Reliability-based optimization of stochastic systems using line search

Comput. Methods Appl. Mech. Engrg. 198 (2009) 3915–3924

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Reliability-based optimization of stochastic systems using line search H.A. Jensen a,*, M.A. Valdebenito b, G.I. Schuëller b, D.S. Kusanovic a a b

Department of Civil Engineering, Santa Maria University, Casilla 110-V, Valparaiso, Chile Chair of Engineering Mechanics, University of Innsbruck, Technikerstr. 13, A-6020 Innsbruck, Austria

a r t i c l e

i n f o

Article history: Received 13 February 2009 Received in revised form 19 August 2009 Accepted 20 August 2009 Available online 28 August 2009 Keywords: Gradient-based algorithm High-dimensional reliability problems Non-linear systems Reliability-based optimization Sensitivity analysis Stochastic excitation

a b s t r a c t This contribution presents an approach for solving reliability-based optimization problems involving structural systems under stochastic loading. The associated reliability problems to be solved during the optimization process are high-dimensional (1000 or more random variables). A standard gradientbased algorithm with line search is used in this work. Subset simulation is adopted for the purpose of estimating the corresponding failure probabilities. The gradients of the failure probability functions are estimated by an approach based on the local behavior of the performance functions that define the failure domains. Numerical results show that only a moderate number of reliability estimates has to be performed during the entire design process. Two numerical examples showing the effectiveness of the approach reported herein are presented. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Structural optimization is concerned with achieving an optimal design while satisfying certain constraints. In this regard, the optimal design can be defined as the best feasible design according to a preselected quantitative measure of effectiveness [3,10,11,14]. In most structural engineering applications, response predictions are based on structural models whose parameters are uncertain. This is due to a lack of information about the value of system parameters external to the structure, such as environmental loads (wind loading, water wave excitation, traffic loading, earthquake excitation, etc.) or internal such as material properties, construction defects and system behavior. Under uncertain conditions the field of reliability-based optimization provides a realistic and rational framework for structural optimization which explicitly accounts for the uncertainties [12]. It is noted that due to uncertain conditions, reliability-based optimization formulations are considerably more involved than their deterministic counterpart. Reliability-based optimization requires advanced and efficient tools for structural modeling, reliability analysis and mathematical programming. Modeling and analysis techniques of mechanical systems based on local approximations are well established and sufficiently well documented in the literature [6,45]. On the other hand, several tools for assessing structural reliability have lately experienced a substantial development providing solution to a number of complex problems [5,7,19–21,23,31,38]. In the field of * Corresponding author. E-mail address: [email protected] (H.A. Jensen). 0045-7825/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2009.08.016

reliability-based optimization several procedures have been recently developed allowing the solution of quite demanding problems. Such procedures are usually based on a combination of approximation concepts with standard deterministic optimization techniques [1,8,9,13,16,17,22,26,29,30], or stochastic search algorithms [24,40,41]. The use of approximate models, i.e. meta-models, in reliability analysis and reliability-based optimization has been proposed in a number of publications [13,32,34,36]. In addition, recent developments of efficient and robust sensitivity analysis techniques are closely related to the construction of metamodels for complex structural systems [4,27,35,42,44]. Stochastic search algorithms have also proved to be useful tools for solving challenging optimization problems. In these approaches the values of the random functions are used directly as inputs to the optimization algorithm [40,41]. The algorithms used in these cases are generally direct search schemes which only use the values of random functions to be optimized as inputs. For a thorough review of the previous and other recent advances in the context of optimization problems considering uncertainties it is referred to, e.g. [37,39]. The use of the above optimization approaches has been found useful in a number of structural optimization applications. However, the application of reliability-based optimization to stochastic dynamical systems remains somewhat limited. For example, on one hand, meta-modeling techniques are not well suited to large scale optimization problems when the number of design variables is relatively large. This is specially prohibitive when considering large scale simulation models. On the other hand, most of the methodologies proposed in the literature for the solution of

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reliability-based optimization problems of stochastic dynamical systems do not possess proven convergence properties. Therefore, there is still much room for further developments in this area. It is the objective of this work to implement a methodology for the solution of reliability-based optimization problems of stochastic dynamical systems with monotonic convergence properties. That is, the purpose of this paper is not in the development of new optimization algorithms but to introduce a general framework for solving a challenging class of structural optimization problems considering uncertainties. The solution of this type of problems is extremely demanding since involves reliability and sensitivity analyses in high-dimensional parameter spaces during the optimization process. Novel aspects of this contribution refer to an effective integration of the algorithm for optimization and reliability assessment. In particular, a new approach for efficient sensitivity estimation of probabilities is presented, which is based on an approximate representation of the structural response; this approach is numerically inexpensive, as it requires a single reliability analysis only, in addition to some structural analyses for estimating the sought sensitivities. The information on sensitivities is used in order to determine search directions in the space of the design variables within the optimization algorithm. In addition, a line search scheme specially designed for handling probabilistic constraints is introduced, in which a polynomial approximation of the probability is generated using information on the function value and its derivative (as suggested in [25,43]). The advantage of the proposed scheme is that it requires a very low number of reliability analyses in order to provide an accurate representation of the probability along the search direction. The structure of this paper is as follows. In Section 2, the mathematical formulation of the reliability-based optimization problem is presented. Section 3 briefly describes the optimization algorithm used in this contribution (which is a well-known firstorder scheme ). Next, Section 4 addresses several implementation issues; among these, salient issues discussed are the approach for reliability sensitivity estimation and the application of the specialized line search scheme. Finally, two application problems are presented to illustrate the performance of the proposed methodology. 2. Reliability-based optimization problem 2.1. Formulation Consider the following structural optimization problem:

Min

CðfxgÞ

subject to hi ðfxgÞ 6 0;

i ¼ 1; . . . ; n;

g i ðfxgÞ 6 0;

i ¼ 1; . . . ; m;

ð1Þ

si ðfxgÞ ¼ fai gT fxg  bi 6 0;

i ¼ 1; . . . ; l;

  xi 2 X i ¼ xi jxli 6 xi 6 xui ;

i ¼ 1; . . . ; nd

ð2Þ

are included in the definition of the linear deterministic constraints si ðÞ. The reliability constraints are written in terms of failure probability functions as

hi ðfxgÞ ¼ PF i ðfxgÞ  P F i 6 0;

i ¼ 1; . . . ; n;

PF i ðfxgÞ ¼

Z XF

f ðfzg=fxgÞ dfzg;

ð4Þ

i

where fzg 2 Xfzg  Rnu is the vector of uncertain variables involved in the problem, f ðfzg=fxgÞ is the probability density function of fzg conditioned on fxg, and XF i is the failure domain of failure event F i in the Xfzg space. The failure domain XF i for a given design fxg is defined in terms of the performance function ji as ji ðfxg; fzgÞ 6 0, that is XF i ¼ ffzgjji ðfxg; fzgÞ 6 0g. Recall that fzg is the vector of random variables that describes all uncertainties involved in the system (model and loading parameters). That is, the components of the vector fzg represent the uncertain structural parameters and the random variables used in the characterization of the stochastic excitation. Therefore, the failure probability functions PF i ðfxgÞ; i ¼ 1; . . . ; n account for the uncertainty in the system parameters as well as the uncertainties in the excitation. Finally, it is assumed that ji is a continuous function with respect to the design variables fxg. 2.2. Application to dynamical systems For systems under stochastic excitation the probability that design conditions are satisfied within a particular reference period provides a useful reliability measure. Such measure is referred as the first excursion probability. In this case the failure events F i ; i ¼ 1; . . . ; n are defined as

F i ðfxg; fzgÞ ¼ Di ðfxg; fzgÞ > 1;

ð5Þ

where

Di ðfxg; fzgÞ ¼ max max

   i  r j ðt; fxg; fzgÞ

j¼1;...;nj t2½0;T



r ij

ð3Þ

ð6Þ

is the normalized demand, ½0; T is the time interval, rij ðt; fxg; fzgÞ; j ¼ 1; . . . ; nj are the response functions associated  critical with the failure event i, and r ij is the corresponding   thresh   old level. In this context the quotient r ij ðt; fxg; fzgÞ=r ij is interpreted as a demand to capacity ratio, as it compares the value of the response r ij ðt; fxg; fzgÞ with the maximum allowable value of  this response r ij . The response functions r ij ðt; fxg; fzgÞ; i ¼ 1; . . . ; n; j ¼ 1; . . . ; nj are obtained from the solution of the equation of motion that characterizes the structural model. With the previous definition of normalized demand, the performance function can be written as

ji ðfxg; fzgÞ ¼ 1  Di ðfxg; fzgÞ

where fxg; xi ; i ¼ 1; . . . ; nd is the vector of design variables, CðfxgÞ is the objective function, hi ðfxgÞ 6 0; i ¼ 1; . . . ; n are the reliability constraints, g i ðfxgÞ 6 0; i ¼ 1; . . . ; m are the deterministic non-linear constraints, and si ðfxgÞ 6 0; i ¼ 1; . . . ; l are the deterministic linear constraints. The deterministic constraints are related to design requirements such as structural weight, geometric conditions and material cost components. The side constraints

fxg 2 X;

where PF i ðxÞ is the failure probability function for the failure event F i evaluated at the design fxg, and P F i is the target failure probability for the ith failure event. The failure probability function PF i ðfxgÞ evaluated at the design fxg can be written in terms of the probability integral

ð7Þ

and the corresponding first excursion probability can be expressed as the multidimensional integral

PF i ðfxgÞ ¼

Z

f ðfzg=fxgÞ dfzg:

ð8Þ

ji ðfxg;fzgÞ60

It is noted that the normalized demand function and the performance function are in general non-smooth and therefore non-differentiable [18]. However, the differentiability of these functions is not required in the present formulation (see Section 4.2). It is also noted that the multidimensional probability integral (8) involves a large number of uncertain parameters (hundreds or thousands) in the context of dynamical systems under stochastic excitation. Therefore, Eq. (8) represents a high-dimensional reliability problem.

H.A. Jensen et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3915–3924

3. Method of solution A first-order optimization scheme based on feasible directions has been selected in the present implementation because of its simplicity and robustness. It is emphasized that the focus of this work is neither the application nor the development of a particular optimization algorithm but rather in the efficient integration of standard optimization strategies and reliability analysis techniques in high dimension. In this regard, the proposed scheme can be extended in a straightforward manner to incorporate more advanced optimization algorithms [3,14]. For completeness and clarity the basic ideas involved in the optimization scheme are presented in what follows. 3.1. General description Starting from an interior point in the design space the first step of the approach is to determine an active design since the optimal solution for most practical applications occurs on the boundary of the design space. An active design is a point on the design space boundary, i.e. one or more inequality constraints are active. Next, a one-dimensional search is carried out along a given direction fdg in order to obtain a new design (better than the previous one) and the corresponding active constraints. The direction is calculated using the gradients of the optimization problem functions at the current design. In the present implementation the direction fdg is computed by solving a linear programming subproblem. The step size computed along fdg needs only the function values. In particular, the step size along the search direction is determined such that a descent or merit function is decreased. The cost function (objective function) is selected as the descent function in this case. The process continues until convergence is achieved. By construction the method generates a sequence of steadily improved feasible designs. The details of the procedure can be found in many publications on optimization [3,14,15]. 3.2. Algorithm properties As previously pointed out, the optimization strategy adopted in this work is based on a local optimization algorithm. Therefore, the optimization process can converge to a local optimum. This situation may occur in structural optimization problems with, for example, non-convex objective functions, or non-convex or disjoint design spaces. Thus, the solution of the optimization process does not ensure the identification of the global optimum. That is, the selection of a particular initial solution can affect the value of the final design. Attempts to find the global optimum (or optima) can be based on physical considerations of the problem or the use of global optimization algorithms. It is noted that, however, global optimization schemes require a large number of function evaluations (i.e. reliability estimations) and therefore they are impractical for the optimization of dynamical systems under stochastic excitation. In this regard the implementation of global optimization strategies is the subject of future research. Finally, it is also noted that the selection of the initial feasible design can affect the efficiency of the method in terms of the number of iterations required for determining an optimum. Some of these issues are addressed in Section 5.

4. Implementation issues 4.1. Reliability assessment The reliability constraints hi ðfxgÞ 6 0; i ¼ 1; . . . ; n of the stochastic optimization problem (1) are given in terms of the first

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excursion probability functions P F i ðfxgÞ; i ¼ 1; . . . ; n. Subset simulation (SS) [5] is adopted in this formulation for the purpose of estimating the corresponding failure probabilities during the design process. In the approach, the failure probabilities are expressed as a product of conditional probabilities of some chosen intermediate failure events, the evaluation of which only requires simulation of more frequent events. The failure probability P F i ðfxgÞ is expressed as the product

PF i ðfxgÞ ¼ PðF i;1 ðfxgÞÞ

mY F 1

PðF i;kþ1 ðfxgÞ=F i;k ðfxgÞÞ;

ð9Þ

k¼1

where F i;mF ðfxgÞ ¼ P F i ðfxgÞ is the target failure event and F i;mF ðfxgÞ  F i;mF 1 ðfxgÞ      F i;1 ðfxgÞ is a nested sequence of failure events. It is seen that, even if P F i ðfxgÞ is small, by choosing mF and F i;k ðfxgÞ; k ¼ 1; . . . ; mF  1 appropriately, the conditional probabilities can still be made sufficiently large, and therefore they can be evaluated efficiently by simulation because the failure events are more frequent. For details of this simulation procedure from the theoretical and numerical viewpoint it is referred to, e.g. [5]. 4.2. Gradient estimation It is clear that the optimization scheme requires the estimation of the gradient of the failure probability functions (determination of feasible direction). The gradient of the ith failure probability function at fxk g can be estimated by means of the limit:

 @PF i ðfxgÞ  @x l

¼ lim fxg¼fxk g

Dxl !0

PF i ðfxk g þ fuðlÞgDxl Þ  PF i ðfxk gÞ ; Dxl

l ¼ 1; . . . ; nd ;

ð10Þ

where fuðlÞg is a vector of length nd with all entries equal to zero, except by the lth entry, which is equal to one. The calculation of this limit is a challenging task, as failure probabilities must be evaluated (in most cases) using simulation techniques. Methods for approximating this limit have been developed, e.g. for the case where xl corresponds to a parameter of a probability distribution (see, e.g. [27,44]); however, these methods are not applicable within the scope of this contribution, as the focus is on a deterministic xl . An approach for estimating probability sensitivity with respect to deterministic variables was introduced in [35]; this approach requires solving the equation ji ðfxg; fzgÞ ¼ 0 for one component of fzg, either by means of analytical or numerical procedures; however, this can be an involved task, specially in stochastic non-linear dynamics, which is precisely the topic of this study. Therefore, in this contribution, an approach that overcomes these issues (which was recently introduced in [42]) is applied. For estimating the limit in Eq. (10), two approximate representations of different quantities are introduced. The first of these approximations involves the performance function. If fxk g is the current design, assume that the performance function ji can be approximated in the vicinity of the current design as:

j i ðfxg; fzgÞ ¼ ji ðfxk g; fzgÞ þ fdi gT fDxg;

ð11Þ

where fxg ¼ fxk g þ fDxg. At this stage, it is assumed that the set of coefficients fdi g is known (the issue on how to determine fdi g is discussed later in this section). Besides the approximation of the performance function proposed in Eq. (11), a second approximation is introduced, in which the failure probability is expressed as a function of a threshold of the normalized demand, i.e.:

  P Di ðfxk g; fzgÞ P Di ¼ PF i ðDi Þ;    PF i Di  ew0 þw1 ðDi 1Þ ; Di 2 ½1  ; 1 þ :

ð12Þ

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In Eq. (12), P½ denotes probability of occurrence, Di is a threshold of the normalized demand, w0 and w1 are real constants and  represents a small tolerance, e.g.  ¼ 0:05. By replacing the approximations introduced in Eqs. (11) and (12) in (10), it can be shown that the limit associated with the gradient of the probability can be approximated by (see Appendix A):

 @PF i ðfxgÞ  @x l

By replacing the approximations introduced in Eqs. (17) and (12) in (16), it can be shown that the limit associated with the directional derivative of the probability can be computed approximately as (cf. Eq. (13)):

 

rPF i ðfxgÞT fyg

fxg¼fxk g

 w1 dil P F i ðfxk gÞ;

l ¼ 1; . . . ; nd ;

ð13Þ

fxg¼fxk g

where dil is the lth element of the vector fdi g. It is important to note that for computing the failure probability and the coefficient w1 , a single reliability analysis suffices.

 w1 di PF i ðfxk gÞ:

ð18Þ

The coefficient di required for the linear expansion in Eq. (17) can be computed using a least squares approach. In particular, the least squares problem to be solved is (cf. Eq. (15)):

ji ðfxk g þ fygnj R; fzj gÞ ¼ ji ðfxk g; fzj gÞ þ di nj R; j ¼ 1; . . . ; Nd ; ð19Þ

4.2.1. Coefficient estimation For determining the coefficients fdi g in Eq. (11), it should be noted that the approximate model of the performance function  i ðfxg; fzgÞÞ is a meta-model of the exact performance function. ðj Thus, the sought coefficients could be adjusted using any appropriate procedure, e.g. in a least square sense. This approach is employed in this contribution and it is applied in two steps:  In a first step, for samples fzj g; j ¼ 1; . . . ; M near the limit state surface, that is, ji ðfxg; fzj gÞ  0, the performance function is evaluated at points in the neighborhood of fxk g. These points are generated as

fxpk g  fxk g ¼ fDxg ¼

fnp g R; kfnp gk

p ¼ 1; . . . ; N ¼ Q M;

ð14Þ

where the components of the vector fnp g are independent, identically distributed standard Gaussian random variables, N and Q positive integers and R is a user-defined small positive number. This number defines the radius of the hypersphere fnp g=kfnp gkR centered at the current design fxk g.  In a second step, the coefficients fdi g of the approximation (11) are computed by least squares. To this end, the following set of equations is generated

ji ðfxpk g; fzj gÞ ¼ ji ðfxk g; fzj gÞ þ fdi gT p ¼ j þ ðq  1Þ M;

fnp g R; kfnp gk

ð15Þ

q ¼ 1; . . . ; Q ; j ¼ 1; . . . ; M:

Since the samples fzj g; j ¼ 1; . . . ; M are chosen near the limit state  i is representative surface, the approximate performance function j of the behavior of the actual limit state surface in the vicinity of the design fxk g. Numerical experience has shown that the approximation introduced in Eq. (13) is adequate in the context with the proposed optimization scheme. Issues such as the number of points required for performing least square (Q and M), and the generation of design points in the vicinity of the current design (calibration of the radius R) are discussed in detail in Ref. [42]. 4.2.2. Directional derivative estimation It should be noted that the approach presented above for gradient estimation of the probability can also be used for estimating the derivative along a direction fyg (directional derivative), i.e.:

 

rPF i ðfxgÞT fyg

fxg¼fxk g

¼ lim

Dx!0

PF i ðfxk g þ fygDxÞ  PF i ðfxk gÞ : Dx

ð16Þ

For calculating this limit, it suffices to generate an approximation of  i in the one-dimensional space defined the performance function j by fyg, i.e.:

j i ðfxk g þ fygDx; fzgÞ ¼ ji ðfxk g; fzgÞ þ di Dx:

ð17Þ

where nj ; j ¼ 1; . . . ; N d are independent, identically distributed standard Gaussian random variables, fzj g; j ¼ 1; . . . ; N d are samples of the uncertain parameters located near the limit state surface (i.e., ji ðfxg; fzj gÞ  0), Nd is a positive integer and R is a user-defined small positive number. The number Nd of evaluations of the performance function required for solving the least squares problem in Eq. (19) is smaller than the number N of evaluations required to solve the problem in Eq. (15). Usually, the ratio between the number of samples required for solving the former and latter problems is 1=nd . It should be noted that the approach for estimating the gradient of the failure probability function and the directional derivative described in this section does not require any additional reliability analyses. Only the evaluation of the performance function ji in the vicinity of the current design is needed. 4.3. Line search For deterministic linear and non-linear constraints the step size problem can be handled by standard techniques such as approaches based on descent or merit functions. For reliability constraints, an approach based on a polynomial approximation of the failure probability functions along the search direction is considered in the present formulation. In particular, the polynomial approximation is constructed using information on probability and its directional derivative; this is due to the fact that directional derivatives can be calculated at small numerical costs once the failure probability has been computed at a given point. Polynomial approximations for performing line search are well documented in the literature of deterministic optimization (see, e.g. [2,14,28]), using either zeroth, first or higher order information of a function. Nonetheless, for the cases analyzed in this contribution, the construction of a polynomial approximation for a probabilistic function is somewhat different from the case of a deterministic function. This is a consequence of the fact that the values of the probability and its sensitivity are known up to an estimator only, which is associated with a certain variability. Therefore, for constructing an approximate representation of the failure probability function along the search direction, an approach based on least squares introduced in [25,43] is applied. The distinct feature of this approach over a conventional least squares procedure is that it incorporates simultaneously information on the function value and the derivative for determining the coefficients of the corresponding regression. The approximation of the failure probabilities along the search direction has the form:

  log P F i ðfxk g þ afdgÞ ¼ ak0 þ ak1 a þ ak2 a2 ;

a > 0;

ð20Þ

where logðÞ indicates natural logarithm. This type of approximation has shown to be appropriate for solving RBO problems [13,16]. The coefficients akl ; l ¼ 0; 1; 2 are calculated in two stages, i.e.:

H.A. Jensen et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3915–3924

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1. Firstly, the failure probability and its derivative (along fdg) are calculated at the current candidate optimal design fxk g and also ki is at a point fxinitial g ¼ fxk g þ aki initial fdg; ainitial ; i ¼ 1; . . . ; n defined as a percentage of the current design norm kfxk gk, that ki ki is, aki initial ¼ C kfxk gk. In this implementation, C 2 ½0:10; 0:25. With the information on probabilities and directional derivatives, the coefficients akl ; l ¼ 0; 1; 2 are determined using the approach proposed in [43] (see Appendix B). 2. With the approximation generated at the previous stage, a new   point fx g ¼ fxk g þ aki  fdg is located, such that P F i ðfx gÞ ¼ P F i . Then, the failure probability and the directional derivative at fx g are calculated. Finally, a new set of coefficients akl ; l ¼ 0; 1; 2 is determined, using the information collected at fxk g; fxinitial g and fx g (see Appendix B). Thus, using the approximation in Eq. (20), it is possible to find the step size along the search direction fdg such that the descent function (objective function) is decreased. Using this information together with the solution of the step size problem for deterministic constraints, the required step size from the current design is obtained. Validation calculations have shown that the number of function evaluations, including reliability analyses, is generally very small during the different cycles of the optimization scheme for the problems considered in the context of this work.

Fig. 1. Bridge structure under earthquake loading.

4.4. Variability of reliability estimates The reliability estimates carried out during the optimization process have associated certain variability. This variability is measured by the estimator coefficient of variation, i.e., the ratio of the standard deviation to the mean of the failure probability. In principle, the variability of the estimates can be controlled by the number of samples used during simulation. However, this alternative can be computationally involved especially when dealing with realistic problems. The average over a number of simulation runs can also be used to smooth the estimates variability during the optimization process. This is particularly important during the last iterations of the optimization scheme, i.e., near convergence. Numerical validations have shown that during the optimization first cycles the average over different simulation runs is not essential since the algorithm global performance is not affected by the variability of the estimates. The strategy proposed in Section 4.3 for approximating reliability constraints at the line search step has shown to be appropriate for coping with the variability of the reliability estimates in the numerical examples investigated in this contribution. As information on failure probabilities and directional derivatives (collected at three points) is used simultaneously, the resulting approximation is of sufficient accuracy. Finally, as in the case of the reliability estimates, an accurate descent-feasible direction is not strictly required during the first iterations. Only an approximate direction is sufficient since the important issue at this stage of the optimization process is to move into the correct region of the feasible design space. 5. Examples 5.1. Example 1 A bridge structure under earthquake loading is considered in the first example problem. The structural system is shown in Fig. 1 and one of the resistant element (C-axis) is illustrated in Fig. 2. Section 2 of the bridge structure is modeled as a single degree of freedom shear system. The corresponding mass is taken as 5:19 105 kg, and the modulus of elasticity of the reinforced concrete columns is equal to E ¼ 2:4 1010 Pa. A 5% of critical

Fig. 2. Schematic illustration of resistant element, C-axis.

damping is assumed in the model. For an improved earthquake resistance, the frame structure shown in Fig. 2, is reinforced with a non-linear friction device that follows the restoring force law

rðtÞ ¼ kd ðuðtÞ  c1 ðtÞ þ c2 ðtÞÞ;

ð21Þ

where kd denotes the initial stiffness of the device, uðtÞ is the relative displacement, and c1 ðtÞ and c2 ðtÞ denote the plastic elongations of the friction device. Using the supplementary variable sðtÞ ¼ uðtÞ  c1 ðtÞ þ c2 ðtÞ, the plastic elongations are specified by the non-linear differential equations [33]

_ _ c_ 1 ðtÞ ¼ uðtÞHð uðtÞÞ



sðtÞ  sy HðsðtÞ  sy Þ Hðsp  sðtÞÞ þ HðsðtÞ  sp Þ ; sp  sy _c2 ðtÞ ¼ uðtÞHð _ _uðtÞÞ

sðtÞ  sy HðsðtÞ  sy Þ Hðsp þ sðtÞÞ þ HðsðtÞ  sp Þ ; sp  sy ð22Þ

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where HðÞ denotes the Heaviside step function, sy is a parameter specifying the onset of yielding, and kd sp is the maximum restoring force of the friction device. The values sp ¼ 0:006 m, sy ¼ 0:0042 m, and kd ¼ 6:0 106 N=m are used in this case. The earthquake induced ground acceleration in the x-direction is modeled as a nonstationary filtered white noise process. The filter is defined by the first-order differential equation

9 9 8 8 0 1 0 0 w1 ðtÞ > > > > > > > > = = < < 2 d w2 ðtÞ X1 2n1 X1 0 0 ¼ 0 0 0 1 > > > > w3 ðtÞ > dt > > ; ; > : 2 : X1 2n1 X1 X22 2n2 X2 w4 ðtÞ 9 8 9 8 w1 ðtÞ > 0 > > > > > > = > < < xðtÞeðtÞ = w2 ðtÞ þ w ðtÞ > > 0 > > > > ; > ; : 3 > : w4 ðtÞ 0

ð23Þ Fig. 3. Iteration history of the design process in terms of the objective function.

and the ground acceleration is defined as

aðtÞ ¼ X21 w1 ðtÞ þ 2n1 X1 w2 ðtÞ  X22 w3 ðtÞ  2n2 X2 w4 ðtÞ;

ð24Þ

where xðtÞ denotes white noise, and eðtÞ is the envelope function defined as

eðtÞ ¼

e0:5t  et ; maxðe0:5t  et Þ

0 < t < 10 s:

ð25Þ

The values X1 ¼ 15:0 rad=s; n1 ¼ 0:6; X2 ¼ 1:0 rad=s, and n2 ¼ 0:9, and white noise intensity I ¼ 15:7 102 m2 =s3 have been used in this example. The sampling interval and the duration of the excitation are taken as Dt ¼ 0:01 s, and T ¼ 10 s, respectively. pffiffiffiffiffiffiffiffiffiffi Then, the discrete-time white noise sequence xðtj Þ ¼ I=Dt zj , where fzg; zj ; j ¼ 1; . . . ; 1001, are independent, identically distributed standard Gaussian random variables is considered in this case. The objective function C is defined in terms of the dimensions of the column elements, that is Cðx1 ; x2 Þ ¼ x1  x2 . On the other hand, the reliability constraint is given with respect to the relative displacement. Failure is assumed to occur when the displacement response reaches some critical level for the first time. A threshold level value equal to u ¼ 0:15 m is considered. The failure event is defined as

Fðx1 ; x2 ; fzgÞ ¼ Dðx1 ; x2 ; fzgÞ > 1;

shown in Fig. 3. The corresponding final design is given in Table 1. The trajectory of the optimizer as well as some objective function contours and iso-probability curves are shown in Fig. 4. At the initial design ðx10 ; x20 Þ ¼ ð1:2; 1:0Þ m, the search direction is just rC (descent direction), since this design is an interior point in the feasible design space. It is seen that starting from an interior feasible design (initial design) the process converges already in about three optimization cycles. In this context, each optimization cycle corresponds to a change in the descent-feasible direction during the optimization process. This change is produced when the algorithm finds an active design (see Fig. 4). At the final design the reliability constraint

Table 1 Final design. Design variables

Initial design

Final design

x1 (m) x2 (m) Probability of failure

1.20 1.00

1.02 0.34

3:48 1012

1:00 102

Target failure probability: 1:00 102 No. of optimization cycles: 3

ð26Þ

where

Dðx1 ; x2 ; fzgÞ ¼

max

t k ;k¼1;...;1001

juðt; x1 ; x2 ; fzgÞj : u

ð27Þ

1

2

x

)=

,x

1 ,x

2

0.

)=

1.0

0

75

)=

2

0.8

50 0.

0.7

2

0.6 (x

C

x [m]

subject to PF ðx1 ; x2 Þ  PF 6 0;

,x

Initial Design

x

1

Cðx1 ; x2 Þ ¼ x1  x2

0.9

C(

Min

(x 1

In addition to the reliability constraint, geometric and side constraints are also considered in the design problem. As previously pointed out, the design variables are the column elements dimensions, with initial design ðx10 ; x20 Þ ¼ ð1:2; 1:0Þ m. The reliabilitybased optimization problem is written as

C(

C

1

2

)=

0.5

0.

x2  3:0x1 6 0;

,x

x1  3:0x2 6 0;

 x1 þ 0:3 6 0;

0.4

P

F

PF

2

and where the level of failure is taken equal to 10 . Note that there is one reliability constraint and six deterministic linear constraints involved in the optimal design problem. The iteration history of the design process in terms of the objective function is

0.2 0.4

.5

=0

F

 x2 þ 0:3 6 0;

F

Final Design P

0.3

x2  1:5 6 0;

P

25

ð28Þ

x1  1:5 6 0;

0.6

0.8

P

=1

0

F

−1

1

=1

0

−3

=1

0 −2

1.2

x [m] 1

Fig. 4. Trajectory of the optimization process in the design space. C = objective function contours; PF = iso-probability curves.

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H.A. Jensen et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3915–3924

1.0

,x )=

2

50

0.

(x

,x

1

2

0.7 x2 [m]

)=

2

C

0.5

P

F

2

)=

)=

0

0.

75

=1

0

25

0.

0.4 0.3

P

P

F

.5

=0

F

4000 200 100

,x

1

10 4 6

0.8

0.6

1

x

Reliability Gradient of probability Directional derivative of probability

x

0.9

1

Average number of simulations required per single analysis

C(

,x

Number of analyses required for solving RBO problem

1

C(

Type of analysis

Initial Design B

(x

Table 2 Summary of numerical efforts required to solve Example 1.

Fig. 5. Probability of failure in terms of the threshold level for the final design.

C

PF ðx1 ; x2 Þ  P F 6 0 and the deterministic linear constraint x1  3:0x2 6 0 are active, that is, PF ðx1 ; x2 Þ ¼ P F and x1 ¼ 3:0x2 . From Fig. 4 it is observed that the optimizer moves along the search directions until the objective function is minimized. The method generates a series of steadily improved feasible designs that move toward the optimum. This property is important from a practical viewpoint since the design process may be stopped at any stage still leading to acceptable feasible design better than the initial feasible estimate. The numerical efforts involved in the solution of the RBO problem are due to the estimation of the reliability (by means of SS) and its sensitivity (both gradients and directional derivatives, see Section 4.2), respectively. Table 2 summarizes these numerical efforts. The first column of this table indicates the type of analysis performed, while the second column shows the number of times the aforementioned analysis was repeated throughout the optimization procedure; finally, the third column indicates the average number of simulations required for performing one particular type of analysis. For example, a total of 10 reliability analyses (using SS) are required for solving the RBO problem; each of these analyses involve (on the average) 4000 simulations. In a next step, the reliability of the final design with respect to the relative displacement maximum acceptable value is investigated. To this end, Fig. 5 shows the probability of failure in terms of the threshold level u . A value up to 0.20 m is considered in this figure. For example, the probability that the relative displacement exceeds 0.1 m is about 2:0 101 , whereas a probability of 2:0 104 is obtained for a threshold level equal to 0.20 m. Thus, the reliability of the final design is highly sensitive to the maximum allowable value of the relative displacement of the system. Finally, the influence of the selection of the initial feasible design for the solution of the RBO problem is investigated. For this purpose, two initial designs (denoted as initial design A and B, respectively) are selected. The initial design A is set equal to  A A    x10 ; x20 ¼ ð1:2; 0:4Þ m and the design B, equal to xB10 ; xB20 ¼ ð0:8; 1:0Þ m. The trajectory of optimization for both cases is shown in Fig. 6. In the case of initial design A, a single iteration is required for determining the optimum; this is due to the fact that the feasible direction determined at this initial design is pointing precisely towards the optimal solution. Thus, the numerical efforts associated with this optimization trajectory are (approximately) a third of the costs indicated in Table 2. In the case of initial design B, two iterations are required for determining the optimal solution, i.e. the numerical efforts associated with this optimization trajectory are (approximately) two thirds of the costs indicated in Table 2. The optimization trajectories associated with initial designs A and B highlight the fact that the selection of an initial, feasible solution can be quite relevant from the point of view of numerical costs. However, for this specific problem, the selection of an initial design does not affect the optimal solution, as the problem possesses a single, global optimum. But in more general cases the selection of a particular initial design may lead to different optimal solutions. In such cases, engineering criteria and the background

0.2 0.4

0.6

0.8 x [m]

−3

Final Design

P

=1

0

F

−1

=1

Initial Design A

0 −2

1

1

Fig. 6. Trajectory of the optimization process in the design space for alternative initial designs A and B. C = objective function contours; PF = iso-probability curves.

knowledge on the problem at hand should be applied in order to select an initial design and to assess the properties of the optimal design. 5.2. Example 2 A 10-storey reinforced concrete (RC) frame including a base-isolation system (composed of non linear hysteretic devices) subject to a stochastic ground acceleration is considered for the second example. A schematic representation of the model is shown in Fig. 7. Each floor of the RC frame is supported by six columns of square shape and a height of 3 m. It is assumed that the beams of the frame are rigid in the axial direction, so each floor can be described by a single horizontal degree of freedom (DOF); moreover, the base of the system is modeled as a rigid slab. Thus, the model involves a total of 11 DOFs. Moreover, it is assumed that the RC frame structure remains linear throughout the duration of the ground acceleration. The Young’s modulus is equal to 3 1010 Pa. The floor mass is equal to 1:5 105 kg and the mass of the base, 3:0 105 kg. A 5% of critical damping is assumed in the model. The base-isolation system is composed of five non-linear devices (NLD); each of these devices is modeled using the equations described in Section 5.1 and with identical properties, except that kd ¼ 2 108 N=m.

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H.A. Jensen et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3915–3924 Table 3 Initial and final designs – Example 2.

5@5 m

Design variables

10@3 m

Initial design

Final design

ðm Þ

5.47

4.75

x2 102 ðm4 Þ

5.47

4.75

x3 102 ðm4 Þ

5.47

4.75

x4 102 ðm4 Þ

5.47

4.69

x5 102 ðm4 Þ

5.47

4.56

x6 102 ðm4 Þ

3.41

3.79

x7 102 ðm4 Þ

3.41

3.57

x8 102 ðm4 Þ

3.41

3.01

x9 102 ðm4 Þ

3.41

2.15

x10 102 ðm4 Þ Probability of failure

3.41

1.50

9:41 107

1:16 103

x1 10

2

4

Target failure probability: 1:00 103 No. of optimization cycles: 8

NLD Fig. 7. Ten-storey RC frame with base-isolation system.

The stochastic ground acceleration is modeled as a non-stationary, filtered white noise (see Eqs. (23) and (24)). The envelope of the white noise signal eðtÞ is defined as:

8 2 > < t =16 eðtÞ ¼ 1 > : ðt10Þ2 e

if t 2 ½0; 4 s; if t 24 s;10 s;

125

120

115

1

2

3

4

5

6

7

8

Optimization Cycle Fig. 8. Iteration history of the design process in terms of the objective function – Example 2.

All parameters describing the stochastic ground acceleration are set equal to those of Section 5.1, except for the white noise intensity, which is set equal to I ¼ 6:3 102 m2 =s3 , and the duration of the excitation, which is taken equal to T ¼ 15 s. The objective function C is defined as the volume of the columns of the frame. The design variables fxg are chosen as the inertia of the columns throughout the height, grouped in ten design variables, i.e. the inertia of the columns of each floor constitutes each of the design groups. The failure event is formulated as a first passage problem during the duration of the ground acceleration; the structural responses to be controlled are the 10 interstorey drift displacements and the roof displacement (w.r.t. the base-isolation system). The threshold values are chosen equal to 0.2% of the floor height for the interstorey drift displacements and 0.1% of the frame height for the roof displacement. The tolerable probability of failure ðP F Þ is set equal to 103 . Additionally, geometric and side constraints are incorporated in the problem. Thus, the reliability-based optimization problem is defined as:

Min CðfxgÞ subject to PF ðfxgÞ  P F 6 0; i ¼ 1; . . . ; 9;

 xi þ 2:1 103 6 0; xi  8:3 102 6 0;

130

ð29Þ

if t 210 s;15 s:

xiþ1  xi 6 0;

Objective function (m2 )

135

ð30Þ

i ¼ 1; . . . ; 10;

three optimization cycles. The details on the optimization procedure for the initial design and the 8th iteration cycle are summarized in Table 3. Similarly as in the case of the first example, the numerical efforts involved in the solution of the RBO problem are due to the estimation of the reliability and the estimation of its sensitivity, respectively. Table 4 summarizes the numerical costs associated with the solution of the RBO problem formulated in Eq. (30). The first column of this table indicates the type of analysis performed, while the second column shows the number of times the aforementioned analysis was repeated throughout the optimization procedure; finally, the third column indicates the average number of simulations required for performing one particular type of analysis. For example, a total of 8 estimates of the gradient of the probability are required for solving the RBO problem; each of these analyses require 1000 simulations for calibrating the approximate model shown in Eq. (11).

Table 4 Summary of numerical efforts required to solve Example 2. Type of analysis

Number of analyses required for solving RBO problem

Average number of simulations required per single analysis

Reliability Gradient of probability Directional derivative of probability

22 8

3364 1000

14

100

i ¼ 1; . . . ; 10;

The initial design is shown in Table 3. The results of the optimization procedure are presented in Fig. 8 in terms of the evolution of the objective function. As in the first example, only a few optimization cycles are required for obtaining convergence. In fact, most of the improvement of the objective function takes place in the first

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H.A. Jensen et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 3915–3924

In addition, the failure probability associated with the final design (see Table 3) is computed for the case where no base-isolation system is considered. The resulting failure probability is P F ¼ 4:50 101 . This result highlights the beneficial effect of the base-isolation system in protecting the superstructure (in this case, the 10-storey RC frame). 6. Conclusions An approach for solving challenging reliability-based optimization problems involving dynamical systems under stochastic excitation has been presented. The stochastic reliability-based optimization problem is formulated as the minimization of an objective function subject to deterministic and reliability constraints. The solution strategy integrates a standard gradient-based algorithm with specialized strategies for estimating reliability sensitivity and performing line search w.r.t. probabilistic constraints requiring reduced numerical efforts. Numerical results have shown that the approach generates a sequence of steadily improved feasible designs. That is, the design process has monotonic convergence properties. This property is important from a practical view point since the optimization process can be stopped at any stage still leading to better designs than the initial feasible estimate. This is particularly attractive for dealing with involved problems such as reliability-based optimization of dynamical systems under stochastic excitation. In these problems, which are the cases of interest in this work, each iteration of the optimization process is associated to high computational costs. Moreover, the monotonic convergence property is also important because most of the algorithms for RBO developed so far do not possess this feature. Numerical experience has also shown that the algorithm converges in a relatively small number of optimization cycles. This in turn implies that only a moderate number of reliability estimates has to be performed during the entire design process. A critical issue for computing descentfeasible directions is shown to be the determination of the gradient of the failure probability functions. The approximation used in this contribution for estimating the gradient proved to be most efficient. With only a few extra evaluations of the performance function (and no extra reliability analyses), it is possible to generate a sufficiently accurate estimate of the sought gradient. Concerning the line search step, the proposed approach for handling reliability constraints is considered to be most adequate, as the total number of reliability analyses per line search is limited to three. Moreover, the use of directional derivatives for constructing an explicit approximation of the failure probability function allows to generate accurate estimates. The numerical examples carried out indicate that this approach allows to cope with the inherent variability of the reliability estimates as well. The application examples and additional numerical validations showed that the proposed approach is not only effective, but also scalable with the number of design variables. However, the validity of this premise needs further validation. Thus, future research directions will aim at expanding the methodology reported herein. Specific issues to be investigated include: the application of the proposed approach to large FE models; the determination of the number of design variables that can be handled efficiently by the method; the possibility of computing improved search directions; and the implementation of more advanced optimization algorithms within the proposed framework. Acknowledgments This research was partially supported by the Austrian Research Council (FWF) under Project No. P20251-N13 and by CONICYT under Grant Number 1070903. This support is gratefully acknowledged by the authors.

Appendix A. Approximate gradient estimation Considering the definition of failure probability in Eq. (8), the limit for calculating the partial derivatives of the failure probability functions can be expressed as:

 @PF i ðfxgÞ P½ji ðfxk g þ fuðlÞgDxl ; fzgÞ 6 0 ¼ lim @xl fxg¼fxk g Dxl !0 Dxl

P½ji ðfxk g; fzgÞ 6 0  ; l ¼ 1; . . . ; nd ; Dxl

ð31Þ

where P½ denotes probability of occurrence. Introducing Eqs. (7) and (11), this limit can be approximated as:

 @PF i ðfxgÞ P½1 þ dli Dxl  Di ðfxk g; fzgÞ 6 0  lim @xl fxg¼fxk g Dxl !0 Dxl

P½1  Di ðfxk g; fzgÞ 6 0  ; l ¼ 1; . . . ; nd : ð32Þ Dxl Thus, the calculation of the sought derivatives reduces to estimating the probability that the normalized demand (for a fixed design variable vector) exceeds two different threshold levels. These probabilities can be readily obtained from a single reliability analysis performed using, e.g. subset simulation. In particular, if the failure probability is approximated as an explicit function of the normalized demand (as shown in Eq. (12)), it is possible to further simplify Eq. (32) to:

 @P F i ðfxgÞ ew0 þw1 dli Dxl  ew0  lim ; l ¼ 1; . . . ; nd ; @xl fxg¼fxk g Dxl !0 Dxl  @P F i ðfxgÞ  w1 dli ew0 ; l ¼ 1; . . . ; nd ; @xl fxg¼fxk g  @P F i ðfxgÞ  w1 dli PF i ðfxk gÞ; l ¼ 1; . . . ; nd : @xl fxg¼fxk g

ð33Þ

Appendix B. Approximate representation of failure probabilities Assume that the value and derivative of the function f ðxÞ have been calculated at np points xi ; i ¼ 1; . . . ; np . The objective is determining the coefficients al ; l ¼ 0; 1; 2 of a quadratic polynomial f ðxÞ which approximates f ðxÞ, i.e.:

f ðxÞ  f ðxÞ ¼ a0 þ a1 x þ a2 x2 :

ð34Þ

The sought coefficients can determined by minimizing the following residual [43]:

HðfagÞ ¼ H0 ðfagÞ þ H1 ðfagÞ:

ð35Þ

The residuals H0 ðÞ and H1 ðÞ are defined as:

Pn p  H0 ðfagÞ ¼ 

i¼1

  2 f ðxi Þ  a0 þ a1 xi þ a2 x2i

2 maxi¼1;...;np ðf ðxi ÞÞ  mini¼1;...;np ðf ðxi ÞÞ Pnp 0 4 i¼1 ðf ðxi Þ  ða1 þ 2a2 xi ÞÞ2 H1 ðfagÞ ¼     ; maxi¼1;...;n ðf 0 ðxi ÞÞ þ mini¼1;...;n ðf 0 ðxi ÞÞ 2 p

ð36Þ ð37Þ

p

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