A Broyden–Fletcher–Goldfarb–Shanno algorithm for reliability-based design optimization

Efficient decoupling-assisted evolutionary/metaheuristic framework for expensive reliability-based design optimization problems

To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

In this investigation, a decomposition approach is employed for reliability-based design optimization of multidisciplinary systems. In the last few years, a variety of different approaches for reliability-based design optimization have been proposed in the literature. Traditionally, reliability-based design optimization is formulated as a nested optimization problem, where the inner loop, generally involves the solution to an optimization problem for computing the probability of failure corresponding to the failure modes. Such a formulation is by nature computationally intensive, requiring a large numbers of function and constraint evaluations. To alleviate this problem, researchers have developed single level reliability-based optimization formulations. The single level approach is expensive for multidisciplinary systems because of the fact that the complex coupled nature of multidisciplinary systems are likewise computationally intensive requiring iterative solvers. To reduce the computational effort of performing reliability-based design optimization in application to multidisciplinary systems, a decomposition approach for optimization is employed. The method of Simultaneous ANalysis and Design (SAND) is employed as an optimization driver within a single level reliability-based design optimization strategy. The methodology is illustrated in application to multidisciplinary test problems.

An enriched performance measure approach is presented for reliability-based design optimization to substantially improve computational efficiency when applied to large-scale applications. In the enriched performance measure approach, four improvements are made over the original performance measure approach: as a way to launch reliability-based design optimization at a deterministic optimum design, as a new enhanced hybrid-mean value method, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability-based design optimization iterations. In reliability-based design optimization, a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first-order method and by carrying out the refined reliability analysis using the enhanced hybrid-mean value method for e-active and violated constraints. The enhanced hybrid-mean value method is developed to handle nonlinear and/or nonmonotonic constraints in reliability analysis. The fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness. Moreover, two numerical examples are provided to compare the enriched performance measure approach to existing reliability-based design optimization methods from a numerical efficiency and stability point of view.

Because deterministic optimum designs obtained without taking uncertainty into account could lead to unreliable designs, a reliability-based approach to design optimization is preferable using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space (X-space) and a reliability analysis in an independent and standard normal random space (U-space). This process requires numerous nonlinear mappings between X- and U-spaces for various probability distributions. Therefore, the nonlinearity of the RBDO problem will depend on the type of distribution of random parameters, since a transformation between X- and U-spaces introduces additional nonlinearity into the reliability-based performance measures evaluated during the RBDO process. The evaluation of probabilistic constraints in RBDO can be carried out in two ways: using either the Reliability Index Approach (RIA), or the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity for RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of the highly nonlinear transformations that are involved. However, PMA is rather independent of probability distributions because it only has a small involvement with a nonlinear transformation.

Enhanced sequential approximate programming using second order reliability method for accurate and efficient structural reliability-based design optimization

<div class="htmlview paragraph">This paper presents an enriched performance measure approach (PMA+) for reliability-based design optimization (RBDO) to substantially improve computational efficiency when applied to large-scale applications. Three aspects of PMA+ are presented: as a way to launch RBDO at a deterministic optimum design, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some RBDO iterations. Unlike deterministic design optimization, a significant computational burden is imposed on the feasibility check of constraints in the RBDO process due to the costs of a reliability analysis. Such difficulties can be effectively resolved by using a mean value (MV) first-order method with an allowable accuracy for the purpose of feasibility identification, and by carrying out the refined reliability analysis using the hybrid mean value (HMV) first-order method for <i>ε</i>-active and violate constraints in the RBDO process. In addition, the fast reliability analysis method is proposed by reusing some of the information obtained at the previous RBDO iteration to efficiently evaluate probabilistic constraints at the current design iteration under the condition of design closeness. Other RBDO methods have recently been developed to enhance numerical efficiency of RBDO while maintaining numerical accuracy and stability. Thus, the PMA+ is compared to existing RBDO methods from a numerical efficiency and stability point of view. For a numerical understanding of the RBDO process, several numerical examples are provided, including a large-scale multi-crash application.</div>

<div class="htmlview paragraph">In this investigation a robotic system's dynamic performance is optimized for high reliability under uncertainty. The dynamic capability equations allow designers to predict the dynamic performance of a robotic system for a particular configuration (i.e.,point design). While the dynamic capability equations are a powerful tool, they can not account for performance variations due to aleatory uncertainties inherent in the system. To account for the inherent aleatory uncertainties, a reliability-based design optimization (RBDO) strategy is employed to design robotic systems with robust dynamic performance. RBDO has traditionally been implemented as a nested multilevel optimization process in which reliability constraints require solution to an optimization problem (i.e., reliability analysis). In this work a robust unilevel performance measure approach(PMA) is developed for performing reliability-based design optimization which eliminates the lower level problem in RBDO. A robotic test problem is used to illustrate the efficacy of the approach.</div>

An efficient approach for reliability-based design optimization (RBDO) of series systems is presented. A modified formulation of the RBDO problem is employed in which the reliabilities of the failure modes of a system are included in the set of the design variables. This allows for an optimal apportionment of the reliability of a system among its failure modes. A sequential optimization and reliability assessment method is used to efficiently determine the optimum design. Here, the constraints on the reliabilities of the failure modes of the RBDO problem are replaced with approximate deterministic constraints. The proposed approach is demonstrated on two example problems that have been solved in previous studies without optimizing the required reliability levels of the failure modes. The first example performs RBDO to a cantilever beam with a rectangular cross section under lateral and vertical loads. The constraints are on the strength and the maximum allowable displacement. The second example performs RBDO to a cable-stayed box girder with five failure modes. Compared to the designs found by previous studies, the new approach finds designs with lower mass but without reducing the system reliability.

Structural analysis and design optimization have recently been extended to stochastic approach to take various uncertainties into account. However in areas whwew it is not possible to produce accurate statistical information, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, since improper modeling of uncertainty could cause greater degree of statistical uncertainty than those of physical uncertainty. For uncertainty with insufficient information, possibility-based (or fuzzy set) methods have recently been introduced in stochastic structural analysis and design optimization. The main advantage of the fuzzy analysis is that it preserves the intrinsic random nature of physical variables through their membership functions and, when used for evaluation of designs, yields more conservative design than those from the probabilistic methods. There are two computational aspects in the fuzzy analysis compared to the probability analysis. First, the input fuzzy variables can be defined easier than the input random variables when no or very limited statistical data are available. Secondly, extended fuzzy operations are much simpler than those required to use probability, especially when a number of variables are involved. For possibility-based design optimization (PBDO), like for reliability-based design optimization (RBDO), the performance measure approach (PMA) with an inverse fuzzy analysis is more appropriate than other approaches, such as a possibility index approach. This paper proposes a new formulation of PBDO using PMA. It is also found that PBDO is more desirable than RBDO, since PBDO can inherently handle a system level possibility unlike RBDO. For the inverse fuzzy analysis, the maximal possibility search (MPS) method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the

Advanced solution strategies for time-dependent reliability based design optimization

An adaptive decoupling approach for reliability-based design optimization

Reliability-based design optimization (RBDO) is a technique used for engineering design when uncertainty is being considered. A typical RBDO problem can be formulated as a stochastic optimization model where the performance of a system is optimized and the reliability requirements are treated as constraints. One major challenge of RBDO research has been the prohibitive computational expenses. In this research, a new approximation approach, termed the single-loop deterministic method for RBDO (SLDM_RBDO), is proposed to reduce the computational effort of RBDO without sacrificing much accuracy. Based on the first order reliability method, the SLDM_RBDO method converts the probabilistic constraints to approximate deterministic constraints so that the RBDO problems can be transformed to deterministic optimization problems in one step. Three comparison experiments are conducted to show the performance of the SLDM_RBDO. In addition, a reliable forearm crutch design is studied to demonstrate the applicability of SLDM_RBDO to a real industry case.

Uncertainties in parameters such as materials, loading, and geometry are inevitable in designing metallic structures for cranes. When considering these uncertainty factors, reliability-based design optimization (RBDO) offers a more reasonable design approach. However, existing RBDO methods for crane metallic structures are prone to low convergence speed and high computational cost. A unilevel RBDO method, combining a discrete imperialist competitive algorithm with an inverse reliability strategy based on the performance measure approach, is developed. Application of the imperialist competitive algorithm at the optimization level significantly improves the convergence speed of this RBDO method. At the reliability analysis level, the inverse reliability strategy is used to determine the feasibility of each probabilistic constraint at each design point by calculating its α-percentile performance, thereby avoiding convergence failure, calculation error, and disproportionate computational effort encountered using conventional moment and simulation methods. Application of the RBDO method to an actual crane structure shows that the developed RBDO realizes a design with the best tradeoff between economy and safety together with about one-third of the convergence speed and the computational cost of the existing method. This paper provides a scientific and effective design approach for the design of metallic structures of cranes.

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, systems-level RBDO can be accomplished with computational expense equivalent to several cycles of computationally inexpensive singleloop RBDO based on second-moment methods. Examples of this new approach applied to series, parallel, and combined systems are provided. Nomenclature