0087 engopt paper jensen

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EngOpt 2008 - International Conference on Engineering OptimizationRio de Janeiro, Brazil, 01 - 05 June 2008.

Reliability-Based Optimization of Structural Systems under StochasticExcitation

Hector A. Jensen, Macarena S. Ferre

Department of Civil Engineering, Santa Maria University, Valparaiso, Chile, [email protected]

1. AbstractThis work presents an efficient methodology to carry out reliability-based structural optimization of non-linear systems under stochastic loading. The optimization problem is formulated as the minimization ofan objective function subject to reliability constraints. The reliability constraints are given in terms offirst excursion probabilities. A sequential approximate optimization scheme based on global approxima-tions of the reliability constraints is implemented in the proposed formulation. The approximations ofthe first excursion probabilities in terms of the design variables are constructed by combining a globalmixed linearization approach with a reliability sensitivity analysis. A numerical example that illustratesthe effectiveness of the method is presented.

2. Keywords: Non-linear systems, reliability-based optimization, sensitivity analysis, simulation meth-ods.

3. IntroductionThe reliability-based structural synthesis of non-linear systems subject to stochastic excitation is asso-ciated with high computational costs. During the optimization process reliability measures need to beevaluated several times before an optimal design can be obtained. This in turns implies the repeatedevaluation of dynamic responses which can be extremely time consuming for complex structural sys-tems. Thus, the direct solution of this class of optimization problems is not feasible from a practicalpoint of view. Under these conditions the use of approximation strategies are needed for an efficientstructural synthesis. Several approximation strategies with various capabilities, accuracy and efficiency

have been suggested in the context of reliability-based optimization [1,2,3,4]. In this work, the use ofa sequential approximate optimization approach based on global approximations is considered for thesolution of structural synthesis problems with reliability constraints. A direct analysis is first used toanalyze an initial design and then to generate information that allows the construction of objective andconstraints approximations. An optimization procedure is then applied to the approximate optimiza-tion problem. Following an optimization based on approximate analyses, an exact analysis is performedat the design point obtained by the approximate optimization and new sensitivities are calculated. Inthis manner, a new approximation for the objective and constraint functions can be constructed. Theprocess is repeated until convergence is achieved. A conservative approximation scheme is used for thepurpose of approximating the objective function and reliability constraints in terms of the design vari-ables. Using this approximation strategy a convex and separable approximate optimization problem isgenerated. The objective of this work is to apply the above general scheme to problems involving thestructural synthesis of non-linear systems under stochastic excitation with multiple reliability constraints.

4. Reliability-Based Optimization ProblemConsider the following structural optimization problem

Min F({x})

subject to

PFj ({x}) PFj

, j = 1,...,nc (1)

with side constraints

{x} X, xi Xi = {xi|xli xi x

ui } , i = 1,...,nd

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where {x}, xi, i = 1,...,nd is the vector of design variables, F({x}) is the objective function, PFj ({x})is the probability of the failure event j evaluated at the design {x}, PFj is the target failure probability

for the event j, xli and xui denote the lower and upper limits for the design variables, respectively, and

nc is the number of reliability constraints. Given a design {x} , the probability of failure PFj ({x}) isdetermined by

PFj ({x}) =

Fj

f({z}/{x})d{z} , j = 1,...,nc (2)

where {z} {z} Rnrv is the vector of uncertain variables involved in the problem, f({z}/{x}) is the

probability density function of {z} conditioned on {x}, and Fj is the failure domain of event j in the{z} space. For systems under stochastic excitation, the probability that design conditions are satisfiedwithin a particular reference period (first excursion probability) provides a useful reliability measure.The failure events Fj , j = 1,...,nc are defined as

Fj({x}, {z}) = {maxi=1,...,njmaxt[0,T] | sij(t, {x}, {z}) |> s

j} (3)

where [0, T] is the time interval, sij(t, {x}, {z}), j = 1,...,nc, i = 1,...,nj are the response functions as-sociated with the failure criterion j, and sj is the corresponding critical threshold level. Note that thefailure probability functions PFj ({x}), j = 1,...,nc account for the uncertainty in the excitation. Then,the components of the vector {z} represent the random variables involved in the characterization of thestochastic excitation. The response functions sij(t, {x}, {z}), j = 1,...,nc, i = 1,...,nj are obtained fromthe solution of the equation of motion that characterizes the structural model.

5. Approximate Reliability ConstraintsDuring the optimization process the reliability constraints must be estimated several times. The esti-mation of these quantities for every change of the optimization variables requires the estimation of theprobability of failure of the structural system. The repeated estimation of the failure probability canbe extremely time consuming for systems of practical interest. In the present formulation the compu-

tational cost problem is addressed by the use of a sequential approximate optimization approach. Thesolution of the reliability-based optimization problem is obtained by transforming it into a sequence ofsub-optimization problems having a simple explicit algebraic structure. The process is repeated untilconvergence is achieved, typically measured by the magnitude of changes in the objective function orthe degree of satisfaction of the optimality conditions.

The failure probability functions PFj ({x}), j = 1,...,nc are represented using approximate functionsdependent on the design variables. In particular, the construction of the approximate optimizationproblems is based on the approximations of the transformed failure probability functions hFj ({x}) =ln[PFj ({x})], j = 1,...,nc. The transformed failure probability functions are approximated globally byusing the mixed linearization

hFj ({x}) = hFj ({x0}) +

(i+j )

hFj ({x0})

xi(xi x

0i ) +

(ij )

hFj ({x0})

xi

x0i

xi(xi x

0i ) , (4)

j = 1,...,nc

where {x0} is a point in the design space, and

(i+j )and

(ij )

means summation over the variables

belonging to group (i+j ) and (ij ), respectively. Group (i

+j ) contains the variables for which

hFj ({x0})

xiis

positive, and group (ij ) includes the remaining variables. The expansion given by Eq.(4) corresponds to

a linear approximation in terms of the direct variables (xi) for the variables belonging to group (i+j ), and

a linear approximation in terms of the reciprocal variables (1/xi) for the variables belonging to group(ij ). The above linearization is called convex approximation since the conservative approximation is aconvex function.

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6. Approximate Primal ProblemUsing the convex linearization for the objective and constraint functions the following approximateprimal problem is generated

Min(i+)

F({x0})

xixi

(i)

F({x0})

xi

(x0i )2

xi

subject to

(i+j )

hFj ({x0})

xixi

(ij )

hFj ({x0})

xi

(x0i )2

xi hj (5)

j = 1,...,nc

where

hj = hFj +(i+j )

hFj ({x0})

xix0i

(ij )

hFj ({x0})

xix0i hFj ({x

0}), (6)

with hFj = ln[PFj

] and side constraints


ui } , i = 1,...,nd (7)

If the objective function F({x}) is explicit in {x} the derivatives F({x0})/xi can be evaluateddirectly. On the other hand, a method to compute the gradients of the transformed failure probabilityfunctions hFj ({x

0})/xi will be described in a subsequent section. The convex approximation intro-duces some convex curvature in the approximate functions and therefore the approximate optimizationproblem does not require in general any control parameters such as move limits [5,6]. Thus, the side

constraints for the approximate primal problem are chosen as in the original optimization problem. Itis noted that the approximate optimization problem (5-7) is convex. This is due to the fact that theapproximations of the objective and constraint functions are convex and therefore the feasible domainof the approximate optimization problem is also convex.

7. ImplementationThe optimization process is implemented as follows:

1) Start with an initial feasible design {xk}, k = 0. If the initial design is unfeasible, the optimizationscheme may have convergence difficulties because of the conservative character of the approximate con-straints. In some applications is difficult to get an initial feasible design because the initial design mayviolate some of the reliability constraints. To cope with this difficulty, an auxiliary initial optimizationproblem must be solved in order to find an initial feasible design for the optimization problem [7].

2) Approximate the objective function F({x}) and the transformed failure probability functionshFj ({x}) = ln[PFj ({x})], j = 1,...,nc using the convex linearization approach about the current designpoint {xk}.

3) Formulate the explicit convex optimization problem

Min(i+)

F({xk})

xixi

(i)

F({xk})

xi

(xki )2

xi

subject to

(i+j )

hFj ({xk})

xixi

(ij )

hFj ({xk})

xi

(xki )2

xi hj (8)

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j = 1,...,nc

where

hj = hFj +(i+j )

hFj ({xk})xi

xki (ij )

hFj ({xk})xi

xki hFj ({xk}), (9)

with side constraints


ui } , i = 1,...,nd (10)

4) Solve the approximate subproblem to obtain an optimal solution {x}. The solution of thisproblem can be obtained in an efficient manner due to the explicitness and separability of the functionsinvolved in the problem in terms of the design variables. In addition, since the approximate optimizationproblem is convex, it has only a single optimum. The new point is used as the current design for thenext cycle, that is, {xk+1} = {x}. Set k = k + 1, and go to step 2.

5) The process is continued until some convergence criterion is satisfied. A convergence criteriondefined in terms of the relative change in the objective function between two consecutive design cyclesis implemented here.

8. Reliability AnalysisIt is seen that the reliability constraints of the structural optimization problem previously defined aregiven in terms of the excursion probabilities PFj , j = 1,...,nc. In this work, a subset simulation technique[8] is adopted and integrated into the optimization process. In this approach, the failure probabilitiesare expressed as a product of conditional probabilities of some chosen intermediate failure events, theevaluation of which only requires simulation of more frequent events. Therefore, a rare event simulationproblem is converted into a sequence of more frequent event simulation problems. For example, thefailure probability PFj ({x}) can be expressed as the product

PFj ({x}) = P(Fj,1({x}, {z}))l1k=1

P(Fj,k+1({x}, {z})/Fj,k({x}, {z})) , (11)

where

Fj,l({x}, {z}) = { maxi=1,...,nj

maxt[0,T]

|sij(t, {x}, {z})| > sj} , (12)

is the target failure event, and Fj,l({x}, {z}) Fj,l1({x}, {z})... Fj,1({x}, {z}) is a nested se-quence of failure events. Equation (11) expresses the failure probability PFj ({x}, {z}) as a productof P(Fj,1({x}, {z}) and the conditional probabilities P(Fj,k+1({x}, {z})/Fj,k({x}, {z})), k = 1,...,l 1.It is noted that, even if PFj ({x}) is small, by choosing l and Fj,k({x}, {z}), k = 1,...,l 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. The intermediate failure eventsare chosen adaptively using information from simulated samples so that they correspond to some speci-fied values of conditional failure probabilities [8]. As previously pointed out, to compute PFj ({x}) oneneed to compute the probabilities P(Fj,1({x}, {z}) and P(Fj,k+1({x}, {z})/Fj,k({x})), k = 1,...,l 1.P(Fj,1({x}, {z})) can be readily estimated by Monte Carlo simulation using, for example, N1 in-dependent and identically distributed samples simulated according to the probability density func-tion of the vector of random variables {z}. On the other hand, the conditional failure probabilitiesP(Fj,k+1({x}, {z})/Fj,k({x}, {z})), k = 1,...,l 1 are estimated by using Nk samples simulated accord-ing to the conditional distribution of{z} given that it lies in Fj,k({x}, {z}). Validation calculations haveshown that the method is robust to dimension size and efficient in computing small probabilities.

9. Reliability Sensitivity EstimationFrom Eq. (4) it is seen that the characterization of the approximate transformed failure probabilityfunctions requires the estimation of the gradients, that is,

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hFj ({x})

xi, i = 1,...,nd, j = 1,...,nc (13)

For this purpose, the excursion probability functions PFj ({x}, j = 1,...,nc are approximated locallyabout the current design point {xk} as

PFj ({x}) = expj0+nd

l=1j

l(xlx

kl ) , j = 1,...,nc (14)

Using this representation of the failure probability functions it is clear that the gradients of thetransformed failure probability functions can be readily estimated as

hFj ({x})

xi ji , i = 1,...,nd , j = 1,...,nc (15)

Therefore, the gradients of hFj ({x}) can be estimated directly from the coefficients {j}, ji , j =

1,...,nc, i = 1,...,nd. These coefficients are determined by considering an augmented reliability problem

where the design variables xi, i = 1,...,nd are artificially considered as uncertain with independentcomponents uniformly distributed. Let ({x}) = ndi=1i(xi) be the uniform joint probability densityfunction of the design variables and i(xi) the one dimensional uniform density function of xi. Theprobability density function of the design variable xi is defined in the vicinity of x

ki with support xi .

The coefficients are determined by solving a set of non-linear equations which is defined by consideringsome statistics of the augmented reliability problem [9]. In particular, the average failure probabilityPaverageFj and the first moments of area m

iPFj

, i = 1,...,nd of the failure probability function PFj ({x})

over the space {x} = x1 x2 ... xnd are given by

PaverageFj =

{x}

PFj ({x})({x})d({x})

miPFj

= {x}

xiPFj ({x})({x})d({x}) , i = 1,....,nd (16)

j = 1,...,nc

Substituting the expression of the approximate failure probability functions PFj ({x}), j = 1,...,nc in

(16) yields a set of algebraic non-linear equations for the coefficients j0 and {j}, which can be solved

directly. The evaluation of the coefficients j0 and {j} requires the estimation of the average and the

first moments of area of the failure probability functions. These quantities can be estimated by using aproper simulation technique [11]. Is it noted that the average failure probability PaverageFj corresponds to

the probability of failure of the augmented reliability problem, that is, both {z} and {x} are consideredas uncertain. Once the non-linear set of equations for the coefficients j0 and {

j} is solved, the estimatesfor the gradients of the transformed failure probability functions are completely determined by Eq. (15).

10. ExampleA six-storey building under stochastic earthquake excitation is considered for analysis. A top view of thebuilding is shown in Fig. (1). Each of the six floors is supported by 35 columns of square cross sectionsmade of reinforced concrete. A Youngs modulus E = 3 1010 N/m2, and mass density = 2.5 103

kg/m3 have been used in this case. A finite element model is considered for the structural system.In order to characterize the dynamic behavior of the building, it is assumed that each floor may berepresented sufficiently accurate as rigid within the x y plane when compared with the flexibility ofthe columns. Hence, each floor can be represented by three degrees of freedom, i.e. two translatorydisplacements ux(t) and uy(t) in the direction of the x and y axis, respectively, and a rotationaldisplacement uz(t) about the z axis. The corresponding equation of motion of the system is obtainedby condensation techniques. The associated masses mx = my and mz are taken as constant for all floors8.64 105 kg and 1.35 108 kg m2, respectively.

A classical damping is assumed in the model so that the first modes have a 3% of critical damping.The floor height of 3.0 m is constant for all floors, leading to a total height of 18 m. The earthquake

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y

D1

D2

D4

P4

24 [m]x

D3

ux

uuz

36 [m]P1

P2P

3

Figure 1: Top view of the building model with hysteretic devices

induced ground acceleration a(t) is modelled as a non-stationary filtered white noise process. The filter

is defined by the first-order differential equation

d

dt

y1(t)y2(t)y3(t)y4(t)

=

0 1 0 021 211 0 0

0 0 0 121 211

22 222

y1(t)y2(t)y3(t)y4(t)

+

0w(t)e(t)

00

, (17)

and the ground acceleration is defined as

a(t) = 21y1(t) + 211y2(t) 22y3(t) 222y4(t) , (18)

where w(t) denotes the white noise process and e(t) denotes the envelope function

e(t) =e0.2t e0.4t

max(e0.2t e0.4t), 0 < t < 10 s (19)

The values 1 = 15.7 rad/s, 1 = 0.8, 2 = 0.3 rad/s, and 2 = 0.995, and white noise intensity I =0.05 m2/s3 are used in this example. The sampling interval and the duration of the excitation are takenas t = 0.01 s and T = 10 s, respectively. The discrete-time white noise sequence (tj) =

I/tzj ,

where zj , j = 1, ..., 1001, are independent, identically distributed standard Gaussian random variables isconsidered in this case. The acceleration process a(t) is applied at 45 degrees with respect to the xaxis. For aseismic design purposes, non-linear hysteretic devices are introduced at selected locations asshown in Fig. (1). In each of the six floors, four devices Di, i = 1, 2, 3, 4, are implemented to provideadditional resistant against relative displacements between subsequent floors. The devices exhibit aone dimensional hysteretic type of non-linearity. The initial inter-story stiffnesses of these non-linearelements are within each floor identical and equal to kd = 6.0 108N/m for all devices. They follow theinter-story restoring force law,

ri(t) = kd

i(t) q1i (t) + q2i (t)

, (20)

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where i(t) is the relative displacement between the (i 1, i)-th floor at the position of the device inthe x direction for devices D1 and D3, and in the y direction for the devices D2 and D4, and q

1i (t)

and q2i (t) denote the plastic elongations of the device placed between the ( i 1, i)-th floor. The plastic

elongations are specified by the non-linear differential equations [10]

q1(t) = (t)g1((t), u(t))

q2(t) = (t)g2((t), u(t)) , (21)

whereg1((t), u(t)) =

H((t))

H(u(t) uy)u(t) uyup uy

H(up u(t)) + H(u(t) up)

, (22)

andg2((t), u(t)) =

H((t))

H(u(t) uy) u(t) uyup uy

H(up + u(t)) + H(u(t) up)

, (23)

where H() denotes the Heaviside step function, u(t) = (t) q1(t) + q2(t) is an auxiliary variable, anduy and up are model parameters for the nonlinear behavior. The non-linear restoring force acts betweenadjacent floors with the same orientation as the relative displacement i(t). The values up = 6.010

3mand uy = 4.2 10

3m are used for all devices. The solution of the equation of motion that characterizesthe building with hysteretic devices is carried out by modal analysis. All eighteen structural modes ofthe corresponding condensed equation of motion are considered for the dynamic analysis in this case.The response of the system is computed at discrete time instants tk = (k 1)t, k = 1,...,nT by asuitable numerical integration scheme.

The objective function for the optimization problem is the total weight of the column elements. Thedesign variables are the dimensions of the square cross section of the column elements. The dimensions

of the columns for a given floor are assumed to be identical, and therefore over the height of the buildingthe columns have six different cross sections with initial design x1 = 0.80 m, x2 = 0.80 m, x3 = 0.75m, x4 = 0.70 m, x5 = 0.65 m, x6 = 0.55 m, and side constraints 0.3 m xi 1.40 m. Six reliabilityconstraints are consider in the optimization problem and they are defined as PFi({x}) 10

5, where

PFi({x}) = P

max

j=1,..,4max

tk,k=1,..,1001|ji(tk, {x})| >

u

i = 1, ...., 6, (24)

with ji(tk, {}) is the relative displacement between the (i 1, i)-th floor at the position of the controlpoint Pj evaluated at the design {x}, tk = (k 1)t , k = 1,..., 1001 are the discrete time instants, andu is the critical threshold level and equal to 0.015 m. The final design is given in Table (1), and thecorresponding iteration history of the design process in terms of the reliability constraints is shown inFig. (2). It is observed that process converges in about 4 iterations. The probability that the interstory

drift ratios (constraints) reach the threshold level 0.015 m is shown in Fig. (3) for the initial and finaldesign. It is seen that all reliability constraints are active at the final design. The fast convergence ofthe optimization process leads to a small number of excursion probability estimations to be performedduring the entire procedure.

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Figure 2: Iteration history in terms of the reliability constraints

Table 1: Final designs.

Design Initial Finalvariable design designx1 (m) 0.80 0.77x2 (m) 0.80 0.76x3 (m) 0.75 0.73x4 (m) 0.70 0.70x5 (m) 0.65 0.64x6 (m) 0.55 0.54

Weight (ton) 802.6 758.8

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Figure 3: Probability of the failure events at the initial and final design

11. ConclusionsThe sequential approximate optimization scheme implemented in the formulation proved to be very

effective for the class of problems considered in this study. That is, structural optimization problemsof non-linear systems under stochastic excitation with multiple reliability constraints. Based on theexample problem and additional numerical experiments carried out by the authors it is concluded thatthe proposed sequential approximate optimization scheme usually converges within few iterations for thetype of problems treated here. In fact, as demonstrated in the example problem no more than five designcycles are sufficient for convergency. This efficiency in the context of reliability-based optimization ofstructural systems under stochastic excitation is quite remarkable.

12. AcknowledgmentsThe research reported here was supported in part by CONICYT under grant number 1030375 which isgratefully acknowledged by the authors.

13. References

[1] Gasser, M. and Schueller, G.I., Reliability-Based Optimization of Structural Systems, MathematicalMethods of Operational Research, 1997, Vol. 46:287-307

[2] Enevoldsen, I., and Sorensen, J.D., Reliability-Based Optimization in Structural Engineering. Struc-tural Safety, 1994, Vol.5 (No. 3):169-196.

[3] Royset, J.O. and Polak, E., Reliability-Based Optimal Design using Sample Average Approximations.Probabilistic Engineering Mechanics, 2004, Vol. 19:331-343.

[4] Jensen, H.A., Design and Sensitivity Analysis of Dynamical Systems Subjected to Stochastic Load-ing, Computers & Structures, 2005, Vol. 83:1062-1075.

[5] Fleury, C. and Braibant, V., Structural Optimization: A New Dual Method Using Mixed Variables,International Journal for Numerical Methods in Engineering, 1986, Vol. 23, No 3:409-428.

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[6] Haftka, R.T. and Gurdal, Z., Elements of Structural Optimization, Third edition, Kluwer AcademicPublishers, The Netherland. 1992.

[7] Jensen, H.A., Structural Optimization of Non-linear Systems under Stochastic Excitation. Proba-bilistic Engineering Mechanics, 2006, Vol. 21:397-409

[8] Au, S.K. and Beck, J.L., Estimation of Small Failure Probability in High Dimensions by SubsetSimulation, Probabilistic Engineering Mechanics, 2001, Vol. 16, No 3:263-277.

[9] Valdebenito, M.A. and Schueller G.I., Reliability-Based Optimization Using a Decoupling Approachand Reliability Estimations. Proc. 6th International Congress on Industrial and Applied Mathe-matics (ICIAM), Zurich, Switzerland, July, 2007.

[10] Schenk, C.A., Pradlwarter, H.J. and Schueller, G.I., On the Dynamic Stochastic Response of FE-Models. Probabilistic Engineering Mechanics, 2004, Vol. 19:161-170.

[11] Jensen, H.A., Valdebenito, M.A. and Schueller G.I: , An Efficient Reliability-Based Optimization

Scheme for Uncertain Linear Systems Subject to General Gaussian Excitation. Computer Methodsin Applied Mechanics and Engineering, in press, 2008

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