Differential evolution for optimization of functionally graded beams

Accepted Manuscript Differential evolution for optimization of functionally graded beams C.M.C. Roque, P.A.L.S. Martins PII: DOI: Reference:

S0263-8223(15)00718-7 http://dx.doi.org/10.1016/j.compstruct.2015.08.041 COST 6739

To appear in:

Composite Structures

Please cite this article as: Roque, C.M.C., Martins, P.A.L.S., Differential evolution for optimization of functionally graded beams, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.08.041

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Differential evolution for optimization of functionally graded beams. Roque, C. M. C. and Martins, P. A. L. S. INEGI, Faculdade de Engenharia da Universidade do Porto Universidade do Porto Rua Dr. Roberto Frias 404, 4200-465 Porto, Portugal.

Abstract Differential evolution optimization is used to find the volume fraction that maximizes the first natural frequency for a functionally graded beam. A formulation using three parameters is used to describe volume fraction. Beams with different ratios of material properties are considered. Two methods are used to compute the natural frequencies, analytical and meshless numerical method. Results show that differential evolution is capable to find distributions for volume fraction that increase the natural frequency of beams. It was also found that the RBF numerical method can be used with differential evolution to solve problems related to maximization of natural frequencies in functionally graded beams. Keywords: differential evolution, functionally graded material, Timoshenko beam, free vibration, optimization 1. Introduction Functionally graded composites differ from traditional composites in the organization of their constituents, presenting a continuous variation on their macroscopic structure in a given direction. Since this variation is continuous, interlaminar stress concentration is eliminated, avoiding delamination phenomena usually encountered in layered composites. In an FGM material properties can vary by changing the volume fractions of its constituents. In order to better design materials, volume fraction and distribution can be tailored trough optimization techniques. FGM are being increasingly used in many engineering fields such as aeronautics, automobile and biomedical Preprint submitted to Composite Structures

August 12, 2015

industries, thus becoming more relevant to understand how to use design to produce enhanced structures [1, 2, 3, 4]. In particular, the maximization of free vibration of structures is a common criterion used in structural optimization since structures with high fundamental frequencies tend to be stiffer for a large variety of loads, static or dynamic [5, 6]. In order to understand the behavior of functionally graded beams (FGBs), many authors developed analytical procedures to study their static and dynamic responses. In particular, for the most used beam theories (Euler Bernoulli and Timoshenko), Aydogdu and Taskin studied free vibration analysis of functionally graded beams with simply supported edges [7]. Li presented an unified approach for analyzing the static and dynamic behaviors of functionally graded beams, reducing the Euler Bernoulli and Rayleigh beam theories from the Timoshenko beam theory [8]. Numerical methods have also been used to analyze the dynamic behavior of FGBs. Xiang and Yang applied the differential quadrature method to the free and forced vibration of a laminated functionally graded beam of variable thickness using the Timoshenko beam theory [9]. Su and Banerjee studied the free vibration of functionally graded Timoshenko beams using the dynamic stiffness method [10]. As production methods improve, the control of volume fraction on functionally graded materials becomes finer, allowing a large diversity of designs [11, 12]. Material can be placed within a prescribed design domain to achieve optimized structural performance [13]. For structures under dynamic loading, an important structural optimization criteria used in mechanical engineering, is the maximum frequency of free vibration. Although there exists a vast amount of work related to optimization of functionally graded plates and shells in the dynamic regime, published papers related to functionally graded beams (FGBs) are much fewer. Goupee and Vel used the element-free Galerkin method to analyze the two-dimensional steady-state free and forced vibration of functionally graded beams [14]. In [15] volume fraction optimization of functionally graded beams is studied for maximizing the fundamental natural frequency by applying a new meta-heuristic nature-inspired algorithm called firefly algorithm (FA) which is based on the flashing behavior of fireflies. Yas et al. applied an imperialist competitive algorithm for the optimization of three-parameter power-law distribution of functionally graded (FG) beam [16]. Particle swarm algorithm was also used for optimization of functionally graded materials [17]. In the present work, differential evolution (DE) is proposed to analyze the 2

behavior of FGBs. Differential evolution is a stochastic optimization technique developed by Storn and Price [18]. DE is a simple population based, stochastic function minimizer that may be initialized by sampling the objective function at multiple, randomly chosen initial points. After initialization DE generates new vectors that are perturbations of existing vectors by using the scaled difference of two randomly selected population vectors, a process called differential mutation. The next step in classic DE optimization strategy is Uniform Crossover. Crossover is a biomimetic strategy that enhances the diversity of solutions by mixing members from the target (original) and donor (mutated) populations. Finally, population members with lower objective functions are selected, maintaining the population size constant. This step is called selection. The process ends when a predetermined value of the objective function is reached, or when a maximum number of generations is produced. DE can be used to find approximate solutions to problems that have objective functions that are non-differentiable, non-continuous, non-linear, noisy, flat, multi-dimensional or have many local minima, constraints or stochasticity. Differential evolution is increasingly being used in the optimization of composite structures. Loja et al. used differential evolution to obtain a deflection profile minimization of magneto-electro-elastic composite structures [19]. Le-Anh et al. used the finite element method and a variation of differential evolution to study folded laminated composite plates [20] and Roque and Martins used differential evolution to improve the meshless radial basis function method in the study of composite plates in bending [21]. 2. Free vibration of Timoshenko functionally graded beam with simply supported ends: analytical solution. The displacement field for the Timoshenko beam theory can be written in the form: u(x, z, t) = u0 (x, t) + zφ(x, t) w(x, z, t) = w0 (x, t)

(1)

where u and w denote the displacement at any point of the beam along directions x and z, u0 and w0 are the axial an transverse displacement of a point on the neutral axis and φ denotes the rotation of the cross section about 3

the y-axis. Considering linear analysis, the strain-displacement relationships are given by: ∂u0 ∂w0 ∂φ + z ; γxz = +φ (2) xx = ∂x ∂x ∂x stresses are of the form: E(z) E(z) xx ; τxz = (3) γxz 2 1−ν 2(1 + ν) Using the principle of virtual displacements, the equilibrium equations can be written as: σxx =

∂Nx ∂ 2 u0 ∂2φ = I0 2 + I 1 2 ∂x ∂t ∂t 2 ∂Qx ∂ w0 = I0 2 ∂x ∂t ∂Mx ∂ 2 u0 ∂2φ − Qx = I1 2 + I2 2 ∂x ∂t ∂t where Nx , Mx and Qx are stress resultants and are given by: ∂u0 ∂φ + B11 ∂x ∂x ∂u0 ∂φ Mx = B11 + D11 ∂x  ∂x  ∂w0 +φ Qx = KA55 B11 ∂x Nx = A11

(4) (5) (6)

(7) (8) (9)

10(1+ν) with shear correction factor K = (12+11ν) . Coefficients A11 , B11 , D11 , A55 are given by:

1 (A11 , B11 , D11 ) = 1 − ν2 A55



h/2

E(z)(1, z, z 2 )dz

(10)

−h/2

1 = 2(1 + ν)



h/2

E(z)dz

and inertia components are given by:  h/2 (I0 , I1 , I2 ) = ρ(z)(1, z, z 2 )dz −h/2

4

(11)

−h/2

(12)

For simply supported beams with length L, the solution is assumed to be of the form:

u0 (x, t) =

∞ 

Un cos

 nπx 

eiωn t

(13)

Wn sin

 nπx 

eiωn t

(14)

 nπx 

eiωn t

(15)

n=1

w0 (x, t) = φ(x, t) =

∞ 

n=1 ∞ 

Φn cos

n=1

L L L

where ωn is the frequency of natural vibration. In the present study, only the first frequency is optimized and therefore n = 1. Substitution equations (13)-(15) in (4)-(6) the following set of equations are obtained: I1 Φn ω 2 L2 − A11 π 2 Un − B11 Φn π 2 + I0 Un ω 2 L2 = 0 I0 Wn ω 2 L2 − A55 KΦn πL − A55 Kπ 2 Wn = 0 I2 Φn ω 2 L2 − D11 Φn π 2 − B11 π 2 Un − A55 KΦn L2 +I1 Un ω 2 L2 − πA55 KWn L = 0

(16) (17) (18)

The system is an eigenproblem of type A − ω 2 B = 0, where ω is a natural frequency. 3. Free vibration of Timoshenko functionally graded beam: meshless RBF solution. When an analytical solution is not known or simply laborious, the problem can be solved by using numerical methods such as finite elements or meshless methods. In the present paper, a meshless RBF collocation method is used. The method is simple and yet accurate. A brief description of the method is here presented. Consider a boundary problem with domain Ω ∈ Rn and with an elliptic differential equation given by, Lu(x) = s(x) x ∈ Ω ⊂ Rn Bu(x) = f (x) x ∈ ∂Ω ⊂ Rn 5

(19) (20)

where L and B are differential operators in domain Ω and in boundary ∂Ω, respectively. Points (xj , j = 1, . . . , NB ) and (xj , j = NB + 1, . . . , N ) are distributed in the boundary and on the domain respectively . The solution u(x) is approximated by u˜: u˜(x) =

N  j=1

βj g (x − xj , )

(21)

and inserting L and B operators in equation (21) we obtain the following equations,

u˜B (x) ≡

N 

βj Bg (x − xj , )

u˜L (x) ≡

N 

βj Lg (x − xj , ) = s(xi );

j=1

j=1

= f (xi );

i = 1, . . . , NB

(22)

i = NB + 1, . . . , N

(23)

where f (xi ) and s(xi ) are the prescribed values on boundary nodes and domain nodes, respectively. Solving the previous system for β, the solution can be interpolated using equation (21). In the present paper, the multiquadric radial basis function is used: g=

√

r 2 + 2

(24)

where r is the Euclidean distance between grid points and  is a shape parameter. 3.1. Three-parameter law for functionally graded materials. The beam is made of two materials with properties P1 , and P2 . Properties vary through the beam thickness, considering the rule of mixtures, accordingly to equation (25). P (z) = (P1 − P2 )V1 + P2

(25)

Possible laws for volume fraction V1 include power law [22], sigmoid law [23], exponential law [24] and three parameter law. The three parameter volume fraction proposed by Viola and Tornabene [25] is given by equation (26): 6

V1 =



1 z + +b 2 h



1 z − 2 h

 c p

(26)

parameters b, c and p control the material variation profile through the thickness of the beam and should be chosen such that V1 + V2 = 1, being V1 and V2 the volume fractions of each material. The above equation allows a more diverse material distribution when compared with the traditional power law for volume fraction. By taking b = 0 the usual power law is recovered. In particular, the Viola-Tornabene formulation allows to chose a mixture of materials at the top or bottom surfaces of the beam, which can be important from a material production perspective. 4. Differential evolution. Differential Evolution (DE) is a nature-inspired metaheuristics algorithm, proposed by Storn and Price for global optimization [18]. Classical DE has four main stages: Initialization, difference vector based Mutation, Crossover / Recombination and Selection [26]. The present approach uses the simplest DE variation known as DE/rand/1/bin. The algorithm is controlled by 3 parameters F , Cr and N P . F is the scaling factor typically between 0 and 1, but not restricted to this interval, that controls the differential mutation process. Cr is the Crossover rate which defines the probability of a trial vector to survive. N P is the current population size i. e., the number of competing solutions on any given generation G. The ith vector of current population G with size D can be described by: 

i,G = [x1,i,G , x2,i,G ; x3,i,G ; . . . ; xD,i,G ], i = 0, 1, . . . , N P − 1 X (27) j = 1, . . . , D 4.1. Initialization DE initialization can be made by randomly generating candidate solutions with N P D-dimensional real valued parameter vectors. xj,i,0 = xj,min + randi,j [0, 1](xj,max − xj,min )

(28)

where randi,j [0, 1] is a random number, 0 ≤ rand[0, 1] ≤ 1 which multiplied by the interval length, (xj,max − xj,min ) ensures a distributed sampling of the parameter’s domain interval [xj,min , xj,max ]. There can be different 7

approaches to generate the initial population although random uniformity is the most common. In the present optimization problems, a possible solution for parameters (b, c, p) is added to the initial population, in order to improve convergence. 4.2. Mutation Differential mutation adds a scaled, randomly sampled, vector difference to a third vector. Mutant vectors V i,G , also called Donors are obtained through differential mutation operation:

ri ,G + F (X

ri ,G − X

ri ,G ) V i,G = X 1 2 3

(29)

where F is a positive real number that controls the rate at which the pop ri and X

ri are sampled randomly form the

ri , X ulation evolves. Vectors X 1 2 3 i i i current population and r1 , r2 , r3 are mutually exclusive integers chosen from interval {1, . . . , N P }. In classical DE small F values are associated with Exploitation, understood as an opportunistic strategy if some of the testing solutions are in the vicinity of the global minimum. Conversely, large F values are associated with Exploration as new mutated trial solutions (Donors) incorporate larger differences in relation to the original population (Targets). 4.3. Crossover Crossover enhances the potential diversity of a population. In the case of binomial Crossover, trial vectors Ui,G are produced according to:

ui,j,G =



vj,i,G if randi,j [0, 1] ≤ Cr xj,i,G otherwise

or j = jrand

(30)

According to Storn and Price [26, Ch. 2] Crossover may be understood as a mutation rate or an inheritance probability between successive generations. There are alternatives to binomial Crossover. The most common is exponential Crossover, proposed by Storn and Price [26, Ch. 2]. Both approaches are valid for every problem although success/improvement of one over the other varies according to the problem considered [27, Ch. 1].

8

4.4. Selection Selection may be understood as a form of competition, in line with many examples directly observable in nature. Many evolutionary optimization schemes such as DE or GAs (Genetic Algorithms) use some form of selection. 

i,G ) ≤ f (X

i,G ) if f (U

i,G+1 = Ui,G (31) X

i,G ) > f (X

i,G )

i,G if f (U X As for a selection operation, the pairwise selection, also called greedy selection or elitist selection, is steadily used in the algorithm. As a stopping criteria, a maximum number of generations Gmax is defined. 5. Optimization results. Functionally graded beams with length L = 1 are considered. Ratio length/thickness is L/h = 10. Different material properties ratios s are considered. Material properties for material 1, Young modulus and density ρ, are the same as aluminum. as propriedades so adimensionais, normalizadas?... E1 = 70; ρ1 = 2702; E2 = sE1 ; ρ2 = sρ1 ; s = (0.1, 0.2, 0.5, 0.8, 2, 5, 10) Two different optimization problems are considered, both concerning the maximization of the first fundamental frequency of functionally graded beams. Parameters b, c and p in volume fraction law in equation(26) are rounded to 1 decimal point. In the first problem (Problem 1) the only restraint on volume fraction is 0 ≤ V1 ≤ 1. In problem 2 a 50% mixture between material 1 and material 2 is considered at the bottom of the beam, V1 (−h/2) = 0.5. All presented frequencies are normalized by: ρ2 2 ω = ωL (32) E2 Optimization parameters are kept constant throughout all examples with Cr = 0.9 and F = 0.1. Optimization problems can be stated as:

9

Problem 1 Minimise − ω s.t. A − ω 2 B = 0 with constraints: 0 ≤ b ≤ 20; 0 ≤ c ≤ 20; 0 ≤ p ≤ 20 0 ≤ V1 ≤ 1

Problem 2 Minimise − ω s.t. A − ω 2 B = 0 with constraints: 0 ≤ b ≤ 20; 0 ≤ c ≤ 20; 0 ≤ p ≤ 20 0 ≤ V1 ≤ 1 V1 (−h/2) = 0.5 5.1. Analytical solutions. Simply supported FGB are considered. Adopted stopping criteria is Gmax = 200 generations. For each ratio E2 /E1 , 25 runs are considered. For problem 1, the best solutions for each ratio E2 /E1 are presented in Table 1. Figure (2) shows the corresponding material profiles, V1 , along the beam thickness. Optimization shows that material profile can be tailored accordingly to ratio E2 /E1 in order to achieve maximum natural frequencies. Figure (3) shows profiles considering the mean of parameters b, c and p over 25 runs. Solutions present higher dispersion for ratios E2 /E1 = 5, 10. For problem 2, a constriction is imposed on the bottom surface, V1 (−h/2) = 0.5, corresponding to a 50/50 material mixture. Figure (4) shows the material profiles for the best solutions for each ratio E2 /E1 . The correspondent frequency ω is tabulated in Table 2. 5.2. Numerical RBF solution. Next, instead of using an analytical procedure to compute the fundamental frequency, a meshless numerical method is used. A convergence study is performed with the RBF method, for a simply supported isotropic beam i.e., b = c = p = 0, considering  = 2/n , being n the number of points in a regular grid, used for discretization. Results are plotted in Figure (5). 10

For subsequent optimization runs, the number of points in the discretization grid is chosen to be n = 21, corresponding to a relative error of about 3%. This choice allows an acceptable error maintaining a reasonable extra computational effort in the overall optimization process. Table 3 shows the best optimization results when using the numerical RBF method. Solutions for ω are compared with analytical solutions and relative errors, in percentage, are presented. For all ratios E2 /E1 , errors are bellow 3%, although a consistent increase in error is observed when E2 /E1 increases. In spite of this error, optimized profiles of volume fraction are equal to those encountered using an analytical solution. This could validate the use of RBF method for simulations with DE optimization when an analytical solution is not available, for example for boundary conditions other than simply supported or complex geometries. In addition to simply supported beams, other boundary conditions were tested, namely clamped and free. As an example, results for E2 /E1 = 0.1 are presented in Table 4. Letters C, S and F stand for clamped, simply supported and free boundary conditions and are given by: free : Mx = Qx = Nx = 0 clamped : u0 = w0 = φ = 0 simply supported : Nx = Mx = w0 = 0

(33) (34) (35)

Parameters b, c and p found by differential evolution were the same regardless boundary conditions. 6. Final comments. Differential evolution is used to optimize volume fraction distribution in a functionally graded beam, using a three law parameter for volume fraction. Different ratios E2 /E1 were tested. The formulation was capable of accommodating different design solutions, as optimized volume fraction profiles can be divided into 2 groups, for E2 /E1 > 1 and E2 /E1 < 1. Optimized solution found for E2 /E1 < 1 corresponds to a sandwich-structured composite, with a smooth transition between face and core properties. Solutions found for E2 /E1 > 1 are closer to power law distribution for volume fraction. In addition to an analytical method to solve the problem of free vibrations in a FGM beam, the meshless RBF numerical method was also tested. 11

Results indicate that RBF method can be used with differential evolution in order to solve optimization problems involving free vibration analysis. Parameters b, c and p found by differential evolution were the same regardless boundary conditions. 7. Acknowledgments The support of Ministerio da Ciencia Tecnologia e do Ensino Superior and Fundo Social Europeu (MCTES and FSE) under programs POPH-QREN and Investigador FCT and grant SFRH/BPD/71080/2010 from FSE are gratefully acknowledged. References [1] L. Marin, Numerical solution of the cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials, International Journal of Solids and Structures 42 (15) (2005) 4338 – 4351. [2] S. Matsuo, F. Watari, N. Ohata, Fabrication of a functionally graded dental composite resin post and core by laser lithography and finite element analysis of its stress relaxation effect on tooth root, Dental Materials Journal 20 (4) (2001) 257–274. ˇ Draˇsar, J. Schilz, W. Kaysser, Functionally graded ma[3] E. M¨ uller, C. terials for sensor and energy applications, Materials Science and Engineering: A 362 (1) (2003) 17–39. [4] W. Pompe, H. Worch, M. Epple, W. Friess, M. Gelinsky, P. Greil, U. Hempel, D. Scharnweber, K. Schulte, Functionally graded materials for biomedical applications, Materials Science and Engineering: A 362 (1) (2003) 40–60. [5] W. P. M. Save, W. H. Warner (Eds.), Structural Optimization, Springer, 1990. [6] M. P. Bendsoe, O. Sigmund, Topology Optimization, Theory, Methods, and Applications, Springer, 2004. [7] M. Aydogdu, V. Taskin, Free vibration analysis of functionally graded beams with simply supported edges, Materials and Design 28 (5) (2007) 1651–1656. 12

[8] X.-F. Li, A unified approach for analyzing static and dynamic behaviors of functionally graded timoshenko and euler-bernoulli beams, Journal of Sound and Vibration 318 (4-5) (2008) 1210–1229. [9] H. Xiang, J. Yang, Free and forced vibration of a laminated fgm timoshenko beam of variable thickness under heat conduction, Composites Part B: Engineering 39 (2) (2008) 292–303. [10] H. Su, J. Banerjee, Development of dynamic stiffness method for free vibration of functionally graded timoshenko beams, Computers & Structures 147 (0) (2015) 107 – 116, cIVIL-COMP. [11] B. Kieback, A. Neubrand, H. Riedel, Processing techniques for functionally graded materials, Materials Science and Engineering A 362 (1-2) (2003) 81–105. [12] M. El-Wazery, A. El-Desouky, A review on functionally graded ceramicmetal materials, Journal of Materials and Environmental Science 6 (5) (2015) 1369–1376. [13] O. Sigmund, K. Maute, Topology optimization approaches: A comparative review, Structural and Multidisciplinary Optimization 48 (6) (2013) 1031–1055. [14] A. Goupee, S. Vel, Optimization of natural frequencies of bidirectional functionally graded beams, Structural and Multidisciplinary Optimization 32 (6) (2006) 473–484. [15] S. Kamarian, M. Yas, A. Pourasghar, M. Daghagh, Application of firefly algorithm and anfis for optimisation of functionally graded beams, Journal of Experimental & Theoretical Artificial Intelligence 26 (2) (2014) 197–209. [16] M. H. Yas, S. Kamarian, A. Pourasghar, Application of imperialist competitive algorithm and neural networks to optimise the volume fraction of three-parameter functionally graded beams, Journal of Experimental & Theoretical Artificial Intelligence 26 (1) (2014) 1–12. [17] Y. Xu, W. Zhang, D. Chamoret, M. Domaszewski, Minimizing thermal residual stresses in c/sic functionally graded material coating of c/c composites by using particle swarm optimization algorithm, Computational Materials Science 61 (2012) 99–105. 13

[18] R. Storn, K. Price, Differential evolution - a simple and efficient adaptive scheme for global optimization over continuous spaces, Tech. Rep. TR95-012 (1995). [19] M. Loja, C. M. Soares, J. Barbosa, Optimization of magneto-electroelastic composite structures using differential evolution, Composite Structures 107 (2014) 276 – 287. [20] L. Le-Anh, T. Nguyen-Thoi, V. Ho-Huu, H. Dang-Trung, T. Bui-Xuan, Static and frequency optimization of folded laminated composite plates using an adjusted differential evolution algorithm and a smoothed triangular plate element, Composite Structures 127 (2015) 382 – 394. [21] C. Roque, P. Martins, Differential evolution optimization for the analysis of composite plates with radial basis collocation meshless method, Composite Structures 124 (2015) 317 – 326. [22] G. Bao, L. Wang, Multiple cracking in functionally graded ceramic/metal coatings, International Journal of Solids and Structures 32 (19) (1995) 2853–2871. [23] S.-H. Chi, Y.-L. Chung, Mechanical behavior of functionally graded material plates under transverse loadpart i: Analysis, International Journal of Solids and Structures 43 (13) (2006) 3657 – 3674. [24] H. Ait Atmane, A. Tounsi, S. A. Meftah, H. A. Belhadj, Free vibration behavior of exponential functionally graded beams with varying crosssection, Journal of Vibration and Control 17 (2) (2011) 311–318. [25] E. Viola, F. Tornabene, Free vibrations of three parameter functionally graded parabolic panels of revolution, Mechanics research communications 36 (5) (2009) 587–594. [26] K. Price, R. Storn, J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, U.S. Government Printing Office, 2005. [27] V. Feoktistov, Differential Evolution: In Search of Solutions, Springer Optimization and Its Applications, Springer, 2006.

14

List of Figures 1 2 3 4 5 6

Differential Evolution main steps . . . . . . . . . . . . . . . Problem 1. Best solution after 25 runs. Cr = 0.9, F = 0.1, 200 generations. . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1. Mean solutions after 25 runs. Cr = 0.9, F = 0.1, 200 generations . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2. Optimized solutions for 1 run, Cr = 0.9, F = 0.1, for 2000 generations, V1 (−h/2) = 0.5. . . . . . . . . . . . . . Convergence study for simply supported isotropic beam using numeric RBF meshless method, b = c = p = 0,  = 2/n. . . . First mode for optimized free vibration, for various boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

. 16 . 17 . 18 . 19 . 20 . 21

Initialization

Mutation

Crossover

Selection

Figure 1: Differential Evolution main steps

16

Solution

Best optimized Solution

0.5

0.4

E2/E1=0.1 E2/E1=0.2 E2/E1=0.5 E2/E1=0.8 E2/E1=2 E2/E1=5 E2/E1=10

0.3

0.2

z/h

0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5 0

0.1

0.2

0.3

0.4

0.5 Volume fraction

0.6

0.7

0.8

0.9

1

Figure 2: Problem 1. Best solution after 25 runs. Cr = 0.9, F = 0.1, 200 generations.

17

Mean optimized Solution

0.5

0.4

E2/E1=0.1 E2/E1=0.2 E2/E1=0.5 E2/E1=0.8 E2/E1=2 E2/E1=5 E2/E1=10

0.3

0.2

z/h

0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5 0

0.1

0.2

0.3

0.4

0.5 Volume fraction

0.6

0.7

0.8

0.9

1

Figure 3: Problem 1. Mean solutions after 25 runs. Cr = 0.9, F = 0.1, 200 generations

18

Optimized Solution

0.5

0.4

0.3

0.2

z/h

0.1

0

-0.1 E2/E1=0.1 E2/E1=0.2 E2/E1=0.5 E2/E1=0.8 E2/E1=2 E2/E1=5 E2/E1=10

-0.2

-0.3

-0.4

-0.5 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Volume fraction

Figure 4: Problem 2. Optimized solutions for 1 run, Cr = 0.9, F = 0.1, for 2000 generations, V1 (−h/2) = 0.5.

19

5

4.5

4

3.5

Relative error, %

3

2.5

2

1.5

1

0.5

0 0

20

50

100

200

300 Number of grid points

400

500

600

Figure 5: Convergence study for simply supported isotropic beam using numeric RBF meshless method, b = c = p = 0,  = 2/n.

20

Optimized Solution

0.12 CF CC CS SS

0.1

0.08

first vibration mode

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08 0

0.1

0.2

0.3

0.4

0.5 L

0.6

0.7

0.8

0.9

1

Figure 6: First mode for optimized free vibration, for various boundary conditions.

21

List of Tables 1 2 3 4

Best solutions after 25 runs for fgm beam, problem 1 . . . . Problem 2. Best solutions for fgm beam. . . . . . . . . . . . Problem 1. Best solution after 25 runs for fgm beam, using RBF method, n = 21,  = 2/n. . . . . . . . . . . . . . . . . . Problem 1. Solution for 1 run for fgm beam with different boundary conditions at x = 0, L, using RBF method, n = 21,  = 2/n. C-clamped; S-simply supported; F-free. . . . . . . .

22

. 23 . 24 . 25

. 26

E2 /E1 0.1 0.2 0.5 0.8 2 5 10

ω 0.3761 0.3493 0.3155 0.3002 0.2963 0.2970 0.2937

b 1 1 1 1 0 0 0.7

c 2.2 2.3 2.6 2.8 0 0 0.9

p 16.2 12.1 7.6 5.9 0.2 0.1 0.1

Table 1: Best solutions after 25 runs for fgm beam, problem 1

23

E2 /E1 0.1 0.2 0.5 0.8 2 5 10

ω 0.2966 0.2971 0.2969 0.2951 0.2923 0.2815 0.2703

b 0.5 0.5 0.5 0.5 0.5 0.5 0.5

c p 3.4 1 3.7 1 4.5 1 5.1 1 0.8 1 0.9 1 1.2 1

Table 2: Problem 2. Best solutions for fgm beam.

24

E2 /E1 0.1 0.2 0.5 0.8 2 5 10

ω 0.3747 0.3457 0.3089 0.2920 0.2876 0.2885 0.2848

b 1 1 1 1 0 0 0.7

c p rel. error(ω) % 2.2 16.2 0.4 2.3 12.1 1.0 2.6 7.6 2.1 2.8 5.9 2.7 0 0.2 2.9 0 0.1 2.9 0.9 0.1 3.0

Table 3: Problem 1. Best solution after 25 runs for fgm beam, using RBF method, n = 21,  = 2/n.

25

E2 /E1 ω 0.1 0.5772 0.7915 0.3605 0.3747

b 1 1 1 1

c 2.2 2.2 2.2 2.2

p Boundary conditions 16.2 CS 16.2 CC 16.2 CF 16.2 SS

Table 4: Problem 1. Solution for 1 run for fgm beam with different boundary conditions at x = 0, L, using RBF method, n = 21,  = 2/n. C-clamped; S-simply supported; F-free.

26