Differential evolution for free vibration optimization of functionally graded nano beams

Accepted Manuscript Differential evolution for free vibration optimization of functionally graded nano beams C.M.C. Roque, P.A.L.S. Martins, A.J.M. Ferreira, R.M.N. Jorge PII: DOI: Reference:

S0263-8223(16)30209-4 http://dx.doi.org/10.1016/j.compstruct.2016.03.052 COST 7350

To appear in:

Composite Structures

Received Date: Accepted Date:

21 December 2015 27 March 2016

Please cite this article as: Roque, C.M.C., Martins, P.A.L.S., Ferreira, A.J.M., Jorge, R.M.N., Differential evolution for free vibration optimization of functionally graded nano beams, Composite Structures (2016), doi: http:// dx.doi.org/10.1016/j.compstruct.2016.03.052

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Differential evolution for free vibration optimization of functionally graded nano beams. Roque, C. M. C., Martins, P. A. L. S., Ferreira, A.J.M., Jorge, R. M. N. INEGI, Faculdade de Engenharia da Universidade do Porto Universidade do Porto Rua Dr. Roberto Frias 404, 4200-465 Porto, Portugal.

Abstract A modified couple stress theory is used to study the influence of a scale parameter in the free vibration of a Timoshenko functionally graded beam. For energy harvesting of micro devices, it is require for the beam to resonate at low frequencies. In order to minimize the free vibration frequency of the beam, differential evolution optimization is used to solve the optimization problem. To describe the volume fraction variation along the beams thickness, a three parameter volume fraction law is chosen. Results show that for the selected volume fraction law, and considering linear analysis, the optimal material distribution across the beam thickness is independent of the scale parameter. Results are insensitive to tested boundary conditions. To Professor Reddy, Professor JN Reddy contributed with an immense body of knowledge to the scientific community. His many contributions to the field of computational solid mechanics are well known and used by many. There is a common ground between his work and the work of others of equal scientific relevance - a respect for the fundamentals, often distilled in the form of introductory texts aimed at providing the best guidance for those at the beginning of their scientific journeys. For all that we are truly grateful as scientists; For his easy manner, his sense of humor and his friendship we are grateful - Thank you JN! Keywords: Differential evolution, functionally graded material, Timoshenko nano beam, free vibration, optimization

1

1. Introduction As production techniques for micro-electro-mechanical devices improve, micro scaled devices are being increasingly used as low frequency vibration energy harvesting devices. Low frequencies are a source of energy that occur naturally in the environment or are produce as a secondary product of many human activities. Example of activities that produce peak accelerations bellow 200 Hz include human heartbeat, human walking or commuter rail cars [1]. The idea behind micro energy harvesting devices is to convert the mechanical energy produced in the deformation of a structure into electrical energy, by piezoelectric, electrostatic or electromagnetic induction. In order to achieve maximum deformation for low frequencies, a structure can be design to resonate at low frequencies. In the present study, a minimization problem is solved in order to find the optimal material distribution of a functionally graded beam. Experimental observations indicate that the mechanical behavior of micro and nano systems cannot be accurately simulated by using classical deformations theories. Due to the presence of scale effects, non classical models such the modified couple stress theory is used for analytical analysis. Unlike classical models, this theory contains an internal material length scale parameter that can capture scale effects at the micro scale. Yang et al. [2] proposed a modified couple-stress theory in which the couple-stress tensor is symmetric and there is only one internal material length scale parameter. The theory has been used by previously by the authors in the analysis of nano beams and plates, using a numerical meshless method [3, 4]. In the present paper, a modified couple stress theory is used to simulate the static bending of a micro functionally graded Timoshenko beam. 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. 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 2

Bernoulli and Timoshenko), Aydogdu and Taskin studied free vibration analysis of functionally graded beams with simply supported edges [5]. 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 [6]. Giunta et al used a generic N-order approximation to study functionally graded beams under bending and torsional loadings [7]. A recent review for various kinematics and theories regarding a more general analysis of composite beams can be found in [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]. 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 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 3

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. Modified couple stress theory The modified couple stress theory proposed by Yang et al. [2], has the advantage over the classical couple stress theory of involving only single scale parameter, . A brief description of the modified couple stress theory is here presented, in which the curvature is taken to be symmetrized rotation-gradient, i.e., the strain tensor, εij and curvature tensor, χij are given by, 1 (ui,j + uj,i ) 2 1 χij = (ωi,j + ωj,i ) 2 εij =

(1) (2)

and the strain energy, 1 U= 2



(σij (ε) εij + mij (χ) χij ) dΩ

Ω

where for isotropic materials

4

(3)

σij = λεkk δij + 2Gεij

(4)

mij = 22 Gχij

(5)

where µ and λ are the Lame parameters, λ=

Eν E ; 2µ = (1 + ν)(1 − 2ν) 1+ν

(6)

3. Free vibration of Timoshenko functionally graded nano beams In this section the equations of motion are presented for the presented a fgm Timoshenko nano beams. Full details on the derivation of presented equations can be found in [22]. The displacement field for a Timoshenko beam is, u1 (x, z, t) = u(x, t) + zφx (x, t) u3 (x, z, t) = w(x, t)

(7) (8)

where u1 and u3 are the x- y- and z- components of displacement at a point (x,y,z) and u, w are the components of displacement on the central axis and φx is the rotation of the cross section about the y-axis. The equilibrium equations derived from the principle of virtual work are given by: ∂2u ∂ 2 φx ∂Nxx = m0 2 + m1 2 ∂x ∂t ∂t 2 ∂Qx 1 ∂ Pxy ∂2w = m + 0 ∂x 2 ∂x2 ∂t2 1 ∂Pxy ∂Mxx ∂ 2 φx ∂ 2u − Qx + = m2 2 + m1 2 ∂x 2 ∂x ∂t ∂t

(9) (10) (11)

with 

Nxx = Mxx =

A

A

5

σxx dA

(12)

zσxx dA

(13)



Qx = Ks σxz dA  A mxy dA Pxy = A  m0 = ρdA A  m1 = ρzdA A  ρz 2 dA m2 =

(14) (15) (16) (17) (18)

A

For simply supported beams, it is possible to find analytical solutions by considering the following expansions for u(x, t), w(x, t) and φx (x, t), with ω denoting the free vibration frequency.

u(x, t) = w(x, t) =

∞ 

Un cos

 nπx 

eiωn t

(19)

Wn sin

 nπx 

eiωn t

(20)

Φxn cos

 nπx 

eiωn t

(21)

n=1 ∞  n=1

φx (x, t) =

∞  n=1

L L L

(22) 3.1. Cantilever beam, meshless RBF solution Many beams used in energy harvesting devices are cantilever beams, with ends clamped-free. 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 [23, 24]. 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 6

(23) (24)

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 , )

(25)

and inserting L and B operators in equation (25) 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

(26)

i = NB + 1, . . . , N

(27)

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 (25). In the present paper, the multiquadric radial basis function is used: g=

√

r 2 + 2

(28)

where r is the Euclidean distance between grid points and  is a shape parameter. 3.2. 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 (29). P (z) = (P1 − P2 )V1 + P2

(29)

Possible laws for volume fraction V1 include power law [25], sigmoid law [26], exponential law [27] and three parameter law. The three parameter volume fraction proposed by Viola and Tornabene [28] is given by equation (30): 7

V1 =



1 z + +b 2 h



1 z − 2 h

 c p

(30)

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 [29]. 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 (31) 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 )

(32)

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 8

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 Vi,G , also called Donors are obtained through differential mutation operation:  ri ,G + F (X  ri ,G − X  ri ,G ) Vi,G = X 1 2 3

(33)

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

(34)

According to Storn and Price [29, 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 [29, Ch. 2]. Both approaches are valid for every problem although success/improvement of one over the other varies according to the problem considered [30, Ch. 1].

9

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 (35) 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. 4.5. Differential evolution optimization for the analysis of free vibration of fgm nano beams In the context of nano and micro beams design for energy harvesting, the optimization to be solved involves the minimization of the free vibration frequency. Parameters b, c and p are design variables for the optimization problem. Bounds for c, b, and p are 0 < c, b, p < 10. Population dimension N P = 90. As a stopping criteria, a maximum number of population Gm ax = 500 is chosen. 5. Results and comments A simply supported fgm Timoshenko beam of thickness h = 17.6µm and length Lx = 20h is considered. Material properties are : Ec = 1.44GPa; ρc = 1.22 × 103 kg/m; Em = 10Ec ; ρm = 10ρc , ν = 0.38 and Ks = 5(1 + ν)/(6 + 5ν).

.bh , with b = 2h. Frequencies are normalized by λ = ωL2x ρm Em Optimization parameters are F = Cr = 0.8, Gmax = 500. Values for F and Cr are chosen using previous experience of authors in the analysis of beams using differential evolution optimization [31]. To study the influence of scale parameter , different ratios /h are used in various optimizations. Results are presented in table 1, for 25 runs. Although vibration frequency is different for different /h ratios, the optimal material distribution is constant, considering the present volume fraction law. Free vibration dependence with /h is presented in figure fig:freq, for simply supported beams.

10

The same analysis is made considering a cantilever beam (clamped-free boundary conditions), with 1 single run. The RBF collocation meshless method is used to simulate clamped beams, with shape parameter  = 4Lx /n, n being the number of grid points. Obtained optimized values are the same as those obtained for simply supported beans, b = 1,c = 4.8, p = 1.4. Figure 3 shows the convergence analysis for various cantilever fgm nano beams, for b = 1,c = 4.8, p = 1.4 when n ranges from n = 51 to n = 151. The optimized volume fraction distribution along thickness is plotted in figure 4, corresponding to optimal parameters b = 1, c = 4.8, p = 1.4, for simply supported and cantilever beams. 6. Final remarks Differential evolution optimization was used to design simply supported and cantilever fgm nano beams in order to obtained the lowest free vibration frequency. Optimization results show an optimal distribution for b = 1,c = 4.8, p = 1.4. Material distribution results were independent of scale parameter  and 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, grant SFRH/BPD/71080/2010 from FSE and project PTDC/EME-PME/110084/2009 are gratefully acknowledged. References [1] J. Deng, K. Rorschach, E. Baker, C. Sun, W. Chen, Topology optimization and fabrication of low frequency vibration energy harvesting microdevices (2015). [2] F. Yang, A. Chong, D. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39 (10) (2002) 2731 – 2743. [3] C. Roque, A. Ferreira, J. Reddy, Analysis of mindlin micro plates with a modified couple stress theory and a meshless method, Applied Mathematical Modelling 37 (7) (2013) 4626–4633, cited By 23. 11

[4] C. Roque, D. Fidalgo, A. Ferreira, J. Reddy, A study of a microstructuredependent composite laminated timoshenko beam using a modified couple stress theory and a meshless method, Composite Structures 96 (2013) 532–537, cited By 34. [5] M. Aydogdu, V. Taskin, Free vibration analysis of functionally graded beams with simply supported edges, Materials and Design 28 (5) (2007) 1651–1656. [6] 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. [7] G. Giunta, S. Belouettar, E. Carrera, Analysis of fgm beams by means of classical and advanced theories 17 (8) (2010) 622–635. [8] H. Hu, S. Belouettar, M. Potier-Ferry, E. Daya, Review and assessment of various theories for modeling sandwich composites 84 (3) (2008) 282– 292. [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.

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[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. [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] H. Ma, X.-L. Gao, J. Reddy, A microstructure-dependent timoshenko beam model based on a modified couple stress theory (2008). [23] E. J. Kansa, Multiquadrics. a scattered data approximation scheme with applications to computational fluid-dynamics. i. surface approximations 13

and partial derivative estimates, Computers & mathematics with applications 19 (8-9) (1990) 127–145. [24] E. J. Kansa, Multiquadrics. a scattered data approximation scheme with applications to computational fluid-dynamics. ii. solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & mathematics with applications 19 (8-9) (1990) 147–161. [25] G. Bao, L. Wang, Multiple cracking in functionally graded ceramic/metal coatings, International Journal of Solids and Structures 32 (19) (1995) 2853–2871. [26] 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. [27] 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. [28] E. Viola, F. Tornabene, Free vibrations of three parameter functionally graded parabolic panels of revolution, Mechanics research communications 36 (5) (2009) 587–594. [29] K. Price, R. Storn, J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, U.S. Government Printing Office, 2005. [30] V. Feoktistov, Differential Evolution: In Search of Solutions, Springer Optimization and Its Applications, Springer, 2006. [31] C. Roque, P. Martins, Differential evolution for optimization of functionally graded beams, Composite Structures 133 (2015) 1191–1197.

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List of Figures 1 2 3 4

Differential Evolution main steps . . . . . . . . . . . . . . . Frequency dependence with /h, for optimal b = 1,c = 4.8, p = 1.4, for simply supported beams . . . . . . . . . . . . . Convergence for CF beams for /h = 0, /h = 0.2, /h = 0.4, /h = 0.6, /h = 0.8, /h = 1.0, for cantilever beams . . . . . Best optimized solution for parameters b,c, p, for /h = 0, /h = 0.2, /h = 0.4, /h = 0.6, /h = 0.8, /h = 1.0, for simply supported and cantilever beams. . . . . . . . . . . . .

15

. 16 . 17 . 18

. 19

Initialization

Mutation

Crossover

Selection

Figure 1: Differential Evolution main steps

16

Solution

5.5

×10 6

5

4.5

¯ λ

4

3.5

3

2.5

2

1.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

l/h

Figure 2: Frequency dependence with /h, for optimal b = 1,c = 4.8, p = 1.4, for simply supported beams

17

7

×10 6

/h = 0 /h = 0.2 /h = 0.4 /h = 0.6 /h = 0.8 /h = 1

6

¯ λ

5

4

3

2

1 40

60

80

100

120

140

160

180

200

number of grid points

Figure 3: Convergence for CF beams for /h = 0, /h = 0.2, /h = 0.4, /h = 0.6, /h = 0.8, /h = 1.0, for cantilever beams

18

Best optimized Solution b=1; c=4.8; p=1.4 0.5

0.4

0.3

0.2

z/h

0.1

0

-0.1

-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: Best optimized solution for parameters b,c, p, for /h = 0, /h = 0.2, /h = 0.4, /h = 0.6, /h = 0.8, /h = 1.0, for simply supported and cantilever beams.

19

List of Tables 1

Best, mean and standard deviation for b,c,p, after 25 runs, for simply supported beams . . . . . . . . . . . . . . . . . . . . . 21

20

/h 0.0 0.2 0.4 0.6 0.8 1.0

b 1.0 1.0 1.0 1.0 1.0 1.0

best c 4.8 4.8 4.8 4.8 4.8 4.8

p b 1.4 1.0±0.0 1.4 1.0±0.0 1.4 1.0±0.0 1.4 1.0±0.02 1.4 1.0±0.0 1.4 1.0±0.0

mean±SD c 4.8±0.17 4.8±0.14 4.7±0.21 4.9±0.35 4.9±0.37 4.9±0.44

p 1.4±0.06 1.4±0.05 1.4±0.08 1.4±0.07 1.4±0.07 1.4±0.13

Table 1: Best, mean and standard deviation for b,c,p, after 25 runs, for simply supported beams

21