International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

A sampling-based RBDO algorithm with local refinement and efficient gradient estimation Lacaze, Sylvain; Missoum, Samy; Brevault, Loïc; Balesdent, Mathieu Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015A Sampling-based RBDO Algorithm with Local Refinement andEfficient Gradient EstimationSylvain LacazePh.D. candidate, Aerospace and Mechanical Engineering, The University of Arizona,Tucson, AZ, USSamy MissoumAssociate Professor, Aerospace and Mechanical Engineering, The University of Arizona,Tucson, AZ, USLoïc BrevaultPh.D. candidate, ONERA – The French Aerospace Lab, Palaiseau, FranceMathieu BalesdentResearch Engineer, ONERA – The French Aerospace Lab, Palaiseau, FranceABSTRACT: This article describes a two stage Reliability-Based Design Optimization (RBDO) algo-rithm. The first stage consists of solving an approximated RBDO problem using meta-models. In orderto use gradient-based techniques, the sensitivity of failure probabilities are derived with respect to hyper-parameters of random variables as well as, and this is a novelty, deterministic variables. The secondstage focuses on the local refinement of the meta-models around the first stage solution using general-ized “max-min” samples. The approach is demonstrated on three examples including a crashworthinessproblem with 11 random variables and 10 probabilistic constraints.1. INTRODUCTIONReliability-based Design Optimization (RBDO) isnow a well established field that has lead to a largearray of techniques (Liu and Der Kiureghian, 1991;Tu et al., 1999; Du and Chen, 2002; Liang et al.,2007). The main hurdles in the development of anefficient RBDO algorithm are:• The need to accurately assess one or sev-eral probabilities of failure using a reasonablenumber of calls to potentially expensive func-tions (e.g., finite element analysis). This dif-ficulty is further exacerbated when the limitstates are nonlinear and the probabilities offailure are small (i.e., rare event).• Because of the curse of dimensionality, the ef-ficiency of a RBDO algorithm will degrade no-ticeably as the dimensionality increases.• Finally, noisy behaviors (e.g., sample based re-liability assessment techniques) present a se-rious challenge for RBDO algorithms sincegradient-based and surrogate techniques be-come limited.In order to partly address these issues, this arti-cle introduces an RBDO algorithm which relies ona sequence of RBDO sub-problems in which limitstates are approximated using meta-models. Eachlimit state approximation is locally refined using adedicated adaptive sampling scheme referred to asthe generalized “max-min" (Lacaze and Missoum,2014). This sampling scheme enables the selec-tion of samples in sparse regions while followingthe distributions of the random variables. Basedon this approximation of the limit states, probabili-ties can be estimated using Subset Simulations (Auand Beck, 2001). In order to use gradient-based ap-proaches, gradient estimates are used. While gra-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015dient estimates with respect to distribution hyper-parameters are well known (Lebrun and Dutfoy,2009; Song et al., 2009; Dubourg et al., 2011; Leeet al., 2011), this work introduces gradient esti-mates with respect to deterministic variables as de-rived in Lacaze et al. (2014).2. BACKGROUND2.1. Reliability-based Design OptimizationIn its most complete form, a Reliability-based De-sign Optimization (RBDO) problem can be ex-pressed as:minz,θC(z,θ) (1)s.t. P[g(i)(X,z)≤ 0]≤ PTi i = {1, . . . ,np}hinj (z,θ)≤ 0 j = {1, . . . ,nc}heqk (z,θ) = 0 k = {1, . . . ,ne}lzm ≤ zm ≤ uzm m = {1, . . . ,nz}lθn ≤ θn ≤ uθn n = {1, . . . ,nθ}X∼ fX(x|θ)where C is some cost function, and g(i) is the ithlimit state involved in the calculation of the associ-ated probabilistic constraint. f j and hk are the jthinequality and the kth equality deterministic con-straints respectively. z represents the deterministicdesign variables, and θ are hyper-parameters of theX random variable distribution fX.The proposed RBDO algorithm is specificallydeveloped for problems characterized by poten-tially highly nonlinear limit states g(i), thus mak-ing moment-based approaches such as First OrderReliability Method inaccurate.2.2. Subset simulationsMonte Carlo Simulation (MCS) is notoriouslyknown to be an accurate and robust approach forprobability calculation such as the one involvedin (1). However, MCS can prove inefficient in in-tensive loop or in case of rare event estimation.Among the variance reduction techniques proposedover the years, the Subset Simulation (SubSim,Au and Beck, 2001) has shown reliable results.SubSim starts from a well-known fact that for agiven probability of failure (e.g., 10−4), a coeffi-cient of variation of about 10% can be achieved us-ing 10− log10(Pf )+2 (e.g., 106) samples. Therefore,if a probability of, for example, 10−4 could be ex-pressed as the product of 4 probabilities of 10−1, anefficient estimation could be achieved using 4×103samples instead of 106.For the sake of clarity and without loss of gener-ality, the derivation are made for a generic probabil-ity of failure P [g(X,z)≤ 0]≡ Pf (z,θ)≡ Pf . Givena failure domain Ω f , let Ω f0 ≡ Ω ⊃ Ω f1 ⊃ ·· · ⊃Ω fm ≡ Ω f be a decreasing sequence of m+ 1 fail-ure domains where, ∀i = {1, . . . ,m}:Ω fi(z)≡Ω fi = {x|gi(x,z)≤ 0} (2)a probability of failure defined as:Pf =∫ΩI [g(x,z)≤ 0] fX(x|θ)dx (3)can be expressed as:Pf =m∏i=1Pfi (4)where:Pf1 =∫ΩI [g1(x,z)≤ 0] fX(x|θ)dx (5)and for ∀i = {2, . . . ,m}:Pfi =∫ΩI [gi(x,z)≤ 0]qi−1(x|θ ,Ω fi−1)dx=∫ΩI [gi(x,z)]≤ 0I [gi−1(x,z)≤ 0]∏i−1j=1 Pf jfX(x|θ)dx=∫ΩI [gi(x,z)≤ 0]∏i−1j=1 Pf jfX(x|θ)dx (6)with qi−1(x|θ ,Ω fi−1)the conditional auxiliaryPDF associated to the failure domain Ω fi (Songet al., 2009).However, although less time consuming thanMCS, SubSim still requires a prohibitive amountof calls to the limit state functions g(i) for compu-tationally expensive models (e.g., Finite Element).In order to address this issue, this work uses meta-models.212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20152.3. Meta-modelsOver the years, meta-models based approacheshave found favor in fields such as optimization(Jones et al., 1998; Basudhar et al., 2012), reliabil-ity assessment (Bichon et al., 2008; Bourinet et al.,2011) and RBDO (Bichon et al., 2009; Dubourget al., 2011).In this work Kriging models (a.k.a., Gaussianprocesses) are used (Rasmussen and Williams,2006). However, the tools developed would be ap-plicable to other meta-models such as Support Vec-tor Machines (Vapnik, 2000). For the scope of thispaper, the cost and deterministic constraint func-tions are assumed analytical. This leads to a newexpression of (1), later referred to as RBDO sub-problem:minz,θC(z,θ) (7)s.t. P[g˜(i)(X,z)≤ 0]≤ PTi i = {1, . . . ,np}...The core of this work stems in a sequential ap-proach where the limit state approximations g˜i arelocally refined around the solution to (7) dbest =[zbest ,θ best](i.e., the current optimum) at each it-eration. The key elements of this approach are de-tailed in the next section.3. METHODOLOGYThe first step is to construct initial limit state ap-proximations (Section 3.1). Then for each itera-tion, the sub-problem (7) is solved as discussed inSection 3.2, using SubSim (Section 2.2). The limitstate meta-models are refined (Section 3.3) arounddbest until convergence (Section 3.4). This is illus-trated on Figure 1.3.1. Design of experimentsThe design of experiments (DOE) refers to an ini-tial arbitrary selection of training data. The DOEis of crucial importance as it is the first explorationof the design and probabilistic spaces. One of themost efficient DOE techniques is called CentroidalVoronoi Tesselation (CVT) (Ju et al., 2002). Essen-tially, CVT applies a k-mean type of clustering to a,StartCompute DOESection 3.1Train limit stateapproximations g˜(i)Section 2.3Solve the currentsub problem (7)dbest =[zbest ,θ best]Section 2.2, Section 3.2Converged?Section 3.4StopRefine g˜(i) locallyaround dbestSection 3.3yesnoFigure 1: Flowchart of the proposed methodology.relatively large, sample of points (ideally represen-tative of the population). An advantage of this prin-ciple is that CVT can be specialized to any givendistribution by simply drawing samples from saiddistribution. Figure 2 highlights this feature.3.2. Sub-problem optimizationAt each iteration, the subproblem (7) must besolved using the current limit state approxima-tions. In this work, gradient-based optimizationtechniques are favored due to their convergenceproperties and their scalability when gradients canbe obtained.3.2.1. Gradient estimationIn the sub-problem (7), the probabilistic con-straints:P [g˜i(X,z)≤ 0]≤ PTi (8)are noisy, as the probabilities are estimated usingSubSim (Section 2.2). Therefore, gradient esti-mates are required. For the sake of clarity andwithout loss of generality, the derivation are made312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015x10 0.2 0.4 0.6 0.8 1x200. 2 C VTDxT-4 -3 -2 -1 0 1 2 3 4xO-4-3-2-101234E	 	  Figure 2: Two CVTs: for uniform distribution and de-terministic design (left) and correlated Gaussian distri-bution (right).for a generic probability of failure P [g(X,z)≤ 0]≡Pf (z,θ)≡ Pf .It is well known that probability of failure sensi-tivities with respect to distribution hyper-parameterθ can easily be obtained for Monte-Carlo simula-tions (Lebrun and Dutfoy, 2009):∂Pf∂θ j=∫ΩI [g(x,z)≤ 0]dln fXdθ jfX(x|θ)dx (9)and SubSim (Song et al., 2009). In this work weintroduce a recent derivation for the estimation ofprobability of failure sensitivities with respect todeterministic variables z.Starting from the definition of Pf :Pf (z) =∫ΩI [g(x,z)≤ 0] fX(x|θ)dx (10)it follows that:∂Pf∂ zk=∫Ω∂∂ zkI [g(x,z)≤ 0] fX(x|θ)dx (11)From the theory of distributions, the derivative ofthe indicator function is:dI [y≥ 0]dy= δ [y] ={+∞ if y = 00 else(12)where δ is the Dirac distribution (“impulse").Hence, (11) becomes:∂Pf∂ zk=−∫Ω∂g∂ zk∣∣∣∣x,zδ [g(x,z)] fX(x|θ)dx (13)A straightforward estimate for MCS follows:∂Pf∂ zk≈−1NN∑i=1∂g∂ zk∣∣∣∣X(i),zδ[g(X(i),z)](14)where X ={X(1), . . . ,X(N)}is a CMC sample ofsize N distributed according to fX.One can show that for SubSim (Lacaze et al.,2014):∂Pf∂ zk= Pf (z)m∑i=11Pfi(z)∂Pfi∂ zk(15)with:∂Pf1∂ zk=−∫Ω∂g1∂ zk∣∣∣∣x,zδ [g1(x,z)] fX(x|θ)dx (16)and for i≥ 2:∂Pfi∂ zk=−∫Ω∂gi∂ zk∣∣∣∣x,zδ [gi(x,z)]qi−1(x|θ ,Ω fi−1)dx−Pfii−1∑j=11Pf j∂Pf j∂ zk(17)Equations (14), (16), and (17) involve the Diracdistribution in their calculation. These estimates aretherefore numerically intractable unless a Dirac ap-proximation is used. Numerical experiments in La-caze et al. (2014) show that a Gaussian distributionshould be favored. The scale parameter σ is cal-culated such that a fraction α of the failure sam-ples, referred to as relevant samples Nr, are within±σ . Lacaze et al. proposed that a value of α = 0.5should be used.3.2.2. Numerical considerationThe RBDO sub-problem (7) requires a certain at-tention to numerical implementation. First of, prob-abilities are logarithmic quantities, which can im-pair the convergence. Therefore, working with thelog of the Pf would be advised. For historic rea-sons, the reliability index β = −Φ−1(Pf ) is typi-cally used. The probabilistic constraints in (7) arere-expressed as:βTi−βi ≤ 0 (18)On the other hand, SubSim is a sample basedtechnique, which induces noise in the constraints.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Furthermore, using gradient estimates in place ofanalytical ones can further halve convergence of theoptimization. For this reason a shift sh in the con-straint in combination with a relaxation of the con-straint violation tolerance is used. Ideally, a shiftequal to the variance of the reliability techniquesshould be used. Although this could be achievedwith MCS, SubSim does not provide an analyticalestimate of its variance. In this work, a shift ofsh= 10% is used. The constraints (18) are rewrittenas:−Φ−1[(1− sh)Pf]−βi ≤ 0 (19)These constraints are more restrictive but the con-straint violation tolerance is set to sh.3.3. Adaptive samplingAt each iteration, as described in Section 3.2, thesub-problem (7) solution (i.e., the current optimum)dbest =[zbest ,θ best]is found. In order to achievea local update, generalized “max-min” samples areused to refine the meta-models, around the afore-mentioned current optimum.The generalized “max-min” sample was intro-duced for reliability assessment (Lacaze and Mis-soum, 2014) and used for RBDO Lacaze and Mis-soum (2013) with hyper-parameters θ only. Thissample is found such that it follows a given distri-bution while maximizing the distance to the closestexisting training sample:maxxfX(x|θ best) 1nx minj∣∣∣∣∣∣w−w( j)∣∣∣∣∣∣ (20)s.t. g˜(i)(x,zbest) = 0where nx is the number of random variables X,w = [zbest ,x], and w( j) = [z( j),x( j)] is the jth train-ing sample at the current iteration.At each iteration, np generalized “max-min” arecalculated (one per probabilistic constraint). Thesesamples are searched such that z= zbest , as it is thecurrent optimum and θ = θ best , as it is the currentoptimal distribution hyper-parameters.3.4. ConvergenceAs the proposed approach is vastly inspired fromsequential optimization techniques, a hard conver-gence criterion is considered:ρH =∣∣∣∣∣C(dbest)−C(dbestlast)C(dbestlast)∣∣∣∣∣≤ εH (21)where dbest is the current optimum[zbest ,θ best]anddbestlast the previous optimum. In this work, a value ofεH = 10−2 is used.In addition, it is crucial to track the local ac-curacy of the meta-models used in the probabilis-tic constraints. Dubourg et al. (2011) introduceda very elegant criterion that is used in this work.Using the prediction variance from the Kriging (orthe Probabilistic SVM value), one can compute a95% confidence interval on the probability Pi =P[g˜(i)(X,z)≤ 0]at dbest . The order of magnitudebetween these two bounds is given by:log10(P+iP−i)(22)where P+i and P−i are the upper and lower boundson Pi respectively (for the limit state approximationg˜(i)). Therefore the following criteria is used:ρK = maxi[log10(P+iP−i)]≤ εK (23)In this work, a value of εK = 1.5×10−1 is used.4. RESULTSIn order to check convergence, an actual optimumis calculated for each examples. In this work, ev-ery optimization problem is solved using the MAT-LAB implementation of SQP in fmincon. That ac-tual optimum is obtained by solving the problem (1)(actual limit states are used) using the elements pre-sented in Section Example 1: 1 probabilistic constraintThe first test case is taken from Aoues andChateauneuf (2009). The problem, as shown in Fig-ure 3(a), is defined as:minzz21 + z22 (24)s.t. P [z1z2X2− ln(X1)≤ 0]≤ 10−20≤ z≤ 15X1 ∼N (5,0.32) X2 ∼N (3,0.32)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015x12 4 6 8 10 12 14x22468101214P [  (z  X) 5 0] 5 10!2Optimum(a) Example 1.31; X10 2 4 6 8 1032;X2012345678910Cg1g2g3Optimum(b) Example 2.Figure 3: Overview of example 1 and 2.The actual optimum for this problem is found atz? = [1.36,1.36]. An initial DOE of 40 points tai-lored to the X distribution is used. z0 = [12,12]is used as starting point to be consistent. Fig-ure 4 shows the evolution of the current optimumalong with the convergence metric presented inSection 3.4. Convergence is achieved in 16 itera-tions (56 function calls). For reference, Aoues andChateauneuf (2009) reported 39 function calls us-ing the Single Loop Approach (SLA).4.2. Example 2: 3 probabilistic constraintsThe second test case is taken from Aoues andChateauneuf (2009) for β = 4. The problem, asshown in Figure 3(b), is defined as:minθθ1 +θ2 (25)s.t. P[g(i)(X)≤ 0]≤ 3.17×10−50≤ θ ≤ 10 i = {1,2,3}X1 ∼N (θ1,0.32) X2 ∼N (θ2,0.32)with:g(1)(X) =X21 X220−1g(2)(X) =(X1−X2−5)230+(X1−X2−12)2120−1g(3)(X) =80X21 +8X2 +5−1The actual optimum for this problem is found atθ ? = [3.62,3.65]. An initial DOE of 10 points isused. θ 0 = [5,5] is used as starting point to be con-sistent. Figure 5 shows the evolution of the currentoptimum along with the convergence metric pre-sented in Section 3.4. Convergence is achieved in 4iterations (42 function calls). For reference, Aouesand Chateauneuf (2009) reported 81 function callsusing SLA.4.3. Example 3: 10 probabilistic constraintsThe final example is a car side impact crash-worthiness analysis. It was initially introduced inGu et al. (2001). The formulation used in this workcomes from Youn et al. (2004) such as:minθW (θ) (26)s.t. P[g(i)(X)≤ 0]≤ 10−1 i = {1, . . . ,10}0.5≤ θ ≤ 1.5X j ∼N (θ j,0.03) j = {1, . . . ,7}Xk ∼N (0.345,0.0062) k = {8,9}Xl ∼N (0,102) l = {10,11}An initial DOE of 70 points is used. Figure 6 showsthe evolution of the current optimum along with theconvergence metric presented in Section 3.4. Con-vergence is achieved in 22 iterations (920 functioncalls). For reference, Zou and Mahadevan (2006)used a decoupled approach (without meta-models)that can be adapted to several reliability techniques.Using Zou and Mahadevan’s approach, 5,256 and31,550 function calls were reported using SORMand AIS respectively.5. CONCLUSIONSThe proposed RBDO approach is based on a se-quence of approximated sub-problems solved usinga gradient-based optimization. One of the key fea-tures of this work is the derivation of the sensitiv-ities of probabilities of failure with respect to bothrandom and deterministic variables. The results onthree analytical test cases show promising conver-gence properties. Future studies involve the testingof the adaptive sampling scheme with different jointdistributions. In addition, the authors will investi-gate modification of the algorithm such as an activeset strategy to further reduce the computational bur-den. Indeed, the approximations of failure domainsleading to very low probabilities of failure mightnot need to be refined.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Iteration0 5 10 15 20Pbest2468101214z1z2(a) Evolution of zbest with respect tothe number of iterations.123Op2ti m2 4 6 8 10 12 14 16 18 203;u05101520253035404550(b) Evolution of ρH with respect tothe number of iterations.123Op2ti m2 4 6 8 10 12 14 16 18 203;012345678910(c) Evolution of ρK with respect tothe number of iterations.Figure 4: Convergence plot for example 1.123Op2ti m0 1 2 3 4 53;XCg0123456789103u3(a) Evolution of zbest with respect tothe number of iterations.123Op2ti m1 1.5 2 2.5 3 3.5 4 4.5 53;u05101520253035404550(b) Evolution of ρH with respect tothe number of iterations.123Op2ti m1 1.5 2 2.5 3 3.5 4 4.5 53;012345678910(c) Evolution of ρK with respect tothe number of iterations.Figure 5: Convergence plot for example 2.123Op2ti m0 5 10 15 20 25 303;XCg0.333(a) Evolution of zbest with respect tothe number of iterations.123Op2ti m5 10 15 20 25 303;u05101520253035404550(b) Evolution of ρH with respect tothe number of iterations.123Op2ti m5 10 15 20 25 303;012345678910(c) Evolution of ρK with respect tothe number of iterations.Figure 6: Convergence plot for example 3.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20156. 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