buckling force of the LCPs is increased by applying the clamped mixed boundary condition.while it is reduced significantly with the free boundary condition.

Table 4 Optimization process of the hybrid adaptive Kriging and IPSO for the LCPs under uniaxial load in Case 2.

Fig.7.Effects of fiber orientation angle of each layer on the buckling factor for plates under: a) uniaxial load; b) biaxial loads.

Table 5 Optimization process of the hybrid adaptive method for the LCPs with three layers.

4.4.Effects of aspect ratio b/a

The dimensional effect of the LCPs as length to width(i.e.aspect ratio b/a) is investigated based on the proposed optimization method for b/a = 1.1.5.2.3.and 4.and their optimal results are presented in Table 9.As seen.the optimal buckling force and the optimal design points change with the change of the ratio b/a.The optimum buckling force is varied from 11.79 to 13.51 for the LCPs under biaxial loads.The fiber orientation angles for second sheet are reduced from 45°to 11°by increasing the aspect ratio from 1 to 4.The ratio b/a affects significantly on the buckling force of the long LCPs with b/a ＞4,however it causes the smaller buckling force for the LCPs with conditions of 1.5≤ b/a ≤2.

4.5.Discussion

The iterative histories of the buckling load factor using the proposed A-Kriging- IPSO for studied examples of the LCPs are presented in Fig.9 for mixed boundary conditions of biaxial load and different layers of uniaxial load patterns and different aspect ratios of plates under biaxial load and different layers.As seen,theacceptable results are obtained after 4 iterations using the proposed hybrid optimization method.The buckling load factor increases when the number of layers increases.Especially.the buckling load factors of the LCPs with the number of layers more than 4 is much higher than those of the LCPs with the number of layers less than 3.In addition.the buckling load factor is effected significantly by the boundary conditions of the LCPs.but not effected much by different aspect ratios given in the studied LCPs.

Table 6 Optimization process of the hybrid adaptive method for the LCPs with four layers.

Table 7 Optimization process of the hybrid adaptive method for the LCPs with ten layers.

The optimum results using the proposed method for various conditions as number of composite layers and mixed boundary conditions are compared with the extracted results from Ho-Huu et al.[13]and traditional design in Fig.10.It can be observed that.the proposed method can provide the optimal results which are close to the extracted results from the literature while the buckling load is enhanced compared to traditional design.For the LCPs with four layers.the proposed method provides the buckling load factor larger more than those extracted from the traditional design and literature.

The computational burden of the proposed method in terms of call functions is presented in Fig.11,and compared with the results extracted from Ref.[13]using DE and modified DE optimization methods.It can be seen that the proposed A-Kriging-IPSO method improves significantly the computational burden for the LCPs with 2-layers and different thicknesses.Although the modified DE provides the efficient optimal results with less call functions than the DE.it is still much less efficient than the proposed method for the LCPs with 10-layers and 2-layer and with different boundary conditions.

5.Conclusions

The paper proposes a hybrid intelligent method for optimization of engineering problems by using an optimization solver combined with a modelling approach.The proposed methodology is introduced to reduce significantly the computational burden of the numerical analysis with high-accuracy of optimization results for the complex engineering problems.In this current work.the improved PSO (IPSO) as optimizer is combined with an adaptivebased active Kriging model to search the optimal solutions of the LCPs.In the IPSO,two random adjusting processes are used to adapt the current positions of particles using the best position and dynamical normal random generated number.In the modelling approach-based Kriging.an active strategy by using an iterative procedure is proposed to create two sample data sets generated by the cell-based smoothed discrete shear gap method(CS-DSG3).The active sample points in the first data is selected from the initial generated points while the adaptive points are generated using radial samples in the second data set.This hybrid optimization modelling is investigated to illustrate the accurate and efficient optimal results of the LCPs.The obtained results of the proposed method for studying engineering problems of the LCPS are withdrawn as follows:

1 The proposed A-Kriging-IPSO method showed the accurate and efficient results of studied problems.

Fig.8.Buckling factors corresponding to fiber orientation angles of different layers for the LCPs under uniaxial load.

Table 8 Optimum results of the 2-layer LCPs with different boundary conditions under biaxial loads.

Table 9 Optimum results of the 2-layer LCPs with different aspect ratios b/a under biaxial loads.

Fig.9.Iterative histories for optimization of the buckling force of the LCPs in different conditions.

Fig.10.Comparative optimal results of different conditions for the studied LCPs.

Fig.11.Call functions of the studied LCPs for the proposed method and results extracted from Ref.[13].

2 This method helps reduce strongly the computational burden compared to the traditional optimization method as the DE or the modified DE.

3 The buckling force is insensitive to the inner composite layers of plates with more than four layers.

4 By applying the clamped mixed boundary conditions and increasing the composite layers.the buckling load factor is increased.

5 The different optimal conditions including fiber orientations of layers and their thickness of the LCPs are obtained for different aspect ratios while the buckling force is not significantly changed for the LCPs under biaxial load and 1 ≤ b/a≤4.

6 Generally.the proposed iterative modelling using the Kriging quickly converged after four iterations.Consequently.this approach can be used for optimization of real complex engineering problems in the future.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02-2019.330.