a generalization of the sequential optimization and ...

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling July 7th to July 11th, 2014, Rouen, France

A GENERALIZATION OF THE SEQUENTIAL OPTIMIZATION AND RELIABILITY ASSESSMENT METHOD FOR RBDO PROBLEMS Andre J. Torii1 , Rafael H. Lopez2 , Leandro F.F. Miguel2 1

2

Center for Optimization and Reliability (CORE), Department of Scientific Computing, Universidade Federal da Paraiba, [email protected] Center for Optimization and Reliability (CORE), Department of Civil Engineering, Universidade Federal de Santa Catarina, [email protected], [email protected]

Abstract. In this paper, a novel approach for Reliability Based Design Optimization (RBDO) is presented. In this approach, we employ the concept of shifting vectors, originally developed in the context of the Sequential Optimization and Reliability Assessment (SORA). However, shifting vectors are found and updated based on a novel strategy. The resulting framework is able to use any technique for reliability analysis, such as Monte Carlo Simulation, among others. The main advantage of the approach is that accuracy of reliability analysis depend on the technique used, and thus can be improved. Several numerical examples are presented in order to highlight the advantages of the proposed method. Keywords. Reliability based design optimization, shift vector, reliability accuracy, sequential optimization and reliability assessment 1

INTRODUCTION

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 of the designed engineering system are converted in design constraints, and uncertainty is addresses indirectly, by means of safety coefficients and conservative assumptions. This approach is inherited from design through design codes, which is essentially non-optimal (since approximations are always conservative). On the other hand, a deterministic optimal design has, naturally, more failure modes designed against the limit. Hence, deterministic optimal designs are potentially less safe than their non-optimal counterparts. In this context, reliability-based design optimization (RBDO) has emerged as an alternative to model the safety-under-uncertainty part of the problem. In RBDO, one usually looks for the minimization of some objective function involving material or manufacturing costs subject to constraints on failure probabilities. The RBDO formulation explicitly addresses the uncertainties affecting system performance and ensures that a minimum specified level of safety is maintained by the optimum design. Since design optimization is an iterative process, the failure probability of each constraint must be evaluated at each iteration. It may be accomplished by different approaches, for instance, simulations methods (e.g. Monte Carlos simulation (MCS), Importance Sampling), the first and second order reliability methods (FORM and SORM, respectively) and full characterization methods (e.g. polynomial chaos based approaches). The choice among these approaches is usually a ratio between accuracy and computational cost. In the last two decades, the most employed method for the reliability analysis in RBDO problems was the FORM. Since it is an optimization procedure itself, the solution of the RBDO is a double-loop strategy, i.e. structural optimization and reliability analysis, which leads to high computational costs. To reduce the computational burden of the RBDO, several researchers focused on the decoupling of the structural optimization and the reliability analysis. These methods have been compared in several benchmark problems by different researchers (Aoues and Chateauneuf, 2010; Lopez and Beck, 2012) and the Sequential Optimization and Reliability Assessment (SORA) (Du and Chen, 2004) was one of the most promising techniques. Despite of all these developments, the issue of accuracy remains an open question, due to the limitations of the FORM. It only provides the exact value of the probability of failure if the limit state function is linear and all the random variables are independent and normal, which is a rather limited case in practice. For this reason, some authors started to pursued the development of decoupling techniques which allow for the utilization of any reliability analysis method, such as Zou and Mahadevan (2006). In this context, we propose an approach for RBDO that allows any reliability analysis technique to be employed. The reliability and the optimization steps are decoupled by using a strategy based on shift vectors of the SORA method, which are found as to enforce the required reliability level. Also, the proposed approach does not require gradient information of the probabilistic constraints due to the use of a least squares approximation. The resulting algorithm is able to solve problems that simply cannot be addressed by RBDO techniques based on the FORM. It should be remarked that the proposed approach was not developed to be more computationally efficient than FORM based strategies such as SORA. The main goal of the proposed approach is to improve accuracy of the reliability analysis step. By allowing any reliability analysis technique to be used, the proposed approach broadly extends the range of application of RBDO. This paper is organized as follows: Section 2 presents the formulation of the general RBDO problem. The concepts of the SORA method are detailed in Section 3 and the proposed RBDO approach is presented in Section 4. Several numerical

A.J. Torii, R.H. Lopez, L.F.F. Miguel A generalization of SORA for RBDO problems

examples are solved in Section 5. Finally, the main conclusions drew from this work are presented in Section 6. 2

THE RBDO PROBLEM A RBDO problem can be stated as Find: d ∈ Rv

(1)

that min f (d)

(2)

subject to Pf = P (G(d, X) > 0) ≤ Pt ,

(3)

where d is the design vector, f is the objective function to be minimized, G is a nonlinear limit state function, the vector X ∈ Rn is comprised by the random variables of the system under analysis and P(·) is the probability of a given event occurring. Here, we assume the case of a single inequality constraint, but the results are extended for other cases later on this paper. In this problem, we search for the design vector d that minimizes the objective function subject to a probabilistic constraint. The probabilistic constraint from Eq. (3) actually states that the failure probability Pf of the original constraint being violated must be smaller than a target failure probability Pt . In RBDO, the constraint from Eq. (3) is frequently rewritten in terms of reliability indexes in order to avoid manipulation of very small numbers. In this case, the constraint can be rewritten as β ≥ βt

(4)

where β is the reliability index related to the limit state function G and βt is a target reliability index, which are defined respectively as β = Φ−1 (1 − Pf ) and

βt = Φ−1 (1 − Pt ),

(5)

where Φ−1 is the inverse cumulated density function of a standard normal random variable. 3

SEQUENTIAL OPTIMIZATION AND RELIABILITY ASSESSMENT

In this section, we first present the basic concepts of SORA and then we present a different view of this method, which we use to developed the proposed approach. 3.1

A brief review of SORA

Here, only the basic concepts of SORA are presented, for a detail explanation the reader is referred to (Du and Chen, 2004). SORA is based on the strategy of serial single loops decoupling the optimization and the reliability analysis in RBDO. At each iteration of the method, the reliability analysis is only conducted after convergence of several loops of the optimization problem. The key concept of the method is to shift the boundaries of the violated equivalent deterministic constraints to the feasible direction based on the reliability information obtained in the previous cycle, which makes the reliability constraints improve progressively. Figure 1 shows an example of the SORA boundary shifting procedure. For the sake of simplicity, we assume in this figure that we have two random variables whose mean values are the design variables of the RBDO problem. The equivalent deterministic constraint at iteration k and the probabilistic constraint that has to be fulfilled are represented. Based on the percentile information obtained through the reliability analysis, the shifting value s is found and the equivalent deterministic constraint is shifted towards the probabilistic constraint (dashed line). The reliability constraint is fulfilled when the dashed line coincides with the probabilistic constraint. The proposed approach uses the concept of a shift vector in order to solve the RBDO problem, originally developed in the context of SORA. In this case, we assume that the original RBDO problem can be substituted by the approximate deterministic optimization problem Find: d ∈ Rv

(6)

that min f (d)

(7)

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling July 7th to July 11th, 2014, Rouen, France

Figure 1: Shifting the boundaries of the violated deterministic constraints in SORA

subject to g(d, µX + s) ≤ 0,

(8)

where g is the equivalent deterministic constraint of G, µX is the vector comprised by the mean values of the random parameters X, and s is a shift vector, which is defined as to ensure that satisfaction of the constraint from Eq. (8) enforces the satisfaction of the constraint from Eq. (3). One of the main advantage of SORA is that it requires the solution of the deterministic optimization problem given by Eqs. (6)-(7). This fact simplifies the computational implementation of the approach, since existing deterministic optimization algorithms can be easily adapted. However, in SORA, the shift vector is built using the concept of the Hasofer-Lind reliability index, and consequently it may be inaccurate in several cases. In fact, SORA is able to reduce the computational effort of the RBDO problem, but it is not able to improve accuracy of the reliability analysis. In this case, the accuracy from the probabilistic point of view is entirely dominated by the accuracy of the FORM. 3.2

An alternative view of SORA

Here, we propose an alternative view of the use of shift vectors for RBDO. In this case, we note that a given shift vector s will ultimately result in a given optimum design vector d after the solution of the minimization problem from (6)-(8). This allows us to build a relation between s and d as follows: O : s → d,

(9)

which is given by solving the deterministic optimization problem from (6)-(8) for a given shift vector s. On the other hand, a given design vector d results in a given failure probability Pf , thus defining the relation R : d → Pf ,

(10)

which is given by reliability analysis using the design vector d. Combining Eqs. (9) and (10) we can build a composite relation J = R · O(s) : s → Pf ,

(11)

which is a functional relating shift vectors to failure probabilities. That is, we assume that every shift vector can be associated to a given failure probability, thus defining a functional. Of course this relation is defined implicitly by means of an optimization problem and a reliability analysis procedure, but this is not important for now. We note that the relation from Eq. (11) can be written in terms of the reliability index as J : s → β,

(12)

A.J. Torii, R.H. Lopez, L.F.F. Miguel A generalization of SORA for RBDO problems

where we assume that the reliability index is as defined in Eq. (5). This leads to the conclusion that the reliability index of the optimum solution depends on the shift vector used and write we β (s). The main idea is to find the shift vector s∗ that gives Pf = Pt . When this is accomplished, the optimum design vector ∗ d that gives the desired reliability can be found from Eqs. (6)-(8) using s = s∗ . We note that this is precisely the idea of SORA, where the Hasofer-Lind reliability index is used in the reliability analysis step and some more developments are made. 4

PROPOSED APPROACH

This section presents the proposed approach to decouple the structural optimization and the reliability analysis, which allows for the utilization of any reliability analysis method. 4.1

Main idea We tackle the RBDO problem taking the shift vector as the primary variable. The RBDO problem can then be stated

as F IND : s

(13)

that min [βt − β (s)]2

(14)

subject to βt − β (s) ≤ 0.

(15)

Once this problem is solved, we can recover the optimum design vector from Eqs. (6)-(8) using s = s∗ . The constraint from Eq. (15) ensures that the reliability index of the design is higher than the target reliability index. Besides, the objective function from Eq. (14) tries to enforce β = βt . In most cases, the algorithm will actually make β = βt . However, we leave the problem as stated here in order to allow the reliability index to be higher than the target if needed . This is important in problems subject to more than one constraint, since in these cases it is common that the reliability index of only a few constraints match βt , i.e. only a few probabilistic constraints are active. Obviously, it is not easy to solve the problem given by Eqs. (13)-(15). In order to evaluate β , it is necessary to solve the optimization problem that defines Eq. (9) and to make the reliability analysis that defines Eq. (12). Thus, β (s) is defined implicitly. However, here we build a least squares approximation for the functional J, which allows the efficient solution of the problem. Such an approximation is detailed in the next section. 4.2

Least Squares approximation for J

Instead of solving the problem given by Eqs. (13)-(15) directly, the problem can be solved for some approximation for J. Here, we use linear approximations n

J(s) ≈ JLS (s) = a0 + ∑ [bi si ] ,

(16)

i=1

and quadratic approximations n n   J(s) ≈ JLS (s) = a0 + ∑ [bi si ] + ∑ ci s2i , i=1

(17)

i=1

where a0 , bi and ci are coefficients found by Least Squares approximation. We note that J(s) must be evaluated for at least n + 1 different shift vectors if a linear approximation is to be made. In the case of the quadratic approximation, 2n + 1 shift vectors are necessary. This least squares approximation may be viewed as a response surface approximation and for the interest reader details of the implementation may be found in Torii and Lopez (2012). The implementation of the proposed approach is given in the next section. 4.3

Algorithm

The problem from Eqs. (13)-(15) can be solved by the following steps. First, since we do not know J(s) explicitly, it is interesting to build some approximation for it. A Least Square approximation can be used for this purpose. We then generate a set of different shift vectors S = {s1 , s2 , ..., sm }

(18)

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling July 7th to July 11th, 2014, Rouen, France

and solve the deterministic optimization problem given by Eqs. (6)-(8) using s = s1 , s2 , ..., sm . For example, if the designer decides to start with a linear approximation, m must be at least equal to n + 1. For each shift vector, we find a design vector d = d1 , d2 , ..., dm . We then pursue m reliability analyses using d1 , d2 , ..., dm , resulting in the reliability indexes β (s1 ), β (s2 ), ..., β (sm ), respectively. We now use the approximation JLS in the place of J in the problem from Eqs. (13)-(15) and find an approximate solution for s. This can be solved using any constrained minimization algorithm. Since JLS is an analytical and simple function, the computational effort required is low. This gives a new shift vector that can be included in the set S from Eq. (18). With this new shift vector, we can find a new design vector d and evaluate the new reliability index. With this information, the approximation JLS can be rebuilt. The procedure is repeated until convergence is achieved. The proposed algorithm can then be stated as: 1. Make k = 0 and generate m = n + 1 shift vectors to build the set S(k) = {s1 , s2 , ..., sm }, where k is the iteration number; 2. For each shift vector from S, solve the optimization problem of Eqs. (6)-(8) and obtain the corresponding design vectors {d1 , d2 , ..., dm }; 3. For each design vector, pursue the reliability analysis using the desired method, evaluating the reliability indexes {β (s1 ), β (s2 ), ..., β (sm )}; 4. Build a Least Squares approximation JLS (s) ≈ J(s) relating the shift vectors to the reliability indexes; 5. Solve the minimization problem from Eqs. (13)-(15) and find a new shift vector sk ; 6. For the new shift vector solve the optimization problem of Eqs. (6)-(8) and obtain the corresponding design vector dk ; 7. For the new design vector dk pursue the reliability analysis, finding the corresponding reliability index β (sk ); 8. Check convergence: if converged, then stop, otherwise update the shift vector set as Sk+1 = {s1 , s2 , ..., sm , sk }, make k = k + 1 and return to step (4). The main goal of the proposed approach is to allow the reliability analysis to be made using any strategy. The algorithm requires the evaluation of the reliability index β (or Pf ), but there is no requirement on which method to use, as it may be seen in the description of the proposed method above. In the numerical analysis section, we employ the crude MCS and FORM, but any other method can be used. Accuracy of the results (from the probabilistic point of view) can be improved by using more accurate reliability methods. We note that most of the other RBDO strategies are not flexible in this part. In fact, the SORA can only me made assuming the FORM approximation, which is known to be very poor in some cases. We also note that the proposed approach solves the deterministic optimization problem given by Eqs. (6)-(7), as occurs in SORA. This simplifies the computational implementation of the approach, since existing deterministic optimization algorithms can be easily adapted. 5

NUMERICAL EXAMPLES

It should be noted that the optimization problem given by Eqs. (6)-(8) for a fixed s can be solved using any deterministic optimization algorithm. The same holds for the problem given by Eqs. (13)-(15) solved for JLS . In this section, a Sequential Quadratic Programming (SQP) is employed in both cases. All examples were solved in MATLAB using a machine with 2.5GHz processor and 2Gb RAM. In the following examples we assume that a negative reliability index indicates that the failure probability is higher than 50%. Besides, when no failures are captured by MCS (i.e. the reliability index is very big but unknown) we write β = ∞. 5.1

Example 1: three random variables in a highly nonlinear limit state function The first example studied here is given by F IND : d = (µX1 , µX2 , µX3 )

(19)

that min f (d) = µX1 + µX2 + µX3

(20)

subject to box constraints 0 ≤ di ≤ 10(i = 1, 2, 3)

(21)

A.J. Torii, R.H. Lopez, L.F.F. Miguel A generalization of SORA for RBDO problems

and limit state functions defined by G1 (d, X) = 1 − X1 ,

(22)

G2 (d, X) = 2 − X2 ,

(23)

G3 (d, X) = 3 − X3 .

(24)

In this problem, we set βt = 2.0 and X as independent and Gaussian random variables whose standard deviation is 0.3. The initial solution is taken as d1 = d2 = d3 = 1.0. First, we solve this example using SORA and the proposed approach employing FORM for reliability analysis. The initial solution is taken as d1 = d2 = d3 = 1.0. The result for both methods is d = (1.6000, 2.6000, 3.6000). This optimum design was then verified with a crude MCS (ns = 500, 000), which led to β1 = 2.0019, β2 = 2.0001, β3 = 1.9952, agreeing with the target reliability index. It should be highlighted that, with the constraints defined in Eqs.(22)-(24), this is a very simple problem with linear limit state functions and Gaussian random variables, where FORM is very accurate. Now, in order to investigate the effect of a very nonlinear constraint, we solve the same problem for the limit state function G(d, X) = max[G1 , G2 , G3 ].

(25)

We note that this limit state function makes sense if we want to ensure that the three conditions from Eqs. (22)-(24) are ensured simultaneously, that is related to reliability of systems. Besides, problems where a set of conditions must be met simultaneously are very common in practice. In this case, neither SORA nor the proposed approach using FORM for reliability analysis converge, because FORM itself failed to converge. However, we note that the solution of this problem using SORA or FORM would be the same as the one found in the previous case. The reliability index approximation used in FORM and SORA is based on a tangent hyperplane defined on the most probable point of failure, but this hyperplane would coincide with the limit state functions defined previously. In this context, the solution of the RBDO problem using the Hasofer-Lind reliability index would be d = (1.6000, 2.6000, 3.6000). The reliability analysis of this design for the limit state function of Eq. (25) using a crude MCS(ns = 500, 000) leads to β = 1.4993, which is far from the target βt = 2.0. This is purely because the Hasofer-Lind reliability index is a very poor approximation for the problem with the limit state function from Eq. (25). The maximum operator transforms the well behaved linear limit state functions into a highly non linear (with discontinuous derivatives) function. Hence, all problems involving limit state functions defined as the maximum among a set of functions should be taken carefully when FORM based approaches are employed. Finally, it is important to note that the set of limit state functions from Eqs. (22)-(24) is not the same as the more strict limit state function from Eq. (25), an assumption frequently made in practice. We thus apply the proposed approach using a crude MCS for the reliability analysis . The results found are presented in Table 1. The reliability index of the optimum design given by the proposed approach is β = 2.0366, which is close to the target reliability index of this problem. We also note that the solution is different from the one found in the previous case, highlighting the fact that the two problems are indeed different. Table 1: Example 1: Results obtained with proposed approach

Iteration 0 0 0 0 1 2

β -MCS -0.8884 -0.8891 -0.8898 -1.1512 2.2842 2.0366

(x1 , x2 , x3 ) (1.1990,2.0001,2.9999) (0.9999,2.2001,2.9991) (0.9994,2.0000,3.1996) (0.9991,1.9994,2.9993) (1.8033,2.8015,3.7989) (1.7378,2.7359,3.7338)

Approximation Linear Linear

In this example, we show that FORM based approaches may be inappropriate for highly nonlinear limit state functions. We also note that a few very simple linear limit state functions may be turned into a highly nonlinear function when the maximum operator is employed, a situation frequently encountered in practice. In this case, the proposed approach is able to converge if a simulation method is used in the reliability analysis step. 5.2

Example 2: 10 bars truss optimization problem

We study here the RBDO of the truss structure presented in Fig. 2 (Yi and Cheng, 2008). The structure dimensions are taken as b = h = 360in. The magnitudes of the two applied forces are independent lognormal random variables with LN(1.0E5,5.0E3)lb. The yielding stress is a normal random variables with N(2.5E4,e.5e3)Psi. All these random

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling July 7th to July 11th, 2014, Rouen, France

variables are grouped into the random vector X. We take the cross sectional areas as design variables and treat them as deterministic variables. The total volume of the structure is to be minimized. Lower and upper bounds on the design variables are defined as 0.01in2 and 25.0in2 . The initial design is taken as di =10.0in2 . The target reliability index is βt = 3.0.

3

4 10 5

7

6

h

8

9 1

2

P1

b

P2

b

Figure 2: Ground structure composed by 10 bars

In this problem, we include several types of limit state functions. First, we define the allowable displacement limit state function as G1 (d, X) = max(| u(d, X) |) − 4.5,

(26)

where u is vector of nodal displacements. The stress limit state functions are defined as G1+i (d, X) =| σi (d, X) | − fy (i = 1, 2, ..., 10),

(27)

where σi is the stress inside bar i and fy is the yielding stress of the material. The problem was first solved using SORA with all limit state functions from Eqs. (26)-(27). The solution found was d=(11.56, 7.64, 15.28, 0.01, 0.01, 0.01, 10.80, 8.27, 0.01, 10.80)in2 , that is essentially the same found in literature (Yi and Cheng, 2008). In practice, it makes sense to consider that all limit state functions from Eqs. (26)-(27) must be respected at the same time and thus impose G(d, X) =

max

i=1,2,3,...,11

Gi (d, X) ≤ 0.

(28)

This is the same as assuming that failure occurs if any of the two conditions are met: i) any stress constraint is violated, ii) the displacement constraint is violated. However, FORM may give a very poor approximation for the failure probability in this case. If we subject the solution found by SORA to reliability analysis using MCS (ns = 1, 000, 000) and using the limit state function G, we obtain the actual reliability index of the structure as β = 2.7178, which is much smaller than the target. We also note that SORA does not converge if the limit state function G is used instead of the set Gi . However, if SORA was able to converge, the solution would likely be inaccurate from the probabilistic point of view, since the Hasofer-Lind reliability index is a poor approximation in the case of the limit state function G. We then applied the proposed approach for the problem using the limit state function G and making linear approximations for β . Since FORM gives a poor approximation in this case, reliability analysis is made using MCS with ns = 1, 000, 000. The reliability index found is presented in Table 2. The solution found is d=(14.69, 7.36, 14.72, 0.01, 0.01, 0.01, 10.40, 10.36, 0.01, 10.40) The solution presents a reliability index that is very close to the target. This puts in evidence that the problem solved for the limit state function G is somewhat different from the one solved for the limit state functions Gi . We also note that FORM based methods will not be able to obtain an accurate solution for G, since FORM is inaccurate in this case. 5.3

Example 3: problem with discrete random parameter The last example is given by

A.J. Torii, R.H. Lopez, L.F.F. Miguel A generalization of SORA for RBDO problems

Table 2: Example 2: Results obtained with proposed approach

Iteration 0 0 0 0 1 2 3 4 5 6 7 8 9

β (ns = 1, 000, 000) -1.1817 -1.2632 -0.3048 -0.4535 1.8368 -0.2148 7.0345 0.2548 0.3784 2.0790 2.8336 2.9635 2.9833

Approximation Linear Linear Linear Linear Linear Linear Linear Linear Linear

F IND : d = (µX1 , µX2 )

(29)

that min f (d) = 3µX21 + 2µX2


2

(30)

subject to the constraint from Eq. (4) and box constraints 0 ≤ di ≤ 10(i = 1, 2),

(31)

with limit state functions G1 (d, X) = (10 + 5X3 ) − (X12 + 2X22 )

(32)

G2 (d, X) = (1 + X3 ) − X1 .

(33)

We note that X3 is not a design variable, but just a random parameter of the problem. The random variable X1 has normal distribution with standard deviation equal to 0.2µX1 . The random variable X2 has uniform distribution with lower and upper bounds given by [0.5µX2 , 1.5µX2 ]. The random parameter X3 is a discrete random variable with a probability of 0.7 of assuming the value 1.0 and a probability of 0.3 of assuming the value 0.0. The target reliability index is βt = 2.0. We note that the discrete random parameter X3 cannot be treated directly by FORM based approaches. The proposed approach was applied using MCS (ns = 1e7) for the reliability analysis. The results are presented in Table 3, where β1 and β2 are the reliability indexes for the two limit state functions. We note that the algorithm converged for a solution whose reliability indexes are close to the target. Table 3: Example 3: Results obtained with proposed approach

Iteration 0 0 0 0 1 2 3 4

(x1 , x2 ) (2.4000,2.3455) (2.0000,2.7452) (1.8000,2.3196) (2.0000,2.3453) (2.8856,3.6051) (3.0284,3.6685) (3.0284,3.6685) (3.0812,3.6191)

(β1 , β2 ) (0.4779,1.0715) (0.5862,0.3806) (0.0986,-0.0034) (0.2277,0.3806) (1.8729,1.7076) (2.0387,1.8597) (1.8729,1.7076) (2.0347,1.9127)

Approximation Linear Linear Linear Quadratic

This example illustrates the flexibility of the propose approach in dealing with complex situations. In particular, we note that FORM cannot be applied directly for discrete random variables. The proposed approach, on the other hand, is able to tackle the problem directly if MCS is used for the reliability analysis step.

Proceedings of the 2nd International Symposium on Uncertainty Quantification and Stochastic Modeling July 7th to July 11th, 2014, Rouen, France

6

CONCLUSIONS

In this work, we presented an approach for RBDO that allows any reliability analysis technique to be used. The reliability and the optimization steps are decoupled by using a strategy based on shift vectors, which are found as to enforce the required reliability level. The resulting algorithm is able to solve problems that simply could not be addressed by RBDO techniques based on FORM. Besides, as occurs in SORA, the approach requires the solution of a deterministic optimization problem, what simplifies its computational implementation. The proposed approach was not developed to be more computationally efficient than FORM based strategies such as SORA or similar methods. The main goal of the proposed approach is to improve accuracy of the reliability analysis step. By allowing any reliability analysis technique to be used, the proposed approach broadly extends the range of application of RBDO. FORM is known to be inaccurate in several cases of practical interest, what impedes the application of FORMbased RBDO strategies such as SORA. The accuracy of the reliability analysis in the proposed approach, on the other hand, can be improved by using more accurate techniques. Even though FORM may be employed in the reliability analysis step of the proposed method, it makes more sense to use SORA, or other efficient technique when FORM is accurate. However, the proposed method allows the utilization of any reliability analysis method, for example, from the crude MCS to full characterization approaches. This allows the solution of much more complex problems, such as the one including a discrete random parameter presented here. It is also possible to solve problems with highly nonlinear limit state functions, such as the ones defined as being the maximum among a set of functions. Due to its novelty, many aspects of the proposed approach should be further investigated. Here we proposed the solution of the problem by building a Least Squares approximation for the reliability index based on the shift vectors. However, other approaches for solving the proposed approach should be investigated, such as direct solution or some kind of Taylor expansion. 7

ACKNOWLEDGEMENTS The authors acknowledge the financial support of the Brazilian agencies CNPq and CAPES.

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REFERENCES

Aoues, Y., and Chateauneuf, A., 2010, “Benchmark study of numerical methods for reliability-based design optimization”, Structural and Multidisciplinary Optimization, 41(2):277–294. Du, X., and Chen, W., 2004, “Sequential Optimization and Reliability Assessment method for Efficient Probabilistic Design”, ASME: Journal of Mechanical Design, 126:225–233. 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. Torii, A.J., and Lopez, R.H., 2012, “Reliabiliy analysis of water distribution networks using the adaptive response surface approach”, Journal of Hydraulic Engineering, 138(3):277-236. Yi, P., and Cheng, G.D., 2006, “Further study on efficiency of sequential approximate programming strategy for probabilistic structural design optimization”, Structural and Multidisciplinary Optimization, 35(6):509–522. Zou, T., and Mahadevan, S., 2006, “A direct decoupling approach for efficient reliability-based design optimization”, Structural and Multidisciplinary Optimization, 31(3):190–200. RESPONSIBILITY NOTICE The author(s) is (are) the only responsible for the printed material included in this paper.