Novel decoupled framework for reliability-based design optimization of structures using a robust shifting technique

Abstract

In a reliability-based design optimization (RBDO), computation of the failure probability (P_f) at all design points through the process may suitably be avoided at the early stages. Thus, to reduce extensive computations of RBDO, one could decouple the optimization and reliability analysis. The present work proposes a new methodology for such a decoupled approach that separates optimization and reliability analysis into two procedures which significantly improve the computational efficiency of the RBDO. This technique is based on the probabilistic sensitivity approach (PSA) on the shifted probability density function. Stochastic variables are separated into two groups of desired and non-desired variables. The three-phase procedure may be summarized as: Phase 1, apply deterministic design optimization based on mean values of random variables; Phase 2, move designs toward a reliable space using PSA and finding a primary reliable optimum point; Phase 3, applying an intelligent self-adaptive procedure based on cubic B-spline interpolation functions until the targeted failure probability is reached. An improved response surface method is used for computation of failure probability. The proposed RBDO approach could significantly reduce the number of analyses required to less than 10% of conventional methods. The computational efficacy of this approach is demonstrated by solving four benchmark truss design problems published in the structural optimization literature.

Keywords

reliability-based design optimization / trusses / sensitivity analysis / shifting technique / cubic B-splines / response surface method

Introduction

Recently, optimization in various fields of engineering sciences has been very much considered [¹–⁸]. While the modeling assumptions have become more elaborate and accurate over the years, the reliability analysis of stochastic structural systems still remains a challenging problem. Reliability-based design optimization (RBDO) implements structural optimization considering simultaneously the uncertainties observed in the structural materials and/or applied loading. The general truss RBDO problem with both deterministic and probabilistic design constraints can be formulated as

(1)\begin{array}{c}\hfill \mathrm{M}\mathrm{i}\mathrm{n}/\mathrm{M}\mathrm{a}\mathrm{x}f\left(\mathit{d}\right)\hfill \\ \hfill \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}:\hfill \\ \hfill P\left(G_i\right(\mathit{d},\mathit{X})\le \mathrm{0})\le P_{\mathrm{f}}^{Tl},i=1,\mathrm{...},N_{\mathrm{P}\mathrm{C}}\hfill \\ \hfill \mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}{\sigma }_j\left(\mathit{d}\right)\le {\sigma }_{\mathrm{a}\mathrm{l}\mathrm{l}},j=1,\mathrm{...},N_{\mathrm{m}}\hfill \\ \hfill \begin{array}{c}\hfill \mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}{u}_k\left(\mathit{d}\right)\le u_{\mathrm{a}\mathrm{l}\mathrm{l}},k=1,\mathrm{...},N_{\mathrm{D}\mathrm{O}\mathrm{F}}\hfill \end{array}\hfill \\ \hfill \begin{array}{c}\hfill \mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}{\mathit{d}}^L\le \mathit{d}\le {\mathit{d}}^U\hfill \end{array}\hfill \end{array}

where \mathit{d}={[d_{\mathrm{1}},d_{\mathrm{2}},\mathrm{...},d_n]}^{\mathrm{T}}is a column vector of n deterministic design variables, X={[x_{\mathrm{1}},x_{\mathrm{2}},\mathrm{...},x_m]}^{\mathrm{T}}is the m-dimensional vector of random variables, f(d) is the objective function, P\left(G_i\right(\mathit{d},\mathit{X})\le \mathrm{0})denotes the failure probability for the ith limit state function G_i(\mathit{d},\mathit{X}), P_{\mathrm{f}}^{Tl}is the target failure probability of ith constraint, and N_PC is the number of probabilistic constraints. In Eq. (1), s_j and u_k are the stress of jth member and the nodal displacement of kth degrees of freedom, respectively. Also, {\sigma }_{\mathrm{a}\mathrm{l}\mathrm{l}.}, u_{\mathrm{a}\mathrm{l}\mathrm{l}.}, {\mathit{d}}^L, {\mathit{d}}^U, N_m and N_DOF are the allowable member stresses, the allowable nodal displacements, the allowable lower and upper bounds of d, the total number of members, and the total number of degrees-of-freedom, respectively. The target failure probability could simply be expressed in terms of the target reliability index as P_{\mathrm{f}}^{Tl}=\mathrm{\Phi }(-{\beta }_{Tl}), where \mathrm{\Phi }is the standard normal cumulated distribution function and {\beta }_{Tl} denotes the target reliability index.

The most straightforward approaches to solve the RBDO problem defined in Eq. (1) can be categorized into three groups: nested, mono-level, and decoupled techniques. In nested or double-loop approaches, a reliability analysis is nested within a constrained optimization loop. This approach is also known as the reliability index approach and is based on first-order reliability method (FORM) or an alternative formulation to the RBDO problem known as the performance measure approach [⁹]. Mono-level approaches [⁹–¹²] attempt to fully reformulate the original RBDO problem into an equivalent deterministic design optimization (DDO) problem. In spite of the evident advantage that a single optimization loop provides, these approaches require the computation of second-order derivatives. Moreover, a recent benchmark study [¹³] indicates that mono-level approaches may suffer instability problems. Decoupling approaches aim to separate the optimization loop from the reliability analyses, so that both can be sequentially performed in an independent manner. Decoupled approaches are referred to as sequential optimization and reliability assessment (SORA) [¹⁴], which transforms the RBDO problem into a sequence of deterministic optimization cycles, including an inverse reliability analysis to determine the most probable target point (MPTP) for shifting the constraints into the feasible region. For problems with a strong non linearity in the limit state function, SORA suffers from the possible non-unicity of the MPTP [¹⁵]. Sequential approximate programming (SAP) [¹⁶] is another well-known decoupling technique that formulates the RBDO as a sub-programming problem that uses an approximate form of the objective function subjected to a set of approximate constraints by means of first order Taylor expansion at the current design point. The problem with SAP is that for a non-linear limit state function approximation, the FORM generates non-suitable substantial errors.

Over the last decade, many researchers have applied double-loop approaches to solving RBDO problems in structural engineering by using surrogate-based or response-surface-based approaches to improve computational efficiency.

One of the most popular approaches to reduce computational cost of the reliability analysis is the response surface method (RSM) and its adaptive versions [¹⁷–²³]. Numerous studies have been done on applying RSM to solve reliability analysis and RBDO problems for structural problems [²⁴–²⁹]. RSM employs a polynomial function to approximate the unknown implicit performance function. An accurate estimate of the failure probability can be obtained if the selected polynomial function is a good fit to the actual limit state.

Recently some studies have been published on reliability based optimization of structures using safety design factors [³⁰–³²]. At the end of such procedures, a certain value is obtained for optimum safety design factor that allows a fair judgement on the design type carried. These methods attempt to decouple the optimization procedure from reliability analysis. Among various features of such methods one may refer to their simplicity and their ease of understanding for engineers.

The approach presented in this study may be categorized as a three-phase decoupling method that introduces a novel method where a simultaneous computation of P_f throughout the optimization procedure is avoided at the early stages of the calculations. First, a solo DDO procedure is applied to the problem, then the probabilistic sensitivity approach (PSA) is applied on the design variables at the achieved optimum solution. This approach may ensure determination of the most reliable optimum design while significantly reducing the number of computations. At the end of the process, a cubic B-spline interpolation technique is utilized to find a reliable optimum point that meets the targeted failure probability.

In this study, the reliability analysis utilizes an improved RSM proposed by Zhao and Qiu [²⁹] which has been shown to be computationally efficient for solving structural RBDO problems [³³].

In the next sections, the proposed three-phase decoupling method is defined, including the vital roles of B-splines, an improved RSM, and the probability-based sensitivity approach. To demonstrate the effectiveness of the proposed methodology, two planar and two spatial benchmark truss problems are studied.

Methodology

Classifying random variables in structural problems

In most structural engineering problems, random variables may be classified without a sensitivity analysis. Thus, one may categorize variables as: desired random variables (DRV), for example, the strength of materials and undesired random variables (URV), such as applied loading.

Desired random variables may be identified as the ones that cause an associated reduction in the failure probability when their values are increased, and vice versa. However, there are some random variables whose effect on the limit state function or the failure probability cannot be easily discerned. Thus, merely to classify them, sensitivity analysis is carried out.

Shifting technique and pseudo-probability density function

One of the features of a probabilistic density function, related to random variables in structural problems, is the insignificant value of the coefficient of variation (COV). A small COV causes mean surrounded values to be selected with a higher probability compared to the end zones. It reasons the need for excessive production of samples when the failure probability is low. Figure 1 illustrates normal and lognormal probability density functions for materials with 10⁴ ksi mean moduli of elasticity for COV values.

In the present work, the following shifting relationship is utilized for mean values of random variables

(2){\mu }_X^{t+\mathrm{1}}={\mu }_X^t(\mathrm{1}\pm \lambda (t\left)\right),

where t is the step number, X is a random variable, \mu is the mean probabilistic distribution of X and l is the shifting step length. The probability density function (PDF) with the new mean value out from Eq. (2) is called the pseudo-PDF of random variable X. The sign in Eq. (2) is negative for desired random variables and positive for undesired random variables. The shifting step length l is defined as

(3)\lambda \left(t\right)=ts\times t,

where ts is a shifting parameter (a value chosen between 0 and 1). In fact, as illustrated in Fig. 2, the defined shifting technique applied on the random variables X, allows moving from the first deterministic optimum point d* found at the start of the procedure, to the first reliable point x* in the safe space d*, a point from which the search toward the globally reliable optimum design with regard to the desired probabilistic density arises.

Cubic B-spline interpolation curves

Splines are used for geometry descriptions as well as for the representation of the unknown fields. Cubic splines are widely used to fit a smooth continuous function through discrete data in many fields of research [³⁴,³⁵]. In general, if the function to be approximated is smooth, cubic splines will perform better than piecewise linear interpolations [³⁶]. The most compelling reason for their use is their C² continuity, which guarantees continuous first- and second-derivatives. The authors recommend de Boor [³⁷] as a reference for information about applications of B-splines.

Reliability analysis with an improved RSM

RSMs are one of the most important advances in structural reliability analysis [¹⁷] and are appropriate for cases where the performance function is not known as a closed form expression; for example, when the numerical methods (like finite element analysis) are employed. The basic idea of a conventional RSM is to approximate an implicit limit state function (LSF) by an equivalent polynomial function. In addition, RSMs utilizes an iterative scheme to construct the final response surface function (RSF). Thus, it is vital to find a good estimate for the design point in the first iteration. In this study, to reduce the computational cost of the reliability analysis, the control point of experimental points is constructed using a method proposed by Zhao and Qiu [²⁹].

Description of the PSA

In structural engineering minimization problems, one may achieve the optimum reliable design point by adding a proportion to the deterministic optimum design variables to ensure safe and reliable performance. It is obvious that some variables are less sensitive to changes than others; therefore, PSA can help identify variables that have the most effect on the probability of failure and help find a global optimum design that is also reliable.

Sensitivity measures of failure probability are made by allowing perturbation of each design variable i with respect to (w.r.t.) its own changes or S_i^P. To do so, a finite difference method was carried out as

(4a)S_i^P={\displaystyle \frac{\overline{P}\left(x_i\right)-\overline{P}(x_i\pm \mathrm{\Delta }{x}_i)}{\mathrm{\Delta }{x}_i}},

where \mathrm{\Delta }{x}_iis the perturbation value by which ith design variable alters \overline{P}\left(\right)denotes an operand for computing P_f for the number of {\overline{N}}_ssamples using the following relationship

(4b)\overline{P}\left(\right)={\displaystyle \frac{{\overline{N}}_{\mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}}}{{\overline{N}}_s}},

where {\overline{N}}_{\mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}} is the number of samples violating any limit state function of the problem.

In Eq. (4a), the sign at the numerator is selected such that it causes modification of P_f. The value for S_i^P can be negative, positive, or zero, depending on the type of the variable and its effect on the objective function. In Eq. (4a) the main idea is not to exactly compute the P_f but rather the rate of change of P_f w.r.t. each variable. One of the features of using Eq. (4a) is that through sensitivity analysis, where in turn only one variable is perturbed by a small amount while all other variables maintain their values, a feasible solution is easily determined. It is important to note that in Phase 2 of an algorithm, where the search is begun from the optimum point d* toward the reliable region, by using sampling methods, the number of analyses will be reduced substantially. To reach the feasible reliable region, the probabilistic normalized sensitivity value (NSV^P) for each variable is computed using:

(5)\mathbf{N}\mathbf{S}{\mathbf{V}}_i^P={\displaystyle \frac{S_i^P}{{\displaystyle \sum _{i=\mathrm{1}}^n{S}_i^P}}}.

The probabilistic sensitivity vector NSV^P may then be determined as:

(6)\mathbf{N}\mathbf{S}{\mathbf{V}}^P={[NS{V}_{\mathrm{1}}^P,NS{V}_{\mathrm{2}}^P,\mathrm{...},NS{V}_n^P]}^{\mathrm{T}}.

An updated design vector x^new is computed as

(7){\mathit{x}}^{\mathrm{n}\mathrm{e}\mathrm{w}}={\mathit{x}}^{\mathrm{o}\mathrm{l}\mathrm{d}}+\mathrm{\Delta }\tau ,

where \mathrm{\Delta }\tauis the updating vector based on the NSV^P given as

(8)\mathrm{\Delta }\tau ={\mathit{x}}^{\mathrm{o}\mathrm{l}\mathrm{d}}·\left(\alpha \mathbf{N}\mathbf{S}{\mathbf{V}}^P\right),

where the symbol ·denotes a scalar product operand. One may express Eq. (7) as

(9){\mathit{x}}^{\mathrm{n}\mathrm{e}\mathrm{w}}={\mathit{x}}^{\mathrm{o}\mathrm{l}\mathrm{d}}(\mathrm{1}+\alpha \mathbf{N}\mathbf{S}{\mathbf{V}}^P),

where α represents an “intelligent self-adaptive correction factor” that is activated after determining the optimum reliable point x* at the end of Phase 2, where a further search is processed around x* to achieve the most reliable optimum point x**. The aforementioned procedure is continued until the P_f is greater than the target value.

The value of α in Eqs. (8) and (9) is different during Phases 2 and 3 as follows:

(10)

The total number of analyses N^{Tot}required to obtain the reliable optimum solution x**, may be found according to

(11)N^{Tot}=N_{\mathrm{Phase\; 1}}^{\mathrm{D}\mathrm{D}\mathrm{O}}+N_{\mathrm{Phase\; 2}}^{\mathrm{PSA}}+N_{\mathrm{Phase\; 3}}^{\mathrm{ISAP}}+N^{\mathrm{Reliability}},

where N_{\mathrm{Phase\; 1}}^{\mathrm{D}\mathrm{D}\mathrm{O}} is the number of analyses required to reach d* in Phase 1, N_{\mathrm{Phase\; 2}}^{\mathrm{PSA}} is the number of samples to compute the sensitivity of P_f in Phase 2, N_{\mathrm{Phase\; 3}}^{\mathrm{ISAP}} is the number of analyses to determine x** at the ISAP of Phase 3, and N^Reliability is the total number of analysis required for the P_f calculation.

Figure 3 shows a summary of the proposed three-phase RBDO procedure.

The proposed RBDO algorithm

Figure 4 illustrates the procedure for reliability-based structural design optimization using both PSA and cubic B-splines. The procedure may be summarized as follows:

Phase 1 – Determine the deterministic optimum solution d*

Find the deterministic optimum point with respect to the mean values of random variables, using ViS-BLAST [³⁸].

Phase 2 – Move from d* to find the first reliable optimum solution x*

Step 1. Set iteration number t= 1; set starting design x^t = d*and P_{\mathrm{f}}^t=P\left(x^t\right).

Step2. Compute the vector of probabilistic sensitivity values NSV^P and the corresponding updating vector \mathrm{\Delta }\taufrom Eqs. (6) and (8).

Step 3. Update current design x^t and determine the new design x^t+1 using Eq. (9).

Step 4. Determine the probability of failure P_{\mathrm{f}}^{t+\mathrm{1}}for new design x^{t+\mathrm{1}} using the RSM and compare it with the target probability of failure P_{\mathrm{f}}^{\mathrm{Target}}; save the point (x^{t+\mathrm{1}}, P_{\mathrm{f}}^{t+\mathrm{1}}).

Step 5. If , then set x* = {\mathit{x}}^{t+\mathrm{1}} and f* = f(x*), otherwise set t= t+1 and go to Step 2.

Phase 3 – An ISAP to find x**

Phase 3 revises x* obtained from Phase 2 to finally advance it to x** by modifying the final updating vector \mathrm{\Delta }\tauprior to entering the region where the targeted reliable optimum point lies.

Step 1. Set the ISAP iteration number s=1; set starting design x^s equal to the last design before x* from Phase 2.

Step 2. Use cubic B-spline interpolation to fit (P_f, f(x)) curve on the data of Phase 2 and new computed data from Step 1.

Step 3. Extract the objective function value {\overline{f}}^sfrom fitted curve corresponding to the P_{\mathrm{f}}^{\mathrm{Target}}.

Step 4. Determine the correction factor {\alpha }^sand design x^s+1 corresponding to the extracted {\overline{f}}^semploying the following inverse equation

(12)f\left({\mathbf{x}}^{s+\mathrm{1}}\right)=f\left({\mathbf{x}}^{t-\mathrm{1}}\right(\mathrm{1}+{\alpha }^s\mathbf{N}\mathbf{S}{\mathbf{V}}^{t-\mathrm{1}}\left)\right)={\overline{f}}^s,

where x^t−1 is the design vector and NSV^t−1 is the probabilistic sensitivity vector previous to the determination of x* at Phase 2.

Step 5. Compute the probability of failure P_{\mathrm{f}}^{s+1}corresponding to design {\mathit{x}}^{s+\mathrm{1}} using the RSM. If multiple LSFs are involved, the approach would focus on the most dominant LSF.

Step 6. Compute the relative distance error of P_f using:

(13)erro{r}^{P_{\mathrm{f}}}={\displaystyle \frac{P_{\mathrm{f}}^{s+1}-P_{\mathrm{f}}^{\mathrm{Target}}}{P_{\mathrm{f}}^{\mathrm{Target}}}}\mathrm{.}

Step 7. If erro{r}^{P_{\mathrm{f}}}is less than the allowable values of error ε, in this case set x** = x^s+1, f** = f(x**), and {\alpha }^{*}={\alpha }^s; otherwise, set s= s+1, add new point (f (x^s+1), P_{\mathrm{f}}^{s+\mathrm{1}}) to curve data and go to Step 2.

Examples

In this section, designs are presented for four different benchmark RBDO structural problems: a 10 and 13-bar planar trusses and a 72 and 25-bar space trusses with 10, 7, 16, and 8 design variables, respectively. The first two examples consider continuous design variables and the second two examples use discrete design variables

The number of samples {\overline{N}}_sfor the PSA is set as 50 for all problems. The targeted ε value as for the failure probability error estimation is fixed at 1%.

Comparisons are presented between RBDO results utilizing the improved RSM and one that uses a Monte Carlo simulation to demonstrate the versatility of the RSM in the proposed approach.

If RBDO utilizes MCS instead of RSM for the reliability analysis and a stochastic-based method instead of ViS-BLAST for optimization, then the computational cost for P_f evaluations could be enormous. Nowak [³⁹] suggested that the number of samples N^{\mathrm{M}\mathrm{C}\mathrm{S}}required for the COV of P_f to be less than or equal to n, would be equal to:

(14)N^{\mathrm{M}\mathrm{C}\mathrm{S}}={\displaystyle \frac{\mathrm{1}-{{\displaystyle P}}_{\mathrm{f}}}{v^{\mathrm{2}}{{\displaystyle P}}_{\mathrm{f}}}}\cong {\displaystyle \frac{\mathrm{1}}{v^{\mathrm{2}}{{\displaystyle P}}_{\mathrm{f}}}}.

Hence, for truss problems attempted here with 10, 13, 72, and 25 members, and the corresponding targeted P_f equals to 6.21 × 10⁻³, 10⁻⁵, 1.35 × 10⁻³, and 1.35 × 10⁻³, respectively, if n is considered 0.01, the number of samples for P_f estimating using MCS would be equal to 1.6 × 10⁶, 1.0 × 10⁹, 7.4 × 10⁶, and 7.4 × 10⁶, respectively. In other side, however the number of samples is expected to produce using the RSM, would be comparatively a tiny fraction of 50–100 samples for all problems attempted here.

A 10-bar planar truss problem

Definition

Figure 5 shows the geometry of the 10-bar truss [²⁹]. The deterministic design variables of the structure are the cross-sectional areas of the members and the random variables include the external loads P₁, P₂, and P₃, module of elasticity E, and the length of the horizontal and vertical members L. Table 1 listed the statistical properties for the random variables

The target reliability index is 2.5, i.e., the target failure probability is 6.21 × 10⁻³. The objective function for the minimization is the total area of truss members defined as

(15)f\left(A\right)={\displaystyle \sum _{m=\mathrm{1}}^{\mathrm{10}}{A}_m},

where A_i is the cross-sectional area of member i. The limit state function G for the problem is set implicitly as the vertical deflection Δ of node 3, which is the 6th degree of freedom given by:

(16)G=\mathrm{0.004}-{\mathrm{\Delta }}_{\mathrm{6}}\left(\mathit{X}\right).

Results obtained

This 10-bar truss problem has been solved by Zhao and Qiu [²⁹] using a double-loop approach containing an improved RSM and by Dizangian and Ghasemi [³²] using a decoupled RBDO framework based on the required safety factor using the “fmincon” Matlab function for optimization and Monte Carlo simulation for doing the reliability. Table 2 presents a comparison between 10-bar truss designs obtained with the proposed PSA-ISAP algorithm and previous studies. Design results indicate that the proposed algorithm is an effective and robust framework for solving the RBDO of truss structures.

Figures 6 and 7 show the convergence history for the 10-bar truss during Phase 2 using linear interpolation. When an equivalent point for the targeted P_f is found, the objective value is 62.13 cm² with P_f equals to 0.0063, which is greater than the target value. This error may be due to the existence of a high variational quiddity of the targeted P_f in this highly sensitive region. In Phase 3, cubic B-spline interpolation functions are used to improve the accuracy. Figures 8 and 9 show the fitted B-spline curves as developed in Phase 3. Table 2 lists the details of the 10-bar design results: an optimum value of the cross-sectional area A = 63.6124 cm² and a corresponding value of P_f = 0.00477 after four P_f calculations and 258 finite element (FE) analyses in Phase 2 (see Figs. 6–7).

As shown in Table 2, the failure probability based on the improved RSM compares well with that using the exact MCS with only a 0.6% discrepancy within the safe margin. In addition, the design using the improved RSM was obtained with eight P_f computations.

Table 3 lists the results of four iterations during Phase 3 of the proposed method. In Phase 3, an ISAP is employed from a starting point of f(A) = 55.0746 cm² which converged to an optimum value of 62.3472 cm² with the corresponding P_f = 0.0062 after four iterations and a total of 8 calculations of P_f with 524 FE analyses (see Figs. 8–9).

Figure 10 shows the convergence history of the objective function for the 10-bar truss problem against the failure probability during Phases 2 and 3. Figure 11 performs a close-up view of Fig. 9 and indicates that the ISAP procedure led to a fast convergence around the targeted failure probability.

A 13-bar bridge truss problem

Definition

Figure 12 shows a 13-bar truss first investigated by Nakib and Frangopol [⁴⁰] and then by Ghorbani and Ghasemi [⁴¹]. The material density and modulus of elasticity were considered as 7850 kg/m³ and 206 GPa, respectively. The target probability of failure is 10⁻⁵ and the target reliability index is 4.2649. To enforce symmetry, members of the truss were set into 7 groups as listed in Table 4. The lower and upper bounds of the cross-sectional area of each member range from 6.4516 × 10⁻³ m² to 6.4516 × 10⁻⁵ m². The stochastic variables are the force P applied to the nodes along the lower span and the material yielding stress F_y. These variables exhibit a normal distribution and their values of consistent statistical parameters are listed in Table 5. Since this structure acts as a series system, its failure is assumed when the stress in each member is greater than the F_y. To be consistent with results published in the literature, failure due to buckling is not considered. The limit state function for each member i is defined as:

(17)G_i=F_{\mathrm{y}}-\left|{\sigma }_i\right|.

Results obtained

Nakib and Frangopol [⁴⁰], Ghorbani and Ghasemi [⁴¹] and Dizangian and Ghasemi [³²] solved this problem using IPF-ABT, ANFS-PSO-MCS, and a decoupled safety factor based fmincon-MCS RBDO approaches, respectively.

Table 6 lists the reliable optimum mass and the corresponding cross-sectional areas for each member obtained by the proposed PSA-ISAP-based method compared to studies reported in the literature. From these results, it is seen that the truss mass obtained by the proposed PSA-ISAP algorithm is slightly better than results reported in other studies. Moreover, the proposed PSA-ISAP algorithm requires less computational effort to achieve a solution.

The results listed in Table 6 indicate the estimated failure probability based on RSM compares well with the MCS-based probability calculations using 10⁷ samples. In addition, these results were obtained with just nine P_f computations.

Figures 13 and 14 show the Phase 2 convergence history of the proposed method. The truss mass of 356.782 kg with a corresponding P_f of 7.124 × 10⁻⁶ obtained at the end of Phase 2 was achieved with five P_f calculations and 394 FE analyses.

Figures 15 and 16 show that a truss mass of 347.536 kg and a corresponding P_f of 9.9817 × 10⁻⁶ was found after the ISAP was utilized; this solution is slightly better than the results reported by other researchers. Figures 15 and 16 show the cubic B-spline interpolation curves that were fitted through nine sample points generated during Phase 2 and 3.

Figure 17 shows the convergence history of the objective function for the 13-bar truss problem against the failure probability extracted from Phases 2 and 3 and indicates that the PSA procedure leads to a fast drive toward the safe region. According to data presented in Fig. 17, the new points determined by PSA exhibit nearly an order of magnitude a reduction in the failure probability as compared to the previous point. Figure 18 shows a close-up view of Fig. 17 and indicates that the ISAP procedure provides fast convergence around the targeted failure probability.

Table 7 summarizes the computational performance of the ISAP. After four ISAP iterations with nine P_f computations and 712 FE analyses, the convergence occurred with a corresponding P_f value smaller than the admissible relative distance error (ε) of 1%.

A 72-bar spatial truss problem

Definition

Figure 19 shows a space truss first investigated by Shayanfar et al. [⁴²] as a RBDO problem. The geometry of the truss with node numbering and grouping of the members are also indicated in Fig. 19. Shayanfar et al. [⁴²] set the design variables as discrete values for the cross-sectional areas to minimize the total truss weight. Design variables are selected from a 23-component semi-discrete catalog list CL= [0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 1.7, 1.9, 2.1, 2.3, 2.5, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7, 3.9, 4.1, 4.3, and 4.5]. Table 8 lists the statistical properties of the random variables. In that table, the mean value for the material density is set as 0.1 lb/in³ (2767.990 kg/m³). To enforce geometrical symmetry, members of the truss are organized into 16 groups (see Table 9). Loading conditions consist of a mean 5 kips (22.2411 kN) force in both the x- and y-directions and a mean -5 kips (-22.2411 kN) in the z-direction applied at node 1. The lower and upper bounds on the cross-sectional areas of the members are 0.1 in² (0.6452 cm²) and 4.5 in² (29.0322 cm²), respectively. The allowable lateral displacement at node 1 is 0.3 in (0.762 cm) in both x- and y-directions. The problem is constrained to two limit state functions G₁ and G₂. The target reliability index is 3, i.e., the target failure probability is 0.00135.

Results obtained

Shayanfar et al. [⁴²] have solved this problem using the conventional double-loop approach employing a genetic algorithm (GA) for optimization and first-order reliability method (FORM) for reliability task. Table 9 lists the results from the proposed PSA-ISAP-based method compared to those of Shayanfar et al. [⁴²]. Although Shayanfar et al. [⁴²] produced an optimum weight of 535.79 lb. (243.03026 kg) for a reliability index of b = 3 corresponding to a failure probability of 0.00135, it is slightly in the unsafe region. The proposed PSA-ISAP-based method generated a minimum truss weight of 538.3454 lb (244.1893 kg) that nearly met the targeted P_f of 0.00135.

Due to the randomized nature of the applied loading at node 1, the behavior of the structure is expected to be non-symmetric. In problems where more than one LSF is involved, it might be possible for one LSF to be dominant. This approach may well effect on reducing the computational cost in obtaining the target P_f value throughout the RBDO procedure. The proof for such claim lies in treating this problem, where there are two limit state functions, and in each of the Phases 2 and 3 of the proposed method, the LSF with the greater failure probability, G₁, was found as dominant throughout the process in all stages. However, the role of G₂ was not ignored along the process, although it was found dominated all the way to the end of the process. Figures 20–25 were illustrated according to the P_f of G₁ function.

Figures 20 and 21 show the Phase 2 convergence history of the proposed method. PSA is utilized to find the first reliable optimum solution with a value of 542.33 lb (245.996 kg) with a corresponding maximum P_f = 8.163 × 10⁻⁴ for G₁. Figures 22 and 23 show how the ISAP is utilized to fit cubic B-splines (truss-weight, P_f) curve through ten sample data. Figures 24 and 25 illustrate the convergence history of 72-bar truss weight against P_f computed by RSM in Phase 3. When the ISAP is employed, the initial weight of 532.63 lb (241.597 kg) converges to an optimum value of 538.3454 lb (244.1893 kg) with a corresponding value of 0.00130 for the P_f. The solution generated by the proposed method met the targeted P_f with a smaller relative distance error than the admissible one.

Comparing the results listed in Table 9 indicate that the RSM-based RBOD failure probabilities are very close to those obtained using MCS. In addition, since the design is located further into the safe margin, the resulting value of objective value is slightly larger (0.47%) than that of Shayanfar et al. [⁴²]. The design developed by the proposed algorithm required just 20 P_f evaluations.

Table 10 lists three iterations of ISAP (Phase 3). Based on the data in this table, ISAP has performed a desirable operation so that after 20 P_f computations of and 1440 FE analyses, the exact targeted failure probability was achieved. This may be favorably compared to the solution by Shayanfar et al. [⁴²] where the solution was obtained after 200 times P_f computations.

A 25-bar space truss problem

Definition

Figure 26 shows the topology of a 25-bar space truss recently studied by Ho-Huu et al. [⁴³] and Dizangian and Ghasemi [³³] as a RBDO problem. Ho-Huu et al. [⁴³] solved this truss problem using discrete design variables while Dizangian and Ghasemi [³³] used continuous values for member cross-sectional areas. In this paper, similar to the work done by Ho-Huu et al. [⁴³], design variables are selected from a 30-component discrete catalog list CL= [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 3.0, 3.2, and 3.4 in²].

For this space truss, the material density was 0.1 lb/in³ (2767.990 kg/m³) and the modulus of elasticity was 10000 ksi (68.950 GPa). Table 11 lists the applied nodal loads. Design constraints are a maximum allowable displacement of ±0.35 in (±8.89 mm) at nodes 1 and 2 in both the x and y directions and an allowable stress for all members of ±40 ksi (±275.89 MPa). The minimum admissible cross-sectional areas of all members were set equally as 0.1 in² (6.45 mm²). For consistency with other studies in the literature, the 25 bar elements were classified into eight groups as given in Table 12. For this problem, the random variables are the cross-sectional area A_i of each group, Young’s modulus E, and the external force P; all considered to be statistically independent and to follow normal distributions. The covariant of all random variables (C.O.V) is 5% of variable values. The LSF for this problem was that the displacements of all nodes should be less than ±0.35 in (±8.89 mm) in all directions. The targeted P_f is set equal to 0.001349 (b = 3).

Results obtained

Ho-Huu et al. [⁴³] and Dizangian and Ghasemi [³³] solved this problem using SORA-ICDE and DAF-RSM RBDO approaches, respectively. Table 12 summarizes the RBDO results of this problem and provides a comparison with previous studies. The results obtained by the proposed PSA-ISAP RBDO approach agree well with those of other publications.

As presented in Table 12, the proposed RBDO approach obtained a reliable global optimum solution with the truss weight of 659.22 lb (298.956 kg) using seven P_f computations and a total of 2333 FE analyses. This truss weight is slightly better than those of the SORA-ICDE [⁴³] and DAF-RSM [³³]. The optimum solution from SORA-ICDE [⁴³] has some tolerance from the targeted P_f while DAF-RSM [³³] has not converged exactly to a solution with the targeted P_f.

Figures 27 shows the convergence history of Phases 1 and 2 for P_f and truss weight; at the end of which, the first reliable optimum solution (x*) with the corresponding P_f equals to 0.00067 was achieved. Figures 28 through 31 show convergence fit plots of Phases 2 and 3 which ultimately lead to a global reliable optimum truss design with the corresponding P_f equals to 0.00134 and a truss weight of 659.22 lb (298.956 kg).

Table 13 lists the results of ISAP (Phase 3) for the 25-bar space truss. After three iterations, the modification factor α converged to 0.8. Also, it can be seen from the results that ISAP provides good performance while the optimum solutions have P_f values close to the target.

Complementary computational costs

Table 14 summarizes the computational costs of P_f evaluations based on RSM for the four problems studied.

Conclusions

In this paper, a new technique based on probabilistic sensitivity analysis and cubic B-spline curves is introduced to solve reliability-based optimization of truss problems. The proposed method significantly reduces the number of analyses required for RBDO compared to conventional methods. The technique may be classified as a three-phase decoupling method. Phase 1 finds a deterministic optimum point based on the mean random variables. In Phase 2, the sensitivity of the failure probability with respect to each design variable is estimated and used to compute the first reliability point around the deterministic optimum point. Phase 3 utilizes an ISAP using cubic B-spline interpolation curves. At this stage, an iterative procedure using only a few P_f computations generates an optimum design point that coincides with the targeted failure probability; a point one may claim as being the most reliable optimum point.

Using the modified RSM in Phases 2 and 3, significantly reduced the P_f computational cost. The proposed RBDO algorithm performance was verified with four structural optimization truss problems: three truss problems utilizing a single limit state function and one problem with two limit state functions.

Design obtained with the proposed RBDO algorithm indicate that the method possesses sufficient speed and accuracy. In all presented examples, after completing all the three aforementioned phases, reliable optimum design points were found with just under 20 P_f computations.

A new technique was applied to the 72-bar truss problem with two limit state functions. In this case, it was found that the LSF with the greater failure probability could be treated as the dominant function throughout the process. It is concluded, that in such cases one may focus on the LSF with the maximum P_f, while not ignoring the existence and possibly the dominancy of other LSFs. In future work, the aim will be to globalize this technique as a reliable approach to handle multiple LSF-based problems with the least computational efforts required.

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