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

Approximating sensitivity of failure probability in reliability-based design optimization Liu, Ke; Paulino, Glaucio H.; Gardoni, Paolo Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Approximating Sensitivity of Failure Probability in Reliability-BasedDesign OptimizationKe LiuGraduate Student, School of Civil and Environmental Engineering, Georgia Institute ofTechnology, Atlanta, USAGlaucio H. PaulinoProfessor, School of Civil and Environmental Engineering, Georgia Institute ofTechnology, Atlanta, USAPaolo GardoniAssociate Professor, Dept. of Civil and Environmental Engineering, Univ. of Illinois,Urbana-Champaign, USAABSTRACT: This paper presents an efficient numerical method for approximating the parameter sen-sitivity of the failure probability with respect to design parameters. The method is computationally in-expensive and the obtained approximations are more accurate than the approximations based on firstorder reliability method (FORM). The method is particularly suitable for applications in reliability-baseddesign optimization (RBDO), including reliability-based topology optimization (RBTO).1. INTRODUCTIONReliability analysis has been considered as a partof the standard design procedure to help engi-neers make safe and reliable designs subject tothe inevitable randomness and uncertainties in na-ture. On the other hand, design optimization au-tomates the design process and guides the deci-sion makers to find efficient use of resources andmaximize performance. Therefore, incorporatingreliability-based design within the framework ofdesign optimization becomes a natural goal for en-gineers and researchers. Reliability-based designoptimization (RBDO) is often formulated as op-timization with probabilistic constraint(s). Thereare a variety of approaches for solving such kindof problems in the literature using either gradient-free or gradient-based algorithms. For problemsthat can be described with differentiable functions,gradient-based routine is preferable due to its com-putational efficiency.The challenge of using gradient-based optimiza-tion algorithm for RBDO is the evaluation of fail-ure probability and its sensitivity with respect to de-sign parameters, which affects the limit state func-tion(s). Although numerical estimation of fail-ure probability has been studied extensively, theliterature on the numerical approximation of theparameter sensitivity is limited. Some analyticalwork can be found around the year of 1990. Ho-henbichler and Rackwitz (1986) developed the pa-rameter sensitivity of the estimated failure prob-ability obtained by first order reliability method(FORM). The FORM-based expression is compu-tationally efficient to evaluate, thus it is widely usedin RBDO, but we should be careful when using thisexpression since the approximation can be quite in-accurate for cases where the limit state function isnonlinear. Breitung (1991) and Uryasev (1994) de-rived the analytical expression for the parametersensitivity of the exact failure probability. How-ever, such expression is in integral form and there-fore, precise numerical evaluation of the integralis not likely to be computationally tractable. Ap-proximations of the sensitivity is needed to performRBDO.There are two major formulations for RBDO112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015in the literature: the Reliability Index Approach(RIA), which explicitly uses the gradient of prob-abilistic constraint(s) in the optimization; and thePerformance Measure Approach (PMA), whichconstructs target performance function(s) as equiv-alent deterministic constraint(s) by the inverse re-liability analysis, thus the gradient of probabilisticconstraint(s) is involved implicitly (Tu et al. 2001;Cheng et al. 2006). Currently, the two approachesare mostly implemented in conjunction with FORMand FORM-based expression for the sensitivity offailure probability. Although FORM-based approx-imations for the sensitivity of the failure probabilityare not directly used in the PMA, the PMA essen-tially shares the same approximations of the sen-sitivity with RIA, if the probabilistic constraint(s)is(are) active when the optimization converges (Tuet al. 2001). Another type of gradient-based ap-proach employs the sample average approximation(SAA) where the failure probability and its sen-sitivity calculation are both carried out based onMonte Carlo simulations (MCS) (Royset and Polak2004). Although MCS-based can yield accurate es-timations, they have very high computational costs.The main purpose of this work is to providean alternative method that is more accurate thana FORM-based approximation and require signif-icantly less computational cost than MCS-basedmethods. The proposed method called Segmen-tal Multi-point Linearization (SML) is developedto estimate the sensitivity of the failure probabil-ity with respect to design parameters in componentreliability analysis. The method can be directly em-ployed in the framework of RIA, enabling gradient-based algorithms to be used in RBDO.2. FORM-BASED APPROACHES FORRBDO, REVISITEDIn this section, the two most used approaches ofRBDO, namely FORM-based RIA and PMA arerevisited via the Karush-Kuhn-Tucker (KKT) opti-mality conditions, in order to demonstrate the im-portance of the accuracy of sensitivity approxima-tion. Consider a generic formulation of RBDOproblems with one reliability component:minxf (x)s.t. Pf =∫G(u,x)<0ϕn(u)du6 Ptf (1)h(x)6 0where Ptf is the target failure probability; x is thevector of design variables; u is the vector of ran-dom variables that are transformed from the origi-nal distribution space to standard normal space bya probability preserving transformation; G(u,x) isthe limit state function in standard normal randomspace; ϕn(·) is the multi-variate standard normalPDF for n random variables; and h(x) is a set ofdeterministic constraints such as lower and upperbounds of x. Equivalently, the constraint on failureprobability can be expressed in terms of the gener-alized reliability index β =Φ−1(1−Pf ).Mathematically, the KKT optimality conditionsof the optimization model as described in (1) wouldbecome:(1) Stationary condition:∇x f +λ∇xPf +∑γi∇xhi = 0(2) Primal feasibility:Pf −Ptf 6 0, hi 6 0 ∀i(3) Dual feasibility:λ > 0, γi > 0 ∀i(4) Complementary slackness:λ (Pf −Ptf ) = 0, γihi = 0 ∀iwhere λ and γi’s are the Lagrange multipliers. TheKKT conditions are necessary for the solution to beoptimal. In RBDO, for most cases both the valueand sensitivity of the probabilistic constraint cannot be evaluated exactly, thus the KKT conditionsare only approximately satisfied at the optimum of anumerical solution. The more accurate the approx-imations are, the closer the solution is to the realoptimum.In the RIA formulation, the reliability constraintis considered directly. The sensitivity of failureprobability with respect to design parameters isused to find the search direction in optimization.212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The FORM-based expression by Hohenbichler andRackwitz (1986) is given as:∇xPf ≈ ∇xPf ,1 =−ϕ(β1)‖∇uG∗‖∇xG∗ (2)where G∗ denotes the limit state function evalu-ated at the design point u∗ as defined in Eq. (3),G(u∗,x), with current design x.u∗ = argminu{‖u‖ | G(u,x) = 0} (3)Thus the KKT stationary condition becomes:∇x f +λRIA[−ϕ(β )‖∇uG∗‖∇xG∗]+∑γi∇xhi = 0(4)This expression is an approximation of the KKTstationary condition, the error of which can be quitelarge when the limit state function G is nonlinear.The FORM-based PMA formulation applies aninverse FORM reliability analysis. The approachdefines a target performance function Gt(x) =G(x,ut) and incorporates it as a deterministic con-straint, where ut is an estimation of the design pointof the optimal design and is updated at each itera-tion asut = argminu{G(u,x) | ‖u‖= β t =Φ−1(1−Ptf )}(5)The KKT stationary condition of the PMA is:∇x f +λPMA(−∇xGt)+∑γi∇xhi = 0 (6)Equations (4) and (6) become the same if theprobabilistic constraint is active and the designpoint is unique, that is, ut = u∗ and ∇xGt = ∇xG∗(Tu et al. 2001). Given that the probability of failureare both approximated by FORM, the KKT condi-tions of RIA and PMA become identical. Hence,although PMA tends to be more robust than RIA, itdoes not improve the numerical result of the opti-mization.Many algorithms, which are developed based onRIA and PMA incorporate SORM, MCS or otherreliability methods to heuristically improve the ap-proximation of Pf (i.e. the primal feasibility con-dition) (Royset et al. 2006; Nguyen et al. 2011),but little attention has been paid to the accuracyof the sensitivity which can be more influential inthe search of the optimal solution. Furthermore,the error in the sensitivity is cumulative because itdetermines the search direction at each iteration ofgradient-based optimization schemes.3. THE METHOD OF SEGMENTALMULTI-POINT LINEARIZATIONIt can be found in Breitung (1991) and Uryasev(1994) that the analytical expression of the sensi-tivity of failure probability with respect to designparameters has the following form:∇xPf =−∫Sϕn(u)‖∇uG‖∇xGdS (7)where S denotes the limit state surface. In mostcases, the surface integral in Eq. (7) is not nu-merically tractable to compute since it is a multi-dimensional surface integral. We propose an effi-cient numerical method for approximation of thissurface integral. The method is developed based onlocal linearization of the limit state surface.If the surface S is composed of a set of pieces ofhyperplanes described by linear functions, Eq. (7)can be analytically simplified to probability evalua-tion problems. The idea of the proposed method isto fit the limit state surface with plane segments ina piecewise manner. Then we can perform the inte-gration on each hyperplane segment without mucheffort and compute their summation as the approx-imation. Denote the limit state surface as S and thepiecewise linear fitting as S¯ where each of the planesegment is denoted as S¯i. For each segment S¯i,which is assumed to be described by a linear func-tion G¯i, the gradients ∇uG¯i and ∇xG¯i will be con-stant vectors on the corresponding surface. Thus,they can be taken out of the integral, which leads tothe expression:∇xPf ,i =−1‖∇uG¯i‖∇xG¯i∫S¯iϕn(u)dS¯i (8)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Because all fitting segments are pieces of hyper-planes, each hyperplane segment then has piece-wise linear boundaries, thus its geometry appears aspolygons. To further simplify Eq. (8), we can thenrotate the coordinates of the standard normal spacesuch that the positive direction of the first axis isalong the opposite direction of the normal directionof the plane. As the function of the hyperplane islinear, the normal direction of the hyperplane is inthe same direction of ∇uG¯ as shown in Fig. 1. Dueto the rotational symmetry of the standard normalspace, we can rewrite Eq. (8) by separating coordi-nate u′1 from the integral, i.e.:∇xPf ,i =−ϕ(bi)‖∇uG¯i‖∇xG¯i∫S¯iϕn−1(uˆ′)duˆ′ (9)1u 1u’uˆuˆ’OiG∇u iSibABAB 1u’uˆ’Figure 1: Illustration of the calculation of Eq. (7) on ahyperplane segment.where bi is the distance from the origin to the plane,and uˆ = [u′2,u′3, ...,u′n]T. The surface integral nowis simplified to the volume integral of probability.Assuming we have a proper piecewise linear fittingof the limit state surface where each piece of planesegment is representative of a portion of the limitstate surface, we should be able to construct an ap-proximation of ∇xPf based on Eq. (9) that has thefollowing form of weighted sum:∇xPf ≈p∑i=1Wi∇xG¯i (10)whereWi =−ϕ(bi)‖∇uG¯i‖∫S¯iϕn−1(uˆ′)duˆ′. (11)In particular, if we use only one hyperplane tofit the limit state surface which is defined by takinga tangent expansion at the design point, then fromEqs. (10) and (9) we can obtain Eq. (2), indicatingthat FORM-based approximation is a special caseof the proposed method. It is important to noticethat, due to the exponential decay of the probabil-ity density in the standard normal space, we onlyneed to focus on the region that is close to the ori-gin where the probability density is high. Accord-ing to Eq. (7), the integrand becomes too small tomake an impact on the overall integration when itis evaluated far from the origin. In addition, as weincrease the number of fitting plane segments, theaccuracy of the approximation can be improved.A proper fitting scheme is essential to the ap-proximation. In general, each plane segment canbe completely defined by a fitting point, the normaldirection of the plane, and the boundary of the seg-ment. A straightforward thought is to let G¯i be thefirst order expansion of the limit state function Gat the corresponding fitting point ui, which ensuresfirst order accuracy. Therefore, ∇xG¯i = ∇xG(ui,x)and ∇uG¯i = ∇uG(ui,x), where x is the current de-sign. This leads to a tangent fitting scheme. If thelocally most central points are selected to be the fit-ting points, the tangent fitting has the same approx-imation of the limit state surface as the multi-pointFORM (Ditlevsen and Madsen 2007), but here theapproximation of the limit state surface is primarilyused to calculate the sensitivity of failure probabil-ity rather than the failure probability itself.However, in high dimensional random space, thetangent fitting scheme makes it quite difficult totrack the boundaries of the planes segments. More-over, it can be affected a lot by possible perturbedlocal information of the limit state function. Toovercome these challenges, we can project the gra-dient of the limit state function at a fitting point inthe random space to a prescribed direction and de-fine the hyperplane segment by the projected gra-dient. Since the design variables x are not in therandom space, the value of ∇xG¯i will remain as the412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015first order expansion of G(u,x). That is, we take∇uG¯i = (nT∇uG(ui,x))n where n is the prescribednormal direction for the plane segment, and stillkeep ∇xG¯i = ∇xG(ui,x). Hence, this compromisewould only affect the computation of the weights.However, if the angle between the gradient vec-tor ∇uG(ui,x)) and normal direction n is large, thecomputed weight can be erroneous. For example, inan extreme case, if the angle becomes 90◦, the cor-responding weight will be infinitely large. Thereare many alternative ways to specify the normals,and different choice of the normals leads to differ-ent fitting schemes.An orthogonal fitting (OF) scheme is developedfor a simple and quick construct of the approxima-tion. The basic idea is to fit the limit state surfacewith plane segments that have normals along an or-thogonal basis of the space. The general procedureis described as follows:(1) Select a reference point on the limit state sur-face;(2) Rotate the coordinates such that the referencepoint lies on the positive part of the first axis ofthe new coordinates;(3) Search for intersection points of the new axesand the limit state surface within radius r =k1b1 from the origin in both positive and neg-ative directions, where k1 is a user defined pa-rameter and b1 is distance from the referencepoint to the origin;(4) Define plane segment i by its fitting point withthe normal ni being the direction of the axis onwhich the fitting point lies;(5) For the half axis ±e′j, that has no intersectionpoint within the search region, a plane segmentwith normal along e′1 direction is fitted at theoff-axis point u j (denote as u j+n for −e′j) withcoordinate ±k2b1e′j +b je′1.The values for k1 and k2 are based on heuris-tic rules. The parameter k1 determines the sizeof the search region for the intersection point,and it is suggested to have the value such thati n+u(u ,x)i+n∇uGiG(u,x) = 0 uSearch region2 1k b2 1ηk bi∇uG(u ,x)1∇uG(u ,x)iu’1u’1uO1bie’1-e’1-e’1 1k bFigure 2: Illustration of orthogonal fitting SML.ϕ(k1b1)/ϕ(b1) = ε where ε is a small value (e.g.ε = 0.1) to ensure that the search region is largeenough while the intersection fitting points are alsoclose to the origin. On the other hand, k2 is toensure that the off-axis fitting points stay not toofar from the origin. It is generally good to setk2 be the minimum of 1 and 3/b1, which is thesame rule as in the point-fitting SORM proposed byDer Kiureghian et al. (1987) except that the refer-ence point is not necessarily the design point. Inaddition, as shown in Fig. 2, a partition coeffi-cient η is used to determine the boundaries of theplane segments determined by the reference pointand off-axis fitting points. In practice, η can takethe value between 0.5 and 1.0.The choice of the reference point also influencesthe accuracy of the SML approximation obtainedby this fitting scheme. Intuitively, the referencepoint should be close to the origin. The design pointis a good candidate for the reference point, how-ever, in some particular cases, other choices of thereference point would actually make the approxi-mation more accurate than using the design pointas the reference point.In addition, it is worth noticing that the methodalso provides an approximation of Pf associatively.Based on the existing approximation of the limitstate surface, computing the approximate failureprobability is a very light task. The output is ob-served to be generally better than FORM. Thus,in an implementation of RBDO, one can get theapproximation of the gradient of failure probabil-ity and the failure probability itself using the seg-512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015mental multi-point linearization method. One mayalso couple other reliability methods, for exam-ple SORM, point-fitting SORM and MCS with theSML method for approximation of the failure prob-ability, and only use SML for approximating thegradient of failure probability.4. NUMERICAL EXAMPLES4.1. Preliminary InvestigationThis example is to show the accuracy of the estima-tion obtained by the proposed method. We considera typical quadratic limit state function defined withoriginal random variables v:g(v,x) = x3− v3− x2v22− x1v21 (12)where x1, x2 and x3 are the design parameters. Weassume that the random variables have the samemarginal standard normal distribution. In addition,v2 and v3 are correlated with a correlation coeffi-cient of 0.2 while the other random variables aretaken to be statistically independent. We will com-pare the estimations of sensitivity by (1) the pro-posed method, (2) FORM-based expression (i.e.Eq. 2) and (3) the MCS-based approximation pro-posed by Royset and Polak (2004). We use themaximum number of simulations as suggested inRoyset and Polak (2004), which equals 25000. Wecan expect that the MCS-based approximations arevery close to the actual values. In addition, the cor-responding approximations of the reliability indexby the three methods are compared as well. Thedesign point is selected to be the reference point inthe SML method using the suggested OF scheme.Because the sensitivity of failure probability withrespect to design parameters is essentially a gradi-ent vector, the vector direction and vector length(L2-norm) of the approximate sensitivity are thetwo factors that determine its accuracy. However,in most optimization problems, the direction ofthe gradient vector is of most interest, because thesearch direction at each iteration in an optimizationprocess is determined by the direction of the gra-dient and most modern gradient-based optimizationalgorithms employ techniques to adaptively find theproper step length (e.g., line search) (Ascher andGreif 2011). We define an angle γ that measuresthe relative angle between the approximations ofthe gradient and the actual gradient. In the exam-ple, the MCS-based approximation is taken as theactual gradient, hence the angle γ for MCS-basedmethod is assumed to be 0. The magnitude ofthe angle γ provides the information about the er-ror in the vector direction. Thus, γ is small whenthe direction of an approximate gradient is accu-rate. Given a nonlinear limit state function g withx3 = 3.0 and x2 = 0.15, at different values of x1,Fig. 3 shows the measures of γ for the approxima-tions by FORM-based expression and the proposedmethod. We can observe that the SML method yieldsmaller error in vector direction of the approximategradient than the commonly-used FORM-based ap-proximation, i.e. Eq. (2). In addition, Fig. 4shows the approximations of reliability index β bythe three methods. Overall, although the accuracyof the approximations by the proposed method isnot as high as MCS-based method, it is much bet-ter than traditional FORM/FORM-based approxi-mation. In terms of the computational cost, theSML method is just a little bit larger than the eval-uation of FORM/FORM-based method, but muchless than MCS.−0.45 −0.3 −0.15 0 0.15 0.3 0.450102030405060708090  FORM(Eq. 2)SML−OF1xγFigure 3: The angle γ between the approximate gradi-ents and the actual gradient of ∇xPf .4.2. Application to RBTOAmong various RBDO problems, the reliability-based topology optimization is one of the mostchallenging task. We conduct the RBTO on a612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015−0.45 −0.3 −0.15 0 0.15 0.3 0.451.822. FORMSML−OFMCSβ1xFigure 4: Approximations of reliability index β .ground structure based elastic formulation wherewe are trying to find the structure with minimal vol-ume by sizing a large number of potential mem-bers (Christensen and Klarbring 2008; Zegard andPaulino 2014). The final topology can be used toindicate the optimal structural layout. The limitstate function appears as a threshold on the totalcompliance, which is reciprocal to the overall stiff-ness of the structure. Due to the nature of topologyoptimization, the structural layout indicated by thesolution can be highly affected by the accuracy ofsensitivity approximation. On the other hand, theevaluation of the gradient of compliance, which isnecessary for the computation of the gradient of thefailure probability, is quite computationally expen-sive since it requires FE analysis, thus MCS-basedapproximation is not applicable.Consider a layout design of a crane in 2D. Thedesign domain (ground structure) and boundaryconditions of the problem is shown in Fig. 5. Thestatistics of the random variables are shown in Ta-ble. 1. The correlations coefficients (C.C.) be-tween loads are included to make the design con-dition more practical. Uncertainty in material prop-erty (i.e. Young’s Modulus) is also considered andmodeled as a random variable with lognormal dis-tribution since negative value of E is not physicallyadmissible.The example is performed using the approxi-mations of Pf and its sensitivity by FORM-basedmethod and the proposed SML method separately.Figure 5: Design domain, initial ground structure andboundary conditions.Table 1: Statistics of Random VariablesVariable Distribution µ σ C.C.V1 Normal -5 1 0.2V2 Normal -3 2H1 Normal 0 3 0.3H2 Normal 0 3E Lognormal 100 10 0.0Again, the design point is the reference point inSML. The target reliability index is 3.0 for bothcases. The obtained topologies are shown in Fig.6a and Fig. 6b. A simple cutoff strategy is adoptedto obtain the members shown in the final topology(Zegard and Paulino 2014). The local stability andequilibrium in the plotted layouts may not be sat-isfied, which is inherent from this kind of topologyoptimization approach. A crude MCS is performedwith c.o.v = 5% to check the actual reliability of thestructure. We can observe that not only the optimalvolumes and actual reliability indices are differentfor the two cases, but the structural layout is alsodifferent. The conclusion is that the RBTO basedon the approximations by SML method yields astructural layout that possesses more efficient use ofmaterial. In topology optimization, the final topol-ogy usually highly depends on the values of sen-sitivities. Thus comparing to traditional FORM-based approach, the SML-based approach is morelikely to produce result that is closer to the exactsolution of a RBTO problem since the sensitivity712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015approximation is more accurate.(a)(b)Figure 6: Optimal topology by RBTO using (a) FORM-based method and (b) SML method. (a) Volume = 650,βMCS = 2.48; (b) Volume = 696, βMCS = 2.91.Cuto f f = 0.01.5. CONCLUSIONSThe proposed method can be used as a general toolfor reliability analysis, and it is suitable for a va-riety of RBDO problems especially when the ac-curacy of the parameter sensitivity is essential forconvergence to an optimal solution (e.g., RBTO).The benefit of low computational cost enables themethod to be applied to practical engineering prob-lems. The theory behind the method is general andother fitting schemes can be developed to better ap-proximate the sensitivity of the failure probabilitywith respect to design parameters.6. REFERENCESAscher, U. M. and Greif, C. (2011). A First Course inNumerical Methods. Society for Industrial and Ap-plied Mathematics, Philadelphia, PA, USA.Breitung, K. (1991). “Parameter sensitivity of failureprobabilities.” Reliability and Optimization of Struc-tural Systems ’90, A. Der Kiureghian and P. Thoft-Christensen, eds., Vol. 61 of Lecture Notes in Engi-neering, Springer Berlin Heidelberg, 43–51.Cheng, G., Xu, L., and Jiang, L. (2006). “A sequen-tial approximate programming strategy for reliability-based structural optimization.” Computers and Struc-tures, 84(21), 1353 – 1367.Christensen, P. and Klarbring, A. (2008). An Introduc-tion to Structural Optimization. Solid Mechanics andIts Applications. Springer.Der Kiureghian, A., Lin, H., and Hwang, S. (1987).“Secondorder reliability approximations.” Journal ofEngineering Mechanics, 113(8), 1208–1225.Ditlevsen, O. and Madsen, H. (2007). Structural Relia-bility Methods. Technical University of Denmark.Hohenbichler, M. and Rackwitz, R. (1986). “Sensitiv-ity and importance measures in structural reliability.”Civil Engineering Systems, 3(4), 203–209.Nguyen, T. H., Song, J., and Paulino, G. H. (2011).“Single-loop system reliability-based topology opti-mization considering statistical dependence betweenlimit-states.” Structural and Multidisciplinary Opti-mization, 44(5), 593–611.Royset, J., Der Kiureghian, A., and Polak, E. (2006).“Optimal design with probabilistic objective and con-straints.” Journal of Engineering Mechanics, 132(1),107–118.Royset, J. and Polak, E. (2004). “Reliability-based op-timal design using sample average approximations.”Probabilistic Engineering Mechanics, 19(4), 331 –343.Tu, J., Choi, K. K., and Park, Y. H. (2001). “Design Po-tential Method for Robust System Parameter Design.”AIAA Journal, 39, 667–677.Uryasev, S. (1994). “Derivatives of probability functionsand integrals over sets given by inequalities.” Journalof Computational and Applied Mathematics, 56(1–2),197 – 223.Zegard, T. and Paulino, G. (2014). “GRAND - groundstructure based topology optimization for arbitrary2D domains using MATLAB.” Structural and Mul-tidisciplinary Optimization, 1–22.8


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