Single-loop Reliability-based Design Optimization Research Articles

A single-loop reliability-based design optimization using adaptive differential evolution

Reliability-based design optimization (RBDO) is an efficient tool for generating reliable and optimal solution under uncertainty of design variables. The major challenges in solving RBDO problems are generating optimal or near optimal reliable solution and higher computational cost. In this paper, a single-loop RBDO formulation is developed for addressing these challenges that uses shifting vector approach for achieving feasibility for violated constraints or performance functions. The formulation also incorporates target and trial vectors of differential evolution (DE) for guiding the algorithm. DE is also made adaptive by designing a heuristic parameter that controls two mutation operators for both exploration and exploitation of search space. The proposed RBDO method is tested on three mathematical and four engineering examples. The reliability of obtained solutions from RBDO methods are verified using Monte Carlo simulations with sample size of one million. The results of the proposed method are compared with various RBDO methods from the literature and a double-loop based DE method. It is found that the proposed method generates the best reliable solution for all examples.

A Hybrid Method for Truss Mass Minimization considering Uncertainties

In real-world structural problems, a number of factors may cause geometric imperfections, load variability, or even uncertainties in material properties. Therefore, a deterministic optimization procedure may fail to account such uncertainties present in the actual system leading to optimum designs that are not reliable; the designed system may show excessive safety or sometimes not sufficient reliability to carry applied load due to uncertainties. In this paper, we introduce a hybrid reliability-based design optimization (RBDO) algorithm based on the genetic operations of Genetic Algorithm, the position and velocity update of the Particle Swarm Algorithm (for global exploration), and the sequential quadratic programming, for local search. The First-Order Reliability Method is used to account uncertainty in design and parameter variables and to evaluate the associated reliability. The hybrid method is analyzed based on RBDO benchmark examples that range from simple to complex truss parametric sizing optimizations with stress, displacements, and frequency deterministic and probabilistic constraints. The proposed final problem, which cannot be handled by single loop RBDO algorithms, highlights the importance of the proposed approach in cases where the discrete design variables are also random variables.

Response surface single loop reliability-based design optimization with higher-order reliability assessment

Reliability-based design optimization (RBDO) aims at determination of the optimal design in the presence of uncertainty. The available Single-Loop approaches for RBDO are based on the First-Order Reliability Method (FORM) for the computation of the probability of failure, along with different approximations in order to avoid the expensive inner loop aiming at finding the Most Probable Point (MPP). However, the use of FORM in RBDO may not lead to sufficient accuracy depending on the degree of nonlinearity of the limit-state function. This is demonstrated for an extensively studied reliability-based design for vehicle crashworthiness problem solved in this paper, where all RBDO methods based on FORM strongly violates the probabilistic constraints. The Response Surface Single Loop (RSSL) method for RBDO is proposed based on the higher order probability computation for quadratic models previously presented by the authors. The RSSL-method bypasses the concept of an MPP and has high accuracy and efficiency. The method can solve problems with both constant and varying standard deviation of design variables and is particularly well suited for typical industrial applications where general quadratic response surface models can be used. If the quadratic response surface models of the deterministic constraints are valid in the whole region of interest, the method becomes a true single loop method with accuracy higher than traditional SORM. In other cases, quadratic response surface models are fitted to the deterministic constraints around the deterministic solution and the RBDO problem is solved using the proposed single loop method.

A single loop reliability-based design optimization using EPM and MPP-based PSO

A reliability-based design optimization (RBDO) incorporates a probabilistic analysis with an optimization technique to find a best design within a reliable design space. However, the computational cost of an RBDO task is often expensive compared to a deterministic optimization, which is mainly due to the reliability analysis performed inside the optimization loop. Theoretically, the reliability of a given design point can be obtained through a multidimensional integration. Integration with multiple variables over the safety domain is, unfortunately, formidable in most cases. Monte- Carlo simulation (MCS) is often used to solve this difficulty. However, the inherit statistic uncertainty associated with MCS sometimes causes an unstable RBDO solution. To avoid this unstable solution, this study transforms a multi-variable constraint into a single variable constraint using an exponential function with a polynomial coefficient (EPM). The adaptive Gauss-Kronrod quadrature is used to compute the constraint reliability. The calculated reliability and its derivative are incorporated with an optimizer such as sequential quadratic programming (SQP) or most probable point particle swarm optimization (MPP-based PSO) to conduct the RBDO task. To ensure the design accuracy, the stability of the RBDO algorithm with respect to the initial point is investigated through several numerical examples.

Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.

Reliability-Based Optimization With Discrete and Continuous Decision and Random Variables

Engineering design problems frequently involve both discrete and continuous random and design variables, and system reliability may depend on the union or intersection of multiple limit states. Solving reliability-based design optimization (RBDO) problems, where some or all of the decision variables must be integer valued, can be expensive since the computational effort increases exponentially with the number of discrete variables in discrete optimization problems, and the presence of both system and component level reliability makes RBDO more expensive. The presence of discrete random variables in a RBDO problem has usually necessitated the use of Monte Carlo simulation or some other type of enumeration procedure, both of which are computationally expensive. In this paper, the theorem of total probability is used to allow for the use of the first-order reliability method in solving mixed-integer RBDO problems. Single-loop RBDO formulations are developed for three classes of mixed-integer RBDO with both discrete and continuous random variables and component and system-level reliability constraints. These problem formulations can be solved with any appropriate discrete optimization technique. This paper develops, for each of the three problem classes, greedy algorithms to find an approximate solution to the mixed-integer RBDO problem with both component and system reliability constraints and/or objectives. These greedy algorithms are based on the solution of a relaxed formulation and require hardly additional computational expense than that required for the solution of the continuous RBDO problem. The greedy algorithms are verified by branch and bound and genetic algorithms. Also, this paper develops three algorithms, which can allow for calibration of reliability estimation with a more accurate reliability analysis technique. These algorithms are illustrated in the context of a truss optimization problem.

Design Optimization With System-Level Reliability Constraints

Reliability-based design optimization (RBDO) of mechanical systems is computationally intensive due to the presence of two types of iterative procedures—design optimization and reliability estimation. Single-loop RBDO algorithms offer tremendous savings in computational effort, but they have so far only been able to consider individual component reliability constraints. This paper presents a single-loop RBDO formulation and an equivalent formulation that can also include system-level reliability constraints. The formulations allow the allocation of optimal reliability levels to individual component limit states in order to satisfy both system-level and component-level reliability requirements. Four solution algorithms to implement the second, more efficient formulation are developed. A key feature of these algorithms is to remove the most probable points from the decision space, thus avoiding the need to calculate Hessians or gradients of limit state gradients. It is shown that with the proposed methods, system-level RBDO can be accomplished with computational expense equivalent to several cycles of computationally inexpensive single-loop RBDO based on second-moment methods. Examples of this new approach applied to series, parallel, and combined systems are provided.

Reliability-Based Wing Design Optimization Using Trust Region-Sequential Quadratic Programming Framework

The computational efficiency of the reliability-based design optimization(RBDO) can be a critical issue when high-fidelity simulation methods are used. An efficient RBDO framework is presented that utilizes a single-loop RBDO algorithm in which reliability analysis and optimization are conducted in a sequential manner by approximating the limit-state function. A further improvement in computational efficiency is achieved by applying the fixed sensitivity of the limit-state function to the reliability analysis and the equivalent deterministic constraint calculation. To guarantee the convergence of the single loop and the fixed sensitivity algorithm, a trust regionsequential quadratic programming RBDO strategy, is developed. The proposed framework validates low-fidelity approximation models in the trust region radius. The framework is examined by solving an analytical test problem to show that the proposed strategy has the computational efficiency over existing methods while maintaining accuracy. The proposed strategy is applied to a three-dimensional aerodynamic wing design problem to get improved results satisfying probabilistic constraints.