Reliability-Based Design Optimization of Structures Using Complex-Step Approximation with Sensitivity Analysis

Abstract

Structural optimization aims to achieve a structural design that provides the best performance while satisfying the given design constraints. When uncertainties in design and conditions are taken into account, reliability-based design optimization (RBDO) is adopted to identify solutions with acceptable failure probabilities. This paper outlines a method for sensitivity analysis, reliability assessment, and RBDO for structures. Complex-step (CS) approximation and the first-order reliability method (FORM) are unified in the sensitivity analysis of a probabilistic constraint, which streamlines the setup of optimization problems and enhances their implementation in RBDO. Complex-step approximation utilizes an imaginary number as a step size to compute the first derivative without subtractive cancellations in the formula, which have been observed to significantly affect the accuracy of calculations in finite difference methods. Thus, the proposed method can select a very small step size for the first derivative to minimize truncation errors, while achieving accuracy within the machine precision. This approach integrates complex-step approximation into the FORM to compute sensitivity and assess reliability. The proposed method of RBDO is tested on structural optimization problems across a range of statistical variations, demonstrating that performance benefits can be achieved while satisfying precise probabilistic constraints.

1. Introduction

Structural systems operate in variable and uncertain environments that impact building performance and integrity. While maintaining a satisfactory level of safety and reliability, structural systems are expected to satisfy their design criteria, such as performance and serviceability. Uncertainties and tolerances are commonly present and even anticipated during the design process and construction. The prediction of building response to these events has a crucial role in the design and optimization of structures. In the optimal design of structural systems, variations in the material, loads, and parameters of a structure are random variables with a certain probability distribution rather than fixed coefficients. With the existence of such uncertainties, the problem of reliability analysis and its reflection must be addressed during the design process of structures. The inclusion of reliability analysis within the design process could drastically reduce the total cost of development and the risk of failure.

Structural design optimization can be viewed as a process that methodically searches for improved engineering solutions and designs while considering requirements and constraints that exist in the field. In deterministic design optimization (DDO) [1], the uncertainties of a structural system, such as material properties, forces, models, and sizes, are taken into account subjectively and indirectly, by means of safety factors or load combinations with coefficients specified in the design codes. The procedures of deterministic design optimization consider the effects of uncertainties in a qualitative way, without incorporation of quantitative characterization of the probabilistic nature of system and design. Non-deterministic optimization has emerged as an alternative form to accurately model the safety-under-uncertainty aspect of the optimization problem. Methods for considering uncertainty in the optimal design process can be categorized into two groups. Reliability-based design optimization (RBDO) aims to meet a specific target failure probability for objectives and/or constraint functions that include uncertainty parameters. Robust design optimization (RDO), on the other hand, minimizes the sensitivity of the objectives and constraints to the uncertainty design parameters.

The growing interest in the identification of feasible, reliable, and robust solutions in structural design optimization has led to significant advances in theoretical and numerical developments, such as simulation and analysis approaches and optimization algorithms. Sandren and Cameron [2] proposed a robust design optimization method for structures with considerations of parameter variation and sensitivity to variation by using genetic and non-linear programming algorithms. Marano et al. [3] developed a robust optimal design criterion for a tuned mass damper device under random vibrations. The robustness of optimal solutions was obtained by solving a multi-objective optimization problem in which the mean and standard deviations of the deterministic objection were minimized. A robust design of a vibration absorber under uncertainties in mass and stiffness of the dynamic system was utilized to demonstrate the robust design approach by Zang et al. [4]. Robust design optimization was performed by a direct first-order perturbation method based on a Taylor expansion. The development of nonlinear programming-based RDO [5,6,7] in various fields with diverse applications and a thorough review of RDO methods are illustrated in the literature, e.g., by Messac and Ismail-Yahaya [8], as well as Zhang et al. [4].

In RBDO, a design at each iteration step is checked to determine whether it satisfies probabilistic constraints by reliability analysis, which requires repetitive computational efforts [9]. To assess the probabilistic constraints in RBDO, various methods have been developed [10,11]. The reliability index approach (RIA) [9] and its inverse, the performance measure approach (PMA) [10,12], utilize the first-order reliability method (FORM) [13] for reliability analysis. RBDO is performed using a nested approach, such that each iteration in the design optimization process involves a sub-loop of iterations for reliability assessment at the current stage of design. This is often called a “double-loop” approach because each step of the optimization process involves another iteration for the reliability analysis [13]. As a result of the repeated iteration process, the computational cost associated with the utilization of the double-loop approach can prohibitively be expensive. To improve the computational efficiency of reliability analysis and optimization procedure, many RBDO approaches, such as decoupling [14,15] and single-loop (SL) methods [16,17,18,19,20], have been proposed. The decoupling method decouples the RBDO procedure by constructing a sequence of cycles with deterministic optimization followed by the reliability analysis of the converged solution in the feasible domain. The SL approach is based on the Karush–Kuhn–Tucker (KKT) optimality conditions to approximate the solution of the sub-loop optimization. As a result, the sub-loop for reliability analysis is replaced by a deterministic constraint, which transforms a double-loop RBDO problem into an equivalent single-loop optimization problem. RBDO methods have been applied to many fields, including structural engineering [17,21], mechanical engineering [22,23], aerospace engineering [24], and material science [25]. A comprehensive review on RBDO can be found in the work of Song et al. [26], Frangopol and Maute [27], and Tsompanakis et al. [28].

In DDO, RDO, and RBDO, sensitivity analysis of probabilistic constraints is crucial to the utilization of efficient optimization algorithms. The dependence on reliability with respect to the design inputs or values can be evaluated using sensitivity analysis. There have been multiple approaches in sensitivity analysis, such as Monte Carlo Simulation-based approaches [29,30], and simulation-based methods [31,32]. However, these approaches generally require high computational power that might restrict their applicability, especially when sensitivity analysis needs to be performed with respect to many parameters. The difficulty of sensitivity analysis increases when the underlying functions and models are complex, or the analytical derivation of gradients is not feasible due to the implicit system. In this paper, the sensitivity analysis of failure probability, a probabilistic constraint of RBDO, is proposed using complex-step (CS) approximation integrated with the FORM. Li [33] showed sensitivity analysis in the FORM-based RBDO using the CS. This paper provides details on the general integration of CS in the FORM and discusses uniform applicability to structural RBDO. The sensitivity analysis is performed using a very small complex value, such that the outcome is not significantly affected by a selection of steps, unlike the finite difference method. In addition, the proposed method can be easily implemented in RBDO and is free from round-off and subtractive cancellation errors when compared with the finite difference method. Furthermore, sensitivity analysis and assessment of the failure probability can be simultaneously performed using the CS approximation simply by taking real and imaginary parts of numerical results so that computational efficiency in analysis and optimization increases. This paper presents the RBDO method utilizing the FORM and presents a sensitivity analysis of probabilistic constraints with respect to design and random variable parameters to use a gradient-based optimization algorithm.

2. Reliability Analysis and Reliability-Based Design Optimization

2.1. Reliability Analysis Using the First-Order Reliability Method

The time-invariant reliability problem can be characterized by an n-vector of random variables $x={\left(x_1,x_2,x_3,\dots ,x_n\right)}^{\mathrm{T}}$ and the subset Ω of their outcome space, which defines the failure events of the limit-state function g(x). The failure probability $P_f$ is given by the n-fold integral

$$\begin{array}{c}{P}_f=P\left(x\in \left\{\mathsf{\Omega }\equiv g\left(x\right)\le 0\right\}\right)={{\displaystyle \int }}_{g\left(x\right)\le 0}^{}k\left(x\right)dx\end{array}$$

(1)

where k(x) is the joint probability density function (PDF) of x. By transforming the random variables into the standard normal space $\left(u=\mathrm{T}\left(x\right)\right)$, the failure probability integral can be written as

$$\begin{array}{c}{P}_f={{\displaystyle \int }}_{G\left(u\right)\le 0}^{}{\phi }_n\left(u\right)du\end{array}$$

(2)

where $G\left(u\right)$ is the limit-state function transformed from the original space into the standard normal space and ${\phi }_n\left(·\right)$ is the n-variate standard normal density function for the random vector. The component reliability problem utilizes a reliability method to compute the failure probability. The FORM linearizes the limit-state function $G\left(u\right)$ at a point u*, which is obtained by solving a constraint optimization problem as

$$\begin{array}{c}\underset{u\ast }{\arg \min }\left\{\Vert u\Vert |G\left(u\right)=0\right\}\end{array}$$

(3)

The point u* indicates a location in the linearized limit-state function, in which the distance from the origin in the standard normal space is the minimum. This point is commonly known as the “design point” or “most probable point” (MPP). The linearized limit-state function is written as

$$\begin{array}{c}G\left(u\right)=\nabla G\left(u^{\ast }\right)\left(u-u^{\ast }\right)\end{array}$$

(4)

where $\nabla G\left(u^{\ast }\right)=\left({\partial }_{u_1^{\ast }}G, {\partial }_{u_2^{\ast }}G,\dots ,{\partial }_{u_n^{\ast }}G\right)$ denotes the gradient row vector. The linearization represents the failure domain $G\left(u\right)\le 0$, that is, the half-space defined by $\beta -\alpha u\le 0$. $\beta =\alpha u^{\ast }$ is the reliability index and $\alpha =-\frac{\nabla G\left(u^{\ast }\right)}{\Vert \nabla G\left(u^{\ast }\right)\Vert }$ denotes the normalized negative vector at the design point (refer to Figure 1). This representation leads to

$$\begin{array}{c}G\left(u\right)=\Vert \nabla G\left(u^{\ast }\right)\Vert \left(\beta -\alpha u\right)\end{array}$$

(5)

The probability content of the half-space, which is defined by the reliability index, gives the approximation of the failure probability as

$$\begin{array}{c}{P}_f\approx \mathsf{\Phi }\left(-\beta \right)\end{array}$$

(6)

where $\mathsf{\Phi }\left(·\right)$ denotes the cumulative distribution function (CDF) of the standard normal distribution.

It should be noted that there are various algorithms for approximation of the design point in Equation (3) and the reliability index [34,35]. The accuracy of the FORM depends upon how well the actual limit-state surface is represented by the linear approximation. For nonlinear limit-state functions having a larger number of random variables, the FORM may not identify accurate design points or encounter a convergence issue while searching for the most probable points for complex systems. The nonlinearity of the limit-state surface is caused by various factors, such as the nonlinear relationship between random variables and the transformation of random variables from the original space to standard normal space. The second-order reliability method (SORM) [13,36,37] can improve the assessment given by FORM by including information about the curvature. The SORM approximates the limit-state surface by using a hyper-paraboloid with tangent hyperplane and main curvatures at the design point. Furthermore, Der Kiureghian and Dakessian [38] showed that a limit-state function might have multiple design points. An optimization algorithm utilized to solve Equation (3) may converge to a local design point that results in a gross approximation error. Several search algorithms [38,39,40] integrating multiple design points have been proposed as an effort to improve the accuracy of reliability analysis.

2.2. Reliability-Based Design Optimization

One of the goals in structural optimization is to find the optimal solutions in a given design domain subjected to tractions and displacement boundary conditions while satisfying given design constraints. When the reliability analysis of a structure under uncertainty is incorporated into structural optimization, it is referred to as RBDO. RBDO aims to achieve the optimal design under given probabilistic constraints, arising from uncertainties in material properties or loads. The component RBDO (CRBDO) approach seeks to satisfy the probabilistic constraint regarding each failure event. A CRBDO problem can be formulated as

$$\begin{array}{ll}\underset{d,{\mu }_x}{\min }& f_{obj}\left(d,{\mu }_x\right)\\ \mathrm{s}.\mathrm{t}.& P_{fi}=P\left[g_i\left(d,x\right)\le 0\right]\le P_{fi}{}^{\mathrm{t}},i=1,\dots ,n_c\\ & d^{\mathrm{l}}\le d\le d^{\mathrm{u}}\end{array}$$

(7)

where $f_{obj}$ is the objective function; d is a vector of deterministic design variables; ${\mu }_x$ is the vector of the means of random variable x; $P\left[g_i\left(d,x\right)\le 0\right]$ is the failure probability of the i-th limit-state function $g_i\left(·\right)$; $P_{fi}{}^{\mathrm{t}}$ represents the target failure probability of the i-th limit-state; $n_c$ is the number of probabilistic constraints; and d^l and d^u are the lower bounds of design variables and upper bounds of design variables, respectively. This problem is solved by generating and solving a sequence of explicit subproblems that are approximations of Equation (7). Optimization algorithms, such as sequential linear programming (SLP), sequential quadratic programming (SQP; [41]), convex linearization (COLIN; [42]), optimality criteria (OC; [43]), and method of moving asymptotes (MMA; [44]), incorporate information about the sensitivity of objective and probabilistic constraint functions to solve subproblems.

2.3. Sensitivity Analysis

Sensitivity analysis of the probabilistic constraint aims to evaluate the dependence of reliability on design parameters or random variables. To utilize an efficient gradient-based optimization algorithm in RBDO, sensitivity analysis is integral. The sensitivity of the failure probability with respect to the design parameter $d$ [45] can be computed by a chain rule as

$$\begin{array}{c}{\nabla }_d{P}_f=\frac{d{P}_f}{d\beta \left(d,x^{\ast }\right)}·{\nabla }_d\beta \left(d,x^{\ast }\right)=\frac{d\mathsf{\Phi }\left(-\mathsf{\beta }\right)}{d\beta \left(d,x^{\ast }\right)}·{\nabla }_d\beta \left(d,x^{\ast }\right)=-\phi \left(\beta \right)·{\nabla }_d\beta \left(d,x^{\ast }\right)\end{array}$$

(8)

where

$$\begin{array}{c}{\nabla }_d\beta \left(d,x^{\ast }\right)=\frac{1}{\Vert \nabla G\left(d,u^{\ast }\right)\Vert }·{\nabla }_{d}g\left(d,x^{\ast }\right)\end{array}$$

(9)

Similarly, the sensitivity of the probability approximation with respect to a set of distribution parameters $\mathsf{\theta }$, such as the mean and standard deviation, is obtained from

$$\begin{array}{c}{\nabla }_{\mathsf{\theta }}{P}_f=\frac{d{P}_f}{d\beta \left(d,x^{\ast }\right)}·{\nabla }_{\mathsf{\theta }}\beta \left(d,x^{\ast }\right)=\frac{d\mathsf{\Phi }\left(-\mathsf{\beta }\right)}{d\beta \left(d,x^{\ast }\right)}·{\nabla }_{\mathsf{\theta }}\beta \left(d,x^{\ast }\right)=-\phi \left(\beta \right)·{\nabla }_{\mathsf{\theta }}\beta \left(d,x^{\ast }\right)\end{array}$$

(10)

where

$$\begin{array}{c}{\nabla }_{\mathsf{\theta }}\beta \left(d,x^{\ast }\right)=\alpha J_{u,\mathsf{\theta }}\left(x^{\ast },\mathsf{\theta }\right)\end{array}$$

(11)

where $J_{u,\mathsf{\theta }}$ is the Jacobian of the $\mathsf{\theta }$ to u transformation.

3. Complex-Step Approximation and Finite Difference Method

The complex-step (CS) derivative method [46] was introduced by Squire and Trapp [47] and has been proven to be more efficient for the first-order derivative calculation than the conventional finite difference method [48]. In the CS derivative approximation, an imaginary number multiplied by the step size h is utilized. The first derivative is approximated by a single function evaluation with it. To derive the CS derivative and identify its associated errors, consider the differentiable function f($·$) and the point y on the real axis ($y \in ℝ)$. The Taylor series expansion of f($\mathrm{y}+\mathrm{i}h$) gives

$$\begin{array}{c}f\left(\mathrm{y}+\mathrm{i}h\right)\approx f\left(y\right)+ih{f}^{\prime }\left(y\right)-\frac{h^2}{2!}{f}^{\left(2\right)}\left(y\right)-i\frac{h^3}{3!}{f}^{\left(3\right)}\left(y\right)+\frac{h^4}{4!}{f}^{\left(4\right)}\left(y\right)+\dots \end{array}$$

(12)

where i is the imaginary unit. Rearranging Equation (12) and taking the imaginary part of both sides yield

$$\begin{array}{c}{f}^{\prime }\left(y\right)\approx \frac{\Im \left[f\left(y+\mathrm{i}h\right)\right]}{h}+O\left(h^2\right)\end{array}$$

(13)

where $\Im$[·] denotes the imaginary part of the function f. When a small h is selected, terms with the order h² and higher can be truncated. Evaluating the function f at the imaginary argument $y+\mathrm{i}h$, and dividing it by h gives an approximation to the first derivative. The CS approximation in Equation (13) does not cause subtractive cancellation and results in calculations without the associated round-off error. Note that the real part of Equation (13) multiplied by h is equal to f(y). Therefore, the first derivative approximation and function value can be computed simultaneously, which will result in increased computational efficiency for analysis and optimization. The level of accuracy in the first derivative using the conventional finite difference method can vary with a change in the step size, which is generally selected arbitrarily. This level indicates that the actual effect of the step size in the finite difference method on the result makes it difficult to predict prior to actual calculation. It is noteworthy that sensitivity analysis using the CS approximation simply requires the evaluation of the function with a complex variable without analytical derivations of gradients. Therefore, the CS approximation allows for sensitive analysis in situations when the analytic derivative of gradients cannot be explicitly expressed or when each analysis becomes computationally expensive.

More accurate finite difference approximations for the first derivative can be derived by combining different Taylor series expansions. For instance, subtracting the two expansions at the points f(x−h) and f(x+h) gives

$$\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{f\left(x+h\right)-f\left(x-h\right)}{2h}+O\left(h^2\right)\end{array}$$

(14)

An error from Equation (14), which is referred to as the second-order central difference method, is proportional to $h^2$. Similarly, second-order forward difference (FDM²) and fourth-order central difference methods method (CDM⁴) for approximating the first derivative are

$$\begin{array}{c}{f}^{\prime }\left(x\right)=\left\{\begin{array}{l}{\mathrm{FDM}}^2:\frac{-3f\left(x\right)+4f\left(x+h\right)-f\left(x+2h\right)}{2h}+O\left(h^2\right)\hfill \\ {\mathrm{CDM}}^4:\frac{-f\left(x+2h\right)+8f\left(x+h\right)-8f\left(x-h\right)+f\left(x-2h\right)}{12h}+O\left(h^4\right)\hfill \end{array}\right.\end{array}$$

(15)

Although the high-order finite difference method leads to less truncation error, which can also be minimized by taking a relatively small h, subtractive cancellations involved in formulas may lead to significant round-off errors, unlike the CS approximation.

4. Proposed Method of Sensitivity Analysis

The proposed method utilizes the CS approximation to compute the sensitivity of a probabilistic constraint with respect to a vector of design variables. This section introduces procedures of the CS derivative approximation for gradients and sensitivity analysis. Consider the vector of random variables $x={\left(x_1,x_2,x_3,\dots ,x_n\right)}^{\mathrm{T}}$ and the vector of design variable $d={\left(d_1,d_2,d_3,\dots ,d_z\right)}^{\mathrm{T}}$. The sensitivity of a reliability index in the FORM with respect to a vector of design variables can be derived using the CS approximation as

$$\begin{array}{ll}{\nabla }_d\beta \left(d,x^{\ast }\right)& =\frac{1}{\Vert \nabla G\left(d,u^{\ast }\right)\Vert }·{\nabla }_{d}g\left(d,x^{\ast }\right)=\frac{1}{\Vert \nabla G\left(d,u^{\ast }\right)\Vert }·{\left[\frac{\partial g\left(d,x^{\ast }\right)}{\partial d_1},\dots ,\frac{\partial g\left(d,x^{\ast }\right)}{\partial d_k},\dots ,\frac{\partial g\left(d,x^{\ast }\right)}{\partial d_z}\right]}^{\mathrm{T}}\\ & =\frac{1}{\Vert \nabla G\left(d,u^{\ast }\right)\Vert }·\frac{\Im {\left[g\left(d+\mathrm{i}h{e}_1,x^{\ast }\right),\dots ,g\left(d+\mathrm{i}h{e}_k,x^{\ast }\right),\dots ,g\left(d+\mathrm{i}h{e}_z,x^{\ast }\right)\right]}^{\mathrm{T}}}{h}\end{array}$$

(16)

where $e_k$ denotes the k-th column of a z-th order identity matrix ($I_z$). Because the limit-state function is defined in terms of the random variables x, it is necessary to carry out these calculations in the original space. Results are then transformed in the standard normal space to perform the FORM. The gradient vector $\nabla G\left(u^{\ast }\right)$ can be rewritten as

$$\begin{array}{c}\nabla G\left(d,u^{\ast }\right)={\nabla }_{x^{\ast }}g\left(d,x^{\ast }\right)J_{u,x}^{-1}\left(x^{\ast }\right)\end{array}$$

(17)

where $J_{u,x}^{-1}$ is the inverse of the Jacobian of the x to u transformation. Note that the Jacobian is dependent on the distribution types of random variables. For instance, the Jacobian of transformation $J_{u,x}$ for statistically independent random variables $x$ is

$$\begin{array}{c}{J}_{u,x}=diag\left[J_{ii}\right], i=1,\dots ,n \\ J_{ii}=\frac{f_i\left(x_i\right)}{\phi \left({\mathsf{\Phi }}^{-1}\left[{{\displaystyle \int }}_{-\infty }^{x_i}{f}_i\left(x_i\right)d{x}_i\right]\right)}\end{array}$$

(18)

where $f_i\left(x_i\right)$ is the marginal PDF of $x_i$ and $\phi$(∙) is the univariate standard normal PDF. If the random variables x are normally distributed with the correlation matrix $R=L{L}^{\mathrm{T}}$, in which $L$ is a lower-triangular matrix obtained by the Cholesky decomposition of R, the Jacobian is

$$\begin{array}{c}{J}_{u,x}=L^{-1}{D}^{-1}\end{array}$$

(19)

where D is the diagonal matrix of standard deviations. Furthermore, the Jacobian matrix of dependent nonnormal random variables can be derived [49] as

$$\begin{gathered} \begin{array}{lll}{J}_{11}& =\frac{f_1\left(x_1\right)}{\phi \left({\mathsf{\Phi }}^{-1}\left[{{\displaystyle \int }}_{-\infty }^{x_1}{f}_1\left(x_1\right)d{x}_1\right]\right)}& \\ J_{ij}& =0& i<j\\ & =\frac{f_i\left(x_i|x_1,\dots ,x_{i-1}\right)}{\phi \left({\mathsf{\Phi }}^{-1}\left[F_i\left(x_i|x_1,\dots ,x_{i-1}\right)\right]\right)}& i=j>1\\ & =\frac{1}{\phi \left({\mathsf{\Phi }}^{-1}\left[F_i\left(x_i|x_1,\dots ,x_{i-1}\right)\right]\right)}\frac{\partial F_i\left(x_i|x_1,\dots ,x_{i-1}\right)}{\partial x_j}& i>j\end{array} \\ \mathrm{where} F_i(x_i|x_1,\dots ,x_{i-1})=\underset{-\infty }{\overset{x_i}{{\displaystyle \int }}}{f}_i(x_i|x_1,\dots ,x_{i-1})d{x}_i \end{gathered}$$

(20)

where $f_i(x_i|x_1,\dots ,x_{i-1})$ denotes the conditional PDF of $x_i$ for given $x_1,\dots ,x_{i-1}$.

The gradient of the limit-state function ${\nabla }_{x^{\ast }}g\left(d,x^{\ast }\right)$ in Equation (17) is then computed as

$${\nabla }_{x^{\ast }}g\left(d,x^{\ast }\right)={\left[\frac{\partial g\left(d,x^{\ast }\right)}{\partial x_1^{\ast }},\dots ,\frac{\partial g\left(d,x^{\ast }\right)}{\partial x_k^{\ast }},\dots ,\frac{\partial g\left(d,x^{\ast }\right)}{\partial x_n^{\ast }}\right]}^T$$

(21)

Application of the CS approximation to Equation (21) yields

$$\begin{array}{c}{\nabla }_{x^{\ast }}g\left(d,x^{\ast }\right)=\frac{1}{h}·\Im {\left[g\left(d,x^{\ast }+\mathrm{i}h{e}_1\right),\dots ,g\left(d,x^{\ast }+\mathrm{i}h{e}_k\right),\dots ,g\left(d,x^{\ast }+\mathrm{i}h{e}_n\right)\right]}^{\mathrm{T}}\end{array}$$

(22)

Note that taking a real part of $g\left(d,x^{\ast }+\mathrm{i}h{e}_k\right), k=1,\dots ,n$ in Equation (22) results in a value of $g\left(d,x^{\ast }\right)$. This indicates that both the value and sensitivity of the limit-state function can be obtained from Equation (22). Using Equation (8), the sensitivity of the first-order probability approximation with respect to a vector of design variables is obtained by computing

$$\begin{array}{c}{\nabla }_d{P}_f=-\phi \left(\beta \right)·\frac{\Im {\left[g\left(d+\mathrm{i}h{e}_1,x^{\ast }\right),\dots ,g\left(d+\mathrm{i}h{e}_k,x^{\ast }\right),\dots ,g\left(d+\mathrm{i}h{e}_z,x^{\ast }\right)\right]}^{\mathrm{T}}}{h\Vert {\nabla }_{x^{\ast }}g\left(d,x^{\ast }\right)J_{u,x}^{-1}\left(x^{\ast }\right)\Vert }\end{array}$$

(23)

Note that the gradients in Equation (21) are also utilized in the FORM to compute a failure probability of a limit-state function.

When the improved Hasofer and Lind algorithm [50] is utilized to solve Equation (3), a sequence of points $u$ by using a recursive formular is identified as

$$\begin{array}{c}{u}^{i+1}=u^i+{\mathsf{\lambda }}^i{s}^i, i=1,2,3,\dots .\end{array}$$

(24)

where ${\mathsf{\lambda }}^i$ is a search scale factor and $s_i$ is the search direction vector obtained by

$$\begin{array}{c}{s}^i=\left(\frac{G\left(d,u^i\right)}{\nabla G\left(d,u^i\right)}+{\alpha }^i{u}^i\right){\left({\alpha }^i\right)}^{\mathrm{T}}-u^i\end{array}$$

(25)

The gradient $\nabla G\left(d,u^i\right)$ can be computed by using the CS approximation as

$$\nabla G\left(d,u^i\right)={\nabla }_{x^i}g\left(d,x^i\right)J_{u,x}^{-1}\left(x^i\right)=\frac{1}{h}·\Im {\left[g\left(d,x^i+\mathrm{i}h{e}_1\right),\dots ,g\left(d,x^i+\mathrm{i}h{e}_k\right),\dots ,g\left(d,x^i+\mathrm{i}h{e}_n\right)\right]}^{\mathrm{T}}{J}_{u,x}^{-1}\left(x^i\right)$$

(26)

The proposed RBDO method using the CS approximation (RBDO−CS) performs the reliability analysis by identifying the design points based on gradients in Equations (22) and (26), and sensitivity analysis using Equation (23) to utilize a gradient-based optimization algorithm.

5. Numerical Applications

5.1. Comparative Study of the Accuracy of Complex-Step and Finite Difference Methods

The effect of step size on the first derivative by finite difference and CS methods is investigated. Consider the following highly nonlinear function:

$$\begin{array}{c}f\left(x\right)=\frac{e^x}{\sqrt{{\sin }^3\left(x\right)+{\cos }^3\left(x\right)}}\end{array}$$

(27)

The four finite difference methods (first-order forward (FDM), second-order forward (FDM²), second-order central (CDM²), and fourth-order central difference (CDM⁴) approximations) and CS approximation discussed in Section 3 are utilized to compute the first derivative at $x=-0.35$ with varying step sizes.

The numerical results in Figure 2a show that the first derivatives by the CS method are not sensitive to the selection of step size. Finite difference methods, however, break down as the step size reaches the last value. Although higher-order finite difference methods result in fewer truncations errors, subtractive cancellation and their associate round-off errors affect the calculations. It should be noted that the number of function evaluations in higher-order finite difference methods is higher than one in the CS approach. Errors between the exact derivatives and approximated derivatives with varying step sizes are plotted in Figure 2b. The comparison results confirm that the CS approximation is more accurate for small step sizes, whereas the finite difference approach cannot achieve full accuracy.

5.2. RBDO of the Short Column under a Probabilistic Strength Constraint

The proposed RBDO−CS approach is applied to identify the optimal member size of a short column subjected to an axial force $P$ and bending moments $M_1$ and $M_2$, as shown in Figure 3. Assume an elastic perfectly plastic material with yield stress ${\sigma }_y$. The limit-state function regarding the failure of the column is defined by

$$\begin{array}{c}g\left(d,x\right)=1-\frac{M_1}{S_1\left(d\right){\sigma }_y}-\frac{M_2}{S_2\left(d\right){\sigma }_y}-{\left(\frac{P}{A\left(d\right){\sigma }_y}\right)}^q\end{array}$$

(28)

where $x={\left(M_1,M_2,P,{\sigma }_y\right)}^{\mathrm{T}}$ denotes the vector of random variables, $q=2$ is the limit-state function parameter, $d$ is the design variable, $A\left(d\right)=1.5d^2$ is the cross-sectional area, and $S_1\left(d\right)=0.375d^3$ and $S_2\left(d\right)=0.25d^3$ are the flexural moduli of the plastic column section. Assume that $M_1,M_2,P,\mathrm{and}{\sigma }_y$ have the Nataf distribution with the marginal distributions and coefficient of variance (c.o.v) listed in

Table 1. Description of random variables for the short column optimization problem.

Random Variables	Marginal Distribution	Mean	c.o.v	Correlation Coefficient
				${\mathit{M}}_1$	${\mathit{M}}_2$	$\mathit{P}$	${\mathit{\sigma }}_{\mathit{y}}$
$M_1,\mathrm{kNm}$	Normal	250	0.3	1.0
$M_2,\mathrm{kNm}$	Normal	125	0.3	0.5	1.0
$P,\mathrm{kN}$	Gumbel	2500	0.2	0.3	0.3	1.0
${\sigma }_y,\mathrm{MPa}$	Weibull	40	0.1	0	0	0	1.0

. The objective function for this problem is taken as the minimization of the cross-sectional area, and the probabilistic constraint is defined in terms of yield strength. The RBDO problem is formulated as

$$\begin{array}{ll}\underset{d}{\min }& A\left(d\right)\\ \mathrm{s}.\mathrm{t}.& P_f=P\left[g\left(d,x\right)\le 0\right]\le {P_f}^{\mathrm{t}}\\ & d^{\mathrm{l}}\le d\le d^{\mathrm{u}}\end{array}$$

(29)

Table 2. Parameters for the target failure probability and optimization.

$\mathbf{Target}\mathbf{Failure}\mathbf{Probability}{\mathit{P}}_{\mathit{f}}{}^{\mathit{t}}$	Initial Design ${\mathit{d}}^0,\mathbf{m}$	Lower Bound ${\mathit{d}}^{\mathit{l}},\mathbf{m}$	Upper Bound ${\mathit{d}}^{\mathit{u}},\mathbf{m}$	Convergence Criterion
0.01	0.25	0.1	2.0	${10}^{-4}$

summarizes the target failure probability and optimization parameters. A sensitivity analysis is performed by the proposed method to utilize a gradient-based optimization algorithm—the method of moving asymptotes (MMA; [44]). The optimization process is repeated until the convergence criteria on the maximum number of interactions or the relative change in the design variable is satisfied. The initial reliability analysis of the limit-state function $g\left(d^0,x\right)$ by the FORM results in $u^{\ast }={\left(-2.0084,-0.9605,-0.3399,0.4619\right)}^{\mathrm{T}}, \beta =-2.2989$, and $P_f=0.9892.$ In this study, the single loop approach using the Karush–Kuhn–Tucker conditions (SL–KKT) [18] is utilized to compare the proposed RBDO–CS method. The CS approximation is also implemented to compute sensitivities in the SL–KKT. The same optimization parameters and convergence criteria are considered for the comparison.

Table 3. Comparison between results from the RBDO−CS and SL−KKT approaches for the column design problem.

RBDO Method	$\mathbf{Optimal}{\mathit{f}}^{*}$$\left({\mathit{m}}^2\right)$	$\mathbf{Optimal}{\mathit{d}}^{*}$$\left(\mathit{m}\right)$	${\mathit{P}}_{\mathit{f}}$		Averaged Computational Time Carried out to Five Iterations, (Seconds)	Total Number of Iterations
			FORM	SORM
RBDO−CS	0.2293	0.3909	0.0100	0.0126	0.0130	11
SL−KKT	0.2293	0.3908	0.0101	0.0127	0.0108	12

represents the optimal area and design variables obtained by the proposed RBDO–CS and SL–KKT methods, as well as the averaged computational time carried out to five optimization iterations (a modern (2020) eight-core 3.8 GHz Intel process (16 GB RAM) is used for the analyzes). Because the SL–KKT approach does not perform reliability analysis during optimization procedures, the failure probability of limit-state function is computed by the FORM with optimal column design. The optimal results from both approaches are in agreement with each other. Figure 4 shows the convergence histories of the RBDO–CS and the SL–KKT methods regarding the objective function, design variable, and failure probability. The proposed RBDO–CS method enables the identification of a feasible solution while satisfying the probabilistic constraint. Although the proposed RBDO–CS method performing reliability analysis requires a slightly increased computational time for five iterations of the optimization procedure when compared to the SL–KKT method, the number of iterations required to identify the converged solution is lower. Of note, when the limit-state function is highly nonlinear, the second-order reliability method (SORM; [13]) leads to a more accurate assessment of reliability. The limit-state function in Equation (28) is nonlinear and the SORM is utilized to assess failure probabilities of the optimized column by the RBDO–CS, and SL–KKT methods. The SORM results in Table 3 indicate that the first-order linearization of the limit-state function may lead to a less conservative design when it has a high degree of nonlinearity.

5.3. RBDO of the Ductile Frame Structure Subjected to Probabilistic Moment Strength Constraints

The frame structure under uncertainty in the magnitudes of external forces and moment capacities is illustrated in Figure 5a. Assume that the frame structure is constructed with ductile members that have plastic moment capacities $m_i, i=1,\dots ,5$, at joints. Under the externally applied forces $h, v$, this frame may fail due to any of the sway, beam, and combined mechanisms illustrated in Figure 5b. Using the principle of virtual work, the limit-state functions regarding these mechanisms are formulated [13] as

$$\begin{array}{l}{g}_1\left(d,x\right)=m_1+m_2+m_4+m_5-h{d}_1\\ g_2\left(d,x\right)=m_2+2m_3+m_4-v{d}_2\\ g_3\left(d,x\right)=m_1+2m_3+2m_4+m_5-h{d}_1-v{d}_2\end{array}$$

(30)

where $x={\left(m_1,m_2,\dots ,m_5,h,v\right)}^{\mathrm{T}}$ is the vector of random variables, and $d={\left(d_1,d_2\right)}^{\mathrm{T}}$ is the vector of design variables ($d_1$: height, $d_2$: width). Assumed distributions and second moments of the random variables are tabulated in

Table 4. Description of random variables for the ductile frame optimization problem.

Random Variables	Marginal Distribution	Mean	c.o.v	Correlation
$m_1,\mathrm{kNm}$	Joint lognormal	150	0.2	${\rho }_{m_i{m}_j}=0.3$ $,i\ne j$
$m_2,\mathrm{kNm}$
$m_3,\mathrm{kNm}$
$m_4,\mathrm{kNm}$
$m_5,\mathrm{kNm}$
$h,\mathrm{kN}$	Gumbel	50	0.4	Independent
$v,\mathrm{kN}$	Gamma	60	0.2	Independent

. The maximization of the height and width of the ductile frame structure and the three limit-state functions in terms of the moment capacities are considered the objective and probabilistic constraint functions, respectively. The RBDO problem of the ductile structure can be stated as

$$\begin{array}{ll}\underset{d}{\max }& f\left(d\right)=d_1+2d_2\\ \mathrm{s}.\mathrm{t}.& P_{f1}=P\left[g_1\left(d,x\right)\le 0\right]\le P_{f1}{}^{\mathrm{t}}\\ & P_{f2}=P\left[g_2\left(d,x\right)\le 0\right]\le P_{f2}{}^{\mathrm{t}}\\ & P_{f3}=P\left[g_3\left(d,x\right)\le 0\right]\le P_{f3}{}^{\mathrm{t}}\\ & d^{\mathrm{l}}\le d\le d^{\mathrm{u}}\end{array}$$

(31)

A target failure probability of 0.003 is assigned to all probabilistic constraints. The initial design and parameters for RBDO are summarized in

Table 5. Target failure probability and parameters for design variables and optimization.

$\mathbf{Target}\mathbf{Failure}\mathbf{Probability}{\mathit{P}}_{\mathit{f}}{}^{\mathbf{t}}$	Initial Design ${\mathbf{d}}^0,\mathbf{m}$	Lower Bound ${\mathbf{d}}^{\mathbf{l}},\mathbf{m}$	Upper Bound ${\mathbf{d}}^{\mathbf{u}},\mathbf{m}$	Convergence Criterion
$P_{fi}{}^{\mathrm{t}}=3\times {10}^{-3}$ $\left(i=1,2,3\right)$	(7.0, 7.0)	(1.0, 1.0)	(10.0, 10.0)	${10}^{-4}$

. Using the FORM with $d^0$, the following values for the three constraints are obtained:

Limit-State Function	$\mathbf{\beta }$	${\mathit{P}}_{\mathit{f}}$
$g_1\left(d^0,x\right)$	1.452	0.073
$g_2\left(d^0,x\right)$	1.435	0.076
$g_3\left(d^0,x\right)$	0.701	0.242

The convergence histories of the objective function, design variables, and failure probabilities given in Figure 6 demonstrate the quick identification of converged solutions from the RBDO−CS method while satisfying the probabilistic constraints. To verify the results of RBDO−CS, the SL−KKT approach is utilized for the ductile structure with the same design and optimization parameters in Table 4 and Table 5. Figure 6 shows the comparison of results from RBDO−CS and SL−KKT methods, which are in agreement. Values of design variables and failure probabilities of the limit-state functions obtained by the FORM and SORM at optimal solutions are listed in

Table 6. Comparison between results from the RBDO−CS and SL−KKT methods for the frame structure problem.

RBDO Method	$$\begin{gathered} \mathbf{Optimal}{\mathit{f}}^{\ast } \\ \left(\mathbf{m}\right) \end{gathered}$$	$$\begin{gathered} \mathbf{Optimal}{\mathbf{d}}^{\ast } \\ \left(\mathbf{m}\right) \end{gathered}$$		${\mathit{P}}_{\mathit{f}\mathit{i}}$		Averaged Computational Time Carried out to Five Iterations, (Seconds)	Total Number of Iterations
				FORM	SORM
RBDO−CS	13.658	(3.362,5.148)	i = 1	0.0004	0.0004	0.105	12
			i = 2	0.0030	0.0028
			i = 3	0.0030	0.0032
SL−KKT	13.667	(3,364,5.151)	i = 1	0.0004	0.0012	0.031	14
			i = 2	0.0030	0.0027
			i = 3	0.0030	0.0033

. Note that the failure probability of $g_1\left(d^{\ast },x^{\ast }\right)$ is less than the target failure probability, which indicates that this constraint is inactive at the local optimum.

The RBDO−CS method results in the identification of more optimal solutions when compared to the results from SL−KKT method. Furthermore, the SORM results indicate that the failure probabilities of limit-state functions at $d^{\ast }$ obtained by RBDO−CS method are closer to the target failure probabilities than by the SL−KKT method.

5.4. RBDO of the Truss Structure under Probabilistic Strength Constraints

The proposed RBDO−CS method is applied to identify the optimal element sizes of an indeterminate truss structure by McDonald and Mahadevan [51] (see Figure 7). The deterministic design variables $d={\left(A_1,A_2,A_3,A_4,A_5,A_6\right)}^{\mathrm{T}}$ are the cross-sectional areas of elements. The applied force $F_A$ and yield strengths ${\sigma }_i, i=1,\dots ,6$ of the elements are the random variables $x={\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_y,F_A\right)}^{\mathrm{T}}$, which are assumed to follow the normal distribution. All random variables are statistically independent of each other, and the means and standard deviations of them are summarized in

Table 7. Description of random variables for the truss optimization problem.

Random Variables	Marginal Distribution	Mean	Standard Deviation	Correlation
$F_A, \mathrm{kips}$	Normal	1000	100	Independent
${\sigma }_1, \mathrm{ksi}$	Normal	36	3	Independent
${\sigma }_2, \mathrm{ksi}$
${\sigma }_3, \mathrm{ksi}$
${\sigma }_4, \mathrm{ksi}$
${\sigma }_5, \mathrm{ksi}$
${\sigma }_6, \mathrm{ksi}$

. In this example, RBDO aims to minimize the volume of the truss structure while satisfying the probabilistic constraints in terms of yield strengths. In this example, the truss element forces are derived in terms of the applied load with the assumption of the two diagonal elements carrying equal forces. Limit-state functions are defined as:

$$\begin{gathered} \begin{array}{lll}{g}_i\left(d,x\right)& =A_i{\sigma }_i-\frac{\sqrt{2}}{2}{F}_A,& i=1,2\\ & =A_i{\sigma }_i-\frac{1}{2}{F}_A,& i=3,4,5,6\end{array} \\ {A}_1,A_2,A_3,A_4,A_5,A_6\ge 0 \end{gathered}$$

(32)

The RBDO problem can be defined as follows:

$$\begin{array}{c}\underset{d}{\min }f\left(d\right)=\sqrt{2}{{\displaystyle \sum }}_{i=1}^2{A}_i+{{\displaystyle \sum }}_{j=3}^6{A}_j\\ \mathrm{s}.\mathrm{t}.P_{fi}=P\left[g_i\left(d,x\right)\le 0\right]\le P_{fi}{}^{\mathrm{t}},i=1,\dots ,6\\ d^{\mathrm{l}}\le d\le d^{\mathrm{u}}\end{array}$$

(33)

The initial design variables $d^0$ are set to 12 in², and the corresponding failure probabilities of the constraints by the FORM are $P_{fi}=\left(0.999,0.999,0.865,0.865,0.865,0.865\right)$. The proposed RBDO−CS method with the parameters listed in

Table 8. Target failure probability and parameters for design variables and optimization.

Target Failure $\mathbf{Probability}{\mathit{P}}_{\mathit{f}}{}^{\mathit{t}}$	Initial Design ${\mathbf{d}}^0,{\mathbf{in}}^2$	Lower Bound ${\mathbf{d}}^{\mathbf{l}},{\mathbf{in}}^2$	Upper Bound ${\mathbf{d}}^{\mathbf{u}},{\mathbf{in}}^2$	Convergence Criterion
$P_{fi}{}^{\mathrm{t}}=1.667\times {10}^{-4}$ $\left(i=1,\dots ,6\right)$	(12, 12)	(5, 5)	(50, 50)	${10}^{-3}$

is able to find an optimal solution quickly, whose convergence history is plotted in Figure 8. Furthermore, the optimal designs are confirmed with the SL−KKT approach.

Table 9. Results of the RBDO−CS and SL−KKT approaches for the truss optimization problem.

$\mathbf{Optimal}{\mathit{f}}^{\ast }\times \mathbf{L}, \mathbf{i}{\mathbf{n}}^3$	$\mathbf{Optimal}{\mathit{A}}_{\mathit{i}}^{\ast }, \mathbf{i}{\mathbf{n}}^2$		${\mathit{P}}_{\mathit{f}\mathit{i}}$
RBDO−CS/SL−KKT	Element	RBDO−CS/SL−KKT	FORM	SORM	MCS
177.46	1	31.37	$1.667\times {10}^{-4}$	$1.667\times {10}^{-4}$	$1.655\times {10}^{-4}$
	2	31.37	$1.667\times {10}^{-4}$	$1.667\times {10}^{-4}$	$1.672\times {10}^{-4}$
	3	22.18	$1.667\times {10}^{-4}$	$1.667\times {10}^{-4}$	$1.684\times {10}^{-4}$
	4	22.18	$1.667\times {10}^{-4}$	$1.667\times {10}^{-4}$	$1.667\times {10}^{-4}$
	5	22.18	$1.667\times {10}^{-4}$	$1.667\times {10}^{-4}$	$1.675\times {10}^{-4}$
	6	22.18	$1.667\times {10}^{-4}$	$1.667\times {10}^{-4}$	$1.654\times {10}^{-4}$

represents the optimal design and failure probabilities of constraints. The results of the SORM and MCS with 10⁸ samples (c.o.v = 0.02) confirm the failure probability $P_{fi}$ of the optimal design by the proposed method. It should be noted that the FORM and SORM approximations of the probability of failure are sufficiently accurate for this problem. It is because the limit-state functions in Equation (32) are linear, and random variables following normal distributions are statistically independent of each other.

5.5. RBDO of the Truss Cantilever Structure Subjected to Probabilistic Displacement Constraints

Consider a truss cantilever structure consisting of 56 bar elements and 21 nodes (Figure 9c,d). The truss cantilever structure subjected to two external forces at nodes 5 and 21 contains three-roller supports as shown in Figure 9a. All of the truss elements have the same modulus of elasticity E. The objective of RBDO is to minimize the weight of the truss cantilever structure and constraints are vertical displacements ($u_{5y}, u_{21y})$ at nodes 5 and 21. Limit-state functions are defined as follows:

$$\begin{array}{l}{g}_1\left(d,x\right)={\overline{u}}_5-u_{5\mathrm{y}}\left(d,x\right)\\ g_2\left(d,x\right)={\overline{u}}_{21}-u_{21y}\left(d,,,x\right)\end{array}$$

(34)

where ${\overline{u}}_5, {\overline{u}}_{21}$ are the maximum allowable displacements in the vertical direction, respectively. Two external forces and modulus of elasticity are assumed to be random variables $x=\left(F_{5y}, F_{21y},E\right)$, and cross-sectional areas are design variables $d=(A_i, i=1,\dots ,56$).

Table 10. Description of random variables for the truss structure optimization problem.

Random Variables	Marginal Distribution	Mean	c.o.v	Correlation
$F_{5\mathrm{y}},\mathrm{kips}$	Joint lognormal	120	0.1	${\rho }_{F_{5y,}{F}_{21y}}=0.3$
$F_{21\mathrm{y}},\mathrm{kips}$	Joint lognormal	140	0.1
$E,\mathrm{ksi}$	Normal	29000	0.2	Independent

provides marginal distributions, mean values, and second moments of the random variables. Parameters for reliability analysis and optimization are listed in

Table 11. Parameters for the target failure probability and optimization.

${\mathit{P}}_{\mathit{f}}{}^{\mathbf{t}}$	${\overline{\mathit{u}}}_5$ (in)	${\overline{\mathit{u}}}_{21}\left(\mathbf{in}\right)$	$\mathbf{Initial}\mathbf{Design}{\mathbf{d}}^0\left({\mathbf{in}}^2\right)$	Lower Bound ${\mathbf{d}}^{\mathbf{l}}\left({\mathbf{in}}^2\right)$	Upper Bound ${\mathbf{d}}^{\mathbf{u}}\left({\mathbf{in}}^2\right)$	Convergence Criterion
0.005	0.1	0.3	5.0	0.5	50.0	${10}^{-3}$

. The RBDO problem can be written in a nested formulation as follows:

$$\begin{gathered} \begin{array}{l}\underset{d}{\min }f\left(d\right)={{\displaystyle \sum }}_{i=1}^{56}{A}_i{L}_i\\ \mathrm{s}.\mathrm{t}.P_{fi}=P\left[g_i\left(d,x\right)\le 0\right]\le P_{fi}{}^{\mathrm{t}},i=1,2\\ \mathrm{with}K\left(d,x\right)u\left(d,x\right)=f\left(x\right)\end{array} \\ {d}^{\mathrm{l}}\le d\le d^{\mathrm{u}} \end{gathered}$$

(35)

where $K$ is the global stiffness matrix of the truss cantilever structure, $u$ is the global displacement vector, and $f$ is the global external force vector. Note that $u$ is an implicit function defined through the equilibrium equations $K\left(d,x\right)u\left(d,x\right)=f\left(x\right)$. The derivative of the displacement vector with respect to a parameter $\theta \left(=d \mathrm{or} x\right)$ is obtained analytically as

$$\begin{array}{c}{\nabla }_{\theta }u\left(d,x\right)=\frac{\partial K^{-1}\left(d,x\right)}{\partial \theta }f\left(x\right)+K^{-1}\left(d,x\right)\frac{\partial f\left(x\right)}{\partial \theta }\end{array}$$

(36)

The sensitivity analysis in Equation (36) requires gradients of inversed stiffness matrix and force vector, which are computationally expensive and implicitly defined. Rather than directly solving Equation (36) for sensitivity analysis, applying the CS derivative approximation to Equation (36) yields

$$\begin{array}{ll}\theta =A_i:& {\nabla }_{A_i}u\left(d,x\right)=\frac{\partial K^{-1}\left(d,x\right)}{\partial A_i}f\left(x\right)=\frac{\Im \left[K^{-1}\left(d+\mathrm{i}h{\widehat{e}}_i,x\right)f\left(x\right)\right]}{h}\\ \theta =F_{5y}:& {\nabla }_{F_{5y}}u\left(d,x\right)=K^{-1}\left(d,x\right)\frac{\partial f\left(x\right)}{\partial F_{5y}}=\frac{\Im \left[K^{-1}\left(d,x\right)f\left(x+\mathrm{i}h{\overline{e}}_1\right)\right]}{h}\\ \theta =F_{21y}:& {\nabla }_{F_{21y}}u\left(d,x\right)=K^{-1}\left(d,x\right)\frac{\partial f\left(x\right)}{\partial F_{21y}}=\frac{\Im \left[K^{-1}\left(d,x\right)f\left(x+\mathrm{i}h{\overline{e}}_2\right)\right]}{h}\\ \theta =E:& {\nabla }_{E}u\left(d,x\right)=\frac{\partial K^{-1}\left(d,x\right)}{\partial E}f\left(x\right)=\frac{\Im \left[K^{-1}\left(d,x+\mathrm{i}h{\overline{e}}_3\right)f\left(x\right)\right]}{h}\end{array}$$

(37)

where ${\widehat{e}}_i$ is the i-th column of a 56-th order identity matrix ($I_{56}$) and ${\overline{e}}_k$ is the k-th column of a 3-rd order identity matrix ($I_3$). Sensitivities of the limit-state functions with respect to design variables can be computed by using Equation (37). Furthermore, sensitivities of limits-state functions with respect to random variables are used in the FORM for reliability analysis. Notably, the displacement vector u and its sensitivity are computed without additional step by taking real and imaginary parts in Equation (37). The initial displacements at nodes 5 and 21 of the truss cantilever structure with mean values of random variables and uniform cross-sectional areas of 5 in² for all elements are 0.1339 in and 0.3557 in and corresponding failure probabilities by the FORM are 0.928 and 0.789.

Figure 10 illustrates the convergence histories of objective and probabilistic constraint functions, optimized element sizes, and optimal cantilever structure. The proposed RBDO−CS method quickly identifies the solutions satisfying probabilistic constraints and then reduces the volume to minimize. Optimization results in Figure 10c indicate that elements connected to the roller support in the middle and the bottom chords are primarily strengthened to satisfy probabilistic constraints.

Table 12. Comparison between results from the RBDO−CS and SL−KKT for the truss cantilever structure design.

RBDO Method	Optimal $f^{\ast }$ ($f{t}^3$)	$P_{fi}$				Averaged Computational Time Carried out to Five Iterations, (Seconds)	Total Number of Iterations
		$i$	FORM	SORM	MCS
RBDO−CS	5.828	1	0.0050	0.00516	0.00516	0.0637	37
		2	0.0050	0.00516	0.00517
SL−KKT	5.827	1	0.0050	0.00517	0.00518	0.0591	175
		2	0.0050	0.00518	0.00519

provides the results from the RBDO−CS method for the truss cantilever structure. The SL−KKT method is utilized to verify optimization results. It is observed that the results from the proposed RBDO−CS and SL−KKT approaches closely match. Failure probabilities of limit-state functions with optimal design $d^{\ast }$ and random variables are estimated by the FORM, SORM, and MCS with ${10}^7$ samples (c.o.v = 0.03). This study confirms that the proposed RBDO−CS method enables the efficient identification of optimal solutions. It is noted that the computational costs of RBDO−CS and SL−KKT methods are similar with a larger number of design variables, however, the proposed RBDO−CS method identifies converged solutions faster than the SL−KKT method. Next, the force random variables $F_{5y},F_{21y}$ are assumed to follow the normal distributions with the same means and standard deviations. This is to investigate the effect of probabilistic distribution types on the optimal design and to demonstrate the general applicability of the proposed method. The minimum objective function value is larger than that of the lognormal distribution case, as illustrated in Figure 11. Furthermore, the impact of statistical correlation between random forces on the optimal design is investigated by altering correlation coefficients ${\rho }_{F_{5y},F_{21y}}$ from 0.0 to 0.6. The optimal volumes obtained by the RBDO−CS method are shown in Figure 11. A positive correlation among the random forces results in higher volume, i.e., more conservative design. This is because positively correlated forces $F_{5y},F_{21y}$ increase the vertical displacements, and thus the failure probabilities.

6. Concluding Remarks

An approach to the RBDO of structures is developed using CS approximation and the FORM. The improvement in the accuracy of probabilistic constraints by the CS derivative and its straightforward implementation are presented in this paper. The proposed approach incorporates component reliability targets in RBDO and eliminates the need to analytically derive gradients of functions with respect to the design and random variables for sensitivity analysis. In addition to accounting for the statistical dependence between two-component events, the approach can compute the sensitivities of the failure probability to various parameters, which facilitate the use of gradient-based optimization algorithms. The first numerical example demonstrates the accuracy of the CS approximation without significant reliance on a step size, compared to finite difference methods. Furthermore, the uniform applicability of the proposed approach to RBDO of structures is confirmed through four numerical examples. This paper focuses on the CRBDO with probabilistic constraints of multiple failure modes of a design. The CRBDO problem is formulated such that the optimal structure satisfies each failure mode with target failure probabilities. In some instances, the design constraint needs to be defined as a “system” event of multiple constraints, (i.e., a logical (or Boolean) function of multiple failure modes), which is commonly termed a system reliability-based design optimization (SRBDO) problem. In these cases, the sensitivity analysis of the system reliability can be more expensive and complicated than a CRBDO problem due to the statistical dependency among failure events and complexity inherent in the formula. The implementation of -step approximation in SRBDO with diverse reliability analysis methods remains to be further explored. To improve the accuracy of reliability analysis and enhance the use of gradient-based optimization algorithms, areas for further study include sensitivity analysis with CS approximation unified with the SORM or multiple design points approaches, and its incorporation into RBDO.