OPTIMIZATION ANALYSIS OF STIFFENED COMPOSITE PLATE BY ADJUSTED DIFFERENTIAL EVOLUTION PHÂN TÍCH tối ưu tấm COMPOSITE GIA CƯỜNG BẰNG TIẾN hóa KHÁC BIỆT cải TIẾN

OPTIMIZATION ANALYSIS OF STIFFENED COMPOSITE PLATE BY ADJUSTED DIFFERENTIAL EVOLUTION PHÂN TÍCH TỐI ƯU TẤM COMPOSITE GIA CƯỜNG BẰNG TIẾN HÓA KHÁC BIỆT CẢI TIẾN Thuan Lam-Phat1a, Son Ngu

OPTIMIZATION ANALYSIS OF STIFFENED COMPOSITE PLATE

BY ADJUSTED DIFFERENTIAL EVOLUTION

PHÂN TÍCH TỐI ƯU TẤM COMPOSITE GIA CƯỜNG BẰNG TIẾN HÓA KHÁC

BIỆT CẢI TIẾN

Thuan Lam-Phat1a, Son Nguyen-Hoai1b, Vinh Ho-Huu2a, Trung Nguyen-Thoi2b

1GACES, HCMC University of Technology and Education, Vietnam

2Division of Computational Mathematics and Engineering (CME), Institute of Computational

Science (INCOS), Ton Duc Thang University, Hochiminh City, Viet Nam

a lamphatthuan@gmail.com; b son55vn@yahoo.com

c hohuuvinh@tdt.edu.vn; d nguyenthoitrung@tdt.edu.vn

ABSTRACT

In this paper, an improved version of the Differential Evolution (DE) called adjusted DE

is adopted to solve for the optimum solution of fiber angle of the stiffened composite plate The structure is made of a composite plate stiffened with composite beam The objective function in this problem is the strain energy of the structure and is computed by finite element analysis using the CS-DSG3 element The numerical examples are given for two cases of square and rectangular plate to verify the accuracy and the effectiveness of the method The results obtained from the adjusted DE algorithm are compared with those of the references showing good agreement

Keywords: Stiffened composite plate, composite beam, differential evolution,

optimization analysis, finite element analysis

TÓM TẮT

Trong bài báo này, một phiên bản cải thiện của Tiến hóa khác biệt gọi là Tiến hóa khác biệt hiệu chỉnh được sử dụng để giải tìm nghiệm tối ưu góc hướng sợi của tấm composite gia cường Kết cấu này được tạo thành từ tấm composite gia cường bởi dầm composite Hàm mục tiêu trong bài toán này là năng lượng biến dạng của kết cấu và được tính bằng phân tích phần

tử hữu hạn sử dụng phần tử CS-DSG3 Những ví dụ số được đưa ra cho hai trường hợp tấm vuông và tấm chữ nhật để kiểm tra độ chính xác và tính hiệu quả của phương pháp Những kết quả đạt được từ thuật toán tiến hóa khác biệt hiệu chỉnh được so sánh với những kết quả của lời giải tham khảo cho thấy tính tương đồng cao

Từ khóa: tấm composite gia cường, dầm composite, tiến hóa khác biệt, phân tích tối

ưu, phân tích phần tử hữu hạn

1 INTRODUCTION

Design optimization is one of the most interesting research directions that brings a lots

of profits in both life and industry And so, together with the increase of optimization problems in practice, methods for design optimization are also quickly developed The optimization methods can be classified into two main groups: gradient-based and popular-based approach Methods popular-based on gradient information is fast but usually stuck in local solution and depend too much on a good initial point to obtain global optimal solution To deal with such disadvantages, population-based global optimization methods are utilized alternatively Among global optimization algorithms, Differential Evolution (DE) firstly introduced by Storn and Price in 1997 [15] was one of the most potential algorithms As many other evolutionary algorithms, the DE is a population-based method, which models the

stochastic evolution processes of nature such as mutation, cross-over, selection, etc process

to iteratively evolve the population to the best solutions The mutation and cross-over phases are to create the necessary diversity for the population, while the selection phase facilitates the exploitation for the better candidates in the search region The DE has demonstrated excellently performance in solving many different engineering problems Le-Anh et al [7] using an adjusted Differential Evolution algorithm and a smoothed triangular plate element for static and frequency optimization of folded laminated composite plates Ho-Huu et al [18] proposed a new version of the DE for shape and sizing optimization of truss with discrete variables Besides, Ho-Huu et al [17] also introduced two new improvement steps to increase the convergence of DE algorithm based on roulette wheel selection (ReDE) The new adjusted

DE algorithm is applied for solving shape-and-size optimization problem of truss structure with frequency constraints

In this paper, the new version of DE is utilized in searching for the optimal fiber angle

of the stiffened composite plate Stiffened composite plates have been widely used in many branches of structural engineering such as aircraft, ships, bridges, buildings etc…for their advantages of larger bending stiffness with less amount of material However, given the layered structure of composite materials, choosing the best design that satisfies the working requirement is difficult In addition, the complex mechanical behavior of composite materials also increases the difficulty of the problems related to their design [16] T Nguyen-Thoi et al [13] used SQP to find the optimal fiber orientations for stiffened composite plate, but the results still depend on the initial to get the exact solution Many meta-heuristic optimization methods such as the genetic algorithm and simulating annealing are also widely used to optimize composite laminates Apalak et al [10] conducted layered optimization for the maximum fundamental natural frequency of composite laminates through a genetic algorithm Javidrad and ouri [5] applied a modified simulated annealing to minimize the function composed of the relative differences of the effective stiffness properties and the weight of the considered laminate by changing the number of layers and the fiber angle of each layer For the geometric design of a composite material-stiffened panel, Marin et al [8] used the genetic algorithm, including the application of elitism, which preserved the use of the Pareto front Falzon and Faggiani [3] applied the genetic algorithm to improve the post-buckling strength

of stiffened composite panels However, the DE approach we used in this research with some modifications in original algorithm can give the better results In this paper, an improved version of the Differential Evolution (DE) is adopted to solve for the solution of fiber angle of the stiffened composite plate The objective function, the strain energy of the structure, is computed by finite element analysis using the cell-based smoothed discrete shear gap technique with triangular elements (CS-DSG3) proposed by T Nguyen-Thoi et al [1,4] The numerical solutions obtained from the method are compared with references to show the effectiveness and the accuracy of the algorithm

2 THEORY FUNDAMENTAL

An optimization problem can be expressed as follows:

i j

f





x

x x

where xis the vector of design variables; h i( )x  0 and g j( )x  0 are inequality and

equality constraints; l, m are the number of inequality and equality constraints, respectively;

( )

f x is the objective function which can be the function of weight, cost etc

Design optimization of a structure is to find optimal values of design variables in design space such that the objective function is minimum [13] Dealing with such problems, many optimization methods are used including gradient-based and population-based approach to

find the solution In this paper, the Differential Evolution is utilized to address the problem of finding optimal fiber orientations of the stiffened composite plate

2.1 Brief on the differential evolution algorithm [17,18]

The original differential evolution (DE) algorithm firstly proposed by Storn and Price [15] has been widely used to solve many kinds of optimization problems The scheme of this algorithm consists of four main phases as follows:

Phase 1: Initialization

Create an initial population by randomly sampling from the search space

Phase 2: Mutation

Generate a new mutant vector vi from each current individual xi based on mutation operations

Phase 3: Crossover

Create a trial vector ui by replacing some elements of the mutant vector vi via crossover

operation

Phase 4: Selection

Compare the trial vector ui with the target vector xi One with lower objective function value will survive in the next generation

To improve the effectiveness of the algorithm, the Mutation phase and the Selection

phase are modified to increase the convergence rate as follow:

In the mutation phase, parent vectors are chosen randomly from the current population

This may make the DE be slow at exploitation of the solution Therefore, the individuals participating in mutation should be chosen following a priority based on their fitness By doing this, good information of parents in offspring will be stored for later use, and hence will help to increase the convergence speed To store good information in offspring populations, the individuals is chosen based on Roulette wheel selection via acceptant stochastic proposed

by Lipowski and Lipowska [2] instead of the random selection

In the selection phase, the elitist operator introduced by Padhye et al [12] is used for the

selection progress instead of basic selection as in the conventional DE In the elitist

process, the children population C consisting of trial vectors is combined with parent population P of target vectors to create a combined population Q Then, best individuals are chosen from the combined population Q to construct the population for the next generation

By doing so, the best individuals of the whole population are always saved for the next generation

The modified algorithm Roulette-wheel-Elitist Differential Evolution is then expressed

as below:

1:

2:

3:

4:

5:

6:

7:

8:

Generate the initial population

Evaluate the fitness for each individual in the population

while <the stop criterion is not met> do

Calculate the selection probability for each individual

for i =1 to NP do {NP: Size of population}

Do mutation phase based on Roulette wheel selection

j rand = randi(1,D) {D: number of design variables}

for j =1 to D do

9:

10:

11:

12:

13:

14:

15:

16:

17:

18:

if rand[0,1] < CR or j == jrand then {CR: crossover control parameter}

u i,j = x r1,j + Fx(x r2,j - x r3,j ) {F:randomly chosen within [0,1] interval}

else

u i,j = x i,j

end if

end for

Evaluate the trial vector ui

end for

Do selection phase based on Elitist selection operator

end while

2.2 Brief on the behavior equation of stiffened composite plate [13]

Stiffened composite plate can be seen as the combination between composite plate elements and the stiffening Timoshenko composite beam elements, as illustrated in Figure 1 The stiffening composite beam is set parallel with the axes in the surface of plate and the

centroid of beam has a distance e from the middle plane of the plate The plate-beam system is

discretized by a set of node The degree of freedom (DOF) of each node of the plate

isd [ , ,u v w, x, y]T, in which u v w, , are the displacements at the middle of the plate and

,

x y

  are the rotations around the y-axis and x-axis The DOF of each node of the beam is

st  u u u r s z  r s

d The centroid displacements of beam are expressed as

u u r r( ) zr( )r ; v z s( )r ; w u r z( ) (2) where u u u r, s, zare respectively centroid displacements of beam and  r, s are the

rotations of beam around r-axis and s-axis

Figure 1 A plate composite stiffened by an r-direction stiffener

* Energy equation of stiffened composite plates

The strain energy of composite plate is given by

1

d 2

U   ε D ε ε D κ κ D ε κ D κ γ D γ A (3) where ε κ γ0, b, are respectively membrane, bending and shear strains of composite plate and are expressed as follows

0  [u v u,x, ,y, ,yv,x] ;T b [x x, ,y y, ,x y, y x, ] ;T  [w,x x,w,yy] T

D ,D ,D ,D are material matrices of plate

The strain energy of composite stiffener is given by

1

  b T b b  s T s s

st l st st st st st st

where ε εb st, s st are respectively bending, shear strain of beam and are expressed as follows

st u r rz   r r r r s r st  u z r r

,

b s

st st

D D are material matrices of composite beam

Using the superposition principle, total energy strain of stiffened composite plate is obtained by

1

si

N

P st i



where N st is the number of stiffeners

3 NUMERICAL RESULTS

Consider an optimization analysis of a composite plate stiffened by a composite beam

according to x-direction as in Figure 2 under simply-supported condition The parameter of the problem are given by a = 254 mm, h = 12.7 mm, cx = 6.35 mm and dx = 25.4 mm The analysis

is carried out with two cases of square (b = 254 mm) and rectangular (b = 508 mm) plate

Figure 2 Model of a stiffened composite plate

Both plate and beam have four symmetric layers The fiber orientation for layers of the plate is a set [1221], and for the layers of the beam is [34 43] The plate and beam are made by the same materials with E1 144.8GPa, E2E3 9.65GPa, G12G13 4.14GPa,

G  , 121323 0.3 The plate is subject to a uniform load f = 0.6895 (N/mm2) The optimization problem is now expressed as

1 min

2

T









where U is strain energy and I is fiber orientation of ith layer

Firstly, static analysis for the case of square plate is carried out to verify the reliability

of the finite element solution using CS-DSG3 [14] The results compared with those by Li Li [9] and M Kolli [11] show good agreement

Table 1 Comparison of central deflection (mm) of the simply-supported square stiffened

composite plates subjected to a uniform load f = 0.6895 N/mm2

Orientation angle for both beam

and plate

[0 / 90 / 90 / 0 ] [45 / 45 / 45 / 45 ]0  0  0 0 Method CS-DSG3 [11] [9] CS-DSG3 [11] Central deflection 1.0917 1.0396 1.0892 2.5049 2.4912 Next, we consider the optimization analysis for two cases of square plate and rectangular plate The results of fiber orientations obtained from DE are presented in Table 2

in comparison with those of Genetic Algorithm method corresponding to the value of strain energy and computational cost to estimate the effectiveness of the algorithm In this analysis, the value of the design variables is chosen to be integer

We can see from the Table that the solutions by the DE agree very well with those by the GA The objective function is better and the costs is also less than those from GA It is also seen that the optimal fiber orientations of the square plate problem are quite different from those of the rectangular plate problem under the same conditions This implies that the geometric parameters of the structures also have influence to the results of the optimization problems

Table 2 The optimal results of two problems Type of

stiffened plate

Method Optimal angle [Degree] Strain energy

(N.m)

Computational cost (seconds)

Square

(a = b = 204

mm)

DE 135 48 0 180 6183.2 1678

GA 134.5 46.6 0.25 179.5 6364.0 2373 Rectangular

(a = 204 mm,

b = 508 mm)

DE 159 37 0 180 30301 1923

GA 160.4 35.7 0 179.7 31471 2475

CONCLUSION

In this paper, the fiber orientations optimization analysis for the stiffened composite plate using adjusted DE is presented The objective function is the minimum of the strain energy with constraints on design variables The design variables are the fiber orientations of plate and composite beam and chosen to be integer values The static respond and the optimal solution for the fiber orientations of the stiffened composite plate are also compared with the reference to verify the correction of the algorithm The results illustrated the efficiency of the adjusted Differential Evolution compared with those of the GA

ACKNOWLEDGEMENT

Thank for the University of Technical Education in HCMC and Ton Duc Thang University to support us in this research

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