an approach for the global reliability based design ...

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling June 23th to June 27th, 2014, Rouen, France

AN APPROACH FOR THE GLOBAL RELIABILITY BASED DESIGN OPTIMIZATION OF TRUSS STRUCTURES Rafael H. Lopez1, André J. Torii2, Leandro F.F. Miguel1, José E Souza de Cursi3 1

Center for Optimization and Reliability (CORE), Department of Civil Engineering, Universidade Federal de Santa Catarina, Brasil [email protected], [email protected] 2 Center for Optimization and Reliability (CORE), Department of Scientific Computing, Universidade Federal da Paraiba, Brasil [email protected] 3 Départment Mécanique, Institut National des Sciences Appliquées de Rouen, France [email protected]

Abstract. This paper presents a methodology to perform global reliability based design optimization (RBDO) of the size and shape of truss structures. This methodology is comprised by the use of a global constraint and the response surface method to deal with the reliability analysis together with the Firefly Algorithm (FA) to carry out the structural optimization. The former is responsible for the reduction of the computational cost required in the evaluation of the probabilistic constraints. The latter overcomes the issues related to the non-convexity and mixed-variables of the optimization problem. Several examples are analysed in order to show the effectiveness of the methodology. All the optima found were checked using a classical first order reliability method (FORM) approach, validating the results provided by the response surface method. Also, the authors noticed that the literature lacks established statistical benchmark criteria to evaluate the performance of metaheuristics in size and shape optimization of trusses, especially in the case of RBDO. Hence, this paper presented the results of the mean value and coefficient of variation over 50 runs for each example providing a statistical basis for further comparison in future works. Keywords. truss optimization ,reliability based design optimization, global optimization, discrete optimization 1 INTRODUCTION The deterministic optimization of truss structures has been widely studied in the literature (Lamberti and Pappalettere, 2011). The optimization of this kind of structure usually leads to a mixed-variable, i.e. discrete and continuous design variables, and non-convex problem in the presence of many local minima (Torii, Lopez and Biondini, 2012; Miguel, Lopez and Fadel Miguel, 2013). In order to overcome these two difficulties, several heuristic algorithms have been employed, being the most widely applied the genetic algorithm (GA). However, the use of GA has presented some drawbacks, mainly related to the long computational time required when dealing with large computational models. Hence, alternative approaches have been developed in order to reduce the computational demand needed. Some methods that have been employed for truss optimization are: the particle swarm optimization (PSO), the simulated annealing (SA), the ant colony optimization, the harmony search, among others (Lamberti and Pappalettere, 2011). However, it is widely acknowledged that deterministic optimization is not robust with respect to the uncertainties which affect engineering design. In deterministic optimization, potential failure modes are converted in deterministic constraints and uncertainty is addressed indirectly by means of safety factors and conservative assumptions. This approach is inherited from design through design codes, but it is essentially non-optimal for two reasons. First, safety factors approaches tend to be conservative, since it must take into account a large number of cases with different characteristics. Besides, designs obtained using the safety factors approach are potentially less safe than required if the problem under study does not match exactly the assumptions made in order to obtain the safety factors used. One of the main approaches for taking into account uncertainties in is the reliability based design optimization (RBDO) (Lopez and Beck, 2012). The main goal of RBDO is to optimize a structure ensuring that its probability of failure is lower than a certain level, chosen a priori by the designer. In the context of truss structures, RBDO has been applied to the topology or size optimization by Nakib (1997) and Thampan and Krishnamoorty (2001), and to shape and topology optimization by Morutsu and Shao (1990), Stocki et al. (2001), Thanpham and Krishnamoorthy (2001) and Torii, Lopez and Biondini (2012). It is worth to highlight that none among these works employed a global optimization algorithm to pursue the RBDO of truss structures. Such global algorithms are required due to the non-convexity of the problem at hand and the presence of discrete design variables. It has been observed that the computational cost of the reliability analysis is one of the main issues in the application of RBDO for real problems. The evaluation of the probability of failure is traditionally made using first and second order reliability methods (FORM and SORM, respectively). However, one reliability analysis problem must be solved for each limit state function of the problem, being most problems composed of several limit state functions (e.g. the

R.H. Lopez, A.J. Torii, L.F.F. Miguel, J.E. Souza de Cursi Uncertainties 2014

stress constraints inside each bar of the structure defines a limit state function). Besides, FORM and SORM require the evaluation of the gradient of the limit state function, what is done using finite differences in most cases, since the limit state functions can be implicitly. Consequently, the evaluation of the probability of failure for a given design vector requires, in general, a large number of calls of some finite element code. This number is drastically increased in the context of RBDO since several design vectors must be checked in order to carry optimization. One approach to overcome these difficulties is the utilization of an analytical approximated model as a surrogate for the reliability analysis, which can be obtained with the help of the response surface methodology (Torii and Lopez, 2012). By choosing an appropriate approximation, one is able to represent the response of the system accurately. This reduces the computational effort involved since the reliability analysis problem is solved for the approximated analytical solution and consequently all the information needed (i.e. gradients) can be evaluated efficiently. Besides, it can be used as a black-box in a non- intrusive manner. In the case of heuristic algorithms to pursue the structural optimization in RBDO, the computational cost is a very delicate matter, since both the heuristic algorithm and the reliability analysis are, in general, computationally demanding. However, the firefly algorithm (FA) (Yang, 2009) has proved to be more accurate and efficient than wellestablished heuristic algorithms such as the GA and the PSO. For this reason, several researchers have focused their attention on solving optimization problems using FA in a growing number of papers (Fister, Yang and Fister, 2014), including the deterministic optimization of truss structures developed by the authors (Miguel, Lopez and Fadel Miguel, 2013). However, its implementation in the field of structural optimization is still fairly recent and requires a substantial amount of further study (Gandomi, Yang and Alavi, 2011). Moreover, at the best of the author’s knowledge, the FA has not been applied to solve any RBDO problem. Thus, in this paper, we propose a methodology to pursue size and shape RBDO of truss structures. This methodology consists in using the FA to perform the structural truss optimization and the response surface method to reduce the computational burden of the reliability analysis. Since we employ the FA as the structural optimizer, the proposed methodology can be considered as an attempt for the global RBDO of truss structures. The paper is organized as follows. Section 2 presents the formulation of the RBDO problem. The methodology used to pursue the reliability analysis using the response surface methodology is detailed in Section 3. A description of the FA is given in Section 4. A series of numerical examples is presented in Section 5, while Section 6 presents the main conclusions of the study. 2 PROBLEM FORMULATION We consider that the structures studied in this paper are subject to random variables of the following type: crosssectional areas of the bars A, nodal coordinates C, load magnitudes F, yielding stress of the material in tension St and compression Sc and material elastic modulus E. The algorithm performs size optimization of the truss by changing the mean value of the cross-sectional areas ( a   ) of the m structural members. Shape optimization is made by modification of the mean value of the nodal m

coordinates of the nodes q’ considered as design variables ( c   ). Notice that we employ capital letters for random variables and small letters for deterministic quantities, such as the mean value of the random variables. The optimization procedure seeks the structure of minimum weight subject to stress, displacement and local buckling reliability constraints. For convenience of notation, the design variables a and c are grouped into the vector q'









d  a1 , , am , c1 ,, cq ' and the random variables are grouped into the vector X  A, C, F, S t , S c , E . The optimization problem can then be posed as: Find

d

Minimise

wd    ρ j  j c a j

m

j 1

Subject to

PGk d, X   0   Pfmax  0, k  1,...., q  m a j  , ,

(1)

j  1,..., m

cimin  ci  cimax , i  1,..., q ' where w is the structural weight, m is the number of bars, ρ is the specific weight of the bar material,  is the length of each bar, P  is the probability of the event in parenthesis to occur, Pf

max

is the maximum allowable probability

of failure for each reliability constraint,  is the set of available discrete cross-sectional areas, ci

min

max

and ci

lower and upper bounds for the nodal coordinate ci, respectively, and Gk are the local limit state functions given by

are

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling June 23th to June 27th, 2014, Rouen, France

Gi d, X    i d, X    i

max

 0, i  1,..., q

G j d, X   S j d, X   S j  0, j  1,..., m

(2)

 , for bars in tension ( S j  0) S tj Sj   c b min( S j , S j ), for bars in compression ( S j  0) where

k

and

 kmax

are the displacement and maximum allowable displacement at node i, respectively, S j is the

t j

c

b

stress of bar j, S and S j are the yielding stresses in tension and compression of bar j, respectively, and S j is the Euler buckling stress of bar j. 3 RELIABILITY ANALYSIS 3.1 First order reliability method The main idea of the FORM consists in replacing each limit state function Gk by a tangent hyperplane in the most probable point of failure (MPP). The reliability of the system is then approximated using the reliability index, which is usually denoted by  . To evaluate the  of each constraint, it is usual to introduce a vector of normalized and statistically independent random variables

U   n and a transformation T so that U  T X  . The mapping

T transforms every realization x of X in the physical space into a realization u in the normalized space. Notice that the 1 ( X)) .

limit state function can also be written in the normalized space as H (d, U )  G (d, T The reliability index for a given d



can be obtained from the following optimization problem in the normalized space:

Minimise

u* d   u

Subject to

H (d, U)  0 .

Find

(3)

The solution of Eq. (3) u d  is the MPP, which is defined as the realization of the random vector U that lies over the limit state surface and that is closer to the origin of the normalized space. The reliability index β is defined as the distance from the origin of the normalized space to the MPP. Finally, the failure probability can be approximated as *

PGk d, X   0      d 





Pfmax     target ,

(4)

where  is the standard Gaussian cumulated function. In this paper, the probability constraints of Eq. (1) are approximated using Eq. (4). It is important to point out that the optimization problem of Eq. (3) must be solved to evaluate the reliability index  k of each constraint. Thus, to check the feasibility of a given design vector d , q + m optimization problems must be solved. This procedure leads to very high computational costs, especially when a heuristic algorithm is employed for the structural optimization, since it usually requires the evaluation of at least thousands of designs in order to converge to the global optimum of the problem. To overcome this issue, we restate the reliability analysis using a global constraint and employ the adaptive response surface strategy to evaluate the resulting reliability index. 3.2 Statement of the reliability problem Here, we take into account all the reliability constraints at once by using a global constraint that is obtained employing the maximum operator. In the case that all the constraints given by Eq. (1) are respect, the following condition is also respected:

R.H. Lopez, A.J. Torii, L.F.F. Miguel, J.E. Souza de Cursi Uncertainties 2014

maxG (d, X)  0 ,

(5)

where we assume that the vector G is composed by all the limit state functions Gk of the problem. Instead of evaluating each probabilistic constraint, we assume that the limit state function of the problem is the one given by Eq.(5). That is, all limit state functions are considered at once by taking the maximum value among them. This new limit state function is called here global limit state function in order to put in evidence that we are considering a set of local limit state functions at once. This limit state function can be written as

G global (d, X)  maxG (d, X)  0 ,

(6)

where Gglobal is the global limit state function in the physical space. In the normalized space, we have

H global (d, U )  Gglobal (d, T 1 ( X))  maxG (d, T 1 ( X))  0

(7)

and the resulting reliability analysis problem is stated as: for a given d Find Minimise Subject to

u * (d) u

(8)

H global (d, U)  0 .

Note that this may not be an efficient approach when the FORM is applied directly to the problem, since one would have difficulties in evaluating the gradient of H global , because of the maximum operator. However, we take advantage here of the use of the adaptive response surface approach since it does not require the evaluation of the gradient of the limit state function, as discussed later. 3.3 The adaptive response surface approach The main idea behind the adaptive response surface approach is to make some approximation for the limit state function and then solve the reliability analysis problem using this approximation (Torii and Lopez, 2012). The approximation is then iteratively refined and the procedure repeated until convergence is achieved. Here, we use a linear response surface of the form n ~ H global (u)  H global (u)  a   bi u i ,

(9)

i 1

~

where H global is the approximation of H global , ui are the variables of the problem, n is the number of random variables and a and bi are coefficients to be determined. Once the closed form approximation for the limit state function from Eq. (9) is built, the reliability analysis problem can be solved efficiently with FORM algorithms. According to Eq. (9), we note that for n random variables the number of coefficients to be determined is n+1. Consequently, in order to uniquely define these coefficients, we must evaluate H global at least in n+1 different

~ (i = 1, 2,…, n+1). In order to generate the points u ~ , we build the approximation sampling points, denoted here as u i i

~ . The sampling points are then obtained as centred at some chosen point u 0 ~ u ~ u 1 0 u ~ u ~  .e 0 1  2 ~ ~  u3  u0   .e 2 ,  ~ ~   .e u n 1  u 0 n

(10)

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling June 23th to June 27th, 2014, Rouen, France

where ∆ is a scalar and ei are canonical basis vectors with all its components equal to zero but component i, which is equal to 1. The coefficients a, bi from Eq. (9) can be found by imposing the conditions

~ H global (u i )  H global (u i )

(i  1,2,..., n  1) ,

(11)

which results in a system of n+1linear equations with unknowns a, b1, b2,…,bn. The adaptive response surface approach is made using the following procedure. At the first iteration, we choose

~

(0)

some initial value for ∆ as used in Eq. (10) and denote it as ∆ (0). We then build an approximation H global centred at the

~ (0) ~ ( 0)  0 . The reliability analysis problem is solved for H origin of the normalized space u 0 global using FORM, thus ~*( 0 ) . The value ∆ is then reduced by the update rule giving the MPP at the first iteration u k 1   k  ,

(12)

where k is the iteration number and  k 1

With the updated value 

  0,1 .

~

( k 1)

, another approximation H global is built, but now centred at the MPP from the last

~ . This problem is solved thus giving an updated MPP u ~ iteration u . This procedure is repeated until some convergence criterion is met and the last MPP found u* is taken as the solution to the problem. The convergence criterion can be checked on the change of the design vector, on the change of the reliability index, or in both quantities. More details on this adaptive response surface approach are described by Torii and Lopez (2012). *( k 1)

*( k )

4 FIREFLY ALGORITHM The Firefly Algorithm (FA) is a very recent heuristic optimization algorithm developed by Yang (2009) and is inspired by the flashing behaviour of fireflies. According to Yang (2009), FA optimization has three idealised rules. (a) All fireflies are unisex, so that one firefly is attracted to other fireflies regardless of their sex. (b) Attractiveness is proportional to brightness, so for any two flashing fireflies, the less bright firefly will move towards the brighter firefly. Both attractiveness and brightness decrease as the distance between fireflies increases. If there is no firefly brighter than a particular firefly, that firefly will move randomly. (c) The brightness of a firefly is affected or determined by the landscape of the objective function. Based on these three rules, the basic steps of the FA can be summarised as the pseudo-code shown in Figure 1. There are two essential components to FA: the variation of light intensity and the formulation of attractiveness. The latter is assumed to be determined by the brightness of the firefly, which in turn is related to the objective function of the problem under study.

begin

Objective function w(d), d = (d1, ..., dm+q’)T Generate initial population of fireflies di (i = 1, 2, ..., n) Light intensity Ii at di is determined by w(di) Define light absorption coefficient  while (t < tmax) for i = 1 : n all n fireflies for j = 1 : d loop over all d dimensions if (Ii < Ij), Move firefly i towards j; end if Vary attractiveness with distance r via exp[− r] Evaluate new solutions and update light intensity end for j end for i Rank the fireflies and find the current global best end while Post-process results and visualization

end Figure 1: Pseudo-code of the Firefly algorithm (adapted from Yang (2009)).

R.H. Lopez, A.J. Torii, L.F.F. Miguel, J.E. Souza de Cursi Uncertainties 2014

As light intensity and attractiveness decrease and the distance from the source increases, the variation of light intensity and attractiveness should be a monotonically decreasing function. For example, the light intensity can be:

I  rij   I 0 e

  rij 2

(13)

in which the light absorption coefficient



is a parameter of the FA and rij, is the distance between fireflies i and j at

d i and d j , respectively, which can be defined as the Cartesian distance rij  d i  d j . Because a firefly’s attractiveness is proportional to the light intensity seen by other fireflies, it can be defined by:

 rij    0 e

  rij2

(14)

in which  0 is the attractiveness at r = 0. Finally, the probability of a firefly i being attracted to another, more attractive (brighter) firefly j is determined by:

d i   0 e

  rij2

d

t j



 d ti  ε i , d ti 1  d ti  d i ,

(15)

where t is the generation number, ε i is a random vector (e.g., the standard Gaussian random vector in which the mean

is 0 and the standard deviation is 1) and  is the randomisation parameter. The first term on the right-hand side of Eq. (15) represents the attraction between the fireflies and the second term is the random movement. In other words, Eq. (15) shows that a firefly will be attracted to brighter or more attractive fireflies and also move randomly. Eq. (15) indicates that the user must set parameters  0 ,  ,  and the distribution of ε i to apply the FA, and also shows that

there are two limit cases when  is small or large. a) If  approaches zero, the attractive and brightness are constants, and consequently, a firefly can be seen by all other fireflies. In this case, the FA reverts to the PSO. b) f  approaches infinity, the attractiveness and brightness approach zero, and all fireflies are short-sighted or fly in a foggy environment, moving randomly. In this case, the FA reverts to the pure random search algorithm. Hence, the FA generally corresponds to the situation falling between these two limit cases. The stopping criterion employed in this paper is the maximum number of iterations itmax. One crucial aspect in the application of heuristic algorithms is the constraint handling. Here, we propose the following multiplicative penalty scheme to handle the probability constraints:

J d   wd 1  Pend  ,

(16)

where J is the new objective function to be minimised and Pen is directly proportional to the level of unfeasibility of the design d . This scheme employs different penalizations if the design fails in the deterministic case or not. First, before running the reliability analysis, the current design is checked in the deterministic case, i.e. the finite element code is run using the mean value of all the random variables of the problem. If the design fails in the deterministic case, the following penalization is employed in the construction of the objective function of the problem:

Pent d   G global (d, X)   t ,

(17)

 

in which  is a positive constant. Thus, if we set   0,1 at the first few generations, the designs that violate the deterministic constraints are not highly penalized and it is possible to maintain their characteristics in the pool of

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling June 23th to June 27th, 2014, Rouen, France

designs. It is interesting to do so in order to keep in the pool low area bars, for instance. However, if the design is feasible in the deterministic check, the reliability analysis code is called in order to evaluate  global d  . If the reliability constraint is not fulfilled, Pt is given by:





   

        ,    2

Pent d    global d    t arg et

in which



(18)

,

stands for the absolute value. Finally, if the reliability constraint is fulfilled

J d   wd  .The constraints on the bounds of the nodal coordinates are addressed by a coding approach. These bounds are imposed by not sampling unfeasible designs in the computer code. Thus, the FA solves the following unconstrained optimization problem: Find Minimise Subject to

d J d 

a j  ,

(19)

j  1,..., m

cimin  ci  cimax , i  1,..., q'. The next section contains a numerical analysis to demonstrate the effectiveness of the proposed methodology for solving the reliability based shape and size optimization of trusses. 5 NUMERICAL ANALYSIS In this section, we adapted three benchmark examples that have been analysed in the deterministic optimization of truss structures. Due to the stochastic nature of the FA, the final result can vary depending on the seed used for the random number generation. Yet there is no established statistical benchmark criterion in the literature to evaluate the performance of metaheuristics in size and shape optimization of trusses, especially in the case of RBDO. With the goal of providing a statistical basis for further comparison, this paper presents the results of over 50 runs for each example. In this way, the average values and coefficients of variation are presented along with the optimal results. The problems are presented in increasing order of complexity, and the following set of parameters are used in all examples:   0.1 ,

 0  1 ,   1 ,   0.5

and a value of ε i that follows a uniform distribution between -0.5 and 0.5. The parameters

used in the adaptive response surface approach were ∆ (0) = 1 and λ = 0.25. 5.1 Ten bar truss The ground structure for this example is presented in Figure 2. Note that the structure is subject to two load cases, given by F1 and F2. The structure has total length equal to 720 cm and total height equal to 360 cm. 5

3

1 7

9 5

6

8

10

3

6

1

2

4

4

F2 = 100kN

2

F1 = 100kN

Figure 2: 2D two load cases ten-bar truss example. The moment of inertia of the bars (used to check the buckling constraints) are obtained from the cross sectional areas by

R.H. Lopez, A.J. Torii, L.F.F. Miguel, J.E. Souza de Cursi Uncertainties 2014

I  0.192a 2  1.261a  6.781 .

(20)

The random variables of the problem are presented in Table 1. For convenience, load uncertainties are modelled by uncertain load factors, which are multiplied by the applied forces F1 and F2. The maximum allowable displacement is δmax = 2.0 cm and the weight density is 78.5 kN/m3. The mean value of the cross-sectional areas are chosen from a set  =( 5.09, 6.91, 9.03, 7.53, 8.70, 11.0, 7.34, 8.73, 10.10, 11.50, 12.80, 14.10, 16.60, 9.35, 12.30, 15.10, 10.60, 12.20, 13.90, 15.50, 17.10, 18.70, 11.80, 13.70, 15.50, 19.20, 22.70,23.20, 27.50, 29.90, 34.80, 43 37.20, 46.10, 55.40, 61.90, 69.10, 76.40) cm2. Table 1: Random variables of 2D two load cases ten-bar truss example Variables Cross sectional areas Elastic modulus Yielding stress (tension and compression) Node coordinates (except supports) Load factors for F1 and F2

Distribution Lognormal Lognormal Lognormal Normal Lognormal

E[X] 210 GPa 210 MPa 1.0

σ[X] 0.05E[A] cm2 10.5 GPa 10 MPa 2.0 cm 0.2

Here, the study is performed to size and shape optimization of the truss, and the target reliability index is



 3.00 . The design variables are the mean value of the area of all the bars and the mean value of the vertical coordinates of nodes 1, 2, 3 and 4, i.e. d  a1 ,  , a10 , y1 ,  , y 4  . Considering the origin of the coordinate system target

the position of node 6, the side constraints for these design variables are 15 cm ≤ y1 ≤ 45 cm, -10 cm ≤ y2 ≤ 10 cm, 15 cm ≤ y3 ≤ 45 cm, -10 cm ≤ y4 ≤ 10 cm. Because the nodal coordinates are continuous and the cross-sectional areas are taken from a set of 38 discrete variables, this problem is a mixed-variable optimization problem in that it deals simultaneously with integer and continuous design variables. The present study uses n = 20 fireflies and tmax = 3000 iterations, resulting in 60000 reliability analysis pursued. The best result over 50 runs achieved by the FA weights 4.703 kN as observed in Table 2 and it is illustrated in Fig. 3. The feasibility of this design was checked using a standard FORM algorithm, which provided the same reliability index. The mean value of the weight obtained with the FA optimization scheme over the 50 runs is equal to 4.929 kN and the coefficient of variation is 2.4 %. The convergence history of a typical run is shown in Fig. 4. We also measure the average number of calls of the finite element code over the 50 runs, which was equal to 9.52x106. It is important to point out that the individual evaluation of all the reliability constraints of Eq.(1) using a standard FORM procedure may present convergence issues and high computational cost, especially in the case of constraints that are too feasible. In this example, we tried to solve the optimization problem using a standard FORM procedure, however several designs diverged due to the cause related above and the computational cost was unviable. Thus, the use of the global constraint and the response surface method not only drastically reduces the computational cost of the reliability analysis, but also avoids one of the main causes of convergence problems of FORM.

Figure 3: Best result for the size and shape optimization of the 2D two load cases ten-bar truss example.

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling June 23th to June 27th, 2014, Rouen, France

Table 2: Optimum size and shape solution for the 2D two load cases ten-bar truss example.

d* a1(cm2) a2(cm 2) a3(cm 2) a4(cm 2) a5(cm 2) a6(cm 2) a7(cm 2) a8(cm 2) a9(cm 2) a10(cm 2) y1(cm)

19.2 7.34 29.9 19.2 7.34 7.34 7.53 29.9 7.53 22.7 150.0

y2 (cm)

71.5

y3 (cm)

297.0

y4 (cm)

15.1

Weight (kN)

4.70



3.700

Figure 4: Convergence history for the size and shape optimization of the 2D two load cases ten-bar truss example. In this example, we tried to solve the optimization problem using a standard FORM procedure, however several designs diverged due to the cause related above and the computational cost was unviable. Thus, the use of the global constraint and the response surface method not only drastically reduces the computational cost of the reliability analysis, but also avoids one of the main causes of convergence problems of FORM. 5.2 2D three load cases tower example The ground structure for this example is presented in Figure 5. Note that the structure is subject to three load cases. The structure has base width equal to 2000 cm, total height equal to 4000 cm. The loads are F1  100kN and

F2  25kN . 5 16 4

F1

8

15

4

10

21

9

20 14

3

7

F1

F1

F1

F1

F1

F2

F2

F2

F2

F2

F2

F2

F2

13 3

8

19 12

2

6

11 2

7

18 10

1 1

5

9 17

a)

6

Load case I Load case II b) c) Figure 5: 2D three load cases tower example

Load case III d)

The moment of inertia of the bars (used to check the buckling constraints) are obtained from the bar areas by

R.H. Lopez, A.J. Torii, L.F.F. Miguel, J.E. Souza de Cursi Uncertainties 2014

I  0.36a 2 .

(21)

The random variables of the problem are presented in Table 3. All random variables have lognormal distributions. For convenience, loads uncertainties are modelled by uncertain load factors, which are multiplied by the applied forces F1 and F2. The maximum allowable displacement is δmax = 1.0 cm and the weight density is 78.5 kN/m3. The mean value of the cross-sectional areas are chosen from a set  =(10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380) cm2. Table 3: Random variables of 2D three load case tower example Variables

E[X]

σ[X]

Cross sectional areas

-

0.05E[A] cm2

Elastic Modulus

210 GPa

21 GPa

Load factor for F1

1.0

0.1

Load factor for F2

1.0

0.3

Node coordinates (except supports)

-

2.0 cm

Yielding stress (tension and compression)

210 MPa

21 MPa

The study is performed to size and shape optimization of the truss, and the target reliability index is

  3.000 . The design variables are the mean value of the area of all the bars and the mean value of the horizontal coordinates of nodes 2, 3, 4, 5, 7, 8, 9 and 10. Symmetry is imposed in the problem so that the number of nodal coordinates considered as design variables is reduced to four and the number of areas is reduced to thirteen. These nodal coordinates considered as design variables are allowed to move 800 cm in each side, considering their initial position as shown in Figure 5. The origin of the coordinate system is located at node 1. It should be pointed out that it also is a mixed-variable optimization problem. The present study uses n = 20 fireflies and itmax = 3000, resulting in 60000 RAP. The best result over the 50 runs achieved by the FA weights 104.46 kN, is illustrated in Figure 6 and detailed in Table 4. The feasibility of this design was checked using a standard FORM algorithm, which provided the same reliability index. The mean value of the tower weight obtained with the FA optimization scheme over these 50 runs is equal to 109.70 kN and the coefficient of variation is 5.54 %. A typical convergence history of this problem is shown in Figure 7. We also measure the average number of calls of the finite element code over the 50 runs, which is equal to 12.9x106. In this example, we also attempted to solve the RBDO problem using a standard FORM procedure. However, several designs diverged due to the cause related above and the computational cost was unviable. Thus, once again, the proposed global RBDO methodology successfully optimized the shape and size of the truss structure under analysis. t arg et

6 CONCLUDING REMARKS This paper presented a methodology to perform size and shape global RBDO of truss structures. This methodology is comprised by the use of a global constraint and the response surface method to deal with the reliability analysis together with the FA to carry out the global structural optimization. The former is responsible for the reduction of the computational cost required in the evaluation of the probabilistic constraints. The latter overcomes the issues related to the non-convexity and mixed-variables of the optimization problem. The examples showed that the proposed methodology is able to perform global RBDO of the size and shape of trusses efficiently. All the optima found were checked using a classical FORM approach, validating the results provided by the response surface method. Also, the proposed methodology not only drastically reduces the computational cost of the global RBDO, but also decreases the convergence issues of the FORM. The authors noticed that the literature lacks established statistical benchmark criteria to evaluate the performance of metaheuristics in size and shape optimization of trusses, especially in the case of RBDO. Hence, this paper presented the results of the mean value and coefficient of variation over 50 runs for each example providing a statistical basis for further comparison in future works. It should be pointed out that although the proposed methodology reduced the computational cost of the reliability analysis and made it possible to couple it with a heuristic algorithm, all the other drawbacks inherent to the FORM still prevail. For example, the accuracy in the approximation of the probability of failure of non-linear limit state functions and non-normal random variables. The coupling of more precise reliability methods, such as full characterization

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling June 23th to June 27th, 2014, Rouen, France

methods (Lopez, Miguel and de Cursi, 2013; Lopez et al., 2014) and simulation methods (Hurtado and Ramirez, 2013), with global optimization algorithms is of high importance and should be subject of future researches. Table 4: Best design of the 2D three load case tower example. Design variables

FA

a1, a5(cm2) a2, a6(cm 2) a3, a7(cm 2) a4, a8(cm 2) a9, a10 (cm 2) a11, a12 (cm 2) a13, a14 (cm 2) a15, a16(cm 2) a17(cm 2) a18(cm 2) a19(cm 2) a20(cm 2) a21(cm 2) x2 (cm) x3 (cm) x4 (cm) x5 (cm) Weight (kN)

100 80 60 40 70 50 40 50 10 10 10 10 10 192.6344 389.8629 573.9016 746.5152 104.46

 global d 

3.000

Figure 6: Best result for the size and shape optimization of the 2D three load case tower example.

R.H. Lopez, A.J. Torii, L.F.F. Miguel, J.E. Souza de Cursi Uncertainties 2014

Figure 7: Convergence history for the size and shape optimization of the 2D three load case tower example. 7 ACKNOWLEDGEMENTS The authors acknowledge the financial support of the Brazilian agencies CNPq and CAPES. 8 REFERENCES Fister, I., Yang, X.S., Fister, D., 2014, “Firefly algorithm: A brief review of the expanding literature”, Studies in Computational Intelligence, 516:347 – 360. Gandomi, A.H., Yang, X.S., Alavi, A.H., 2011, “Mixed variable structural optimization using Firefly Algorithm”, Computers and Structures, 89:2325 – 2336. Hurtado, J.E. and Ramirez, J., 2013, “The estimation of failure probabilities as a false optimization problem”, Structural Safety, 45:1-9. Lamberti, L. and Pappalettere, C., 2011, "Metaheuristic Design Optimization of Skeletal Structures: A Review", Computational Technology Reviews, 4:1-32. Lopez, R.H., and Beck, A.T., 2012, “RBDO methods based on FORM: a review”, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 34(4):506–514. Lopez, R.H., Miguel, L.F.F., Belo, I.M., Souza de Cursi, J.E., 2014, “Advantages of employing a full characterization method over FORM in the reliability analysis of laminated composite plates”, Composite Structures, 109: 635-642. Lopez, R.H., Miguel, L.F.F., Souza de Cursi, J.E., 2013, “Uncertainty quantification for algebraic systems of equations”, Computers and Structures, 128:189-202. Miguel, L.F.F., Lopez, R.H., Fadel Miguel, L.F., 2013, “Multimodal size, shape, and topology optimisation of truss structures using the Firefly algorithm”, Advances in Engineering Software, 56:23-37. Murotsu, Y., and Shao, S., 1990, “Optimum shape design of truss structures based on reliability”, Structural Optimization, 2:65-76. Nakib, R., 1997, “Deterministic and reliability-based optimization of truss bridges”, Computers & Structures, 65(5):767-775. Stocki, R., Kolanek, K., Jendo, S., Kleiber, M., 2001, “Study on discrete optimization techniques in reliability-based optimization of truss structures”, Computers & Structures, 79:2235-2247. Thampan, C.K., and Krishnamoorthy, C.S., 2001, “System reliability-based configuration optimization of trusses”, J Struct Engrg 27(8):947-956. Torii, A.J., Lopez, R.H., Biondini, F., 2012, “An approach to reliability-based shape and topology optimization of truss structures”, Engineering Optimization, 44:37-53. Torii, A.J., Lopez, R.H., Luersen, M., 2011, “A local-restart coupled strategy for simultaneous sizing and geometry truss optimization”, Latin American Journal of Solids and Structures, 8:335-349. Torii, A.J., Lopez, R.H., 2012, “Reliability Analysis of Water Distribution Networks Using the Adaptive Response Surface Approach”, Journal of Hydralical Engineering, 138:227-236. Wiener, N., 1938, “The homogeneous chaos”, American Journal of Mathmatics, 60(23–26):897–936. Yang, X.-S., 2009, “Firefly algorithms for multimodal optimization”, in: Stochastic Algorithms: Foundations and Applications, Lecture Notes in Computer Sciences, 5792:169-178. RESPONSIBILITY NOTICE The following text, properly adapted to the number of authors, must be included in the last section of the paper: The author(s) is (are) the only responsible for the printed material included in this paper.