Structural System Reliability: Overview of Theories and Applications to Optimization

This section reviews SSR methods that are applicable directly to systems in Eq. (3) or Eq. (4).

This section reviews SSR methods directly applicable to general systems in Eq. (5), including series and parallel systems in Eqs. (3) and (4).

Several studies addressed challenges of SSR problems involving sequential failures by deriving analytical formulations. For example, Stahl and Geyer (1984) derived mathematical formulations to estimate the probability of fatigue-induced sequential failures for the Daniels (1945) system problem in which the reliability of a bunch of wires is investigated while considering the load-redistribution effects. The fatigue life was derived to follow a lognormal distribution. The correlations between structural members also were derived analytically. These analytical formulations made it possible to calculate the probability of a sequential failure event, and the failure probability of the Daniels system then was estimated. Although this method can provide an exact solution for the system reliability of a structural system with consideration of sequential failures, its application is limited to fatigue problems in parallel systems in which the random variables are assumed to follow lognormal distributions.

Some researchers modeled the problem using LP. For example, Ditlevsen and Bjerager (1984) expressed the failure of structural members as linear functions and calculated the system failure probability by solving the corresponding LP problem. The proposed method was applied to frame structures to obtain the lower and upper bounds of the system reliability. Bjerager (1989) also introduced the use of LP into modeling such SSR problems, and estimated the system-level reliability of truss structures. However, the application of these LP-based methods is limited to cases in which the failures of structural members are described by linear limit state functions, and the associated random variables are assumed to be normally distributed.

MCS (Rubinstein and Kroese 2016; Shinozuka 1983), which involves repeating computational simulations for many scenarios based on randomly generated values of uncertain parameters (so-called samples), is advantageous in dealing with sequential failures because of its straightforward applications to general reliability problems. Some researchers have combined MCS-based methods with failure path algorithms to consider the order of failure events of structural members leading to system collapse. This failure sequence description helps to provide more accurate estimates of system reliability. This section introduces such efforts, with a focus on MCS. More details on failure path algorithms are provided in the section “Failure Path Approaches.”

Yang and Younis (2005) estimated the failure probability of a power plant system considering the load redistribution that may occur during sequential failures. They calculated the probabilities of component events constituting a failure path using FORM and SORM (Der Kiureghian 2005), and estimated the system failure probabilities using MCS. Similarly, Dey and Mahadevan (1998), Enright and Frangopol (1999), and Mahadevan and Raghothamachar (2000) employed adaptive importance sampling and a failure path approach to calculate the system failure probabilities of a truss, frame, and concrete bridge, respectively. Mahadevan and Xiao (1995) used the failure path algorithm with the Markov-chain Monte Carlo (MCMC) method to estimate the system reliability of a truss.

Fig. 3 illustrates the concepts of standard MCS, advanced MCS, and advanced MCS with the failure path algorithm. As mentioned previously, a standard MCS [Fig. 3(a)] may require a large number of samples and involve a huge computational cost because most of the randomly-generated samples are likely to fall within the survival domain, whereas a small number of samples are likely to fall within the failure domain. To resolve this issue and increase the efficiency of SSR analysis, various advanced approaches to MCS have been introduced and combined with failure path algorithms so that more samples can fall within the failure domain [Figs. 3(b and c)]. These advanced MCS-based methods have been applied successfully to various structural systems, such as trusses, frames, and bridges. However, application of such methods to more complex structural systems still involves considerable computational cost, so the computational efficiency of advanced MCS-based methods needs to be improved further.

Failure path approaches are initiated by a deterministic search approach, which performs a deterministic structural analysis using the mean values of the random variables to identify a sequence of failures leading to structural collapse. The most representative method of deterministic search is the incremental load method (Moses 1977, 1982). This method identifies a failure mode by following a load path from initial component failure to system collapse, which leads to a linear equation describing failure. The method has been developed for use in identifying collapse mode expressions for large structural systems consisting of both ductile and brittle components. The method often utilizes existing structural analysis programs with incremental loadings and repeats structural analyses following the sequences of component failures. The array of different potential collapse modes is determined by a systematic strategy of examining changes in component utilization ratios following element failures. The analysis procedure is summarized as follows:

The deterministic approach is prone to missing critical failure sequences because the sequences identified by this approach are not necessarily the most probable ones. In addition, the deterministic search can identify only one failure path for a given load because the method relies on deterministic structural analysis. To obtain additional sequences, the values of some variables need to be modified, and the deterministic analysis must be repeated. For example, one can strengthen structural members involved in the identified sequence to search for additional failure sequences. For this reason, deterministic search may not be appropriate for use in probabilistic evaluation of the risk of sequential failures.

This section deals with a probabilistic extension of the deterministic search in the section “Deterministic Search.” The first step is to identify the member that is most likely to fail in an intact structure. A new structural analysis model is constructed to reflect the damage or failure of the identified member. Through component reliability analysis using the model, the member that is most likely to fail under the damage scenario is identified. This process is repeated until a system failure such as collapse is observed.

One of the most representative methods in this category is the truncated enumeration method. For structural systems with nonlinear member behavior, the truncated enumeration method can be used to identify sequence-dependent failure modes (Melchers and Tang 1984; Tang and Melchers 1987). The method is derived from consideration of the exhaustive enumeration of the event tree containing all combinations of structural member failures. This is performed by imposing a truncation criterion to delete those modes of failure which would make a negligible contribution to the probability of total system failure.

Another widely used method in this category is the $\beta$-unzipping method (Hashemolhosseini 2013; Lu et al. 2018; Thoft-Christensen and Murotsu 2012). This method first requires a component-level reliability analysis of each structural element, and selection of the minimum reliability index. Potential failure components with reliability index values within a specified interval then are identified. The initial system then branches into different subsystems, according to a system updating criterion, by removing the component that is identified to have failed. These procedures are repeated until the structure collapses or fails to satisfy a specified performance requirement. Finally, the reliability indexes of the identified components are combined into a parallel-series system that is similar to an event tree, whereby the system reliability can be eventually evaluated.

However, the sequences identified by a local search approach may not be the most critical sequences overall. For example, there may be a member with the failure probability that is lower than that of the most likely to fail member, but the conditional probability of structural collapse given its failure can be fairly high. This important sequence could be missed if a search scheme focuses on the most likely to fail members at each step.

One of the most widely used probabilistic approaches for global search is the so-called branch-and-bound method (Guenard 1984; Lee and Song 2011; Murotsu 1984), which was introduced to identify critical sequences with significant likelihood in an efficient manner. Although many research efforts have developed risk analysis methods based on the branch-and-bound approach, these methods still remain either time-consuming or prone to missing critical failure sequences.

The branch-and-bound method is considered to be more accurate than the search methods introduced in the sections “Deterministic Search” and “Probabilistic Local Search.” To identify system failure sequences that globally are most likely to occur, the method compares the probabilities of all the failure sequences that have been investigated during the search process and assumes that the most likely sequence of further damage occurs. When the system failure of interest, such as structural collapse, is observed, the particular sequence is identified as a system failure sequence. Unless heuristic rules are employed to truncate apparently insignificant sequences, the branch-and-bound method can identify system failure sequences in the decreasing order of their probabilities. This makes it possible to terminate the search process without ignoring significant system failure sequences.

For example, Karamchandani et al. (1992) applied the branch-and-bound search to offshore platforms to identify important sequences of member failures and calculate the system-level failure probability in terms of a lower bound. In the analysis, the branch-and-bound search introduces more reasonable rules for the determination of structural collapse than do heuristic rules, and reduces the required number of failure sequences and the computational cost.

Although this selective search approach based on the probabilities of sequences helps to reduce the number of sequences to explore, one still might need to explore a large number of sequences to obtain a reliable estimate of the risk of structural system failure (Lee and Song 2011). This is due to the lack of reasonable criteria that would help terminate the search without underestimating the system failure risk. A structural system failure event often is described as the union of the identified system failure sequences. During a search process, one can obtain a lower bound of the system failure probability by means of a system reliability analysis that employs the identified failure sequences, which can be continuously updated as new failure sequences are identified. Because the upper bound usually is unknown, the trend of the lower-bound updates alone cannot provide accurate termination criteria. This is because the size of the updates of the lower bound caused by newly identified system failure sequences does not decrease monotonically, due to the statistical dependence between identified failure modes. This is the case even if the likelihood of identified sequences decreases monotonically. Therefore, a termination criterion based solely on the apparent convergence of the lower bound may underestimate the system risk. Furthermore, a new system reliability analysis needs to be conducted each time the lower bound is updated.

To overcome these challenges, a branch-and-bound method employing system reliability bounds, termed the ${\mathrm{B}}^3$ method (Lee and Song 2011, 2012), was developed. As in other existing approaches based on the branch-and-bound method, the ${\mathrm{B}}^3$ method employs a systematic search scheme in which branching and bounding are repeated (Fig. 6). Unlike other existing approaches, however, the ${\mathrm{B}}^3$ method identifies disjoint failure sequences in order to (1) obtain both the lower and upper bounds of the system failure probability, (2) achieve a monotonic decrease in the updates of the lower bound as the search process proceeds, and (3) update both bounds of the system failure probability without performing additional system reliability analysis. The updated bounds provide reasonable criteria for terminating the branch-and-bound search without missing critical sequences or estimating the system-level risk inaccurately. The ${\mathrm{B}}^3$ method was improved after its initial development and successfully demonstrated by applications to truss and continuum structures. However, applications of the ${\mathrm{B}}^3$ method have been limited to the problem of fatigue-induced sequential failure. Further research is needed to extend its applications to general SSR problems with sequential failures.

Another notable probabilistic global search is a selective search technique by Kim et al. (2013), which identifies sequential failure modes using a genetic algorithm. By identifying the domains representing sequential failures near the origin of the standard normal space with a priority, the method facilitates identifying those with dominant contributions to the system failure probability. The probabilities of the identified modes are calculated by the MSR method (Kang et al. 2008, 2012; Song and Kang 2009). This method was applied to complex offshore structural systems (Coccon et al. 2017) through combination with multiscale MSR analysis.

Table 6 compares the SSR methods reviewed in this section in terms of their implementation, efficiency, and accuracy, and provides other remarks. Direct comparison of these methods is challenging, and may not be reasonable because the associated problems differ in terms of the aspects of structural types, level of structural complexity, failure modes and definitions, and statistical modeling of random variables. More attention should be paid to the ongoing and further improvement of these methods, which is related closely to the development of SSR methods for Boolean system events (section “SSR Methods for Boolean System Events”).

RBDO problems typically are formulated with an objective function and deterministic and/or probabilistic constraints. Examples of the use of the objective function in RBDO include minimizing the failure probability of a structural system, minimizing cost or weight, and maximizing the structural performance. Design considerations, requirements, or other needs can be imposed in the form of constraint functions including probability terms. For a mathematical description of RBDO, consider design variables $\mathbf{d}$ and random variables $\mathbf{X}$ included in the objective and/or constraint functions. It is assumed that the design variables $\mathbf{d}$ can be controlled to minimize or maximize the objective function throughout the optimization process. In the literature, the mean vector of selected random variables often is included as a design variable when the design optimization process can control the mean of the random variables.

A structural system typically is designed with a level of redundancy and fails as a result of the occurrence of multiple failure events or their combined effects. The failure domain for the system failure can be described aswhere $c_k$ = index set which includes elements of the $k$th cut-set. The cut-set formulation in Eq. (9) can cover series and parallel system events in Eqs. (3) and (4) as well. Alternatively, the link-set formulation in Eq. (5) can be used to define the failure domain. Incorporating the system failure domain in Eq. (9) into the reliability constraint, SRBDO can be formulated. For example, the SRBDO formulation of a linear structure under static loads iswhere $\mathbf{K}$, $\mathbf{u}$, and $\mathbf{f}$ = global stiffness matrix, displacement vector, and load vector, respectively; $P_{\mathrm{sys}}$ = system failure probability; $P_{\mathrm{sys}}^t$ = target failure probability; $g_i(·)$ denotes the $i$th component limit state function whose negative value indicates occurrence of failure; and ${\mathbf{d}}^l$ and ${\mathbf{d}}^u$ = lower and upper bounds of design variables, respectively.

Topology optimization (Bendsøe and Sigmund 2004) aims to determine the optimal distribution of material within the design domain while minimizing the objective function and satisfying the given constraints. Topology optimization can be categorized into two types, continuum and discrete topology optimization, which are determined by the type of the design domain of the system. In a continuum TO, external and internal shapes, as well as inner void areas, are optimized simultaneously. In discrete TO, the procedure determines the optimal connectivity, size, and position of structural members. Deterministic topology optimization (DTO) considers system modeling variables and structural characteristics such as geometry, loading, and material property to be deterministic. Critical reviews of deterministic TO approaches and methods were given by Rozvany (2009) and Sigmund and Maute (2013). Reliability-based topology optimization (RBTO) considers inherent uncertainty, randomness, and variability in the design and manufacturing process by incorporating reliability or safety criteria. The review in this paper focused on SRBTO methods.

The double-loop approach is often performed with either the reliability index approach (RIA) (Nikolaidis and Burdisso 1988) or the performance measure approach (PMA) (Tu et al. 1999), because they are equivalent in describing the probabilistic constraint. In RBDO/TO, the RIA approach directly uses the MPP ${\mathbf{u}}^{*}$ obtained by using FORM, i.e., by solving the optimization problemwhere $\mathbf{u}=\mathbf{u}(\mathbf{x})$ = outcome of uncorrelated standard normal variables that are transformed from the original random vector $\mathbf{x}$; and $G(·)$ = limit state function defined in terms of the standard normal random variables. The reliability index $\beta$ in the standard normal space is obtained as

Alternatively, the PMA approach checks the violation of the probabilistic constraint using the minimum value of the limit state function on the surface of the hypersphere with the radius ${\beta }^{\mathrm{target}}$, which is the generalized reliability index corresponding to the target failure probability, i.e., ${\beta }^{\mathrm{target}}=-{\mathrm{\Phi }}^{-1}(P_f^{\mathrm{target}})$. The performance measure of the limit state function is evaluated by solving the optimization problem

The double-loop (nested) algorithm is the traditional and direct approach consisting of outer and inner optimization cycles. As the outer optimization algorithm minimizes the objective function, the inner optimization cycle in Eqs. (13) or (14) finds the reliability index or the performance measure to check if the design considered by the outer loop satisfies the probability constraints. The nested iterations are repeated until feasible solutions converge to the prescribed thresholds or the termination criteria are satisfied.

Mogami et al. (2006) formulated the probabilistic constraint as a series system in discrete SRBTO which considered uncertainties in the structural stiffness and eigenfrequency. The upper-bound formula by Ditlevsen (1979b) and FORM are used to compute the reliability index of a series system for the RIA-based SRBTO. The derivatives of the system reliability index with respect to design variables are obtained by the direct differentiation with chain rules. Because only two limit state functions are considered in the framework, the direct differentiation for sensitivity analysis is not computationally expensive. Agarwal and Renaud (2004) employed the second-order response surface approximations (RSAs) of limit state functions governing the failure of system. The samples from a region defined within 3 standard deviations from the MPP obtained at the previous design point were used for the approximation. Based on this system reliability analysis, an approximate subproblem was constructed and solved for reliability constraint evaluation. The use of RSAs within the sublevel optimization of RBDO reduces the computational cost of performing double-loop RBDO. Chun et al. (2019b) developed a method for topology optimization of structures subjected to stochastic excitations. The proposed optimization framework incorporates random processes into the SRBTO framework by the discrete representation method (Der Kiureghian 2000). For linear structures subject to Gaussian processes, the probabilistic constraints on the first-passage probability can be constructed efficiently without performing FORM or SORM. The analytical sensitivity was derived for the first-passage probability constraint using an adjoint method and the parameter sensitivity of the system reliability calculated based on the sequential compounding method (Chun et al. 2015).

To improve the efficiency of SRBDO/TO, various decoupling algorithms have been proposed to eliminate the inner reliability analysis loop. This approach decouples the RBDO procedure, which leads to a sequence of cycles constructed by deterministic optimization followed by the assessment of the reliability of the converged solution in the feasible domain. In general, an equivalent deterministic formulation is used in design optimization. If the system reliability evaluated after DO does not meet the target failure probability, the constraint is pushed in the feasible domain using a shift-vector.

Du and Chen (2004) proposed using the reliability information obtained in the previous cycle to shift the violated deterministic constraints in the feasible domain. Sequential approximate programming was proposed by Cheng et al. (2006). This approach identifies the optimal design solution by solving a sequence of subproblems consisting of the approximate objective and constraint functions by a Taylor series–based linear approximation of the reliability. A recurrence formula based on the optimality conditions for MPPs is applied to find an approximate reliability index and its sensitivity.

Aoues and Chateauneuf (2006) proposed an SRBDO method which formulates SRBDO problems in terms of component constraints and their target reliability indexes. The component reliabilities are pulled down to the level of the target system probability by the least-squares minimization of the gap between the component and the target level. The subupdating procedure ensures that the target system reliability is satisfied by adjusting the target reliabilities of components. Aoues and Chateauneuf further developed a decoupling algorithm to improve computational efficiency by transforming the original double-loop optimization framework, including the nested optimization loop for the subupdating procedure, into an equivalent SRBDO formulation. The finite-difference approximation is used for sensitivity analysis of constraints and the sequential quadratic programming (Schittkowski et al. 1994). The proposed method aimed to find the best compromise between satisfying the target system reliability and optimizing the component performance.

Zou and Mahadevan (2006) developed a simulation-based method to decouple reliability analysis iterations from the nested formulation in RBDO. The system reliability constraint is approximated by a Taylor series expansion and is decoupled from the optimization loop. Reliability and sensitivity analysis, used in the Taylor series expansion, are conducted, and then deterministic optimization is performed using the approximated constraints. To achieve higher accuracy in reliability analysis in the proposed RBDO method, the multimodal adaptive importance sampling approach is used. The MPP of a limit state function is determined by FORM or SORM. A new sampling density function is constructed using the weighted sum of the densities at the representative points. The MPP is used to identify the representative points, which are close to the failure region.

Royset et al. (2001) developed a method for approximately solving RBDO problems for series structural systems. The method reformulates the RBDO problems as deterministic and semi-infinite optimization problems (Terlaky 1998) by replacing the reliability indexes of component events in the series system with a corresponding limit state function within balls in the standard normal space. The radii of balls are determined through iterative schemes with reliability computations. The reformulated deterministic, semi-infinite optimization problem is solved by a robust optimization algorithm, and the system failure probability is assessed by a reliability method. This SRBDO approach was applied to the design of a rectangular column, offshore jacket structure, and a frame structure with multiple failure modes.

Ba-Abbad et al. (2006) decoupled the reliability analysis from the design optimization by replacing a probabilistic constraint with an equivalent deterministic constraint constructed with reliability-based factors of safety. The system reliability is approximated by the first-order upper-bound method (Ditlevsen 1979b), and the target failure probability in SRBDO is considered as a design variable. The MPP is identified approximately as a function of the reliability index obtained by the PMA reliability analysis. Ba-Abbad et al. recommended the use of the truss region optimization algorithm to solve the proposed optimization formulation to ensure the validation of the approximated MPP. Thus, the optimization algorithm solves the approximate deterministic optimization problem and simultaneously finds optimal design variables and the minimum allowable reliability indexes of failure events.

Lee et al. (2008) proposed a decoupling method for RBDO of multidimensional systems using an inverse reliability analysis that employed an MPP-based dimension reduction method (DRM). This method calculates the probability of failure using the constraint shift. The reliability index from the probability calculation then is updated to the new reliability index by a recursive formula to ensure that the probability of failure using the MPP-based DRM is the same as the target failure probability. The MPP then is updated approximately by taking the same radial direction of the current MPP with the change rate of the reliability index from the previous iteration. The updated MPP subsequently is used for the next design iteration in RBDO to obtain the optimum design.

Valdebenito and Schuëller (2011) replaced the nested reliability analysis loop in SRBDO by approximating the failure probability as an explicit function of the design variables. The RBDO subproblem is solved by starting an optimization procedure while constructing a local approximation of the failure probability around a candidate optimal design within the subdomain. A sequence of candidate optimal designs obtained by repeated solving of the RBDO subproblem is expected to converge toward the solution of the original RBDO problem. A line search strategy and weighted approximations of reliability sensitivity analysis are introduced to improve computational efficiency. The finite-difference method is used to compute the parameter sensitivity of the reliability. This method was applied to nonlinear structures subject to dynamic loading, considering the first passage probability.

In a single-loop approach, the probabilistic constraints are converted into deterministic constraints approximated by exploiting the Karush–Kuhn–Tucker (KKT) optimality conditions. The position vector computed at the MPP can be approximately collinear to the gradient of the performance function when the KKT conditions are satisfied. Optimization is performed in the outer loop with an equivalent constraint along with the KKT conditions. The single-loop approach approximates the MPPs and the target reliability index in relation to the KKT optimality condition. By considering the target failure probability approximated by the procedure described previously as design variables, the necessity of reliability analysis at each iteration can be removed.

Meng et al. (2018) applied the chaos control theory to evaluate approximately the MPPs rather than employing the KKT optimality conditions. This was to address the numerical instability of the single-loop approach that may occur during iterative computations in highly nonlinear problems. Liang et al. (2007) proposed an SRBDO for series systems by transforming a double-loop optimization problem into a single-loop deterministic optimization problem using an approximation of the MPP of each constraint. The KKT condition is imposed to find the approximated MPP rather than solving the nested optimization problem in each iteration. The target failure probabilities of limit state functions are considered as design variables, and the system failure probability is estimated by the bicomponent theoretical bounding formula (Ditlevsen 1979b). The single optimization loop with the approximated MPP improved the efficiency of SRBDO.

Nguyen et al. (2010) integrated the MSR method (Song and Kang 2009; Kang et al. 2012) into a single-loop SRBDO. The MPP of each constraint is approximated by scaling the negative normalized gradient vector with the target reliability index. The failure probability of a system then is estimated by the MSR method efficiently and effectively. By employing the MSR method, applications of the single-loop SRBDO approach were extended to nonseries system events. Furthermore, the MSR method provides a systematic way to compute the sensitivity of the system failure probability with respect to design variables to facilitate the use of gradient-based optimization algorithms in SRBDO. The single-loop approach with the MSR method was extended further and applied to SRBTO. Nguyen et al. (2011) proposed a single-loop SRBTO approach considering statistical dependence between limit state functions of a highly nonlinear problem. The system failure probability and its sensitivity to statistically dependent limit states are computed using the MSR method. To improve the accuracy of the single-loop approach, the following scheme was proposed to address the error by FORM. After approximated MPPs of limit state functions are obtained by employing the KKT conditions in the single-loop approach, the reliability index for each component is evaluated by SORM in each iteration. The radius of the scaling is updated by taking the ratio of the target reliability index to the reliability index obtained by the SORM analysis and multiplying the ratio by the target reliability index in the previous step. As a result, the approximated MPP can be improved based on the updated radius and the normalized gradient vector in each iteration of optimization.

Reliability analysis of a system problem defined by multiple failure events often employs simulation techniques such as MCS to avoid difficulty in evaluating multifold integrals in system reliability analysis. The accuracy of MCS is affected significantly by the number of samples and regions describing the system failure event.

The probabilistic sufficient factor approach that combined the concepts of the failure probability and the safety factor was proposed by Qu and Haftka (2004) to improve design response surface approximation for RBDO. After completing the probability calculation of a probabilistic constraint by MCS, the probabilistic sufficiency factor (PSF) is obtained from the target probability and MCS sample size, which is not a computationally expensive process. The proposed method showed that design response surface approximation based on the PSF increases accuracy in regions of a low probability of failure and achieves faster convergence in SRBDO. A simulation-based RBTO of a structure subjected to stochastic excitation was proposed by Bobby et al. (2017). The reliability analysis of the probabilistic constraint in terms of the first-passage probability is computationally expensive in general. In the proposed method, the inverse constraint, which is equivalent to the original probabilistic constraint, is considered along with the damage measure threshold corresponding to the target failure probability. The damage measure threshold is the inverse of a parameter estimated directly from the MC samples of structural damage state. An approximate optimization subproblem then is defined from the simulation-based probabilistic analysis to decouple the probabilistic analysis from the optimization loop. Solutions of the sequence of subproblems in the deterministic form are obtained without performing additional probabilistic analyses, and are expected to meet the original probabilistic constraint associated with the first-passage probability. Papadrakakis and Lagaros (2002) solved RBDO problems using MCS, incorporating the importance sampling technique for reliability analysis. In this method, neural networks (NNs) (Rogers and Mcclelland 2014) are trained and used to reduce computational cost in the evaluation of limit state functions in RBDO. The evolution strategies were used for optimization in conjunction with the NN which was more robust and had more-global behavior than mathematical programming.

Alternatively, an approximate representation of the probability in RBDO with an explicit function of the deterministic variables was studied. Gasser and Schuëller (1997) decoupled the reliability analysis loop from RBDO by applying variance-reducing MCS techniques, in which the limit state function was constructed explicitly by interpolation. Taflanidis and Beck (2009) developed a stochastic subset optimization (SSO) algorithm for reliability optimization problems involving system reliability as the objective function and sensitivity analysis. An augmented problem was formulated with design variables that were considered artificially as uncertain. MCS with samples of the model parameters and the design variables that lead to system failure are performed to obtain information that is used to identify a subset of the design space. The identified design space has the highest likelihood of containing optimal design variables. An adaptive iterative approach using MCMC simulation was applied to reduce the size of subsets until the original design space, including near-optimal design variables, was identified. In addition, SSO was used to perform sensitivity analysis with respect to the uncertain parameters and design variables. The SSO algorithm was extended further to nonparametric stochastic subset optimization (NP-SSO) (Jia and Taflanidis 2013). NP-SSO uses kernel density estimation to approximately evaluate the objective function. Candidate optimal points for the global RBDO minimum then are identified using the approximation of the objective function.

The purpose of employing surrogate models is to replace computationally intensive or expensive assessment of functions due to high nonlinearity and dimensionality with surrogate models which are cheap and easy to evaluate. Various surrogate models (Sudret and Der Kiureghian 2002; Papadrakakis and Lagaros 2002; Bichon et al. 2008) have been used for the reliability analysis in RBDO. Moustapha and Sudret (2019) surveyed surrogate-based RBDO and benchmark studies on computational efficiency and effectiveness.

Li and Wang (2020) proposed a reliability-based multifidelity optimization method that deals with design problems involving low- and high-fidelity data. This method utilizes a Gaussian process modeling approach, in which a limit state function is considered to be a realization of a Gaussian process, to construct a surrogate model by fusing data points with different fidelity levels. The generated surrogate model subsequently was employed to perform reliability and sensitivity analyses in solving RBDO problems.

Dubourg et al. (2011) proposed an adaptive surrogate-based RBDO approach which utilized a refinement technique for reliability estimation. The kriging metamodeling technique is adopted to develop surrogates of the performance functions in which errors in the surrogate models are controlled using the adaptive refinement technique.

Acar et al. (2009) performed SRBDO for automobile structures. To handle computationally expensive simulations without compromising accuracy, the metamodels representing a mathematical relationship between the critical responses of system, design variable, and random variables were constructed using quadratic polynomial response surface, radial basis function, and Gaussian process. The system reliability index and sensitivity analysis of system reliability were computed using MCS with the metamodel. Khatibinia et al. (2013) developed a discrete gravitational search algorithm for SRBDO of RC structures while considering soil–structure interaction effects using a metamodel-based MCS. The metamodel consisting of a weighted least-squares support vector machine and wavelet kernel function was incorporated into RBDO in which system reliability analysis was performed by MCS. The discrete gravitational search algorithm proposed to find the optimum design of structures was enhanced using a passive congregation (He et al. 2004). The global surrogate modeling for RBDO was investigated by Bichon et al. (2011). The efficient global reliability analysis (Bichon et al. 2008) was used to create a Gaussian process model of response functions with a small number of samples. Thus, using the inexpensive surrogate model, the probability of failure can be evaluated with a reduction of computational cost but without significant loss of accuracy. Efficient global optimization (EGO) (Jones et al. 1998), which originally was developed for solving unconstrained optimization problems, was extended to establish a new formulation for enforcing constraints in EGO based on an augmented Lagrangian for RBDO problems. Jensen et al. (2020) proposed a method of reliability-based global design optimization for structures under stochastic excitation. A kriging meta-based surrogate model was employed to assess system failure probability. A stochastic simulation scheme with a transitional Markov-chain Monte Carlo method was utilized to solve an optimization problem.

Heuristic optimization algorithms have been adopted and utilized to obtain optimal solutions of problems that may not have explicit mathematical forms or that may make sensitivity analysis difficult due to implicit or nonsmooth manifold complexity. Greiner and Hajela (2012) proposed an approach of coevolutionary multiobjective formulations for discrete topology optimization considering the randomness of applied loads and material properties. The multiobjective optimization problem was formulated to maximize a system reliability index in terms of design interest while minimizing the deterministic volume of a discrete truss structure. Component reliability indexes of structural elements are evaluated using the first-order second-moment method, and a system reliability analysis with component reliability indexes is computed using the method of Park et al. (2004). A Pareto nondominance criterion was obtained using a coevolution strategy that identifies a complete nondominated solution front with increased search efficiency in the evolutionary algorithm of optimization. Thampan and Krishnamoorthy (2001) developed a modified branch-and-bound (MBB) algorithm to assess the system reliability of discrete structures and used a genetic algorithm (GA) to find an optimal truss structure under uncertainties in loads and strength parameters of the material. The MBB approach imposes bounding conditions of prime member group assignment in the subsequent element failures on the basic bounding technique. The GA based on the fitness evaluation for each candidate solution is implemented to solve RBDO problems of parallel, series, and series of parallel subsystems.

Table 7 compares the aforementioned SRBDO/TO approaches, and flowcharts of their processes are given in Fig. 8. A fair comparison of SRBDO/TO approaches is challenging because of different types of optimization problems; target reliability; and implementations of reliability methods, optimization algorithms, and convergence criteria. However, a few examples of comparisons are available for review in the published literature, such as computational cost comparison between the double- and single-loop approaches by Liang et al. (2014). To solve RBDO problems, the double-loop approach required 1.5–3 times more iterations to find a converged solution than did the single-loop approach. Furthermore, the number of function evaluations in RBDO with the double-loop method was 50–380 times higher than that of the single-loop approach, depending on problem types and the number of design variables. Liang et al. indicated that the efficiency of the single-loop method could be comparable to deterministic optimization. Another such example was provided by Bichon et al. (2013), who performed computation experiments with double-loop, decoupling, and surrogate-based approaches for a short-column RBDO problem. The results showed that the surrogate-based method in RBDO could require less than half the number of function evaluations needed for the decoupling method. The decoupling method for the same problem required one-third fewer function evaluations than the double-loop method. The decoupling-based RBDO generally is more efficient than the double-loop formulation, but more computationally expensive than the single-loop formulation. Furthermore, the computational efficiency of a surrogate-based approach in RBDO and of others was illustrated using numerical examples by Moustapha and Sudret (2019).