Reliability-based design optimization using a generalized subset simulation method and posterior approximation

The design of reinforced concrete (RC) structures involves several kinds of uncertainties, which are usually considered through the partial safety factors prescribed in the codes of practice. The traditional design optimization of RC structures uses deterministic information of the problem. The partial safety factors are used to consider loading fluctuation and the variability of material properties. The use of these safety factors in deterministic optimization usually leads to over-designing structures, as these safety factors are calibrated for a large class of structures. In the deterministic optimization procedure the reliability cannot be controlled. For this reason, the Reliability-Based Design Optimization (RBDO) is devoted to design economical and reliable structures. However, the RBDO problem involves the evaluation of probabilistic constraints performed by the reliability analysis. The most common RBDO formulations are based on a nested optimization problem, with an outer loop for the design optimization and an inner loop for the reliability analysis. Therefore, an expensive computation effort is required to solve the RBDO problem. In this paper, a new RBDO method of RC structures is proposed. The RBDO problem is decomposed to several cycles of deterministic design optimization (DDO) based on new safety factors called Optimal System Safety Factors (OSSF). At each cycle of the DDO, the system reliability analysis is performed to verify the reliability of the optimal design, the OSSF are computed by a probabilistic method on the basis of the previous system reliability analysis, then the updated OSSF are provided for the next cycle of the DDO. The application to the design of an RC structure shows the interest and the efficiency of the proposed method.

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.

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

A simulation-based method for reliability based design optimization problems with highly nonlinear constraints

An efficient, fully decoupled approach is proposed to solve the structural reliability-based design optimization (RBDO) problem. The proposed approach utilizes augmented space integral and importance sampling (ASI-IS) to efficiently evaluate a functional relationship between the probability of failure and the design parameters, namely the so-called failure probability function. ASI-IS allows for the avoidance of both the intractable density fitting task associated with augmented space methods and the time-consuming repeated reliability evaluations associated with surrogate model approaches. The resulting functional relationship can be used to completely decouple the original RBDO problem into a deterministic one. Then, an iteration mechanism is constructed with gradually smaller design domains to enhance the efficiency of the optimization process. Furthermore, a sample reuse algorithm is proposed to improve the performance of the proposed approach by collecting the samples generated in previous iterations and reusing them in the current iteration in order to produce a better estimator of the failure probability function. Numerical and engineering examples, including a turbine blade and an aircraft inner flap, are given to demonstrate the efficiency and feasibility of the proposed approach.

Efficient reliability-based design optimization for hydraulic pipeline with adaptive sampling region

This paper presents a new method for performing reliability-based design optimization (RBDO) of structures based on sequential optimization and reliability assessment (SORA). SORA is an effective method for solving RBDO that separates uncertainty analysis from optimization loops for reducing computational cost. However, SORA has some limitations, such as an inability to deal with problems involving discrete design variables or discontinuity in a domain, the dependency of solutions on the starting point of numerical solutions, and the lack of guarantee that global optimum solutions will be found. In this paper, a global decoupling method is proposed to tackle these limitations and improve the performance of SORA. This method links SORA enhanced with modified chaos control (ESORA) to an improved differential evolution (IDE) to perform RBDO. To deal with RBDO problems with discrete design variables, a rounding method is integrated into IDE. IDE also utilizes an adaptive selection scheme in a mutation step and an elitist strategy in the selection phase. To improve the efficiency of the method for RBDO problems with highly nonlinear performance functions and nonnormal random variables, a modified chaos control is employed to assess reliability constraints. Five numerical examples are considered to investigate the strength of the proposed method, illustrating its appropriate efficiency and accuracy. The proposed method is also extendable to more complex problems such as system reliability-based structural optimization and RBDO of nonlinear structures.

<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>

In the engineering problems, the randomness and the uncertainties of the distribution of the structural parameters are a crucial problem. In the case of reliability-based design optimization (RBDO), it is the objective to play a dominant role in the structural optimization problem introducing the reliability concept. The RBDO problem is often formulated as a minimization of the initial structural cost under constraints imposed on the values of elemental reliability indices corresponding to various limit states. The classical RBDO leads to high computing time and weak convergence, but a Hybrid Method (HM) has been proposed to overcome these two drawbacks. As the hybrid method successfully reduces the computing time, we can increase the number of variables by introducing the standard deviations as optimization variables to minimize the error values in the probabilistic model. The efficiency of the hybrid method has been demonstrated on static and dynamic cases with extension to the variability of the probabilistic model. In this paper, we propose a modification on the formulation of the hybrid method to improve the optimal solutions. The proposed method is called, Improved Hybrid Method (IHM). The main benefit of this method is to improve the structure performance by much more minimizing the objective function than the hybrid method. It is also shown to demonstrate the optimality conditions. The improved hybrid method is next applied to two numerical examples, with consideration of the standard deviations as optimization variables (for linear and nonlinear distributions). When integrating the improved hybrid method within the probabilistic model variability, we minimize the objective function more and more.

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.

This study performed reliability-based robust design optimization (RBRDO) of stator and rotor shape of the IPM type electric motor to reduce the torque ripple considering uncertainties. A metamodel based on the design of experiments results, which was calculated by using 2-D electromagnetic finite element analysis, was used to calculate manufacturing uncertainties and material property distribution. As a result, the mean torque ripple of the optimum electric motor design was reduced by 56.4 % and the robustness was increased by 61.8 % compared with the initial design. The RBRDO design was compared with the designs of deterministic design optimization and reliability-based design optimization, and it was shown that, by using the RBRDO method, the optimum IPMSM can be designed to satisfying not only the minimum torque ripple but also the robustness and the reliability of output torque and torque ripple.

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.

Involving inevitable uncertainties in an engineering system design optimization is a necessity and requirements to achieve a reliable design which reliable design optimization approaches are expressed as Reliability-Based Design Optimization (RBDO) methods. Despite tremendous efforts have been made in the field of RBDO, computational efficiency improvement is still a significant challenge. Sequential Optimization and Reliability Assessment (SORA) has made great efforts to improve computational efficiency by decoupling RBDO problem into sequential deterministic optimization and reliability assessment as a single-loop method. In this paper, in order to further improve computational efficiency and extend the application of the current SORA method, an Augmented SORA (ASORA) is proposed by refraining from reliability assessment for satisfied probabilistic constraints in each cycle until all probabilistic constraints are be satisfied. The accuracy and efficiency of the proposed method compare to the SORA method is illustrated through some single and multi-disciplinary mathematical examples and a multi objective example.

With the rapid advancement of computing technology, mathematical modeling and computer simulations have become common and standard methods for design of complex systems. To exercise these models intelligently and eliminate the burden of manual iteration, manipulating inputs and reviewing outputs, optimization strategies are applied in a simulation-based design environment. In reality, the knowledge that engineers have about the design problem is imperfect and incomplete, uncertainty always exists in simulation-based design. The conventional deterministic optimization techniques do not consider the impact of such uncertainties. If the system responses driving the design decisions are very sensitive to these uncertainties, serious problems might confront the designer. Performance of the actual system may differ greatly from expected optimum performance. Again, if the design solution is near one or more constraint boundaries, then even slight uncertainties or changes in the operating environment could produce failed, unsafe designs, and/or result in substantial performance degradation. In recent years, the concept of reliability and robust design has become very popular. Probabilistic design analysis and statistical modeling allow the identification of designs that qualify as not only feasible, but as consistently feasible in the face of uncertainty. Reliability based topology optimization (RBTO) extends the reliability notion to the area of structural topology optimization. Although reliability based design optimization (RBDO) is a well established research area, very few work has been presented in the area of RBTO. In this study, we give an overview of the various works done in the field of RBTO. Ever since Bendsoe and Kikuchi introduced the topology optimization using a homogenization method, many topology optimization methods have been developed for the both linear and nonlinear structures. Traditional topology optimization methods drive the topology of a structure to an optimum design based on the constraint on mass. Kharmanda et al. introduced reliability constraints in deterministic topology optimization problem. They proposed a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight. But the limit state function used by them was not based on failure criteria for the structure. Their formulation considered uncertainty with respect to ∗Design Automation Laboratory †Advisor geometrical dimension and applied load only. Also their reliability analysis seems to be independent of the boundary and loading condition, so their results showed similar values for the uncertain variables for different structures. Tovar et al. have developed the Hybrid Cellular Automaton (HCA) method for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy. In Agarwal, a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis. Patel et al. showed the use of RBTO using the gradient free Hybrid Cellular Automata (HCA) method. Their formulation incorporates uncertainty with respect to material property also. They considered limit state function based on failure modes on the output displacements. This study aims at giving a comparative study on these methods. It points out the advantages and disadvantages of the methods, and their significance in the field of RBTO. As large numbers of design variables are associated with continuum topology optimization problems, RBTO methods are inherently computationally expensive because of additional required system analysis associate with RBDO. Based on the observations, this study presents an efficient method for incorporating reliability analysis in topology optimization.

To reduce the cost of prototype and physical test, CAE analysis has been widely used to evaluate the vehicle performance during product development process. Combining CAE analysis and optimization approach, vehicle design process can be implemented more efficiently with affordable cost. Reliability based design optimization (RBDO) formulation considers variations of input variables, such as component gauges and material properties. As a result, the design obtained by using RBDO is more reliable and robust compared to those by deterministic optimization. The RBDO process starts from running simulation at DOE sampling data points, generating surrogate models (response surface) and performing robust and reliability based design optimization on the surrogate models by using Monte Carlo simulation. This paper presents a RBDO framework in Excel enviroment. Within this framework, the engineer can perform DOE sampling, surrogate modeling, robustness assessment, Monte Carlo simulation, robust design and reliability-based optimization with any type of distribution.