A two-level Kriging-based approach with active learning for solving time-variant risk optimization problems
AA Two-Level Kriging-Based Approach with Active Learning forSolving Time-Variant Risk Optimization Problems
H.M. Kroetz , M. Moustapha , A.T. Beck , and B. Sudret Structural Engineering Department, University of S˜ao Paulo, Av. Trabalhador S˜ao-Carlense, 400,13566-590 S˜ao Carlos, SP, Brazil Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Stefano-Franscini-Platz 5, 8093Zurich, Switzerland
Abstract
Several methods have been proposed in the literature to solve reliability-based optimizationproblems, where failure probabilities are design constraints. However, few methods address theproblem of life-cycle cost or risk optimization, where failure probabilities are part of the objectivefunction. Moreover, few papers in the literature address time-variant reliability problems in life-cycle cost or risk optimization formulations; in particular, because most often computationallyexpensive Monte Carlo simulation is required. This paper proposes a numerical framework forsolving general risk optimization problems involving time-variant reliability analysis. To alleviatethe computational burden of Monte Carlo simulation, two adaptive coupled surrogate models areused: the first one to approximate the objective function, and the second one to approximate thequasi-static limit state function. An iterative procedure is implemented for choosing additionalsupport points to increase the accuracy of the surrogate models. Three application problemsare used to illustrate the proposed approach. Two examples involve random load and randomresistance degradation processes. The third problem is related to load-path dependent failures.This subject had not yet been addressed in the context of risk-based optimization. It is shownherein that accurate solutions are obtained, with extremely limited numbers of objective functionand limit state functions calls.
Keywords : Risk-Based Optimization; Time-Dependent Reliability; Adaptive Kriging
Different approaches have been proposed in the literature to solve design optimization prob-lems considering structural reliability. In reliability-based design optimization or RBDO (Hilton a r X i v : . [ s t a t . C O ] J u l nd Feigen, 1960; Frangopol, 1985; Lopez and Beck, 2012a; Hu and Du, 2014), a determinis-tic objective function involving material and manufacturing costs is minimized under reliabilityconstraints. This approach is a natural extension of deterministic design optimization, wheredeterministic constraints are replaced by probabilistic design constraints. A different problemis obtained when structural reliability is part of the objective function. In life-cycle cost orrisk optimization (Moses, 1977; Enevoldsen and Sorensen, 1994; Beck and Gomes, 2012; Toriiet al., 2019), the objective function is formulated in terms of total expected costs, which includesexpected costs of failure. These, in turn, are given by the product of failure costs by failure prob-abilities. Risk optimization allows one to find the optimal point of balance between safety andeconomy in structural designs. Risk optimization also allows different failure modes to competewith each other.Comprehensive literature reviews (Beyer and Sendhoff, 2007; Schu¨eller and Jensen, 2009;Aoues and Chateauneuf, 2010; Lopez and Beck, 2012a) reveal that the RBDO problem hasreceived much more attention than the life-cycle cost or risk optimization problems. Severalvery efficient methods have been proposed for solving RBDO. In particular, several methods weredesigned to overcome the nested optimization loops arising from the use of First Order ReliabilityMethod (FORM) for structural reliability evaluation. In contrast, not much is found in theliterature about solving risk optimization problems. Moreover, it is worthwhile to emphasize thatthe underlying reliability problems are time-variant , due to the presence of stochastic loading,strength degradation (corrosion, fatigue), consideration of inspection and maintenance, etc.,which adds another level of complexity.Assessing the reliability of engineering structures under random load processes, and withconsideration of resistance degradation, requires time variant reliability formulations. Unfortu-nately, analytical or semi-analytical solutions of time-variant reliability problems are limited tovery specific cases (Melchers and Beck, 2018). The up-crossing rate solution is limited to scalarloads with Gaussian distribution. The out-crossing rate solution is limited to polyhedral failuredomains. Fast probability integration is subject to convergence problems. Load combinationsolutions are mainly limited to discrete (pulse-like) processes. Time integrated or extreme valuesolutions neglect resistance degradation, and so on. Hence, most often time-variant reliabilityproblems have to be solved by Monte Carlo simulation. This has a significant impact in com-putational costs, which makes the outer optimization loop impractical. Hence, general methodsfor solving time-variant risk optimization problems shall involve: a) speeding Monte Carlo simu-lation via dedicated techniques; and/or b) using surrogate models to simplify (approximate) theunderlying time-variant reliability problem.With respect to the first point, Gomes and Beck (2016) proposed a Monte Carlo-basedmethod which involves finding the roots of the limit state function, in the design space, for each ample. Rashki et al. (2014) and Okasha (2016) proposed efficient solutions for risk optimizationinvolving random design variables. These solutions are based on the ranked weighted averagesimulation of Rashki et al. (2012). Regarding the second point, Echard et al. (2011) proposed anactive learning method, combining Kriging and Monte Carlo simulation, for reliability analysis.A similar approach was employed in RBDO by Dubourg et al. (2011a), Moustapha et al. (2016).Wang and Chen (2016) presented an equivalent stochastic process transformation approach forsolving general time-variant reliability problems. This approach was employed by Li et al. (2018)to solve RBDO problems.Based on the above observations, this paper proposes a general procedure for solving time-variant risk optimization problems, based on adaptive Kriging (Jones et al., 1998; Echard et al.,2011; Sch¨obi et al., 2017). The proposed scheme has some similarities with Li et al. (2018);however, herein it is applied for solving time-variant risk optimization problems. Moreover,equivalent stochastic process transformation is not employed herein. Instead, stochastic pro-cesses are explicitly evaluated as time series, allowing different features of time variant reliabilityproblems to be addressed. This includes load-path dependency, where failure is not character-ized by a point in random load space, but by the whole trajectory of the loads. The drawbackof this approach is that Monte Carlo simulations are necessary. To alleviate the computationalburden, the proposed approach includes construction of two levels of adaptive Kriging surrogatemodels. The first level approximates the objective function, allowing for the consideration ofdifferent cost terms. The second level approximates the limit state function associated to eachcost term of the objective function. This two-stage modeling allows different time-variant risk-optimization problems to be addressed, as shown in the examples section. Different expectedimprovement functions are conveniently employed to select additional support points for eachsurrogate model. The efficiency and accuracy of the proposed technique are illustrated in typicaltime-variant reliability problems, which include random loading, random strength degradation,with discrete or continuous random processes, system-reliability and load path-dependency. In a context where structures degrade in time, or when loads are described as stochastic pro-cesses, it may be important to calculate not only instantaneous probabilities of failure, but theprobability of a failure occurring within a certain time interval, sometimes referred to as the cumulative probability of failure in the literature. Consider a set X ( t, ω ) of M = p + q elementsrepresenting the uncertainties of a given problem, where X j ( ω ), j = { , . . . , p } are randomvariables, typically describing geometric characteristics and material properties, and X k ( t, ω ), k = { p + 1 , . . . , p + q } are random processes. In this notation, ω is the outcome in the space of utcomes Ω. Moreover, let d be a vector that gathers together all the system’s design param-eters. It may include parameters describing moments of random variables, in case toleranceson design dimensions are included in the analysis (Moustapha, 2016). A limit state function g ( d , t, X ( t, ω )) defines, for a given d , safe states if it is greater than zero and failure if it issmaller than zero, so that the boundary between desirable and undesirable structure responsesis given by the limit state surface of equation g ( d , t, X ( t, ω )) = 0: D f ( d , t ) = { d , X ( t, ω ) : g ( d , t, X ( t, ω )) ≤ } is the failure domain, D s ( d , t ) = { d , X ( t, ω ) : g ( d , t, X ( t, ω )) > } is the safe domain. (1)For each limit state of the problem, the instantaneous probability of failure P f i at a time t = τ is calculated as: P f i ( d ; τ ) = P ( g ( d , τ, X ( τ, ω )) ≤
0) = (cid:90) D f ( d ,τ ) f X ( x )d x , (2)where P ( • ) indicates the probability of the event • and f X is the joint probability densityfunction of the random variables X for a given configuration d at a time τ .In the problems studied herein, the quantity of interest is the so-called cumulative probabilityof failure P fc ( t , t ) which is defined for a given configuration d as the probability of occurrenceof a structural failure within the time interval [ t , t ]: P f c ( d ; t , t ) = P ( ∃ τ ∈ [ t , t ] : g ( d , τ, X ( τ, ω )) ≤
0) (3)Solutions to time-variant reliability problems include (Melchers and Beck, 2018): the out-crossing approach, the time-integrated approach, the fast probability integration, the nestedFORM approach, directional simulation, and specific load combination solutions for discretepulse-like load processes. These solutions are either approximate, or very specific to particu-lar configurations of the problem. The out-crossing approach is analytical only for Gaussianload processes; for general continuous processes, out-crossing rates need to be approximatedby a parallel system sensitivity formulation (Andrieu-Renaud et al., 2004; Sudret, 2008). Thefast probability integration of Wen and Chen (1987) leads to FORM-like solutions, which arepotentially unstable due to very small conditional failure probabilities. The nested FORM ap-proach of Madsen and Tvedt (1990) applies only to linear or polyhedral failure domains. Thedirectional simulation approach (Melchers, 1992) requires derivation of conditional strength dis-tributions, which are difficult to evaluate. The time-integrated approach involves extreme-valueanalysis which is valid only for scalar load processes, and which neglects any strength degrada-tion. Moreover, the approaches named above do not address solution of load-path dependentproblems. In these problems, failure at a given time τ does not depend on the specific combina-tion of loads for time τ , but on the whole time-trajectory of all loads up to time τ (see Melchers nd Beck (2018) for details). Thus, the only general method for solving time-variant reliabilityproblems is plain, or brute force Monte Carlo Simulation (MCS). But MCS is known to leadto very high computational burden, for problems involving small failure probabilities. Hence,in order to make plain MCS viable in the solution of time-variant optimization problems, thispaper proposes extensive use of surrogate modeling. The adopted simulation approach basically consists in drawing sample trajectories of the limitstate function over the time interval of interest, and then counting the number of such trajec-tories for which failure occurs within each time step. In order to do so, the random processesinvolved in the problem must first be discretized, i.e. represented by a finite set of correlatedrandom variables (Sudret and Der Kiureghian, 2000). In this work, the expansion optimal linearestimation (EOLE) method, after Li and Der Kiureghian (1993), is employed.Let X ( t, ω ) be a scalar Gaussian random process, with mean m ( t ), standard deviation σ ( t )and autocorrelation coefficient function ρ X ( t , t ). An arbitrary number of time points P areselected in the interval [0 , T ], so that t = 0 and t P = T . The EOLE expansion is then givenby: X ( t, ω ) ≈ m ( t ) + σ ( t ) r (cid:88) i =1 ξ i ( ω ) √ λ i φ Ti C t,t i ( t ) , (4)where { ξ i ( ω ) , i = 1 , . . . , P } are independent standard normal variables, { φ i , λ i , i = 1 , . . . , r } arethe eigenvectors and eigenvalues of the correlation matrix C sorted in decreasing order, with C ij = ρ X ( t i , t j ) , i, j = { , . . . , P } . The order of the expansion is defined by the number of terms r ≤ P that are kept after truncating the series. One usually chooses r in such a way that asignificant part of the spectrum of C is retained, i.e. for an ε (cid:28) r = min k ∈ [1 ,..,P ] (cid:110) k, k (cid:88) i =1 λ i ≥ (1 − ε ) tr C (cid:111) (5)where tr C = (cid:80) Pi =1 λ i is the trace of the correlation matrix.Once the random processes are discretized, it is possible to sample trajectories of the limitstate function itself. Consider the limit state g ( d , t, X ( t, ω )) for a given d in the time interval[0 , T ]. Samples of the random processes X k ( t, ω ) , k = { p + 1 , . . . , p + q } are built from the EOLEexpansions, and the time independent random variables X j ( ω ) , j = { , . . . , p } are sampled onceand remain the same throughout the whole trajectory. Let G be an array of length N , where N is the number of time instants in which the continuous time is discretized. The values obtainedin the simulation are stored in this array, where each position i = 1 , ..., N corresponds to a time t i = ( i − · ∆ t , where ∆ t = T N − is the sampling step, considering a uniform discretization. Foreach time interval [ t i , t i +1 ], a counter k i +1 is defined. Every time g presents the first outcrossing n the interval [ t i , t i +1 ], all the counters k n , with n = i + 1 , . . . , N are increased (i.e. all theremaining counters after the outcrossing are increased). A brute Monte Carlo estimation for thecumulative probability of failure until an arbitrary instant t i , i.e. P fc MC (0 , t i ), is given by: P fc MC (0 , t i ) = k i + k N MC , (6)where k counts the number of failures at t = 0. Defining a structural configuration which is safe and cost efficient at the same time is a challengefor the structural designer. Unfortunately, structures will always be associated with a probabilityof failure. When the optimal structural configuration is sought, it is important to optimize insuch a way that structural safety is not compromised. This is in general a demanding task, sincereducing the dimensions of structural elements tends to reduce structural reliability. Many ap-proaches have been proposed in this context, such as Deterministic Design Optimization (DDO)and Reliability Based Design Optimization (RBDO). Although these two formulation are mostoften addressed in the literature, they are known to be less general than Risk Optimization, wherelife-cycle costs are considered throughout the useful lifespan of a structure. A broader reviewand a comparison between the three approaches can be found in Beck and Gomes (2012). Liter-ature review papers (Aoues and Chateauneuf, 2010; Valdebenito and Schu¨eller, 2010; Lopez andBeck, 2012b) show that many shortcuts have been devised for solving nested optimization loopsin FORM-based solutions of RBDO problems. For Risk Optimization problems, however, onlytwo recent papers are known to address efficient solution schemes (Gomes and Beck, 2016; Toriiet al., 2019). In this work, a novel surrogate assisted strategy for the solution of time-dependentRisk Optimization problems is developed, so that general cases can be addressed.
This section aims to describe some important contributions in Risk Optimization, and to em-phasize the lack of efficient surrogate-based techniques addressing the specific complexities ineach step of this general structural optimization approach.For a complete representation of a maintenance planning scheme, the structural degradationthat materials tend to suffer over time must be modeled. The most important phenomena to betaken into account in this context, are corrosion caused by chemical agents in the case of concretestructures, and rust and fatigue in case of steel structures. Joanni and Rackwitz (2008) developedtools for optimizing design and maintenance strategies of aging structural components. Theydefine time-dependent failure models for deteriorating structural components, which are used n inspection and repair strategies considered in the life-cycle optimization objective function.The authors extend these results to optimal repair and retrofit of existing structures in Streicheret al. (2008), and present a renewal model for cost-benefit optimization including three differentmaintenance strategies (Rackwitz and Joanni, 2009).An interesting application was studied in Holicky (2009), where a parametric life-cycle opti-mization is performed for the design of road tunnels. The number of escape routes is optimizedin a Bayesian network framework, and some insights are given in the quantification of societaland economic consequences. The optimization makes use of societal risk, defined as a functionof an expected value of statistical life (SVSL), which is claimed to be an acceptable compensa-tion cost for one fatality. The life time of the structure and the expected number of causalities(dependent on parameters like the number of escape routes in the tunnel) are also considered.In this study, only one limit state and no maintenance or inspection costs are considered.Biondini and Frangopol (2009) perform optimization considering the corrosion of reinforce-ment of concrete structures in aggressive environments. The steel corrosion is modeled as afunction of time. The problem statement resembles that of RBDO problems: a deterministic ob-jective function regarding weight of concrete and steel, subjected to probability constraints. Thelife cycle meaning is given by the fact that constraint probabilities of failure are regarded as timedependent, and must be satisfied during the whole life time of the structure. The optimizationis performed with a gradient-based method, and the reliability constraints are evaluated withMonte Carlo simulation.Okasha and Frangopol (2009) proposed a procedure for optimizing not only maintenanceintervals, but also the choice between different maintenance actions. In a multi-objective opti-mization context, two performance indicators are taken into account: the system reliability indexand a measure of system redundancy. A digital code is generated for each maintenance scenario,including maintenance type, the structural component to which it applies, and a binary digitrepresenting the application or non application of such action. A genetic algorithm is then runto select the most favorable scenario. The time between each intervention is also optimized. Thestructures initial costs and the costs of failure are not optimized, since material and geometricalparameters of the structures are not considered to be design variables.A broad framework is presented by Taflanidis and Beck (2009), who propose techniques forthe estimation and optimization of the life-cycle cost for dissipative devices. The consideredlife-cycle costs include performance-based earthquake engineering considerations, initial, repairand replacement costs. A simulation approach called Stochastic Subset Optimization is proposedto establish a global sensitivity analysis for the performance using a relatively small number ofsystem analyses. A second step is suggested for the refinement of the solution, utilizing classicaloptimization algorithms. n 2010, another comprehensive review concerning structural optimization under uncertain-ties was published (Valdebenito and Schu¨eller, 2010). Surrogate models are entitled a section inthis review, where works with artificial neural networks, support vector machines and Krigingare cited. There is no mention of risk optimization, and all works regard the classic RBDOapproach.Papers without developments in the theory itself, but with life-cycle optimization applicationsare easily found in the literature. Wen and Kang (2001b) studied a nine-story office buildingsubjected to earthquake and wind loading. The hazards are described by a Poisson process, andthe methodology described in Wen and Kang (2001a) is employed. Saad et al. (2016) presenteda comprehensive application to reinforced concrete bridges, considering degradation over timeand use-associated costs. Li and Hu (2014) employed metaheuristics to solve a multi objectiverisk optimization problem. The problem is analyzed in the context of performance based windengineering. Elements of risk optimization theory are often used in wind turbines engineering,where optimum maintenance strategies are a key element for total costs reduction. An overviewof RO applications to this subject can be found in Nielsen (2014).Kroetz et al. (2019) developed a time-dependent procedure for the reliability assessment ofreinforced concrete beams subjected to corrosion. Even with application of efficient modelingtechniques, such as the boundary element method, the authors concluded that surrogate modelscould be necessary for more complex studies. The idea of utilizing surrogate models to aid inoptimization problems under uncertainty is not new (e.g. Adaptive Kriging employed in thesolution of RBDO problems (Bichon et al., 2013; Dubourg et al., 2011b)), but only recentlysurrogate models have been employed for the specific problem of risk optimization.In Gomes and Beck (2013), Artificial Neural Networks are trained to represent limit statefunction associated with structural failures, in a context where the optimization is carried outby a hybrid algorithm. Particle swarm optimization (a zero-order heuristic global optimizationalgorithm) scans the problem domain, and a gradient based algorithm is used to refine theresult. Only initial and expected failure costs are considered. The disregard of other costs,such as inspection and repair costs, seems to be a common practice in stochastic optimization.Aissani et al. (2014) explain that most of these costs can be more or less well estimated, butthe failure cost is particularly important, because it is difficult to evaluate and greatly affectsoptimal solutions. Therefore, other costs can be considered constant in some cases, which meansthat they do not affect the optimization process. In the same work, the idea of consideringnonlinear costs of failure is presented. In the proposed model, failure costs increase linearly withthe probability of failure up to a certain threshold, from which the costs increase exponentially.Ideally, this accounts for catastrophic failures which imply in further social or environmentaldamage. urrogate models were also employed by Carreras et al. (2016). A multi-objective problemconsidering a prototype of a building which is retrofitted through installation of insulation ma-terials is considered. Insulation material and thickness of the insulation layer are optimized, sothat the economic and the environmental performance of the building are optimum. An ob-jective reduction strategy is proposed, and cubic spline surrogates are employed to reduce thecomputational burden of the analysis. Life cycle optimization targeting for energy efficiencyhave also been performed with the aid of Support Vector Machines by Eisenhower et al. (2012).A general surrogate assisted stochastic optimization is proposed by Zhang et al. (2017). Asequential approximation optimization strategy is adopted, where the initial design problem isdecomposed into cycles of Kriging-based, sub-region constrained optimization problems. Adap-tive Kriging models are built from an augmented design of experiment (which considers bothdesign and random variables). This idea was extended to multi-objective stochastic optimizationproblems in Zhang and Taflanidis (2017).The explicit simulation of limit state equations related to the risk optimization problemthroughout the whole time-series of the analysis was not studied in any of the references, possiblybecause of the difficulties regarding the high computational cost of such procedure. In this work,specific adaptive surrogate models are efficiently built to address both the complexities presentin the steps of structural reliability analysis and global optimization. It is well known that Risk Optimization is more general than alternative approaches such asdeterministic design optimization and reliability based design optimization (Beck and Gomes,2012). When the total life-cycle cost of a structure is of interest, a comprehensive approachshould account for the expected cost of failure. Hence, risk optimization is employed herein. Inthis context, the function to be minimized is the total life cost C T ( d ), defined by: C T ( d ) = C I ( d ) + C O ( d ) + C I & M ( d ) + C EF ( d ) , (7)where d ∈ D is a given design configuration. This cost is composed of various terms, namely the I nitial design costs C I , O peration costs C O , I nspection and M aintenance costs C I & M , and the E xpected cost of F ailure C EF , defined as: C EF = N ls (cid:88) j =1 P f j C f j , (8)where j = { , . . . , N ls } enumerates different limit states associated with a possible failure thatoccurs with a probability P f j and whose cost is C f j . Design and reliability constraints can also e considered, so that the optimization problem can be cast as: d ∗ = arg min d ∈ D C T ( d ) , subject to: P f j ≤ ¯ P f j , j = { , . . . , N ls } , (9)where ¯ P f j are target failure probabilities that shall not be exceeded, for each limit state.Constraints are often unnecessary in this type of problem, since the probabilities of failure aredirectly defined in the objective functions. Although reliability constraints can be considered inorder to comply with standards, the solution domain D may also be limited by bound constraints.Ideally, the four cost terms in Eq. (7) should be addressed simultaneously, as for some struc-tures, the initial costs also depend on inspection and maintenance policies (Gomes et al., 2013;Gomes and Beck, 2014). For other structures, initial or construction costs have low dependencywith inspection and maintenance costs (Lee et al., 2004). Without loss of generality, and tomaintain simplicity, the problems addressed in this paper do not consider inspection and main-tenance costs. Expected failure costs, however, have strong dependency with initial costs, asconstruction costs affect both the fixed cost of failure and the failure probability. Expected costsof failure are largely dependent on the use of the structure and the surrounding environment;hence, they are very application dependent. In this paper, for simplicity and without loss ofgenerality, costs of failure are assumed as a multiple of initial costs. Initial or constructions costsvary with the amount and quality of structural materials, as shown in the examples section.Cost of workmanship may also be considered.Since the life cycle of a structure may be considered to span over years or decades, thecosts to be optimized cannot be directly treated. Economic changes over time would make theconsidered values unrepresentative. In order to account for this effect, the structural life timecan be discretized, and all costs brought to present value considering discount rates over eachperiod (e.g. yearly discount rates). This way, cumulative failure probabilities associated witheach given period can be considered to compose the expected cost of failure as follows (Saadet al., 2016): C P VEF ( T ) = N ls (cid:88) j =1 T (cid:88) n =1 P fc jn C f jn (1 + η ) n (10) C P VEF is the expected cost of failure in present value, P fc jn and C f jn are, respectively, the so-called cumulative probability and cost of failure of the j -th limit-state in year n , and η is thediscount rate, herein adopted as 1% per year. In the remainder of this paper, instead of Eq. (8),Eq. (10) will be used to compute the expected cost of failure in Eq. (7). Surrogate-aided optimization
The solution of the problem in Eq. (7) relies on optimization techniques which would usuallyrequire thousands of calls to the objective function C T . Furthermore, the evaluation of a singlecost C T ( d ) requires to solve a time-variant reliability analysis using Monte Carlo simulation.The associated cost amounts to millions of calls to the limit-state function. Solving naivelythis problem as introduced above would therefore be extremely time-consuming. This becomeseven more intractable when the limit-state function involves expensive-to-evaluate computationalmodels.To address this challenge, surrogate modeling is used in this paper. The basic idea is toreplace a time-consuming black box model by an analytical proxy that can be evaluated millionsof times at practically no cost. Several surrogate modeling techniques have been introduced inthe literature to solve optimization and reliability analysis problems, e.g. response surface models(Foschi et al., 2002), polynomial chaos expansions (Blatman and Sudret, 2008, 2010, 2011; Kroetzet al., 2017), support vector machines (Bourinet, 2018; Deheeger and Lemaire, 2007), artificialneural networks (Papadrakakis and Lagaros, 2002; Kroetz et al., 2017; Lehk´y et al., 2018) orKriging (Picheny et al., 2008; Viana et al., 2009; Dubourg et al., 2011a; Moustapha et al., 2016;Kroetz et al., 2017). In this work, we are interested in Kriging as it features a built-in errormeasure that arises from epistemic uncertainty and which allows for the development of activelearning techniques. Such techniques allow one to reduce the computational cost of building thesurrogate model by controlling its accuracy only in confined regions of the input space. A briefliterature review about Kriging and well-known adaptive bulding strategies is presented in thefollowing section, and the techniques adopted in this work are detailed in Appendix A. The idea of Kriging dates back to the decade of 1950, where it was first applied for improvementpredictions in the context of geostatistics. A historical review of the method is presented inCressie (1990). This technique has been used to aid in the solution of structural reliabilityproblems in the works of Romero et al. (2004) and Kaymaz (2005), and found many differentapplications since. Echard et al. (2011) proposed an adaptive learning method where the designof experiment from which the Kriging surrogate model is built is updated depending on a discreteevaluation of the interest region in reliability space. Reliability-based design optimization usingadaptive Kriging was addressed in the work of Dubourg et al. (2011a). In order to furtherreduce computational costs, Echard et al. (2013) coupled importance sampling Monte Carlo withKriging, in solution of structural reliability problems, showing that the technique is suitableeven when small probabilities of failure are present. A broad review about different Kriging pplications in the context of structural reliability can be found in Gaspar et al. (2014).The solution of global optimization problems requires several evaluations of objective func-tions, whose analysis can be time-consuming. Hence, such problems can also benefit from sur-rogate modeling. In this context, Efficient Global Optimization (EGO) was proposed by Joneset al. (1998), where a Kriging surrogate is adaptively built in such a way that exploration ofdesign space and exploitation of promising regions are balanced. Updated information regardingthe values of objective functions in a-priori selected points and the built-in Kriging varianceare considered, when selecting new points to compose the design of experiment of the surrogatemodel. Thus the so-called Expected Improvement Function is defined (See Appendix A for de-tails). Many recent studies on the subject derive from or directly apply the concept of EGO.Bouhlel et al. (2018) combined EGO with the partial least-squared method in order to addresshigher-dimensional problems. Roy et al. (2019) combined EGO with partial least squares and agradient-based method to develop an optimization algorithm, which was then utilized to solve acomplex topology optimization problem. Guzman Nieto et al. (2019) addressed design optimiza-tion of airframes using EGO in a multidisciplinary optimization context, where critical dynamicaeroelastic loads are estimated and the so-called modal strain energy coefficient is studied as anindicator of the necessity of further exploring the design space, should dynamic aeroelastic loadssignificantly change. Ariyarit et al. (2017) performed EGO with a multi-fidelity optimizationtechnique applied to design optimization of helicopter blades, searching for maximum blade effi-ciency. Carraro et al. (2019) proposed an adaptive scheme for selecting target variances, whichare then used as parameters to perform EGO in the design of a tuned mass damper.The consideration of uncertainties further complicates the problem, rendering even highercomputational costs. In order to address the complexity of reliability analysis, Bichon et al.(2008) adapts the general procedure of EGO in such a way that limit state equations are replacedby an adaptive Kriging metamodel. Reliability space is efficiently explored, so that the designof experiments is enriched considering both the proximity of candidate points to the limit stateequation and the variance of the Kriging surrogate. The resulting technique, known as EfficientGlobal Reliability Analysis (EGRA) was applied in the solution of RBDO problems (Bichonet al., 2009; Bichon, 2010). This strategy is thus far utilized in several reliability analysis andoptimization studies, either directly or as a key part of novel proposed techniques (Liu et al.,2019; Chaudhuri et al., 2019; Sadoughi et al., 2018; Wang and Ma, 2018).Li et al. (2018) present an interesting application of adaptive Kriging in time-dependentstructural optimization. Despite some similarities with this work, the authors perform RBDOthrough Stochastic Equivalent Transformation (Wang and Chen, 2016).In the present paper, Risk Optimization is performed in time-dependent problems. The entiretrajectories of limit state equations are considered, so that complete information is obtained from he analysis instead of only extreme values. There is no previous work, to the best knowledge ofthe authors, where adaptive Kriging is used in this context. Thus, a new strategy is proposedherein, where coupled adaptive Kriging surrogate models are iteratively built. The design ofexperiment of each metamodel is enriched according to different strategies, depending on thenature of model to be surrogated. In this paper, a nested adaptive surrogate modeling approach is proposed, for the first time, forsolving time-variant risk optimization problems. The solution scheme employs well-known EGOand EGRA formulations, described in the Appendix, but the application to time-variant riskoptimization problems is novel. Moreover, some of the problems addressed herein have not beensolved before, as they address stochastic load processes and stochastic strength degradation.The strategy addressed herein involves two nested loops: the inner loop involves determinationof cumulative probabilities of failure, which are used as inputs for the objective function, itera-tively evaluated as the outer loop searches for the optimum point. The result is a comprehensiveRisk Optimization solution framework, suitable to account for time-dependent loads, load-pathdependency and structural degradation. In this context, the strategy adopted for solving relia-bility problems is a key aspect. Considering that most known reliability assessment techniquesare imprecise or inadequate in this case, as detailed in Section 2, a Monte Carlo estimation forcumulative probability of failure is adopted, as given in Equation (6). The downside of thisapproach is increasing the computational cost involved in reliability analysis, which is here justone iterative step of the broader optimization algorithm. Thus, EGRA is adopted in this phasewith a tight convergence criterion, enforcing accuracy and alleviating the computational burden.This Kriging surrogate is built in the so-called augmented space, which combines both the designand random variables space, so that one single metamodel can be used to compute the failureprobability regardless of the current value of design parameters. This is particularly convenientsince reliability analyses must be performed for different design configurations, as another levelof surrogate model is assembled in the search for objective function minimum.The nature of variables which compose the total life-cycle cost in a Risk Optimization problemis very diverse: monetary costs, discount rates, probabilities of failure, material properties, andso on. Hence, the behavior of the objective function tends to be problem-specific. A generaltechnique to address this type of problems must be able to efficiently scan the design space,especially because each design point evaluation defines new limit state configurations. Thus,EGO is adopted to search for the global minimum. This way, an efficient multi-level application ofadaptive surrogate models is proposed to address the complexity of time-variant risk optimizationproblems. The proposed framework is illustrated in Figure 1 et value of parameters N EGRA , N EGO , N, 𝒯, max (cid:3084)
𝐸𝐼, max ℱ 𝐸𝐹𝐹Generate large sampleof 𝑁 (cid:3006)(cid:3008)(cid:3019)(cid:3002) points 𝒲
EGRA
Generate initial design 𝒳and evaluate exact LS 𝒢Train Kriging model 𝑔(cid:3556)Evaluate feasibility of candidate set 𝒲Chose next best point 𝑤 ∗ (cid:3404) arg max(cid:4666)ℱ(cid:4667)Check errorConverged?N Generate large sampleof candidate points 𝒮 ∈ 𝔻 EGO
Generate initial design 𝒟and evaluate 𝐶 (cid:3021) 𝑠 using 𝑔(cid:3556)Train Kriging model 𝐶(cid:4634) (cid:3021)
Evaluate expected improv. of candidate set 𝒮Chose next best point 𝑠 ∗ (cid:3404) arg max(cid:4666)ℰ(cid:4667)Check errorConverged? NY YFind minimum of 𝐶(cid:4634) (cid:3021) surrogate E v a l u a t e 𝑔 𝑤 ∗ i n a u g m e n t e d s p a c e a n d a dd t o D O E E v a l u a t e 𝐶 (cid:3021) 𝑠 ∗ u s i n g 𝑔 (cid:3556) a n d a dd t o D O E Generate random process models and discretize time
Figure 1: Proposed Framework
As shown in Figure 1, the proposed scheme is a combination of EGRA and EGO, with aMonte Carlo scheme for solving the time-variant reliability problem. Hence, the formulations inSections 2 and 3 are an inherent part of the solution scheme proposed in this paper.An important feature of the proposed framework is that the complete time series of involvedstochastic processes is obtained by simulation, with the only simplification being time discretiza-tion. This allows problems like load-path dependency to be addressed, as illustrated in Example3. In the next section, three novel examples of risk optimization considering random load pro-cesses and random strength degradation processes are addressed. To the best of the authorsknowledge, no similar examples have been solved before in a context of risk optimization. Ex- mples 1 and 2 are based on the literature, but the original references only address time-invariant(Blatman and Sudret, 2010) or time-variant (Sudret, 2008) reliability analysis. Consider a steel bending beam with rectangular cross-section { b , h } T and length L = 5 m ,submitted to dead loads ρ st b h (Nm − ), where ρ st = 78 . F( ω,t ) 𝑐 corroded area L sound steel Figure 2: Corroded beam under a midspan load, after (Sudret, 2008)
The beam is also subjected to corrosion, in such a way that the corrosion depth d c in all thefaces of the beam increases linearly with time, i.e. d c = κt . Moreover, it is assumed that thecorroded areas have lost all mechanical stiffness. The limit state function which describes theformation of a plastic hinge at midspan reads: g ( d , t, X ) = ( b − κt )( h − κt ) f y − ( F L ρ st b h L , (11)where the yield stress is denoted by f y . The analysis is carried out considering the time interval[0, 10] years. Three corrosion scenarios are considered. In the first one, corrosion kinetics iscontrolled by deterministic κ = 1 mm year − . The second scenario considers the corrosion rateas a random variable with mean of 1 mm year − and a coefficient of variation of 30%. In thethird scenario, the corrosion rate is considered as a discrete pulse process, with annual renewal,and mean intensity of 1 mm year − and coefficient of variation of 30%. In all scenarios, theload F is modeled as a Gaussian random process with mean 6 kN , coefficient of variation 0 . λ = 1 month. For all scenarios,the same inner surrogate model (cid:101) g is used, i.e. a part of the computational cost for obtaining asolution for different corrosion scenarios can be reduced. The random parameters are gathered n Table 1. The risk optimization problem is defined by Eq. (12) C T = C I + (cid:88) i =1 C f P fc i s.t. 0 . ≤ b ≤ . . ≤ h ≤ .
06 (12)The initial costs are related to the cross section of the beam C I = νb h , with ν = 1 / ,
000 times higher, i.e. C f = 1 , C I . In thisacademic example, a very high cost of failure is adopted, penalizing the unsafe regions of thedesign space, generating a total cost function with two very distinct regions (a high C T valuesregion and low C T near-plateau one), separated by a very steep transition, as shown in Figures3a, 3b and 3c. Approximating this objective function is a significant challenge to the Krigingapproximation. A monthly discount rate of 1% is also considered. The optimization problemconsists in determining d = { b , h } T that minimizes the total cost C T ( d ). Table 1: Corroded beam random variables and parametersParameter Distribution Mean COVSteel yield stress (MPa) Lognormal 240 10%Beam breadth (m) Lognormal b h Figures 3a, 3b, and 3c show contour plots of the cost functions for the three corrosion sce-narios. Note that the plane regions have very few contours, while in the steep regions theconcentration of contours is very high. The red squares, triangles and circles are the results of30 optimization runs for each case. A Particle Swarm Optimization (PSO) (Bansal et al., 2011)is also performed on the problem without the aid of surrogate models, considering 30 particlesper iteration in order to compare the results. The stopping criteria for the PSO is a toleranceof 10 − in the change of the value of all design variables. This result is represented by the greendiamond and serves here as reference. On average, for fixed, random variable and stochasticprocess corrosion rate, the objective function was called 23, 25 and 27 times, respectively, in thesolutions using the EGO approach, and 480, 510 and 510 times in the solutions using the PSOapproach. The results for total costs are compared in Figure 4. The box-plots show the optimalresults for C T obtained in the 30 runs of each case. Fairly precise results can be obtained usingthe proposed methodology, with a much smaller number of objective function evaluations. Thestochastic corrosion process version of this problem had never been solved before. a) Fixed κ (b) κ as a random variable (c) κ as a stochastic process Figure 3: Contour plot of the total costs for different design configurations. Red marks representthe solutions for 30 replications considering different cases for the beam example, while green marksrepresent the PSO solutions for each case.
Fixed Random Variable Stochastic Process11.11.21.31.41.51.61.7
Figure 4: Box-plot for total costs considering different modeling of the corrosion rate κ . (30 repli-cations of the analysis) A 2D truss structure is considered, as shown in Figure 5. It is composed by 23 bars and 13 nodes,and subjected to six vertical time-varying loads applied on the upper nodes. The magnitudesof all vertical loads are cast as a single stationary Gaussian process with mean value 50 kN,standard deviation 7 . λ = 1 year. There are three types of bars, with different cross-sectional areas andmaterials, as indicated in Figure 5. Circular bars are considered, and the design variables d , d and d are the radius of the three types of bars. The bars are subjected to corrosion, so thatthe radius of the cross section is decreased over time, following r c = κt . The radius at t = 0 is r i , and the current radius at any given time is r ( t ) = r i − r c ( t ) as shown in Figure 6. Figure 5: Corroded beam under a midspan load, adapted from (Blatman and Sudret, 2010) corroded area 𝑖 𝑐 Figure 6: Cross section
Three scenarios are considered for corrosion rate: in case κ = 10 µm year − is considered. In case κ RV , κ RV and κ RV are correlated random variables with mean 10 µm year − and COV of 30%.The correlation coefficient is set to 0.8. In case µm and COV of 30% and correlation coefficient between them of 0.8.Table 2 describes the remaining random variables of the problem. The limit state equation isdefined implicitly by a finite element model, and is written in terms of the vertical displacementof the mid-span node, herein denoted by V . A maximum allowed displacement of 0 . g ( d , t, X ) = 0 . − V ( d , t, X ) . (13)The time interval in which the analysis is carried out is [0, 30] years, so that the formulation of he risk optimization problem reads: C T = C I + (cid:88) i =1 C f P fc i , s.t. 0 .
02 m ≤ d ≤ .
04 m0 .
02 m ≤ d ≤ .
04 m0 .
02 m ≤ d ≤ .
04 m (14)The design costs are proportional to the area of the bars, i.e. C I = 10 ( d + d + d ), andthe cost of failure is obtained as C f = 10 C I . An annual discount rate of 1% is also considered.The results of optimum cost for 20 analyses are summarized in Figures 7 and 8. Figure 7 showsthe boxplot for each design variable on each case. Figure 8 gathers the corresponding optimumtotal costs. The methodology provides consistent solutions. On average, only 13, 13 and 14 costfunction evaluations were needed to reach a solution for each corrosion rate scenario, respectively. Table 2: Corroding truss random variables and parametersParameter Distribution Mean COV E ( M P a ) Lognormal 210,000 10% E ( M P a ) Lognormal 210,000 10% E ( M P a ) Lognormal 210,000 10% A ( cm ) Lognormal πr A ( cm ) Lognormal πr A ( cm ) Lognormal πr (a) Case Figure 7: Design solutions with 20 replications considering different cases for the truss example.19 ase
Figure 8: Optimum costs obtained for 20 runs of each corrosion rate case.
Consider the truss composed by circular bars 1 and 2, as shown in Figure 9. Two time-variantloads H ( t ) and V ( t ) are applied on the upper node. Three failure modes are considered: tensilerupture of bar 1 ( g t ), buckling of bar 1 ( g b ), and buckling of bar 2 ( g b ). Thus, a time-variantsystem reliability problem is defined considering the limit state equations associated to thesefailure modes: L HV L Figure 9: Two-bar truss scheme20 t ( X , t ) = A σ u − (cid:34) H ( t )2 cos α − V ( t )2 sin α (cid:35) g b ( X , t ) = π EI L − (cid:34) − H ( t )2 cos α + V ( t )2 sin α (cid:35) g b ( X , t ) = π EI L − (cid:34) H ( t )2 cos α + V ( t )2 sin α (cid:35) g sys ( X , t ) = min( g t , g b , g b )where A i is the area of the i -th bar in m and L is the length of the bars in m . The trussis symmetric. The two bars have the same Young Modulus E , defined as a normal randomvariable with µ E = 70GPa and COV E = 0 .
03, and the same ultimate tensile strength, definedas a normal random variable σ u , with µ σ u = 24 . COV σ u = 0 .
1. This value ofultimate stress was set so as to result in a tight compromise between the three different failuremodes. The probability that random variables reach negative values is very small and can beneglected in this example. This problem is load-path dependent, i.e. the structure can violatedifferent limit states or fail at different times depending on the trajectory that the loads followin time. To illustrate the load path dependent problem, consider that the radius of the crosssections are r = 4mm for the first bar, and r = 5 . µ X . Supposethat at time t = t the loads are at point A , and at t = t f > t , the loads correspond to point B . If the loads follow Path 1, the structure fails due to buckling of the first bar. If the loadsfollow Path 2, the horizontal load is increased first, and the structure fails by tensile rupture ofbar 1. Now, if the loads follow Path 3, which corresponds to a concomitant increase in bothloads, point B is safely reached, and there is no failure. Thus, the load-path dependency of theproblem is demonstrated. A B
Path 1 Path 2 Path 3
Figure 10: Load Paths
When the loads are stochastic processes, there is an infinite number of possible trajectories,and evaluating structural reliability depends on considering such trajectories, which only addscomplexity to the problem. Load-path dependent problems cannot be solved by usual techniques,such as time-integration (extreme value analysis) or load combination, as discussed in Melchersand Beck (2018). However, load path-dependent reliability problems can be solved by explicitsimulation of load process realizations, as proposed in this work.Consider now that one is interested in the optimal areas for the two bars, aiming at minimizingtotal costs in a risk-optimization scenario. Forces V ( t ) and H ( t ) are stochastic Gaussian processeswith means 1 kN and 2 kN, respectively. Both loads have a COV of 0.2 and a correlation lengthof λ V = λ H = 1 month. The auto-correlation function of the random processes is given by: R ( x, λ ) = exp (cid:20) − (cid:16) xλ (cid:17) (cid:21) (15)The loads are independent of each other and of the other random variables. A time interval f 10 years is studied, so that the objective function of the problem can be stated as: C T ( r , r ) = C I ( r , r ) + (cid:88) i =1 C f P fc i ( r , r )s.t. 4 mm ≤ r ≤ ≤ r ≤ C I ( r , r ) = 10 (cid:0) A ( r ) + A ( r ) (cid:1) L , and the cost of system failure is 10 times higher. An annual discount rate of 2% isalso considered. Different failure costs could be associated to different limit states, without anychange in the solution procedure.Table 3 shows the results for the optimization problem, comparing 10 runs of the approachproposed in this work (denoted by ’EGO’) and a reference obtained with 20 generations of 30particles of a PSO algorithm, performed without the aid of surrogate models. The standarddeviations of the obtained results are denoted between parenthesis. Table 3: Mean and COV of optimization results and reference r (mm) r (mm) C T N calls EGO 4.37(0.01) 5.32(0.01) 5.16(0.01) 17(5.1)PSO 4.35 5.29 5.13 600
As seen from table 3, the results obtained with both methodologies are remarkably consistent,with less than 1% discrepancy between the optimum design radii and associated total cost.
Expected life-cycle cost, or risk optimization, allows one to find the optimal points of compromisebetween safety and economy in structural desing. Typically, the underlying reliability problem istime-variant, and its solution is far from trivial. Problems involving strength degradation or load-path dependency usually require solution by Monte Carlo simulation, with a large computationalburden, especially in an optimization context. To address efficiently and accurately this type ofproblem, a nested Kriging approach with active learning is proposed in this paper. The strategyis based on constructing two adaptive Kriging surrogates. One surrogate is built so as to mimicthe objective (cost) function, starting from a design of experiment built with LHS in the spaceof the design variables, which is further enriched as the optimization problem is solved using theEGO approach. Another Kriging surrogate model is built for each limit state function, starting rom a first design of experiment built with LHS in the augmented space of both design andrandom variables. The surrogate is then enriched using the EGRA strategy.Three novel risk optimization problems have been addressed, involving stochastic processstrength degradation and stochastic process loading. These problems considered analytical andnumerical (finite element) limit states. A complex load-path dependent problem was also ad-dressed for the first time in an optimization context. Satisfactory accuracy and convergence wasobserved in all examples, with a few calls to the objective function. Solution cost was shown tobe approximately the same for three different models of degradation involving a deterministiccorrosion rate, a rate modeled by random variables, and by a random process.On the other hand, the number of evaluations of the inner surrogate model was found to beexcessively large for this strategy to be applied in problems that combine extremely low failureprobabilities together with time series that require a large number of discretization points. Fur-ther studies are necessary in order to adapt the method to this kind of problems, and to increasethe scope of the solution to involve dynamic problems that cannot be represented by pointwisesurrogates of the limit state equations. The proposed technique was capable of solving problemsinvolving stochastic strength degradation, time-dependent loads and load-path dependency. The first author thanks the Brazilian Council for Higher Education (CAPES) for funding a six-month research term at ETH Zurich ”Programa de Doutorado Sandu´ıche no Exterior”, throughgrant number: 88881.133186/2016-01. The third author aknowledges funding by CNPq (grantn. 306373/2016-5) and FAPESP (grant n. 2017/01243-5).
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A Kriging Basics
Kriging (Santner et al., 2003), also known as Gaussian process regression in machine learning(Rasmussen and Williams, 2006), is an emulator that considers the computational model toapproximate as one realization of an underlying Gaussian process: M ( x ) = p (cid:88) j =1 β j f j ( x ) + Z ( x ) , (17)where the two summands are a deterministic mean known as the trend and a zero-mean station-ary Gaussian process, respectively. The trend can take multiple forms yielding different typesof Kriging. Universal Kriging corresponds to the most general case when the trend is cast as alinear combination of a collection of p weights β = { β j , j = 1 , . . . , p } and regression functions f = { f j , j = 1 , . . . , p } . Ordinary Kriging, which corresponds to the special case when p = 1and f ( x ) = 1, is considered here. The Gaussian process Z ( x ) is defined by its auto-covariancefunction Cov [ Z ( x ) , Z ( x (cid:48) )] = σ R ( x , x (cid:48) ; θ ) where σ is the Gaussian process variance and R is an auto-correlation function with hyperparameters θ . The auto-correlation function is chosenbased on some assumptions about the degree of smoothness and regularity of the underlyingmodel. In this work, the Mat´ern 5 / R ( x, x (cid:48) ; θ ) = (cid:32) √ | x − x (cid:48) | θ + 53 ( x − x (cid:48) ) θ (cid:33) exp (cid:18) −√ | x − x (cid:48) | θ (cid:19) . (18) or multidimensional problems, the auto-correlation function is obtained as a tensor product ofthe unidimensional functions.The training of a Kriging model is first based on building an experimental design whichconsists of a set of input realizations X = (cid:8) χ ( i ) , i = 1 , . . . , n (cid:9) and their corresponding modelevaluations Y = (cid:8) Y ( i ) = M (cid:0) χ ( i ) (cid:1) , i = 1 , . . . , n (cid:9) . Given these data, a generalized least-squareestimate of the weights: (cid:98) β ( θ ) = (cid:0) F T R − F (cid:1) − F T R − Y (19)and the variance estimate: (cid:98) σ ( θ ) = 1 N (cid:16) Y − F (cid:98) β (cid:17) T R − (cid:16) Y − F (cid:98) β (cid:17) (20)can be derived. In these equations, F is a matrix gathering the regression functions evaluatedon the training points, i.e. F ij = f j ( χ ( i ) ) and R is the auto-correlation matrix defined such that R ij = R (cid:0) χ ( i ) , χ ( j ) ; θ (cid:1) .Once the model is trained, the prediction for any given new point x follows a normal distri-bution, i.e. (cid:102) M ∼ N (cid:16) µ (cid:102) M ( x ) , σ (cid:102) M ( x ) (cid:17) where the mean and variance respectively read: µ (cid:102) M ( x ) = f T ( x ) β + r T ( x ) R − (cid:0) y − F T β (cid:1) , (21a) σ (cid:102) M ( x ) = σ (cid:16) − r T ( x ) R − r ( x ) + u T ( x ) (cid:0) F T R − F (cid:1) − u ( x ) (cid:17) , (21b)with r ( x ) = (cid:2) R (cid:0) x , χ (1) (cid:1) , . . . , R (cid:0) x , χ ( n ) (cid:1)(cid:3) and u ( x ) = F T R − r ( x ) − f ( x ).It remains now to estimate the hyperparameters of the auto-correlation function. Variousmethods have been proposed to achieve this goal, among which cross-validation and maximumlikelihood estimation (Bachoc, 2013). In this work we consider the latter, which in fine consistsin solving the following optimization problem (Dubourg, 2011): (cid:98) θ = arg min θ ∈ n θ Ψ ( θ ) = (cid:98) σ ( θ ) | R ( θ ) | n , (22)where Ψ is the so-called reduced likelihood function , n θ is the number of hyperparameters tocalibrate and |•| here denotes the determinant operator.One of the most important aspects of Kriging is its variance (Eq. 21b) which can be seen asa local measure of the accuracy of the surrogate model. Typically, such information can be usedin active learning when one attempts to build a surrogate model by adaptively refining it so asto ensure sufficient accuracy in some regions of interest.Such regions depend on the analyst’s aim. In the case when the Kriging model is used toemulate an objective function in an optimization process, the regions of interest are areas oflocal minima (or maxima). In the case when the analyst is interested in reliability analysis, theKriging model replaces the limit-state function. The region of interest here corresponds to the icinity of the limit-state surface. Both cases have been widely studied in the literature under theframeworks of efficient global optimization and efficient global reliability analysis , respectivelyBichon et al. (2008, 2011). In this paper, both are of interest to us and are briefly described inthe sequel. A.1 Efficient Global Optimization (EGO)
Efficient global optimization has been introduced by Jones et al. (1998) as a means to solveoptimization problems while replacing the objective function with a Kriging surrogate. Thebasic idea is to make use of the Kriging variance so as to balance exploitation of areas wherethe surrogate is minimized and those where its variance is high (due to the lack of data). Thealgorithm starts by fairly sampling the input space and building an initial Kriging model. Thena merit function is used to decide the next point to add in the experimental design so as tobring in the most useful information for the location of the global minimum. Various meritfunctions have been introduced in the literature. We are interested here in the function used inthe contribution of Jones et al. (1998) and originally introduced by Mockus (1974), the so-called expected improvement function, which reads: EI ( x ) = ( y min − µ ˜ M ( x ))Φ (cid:32) y min − µ ˜ M ( x ) σ ˜ M ( x ) (cid:33) + σ ˜ M ( x ) ϕ (cid:32) y min − µ ˜ M ( x ) σ ˜ M ( x ) (cid:33) (23)where ϕ and Φ are the standard Gaussian PDF and CDF and y min is the current known minimum.This function is made of two complementary parts: the first part relates to the probability ofimprovement while the second one is proportional to the Kriging variance. By combining thesetwo aspects the expected improvement function achieves its goal of both exploiting and exploringthe design space. Hence, the next point to add in the experimental design is chosen as the onethat maximizes this function. In the original paper, Jones et al. (1998) uses an optimizationalgorithm, namely the branch-and-bound algorithm, to locate the global maximum. Here wesimply rely on an approximate stochastic (discrete) search. The EGO procedure is then thefollowing:1. Generate a large sample set N EGO of candidates for enrichment S = (cid:8) s (1) , . . . , s ( N EGO ) (cid:9) ,where s ( i ) ∈ D ;2. Generate an initial experimental design D = (cid:8) d (1) , . . . , d ( m ) (cid:9) and evaluate the correspond-ing costs C = (cid:8) C T (cid:0) d (1) (cid:1) , . . . , C T (cid:0) d ( m ) (cid:1)(cid:9) ;3. Train a Kriging model (cid:101) C T using the experimental design {D , C} ;4. Evaluate the expected improvement function using the candidate set S : E = (cid:8) EI (cid:0) s (1) (cid:1) , . . . , EI (cid:0) s ( N EGO ) (cid:1)(cid:9) ; . Choose the next best point as the one that maximizes EI on the set S : s ∗ = arg max s ∈S E ; (24)6. Check if the convergence criteria are met. If they are, skip to 8, otherwise continue withstep 7;7. Evaluate C T ( s ∗ ) and add the couple { s ∗ , C T ( s ∗ ) } to the experimental design {D , C} .Return to step 3;8. End the algorithm.For the applications in this paper, N EGO is set to 10 , and points in S are generated using LatinHypercube Sampling over the design space. The algorithm stops when max E EI ( s ) is lower thana given threshold, as suggested in Bichon (2010). In this paper, a threshold of 10 − was foundto be adequate for all studied examples. A.2 Efficient global reliability analysis (EGRA)
Even when using EGO, time-variant risk optimization problems may still be computationallyintractable. This is due to the necessity of running a full time-variant reliability analysis foreach cost evaluation C T ( d ) that relies on a possibly expensive-to-evaluate limit-state function.The direct approach to cope with this issue is to introduce a meta-modeling at this level, too.In the most general case, this could involve dynamic problems for which metamodeling is still achallenging issue (See for instance Mai et al. (2016) and Mai and Sudret (2017)). In this work,we limit our scope to time-variant problems where the limit-state function is characterized bytime-independent models (e.g. quasi-static problems). In such a case, a Kriging model (cid:101) g can bedirectly used to surrogate the limit-state function, hence further improving the efficiency of thesolution.The reliability counterpart of EGO has been introduced by Bichon et al. (2008) under thename of efficient global reliability analysis . It consists in adaptively building a surrogate model soas to ensure accuracy in the vicinity of the limit-state surface. Similarly to EGO, various meritfunctions, herein known as learning functions, have been proposed in the literature. For Krigingthis includes, among others, the expected improvement for contour estimation (Ranjan et al.,2008), the deviation number (Echard et al., 2011) or the margin probability function (Dubourget al., 2012). In this work, we consider the so-called expected feasibility function proposed by ichon et al. (2008), which reads: EF ( x ) = µ ˜ g ( x ) (cid:34) (cid:18) µ ˜ g ( x ) σ ˜ g ( x ) (cid:19) − Φ (cid:18) − σ ˜ g ( x ) − µ ˜ g ( x ) σ ˜ g ( x ) (cid:19) − Φ (cid:18) σ ˜ g ( x ) − µ ˜ g ( x ) σ ˜ g ( x ) (cid:19)(cid:35) − σ ˜ g ( x ) (cid:34) ϕ (cid:18) µ ˜ g ( x ) σ ˜ g ( x ) (cid:19) − ϕ (cid:18) − σ ˜ g ( x ) − µ ˜ g ( x ) σ ˜ g ( x ) (cid:19) − ϕ (cid:18) σ ˜ g ( x ) − µ ˜ g ( x ) σ ˜ g ( x ) (cid:19)(cid:35) +2 σ ˜ g ( x ) (cid:34) Φ (cid:18) σ ˜ g ( x ) − µ ˜ g ( x ) σ ˜ g ( x ) (cid:19) − Φ (cid:18) − σ ˜ g ( x ) − µ ˜ g ( x ) σ ˜ g ( x ) (cid:19)(cid:35) . (25)This function behaves in a similar way as the expected improvement: it takes high values whenthe evaluated point is close to the limit-state surface and/or when the Kriging variance is high.The next best point to add in the experimental design in order to refine the limit-state surface istherefore the one that maximizes Eq. (25). Instead of directly solving this optimization problem,we rely on an approximate discrete procedure as proposed by Echard et al. (2011) under theframework of active Kriging - Monte Carlo simulation (AK-MCS) (see also Sch¨obi et al. (2017)).The algorithm is as follows:1. Generate a large sample set N EGRA of candidates for enrichment W = (cid:8) w (1) , . . . , w ( N EGRA ) (cid:9) .It is worth mentioning here that these samples are drawn in the so-called augmented space ,a space that encompasses both design and random variables (Kharmanda et al., 2002; Au,2005; Taflanidis and Beck, 2008; Dubourg et al., 2011a). This allows us to build one sin-gle global metamodel that can be used for the reliability analysis regardless of the designchoice. Details on how such an augmented space is built here can be found in Moustaphaet al. (2016) ;2. Generate an initial experimental design X = (cid:8) x (1) , . . . , x ( N ) (cid:9) and evaluate the corre-sponding limit-state responses G = (cid:8) g (cid:0) x (1) (cid:1) , . . . , g (cid:0) x ( N ) (cid:1)(cid:9) ;3. Train a Kriging model (cid:101) g using the experimental design {X , G} ;4. Evaluate the expected feasibility function using the candidate set W : F = (cid:8) EF (cid:0) w (1) (cid:1) , . . . , EF (cid:0) w ( N EGRA ) (cid:1)(cid:9) ;5. Choose the next best point as the one that maximizes EF on the set W : w ∗ = arg max w ∈W F ; (26)6. Check if the convergence criteria are met. If they are, skip to 8, otherwise continue withstep 7;7. Evaluate g ( w ∗ ) and add the couple { w ∗ , g ( w ∗ ) } to the experimental design {X , G} . Re-turn to step 3;8. End the algorithm. n this paper N EGRA is set to 10 and the points are selected using Latin Hypercube Samplingover the augmented space. Convergence is assumed when max F EF F ( w ) is lower than a giventhreshold set to 10 − ..