An approximate sequential optimization and reliability assessment method for reliability-based design optimization

Sequential optimization and reliability assessment (SORA) is one of the most popular decoupled approaches to solve reliability-based design optimization (RBDO) problem because of its efficiency and robustness. In SORA, the double loop structure is decoupled through a serial of cycles of deterministic optimization and reliability assessment. In each cycle, the deterministic optimization and reliability assessment are performed sequentially and the boundaries of violated constraints are shifted to the feasible direction according to the reliability information obtained in the previous cycle. In this paper, based on the concept of SORA, approximate most probable target point (MPTP) and approximate probabilistic performance measure (PPM) are adopted in reliability assessment. In each cycle, the approximate MPTP needs to be reserved, which will be used to obtain new approximate MPTP in the next cycle. There is no need to evaluate the performance function in the deterministic optimization since the approximate PPM and its sensitivity are used to formulate the linear Taylor expansion of the constraint function. One example is used to illustrate that the approximate MPTP will approach the accurate MPTP with the iteration. The design variables and the approximate MPTP converge simultaneously. Numerical results of several examples indicate the proposed method is robust and more efficient than SORA and other common RBDO methods.     

Threshold shift method for reliability-based design optimization

Structural and Multidisciplinary Optimization - We present a novel approach, referred to as the “threshold shift method” (TSM), for reliability-based design optimization (RBDO). The...     

An inverse-measure-based unilevel architecture for reliability-based design optimization

Reliability-based design optimization (RBDO) is a methodology for finding optimized designs that are characterized with a low probability of failure. Primarily, RBDO consists of optimizing a merit function while satisfying reliability constraints. The reliability constraints are constraints on the probability of failure corresponding to each of the failure modes of the system or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. During the last few years, a variety of different formulations have been developed for RBDO. Traditionally, these have been formulated as a double-loop (nested) optimization problem. The upper level optimization loop generally involves optimizing a merit function subject to reliability constraints, and the lower level optimization loop(s) compute(s) the probabilities of failure corresponding to the failure mode(s) that govern(s) the system failure. This formulation is, by nature, computationally intensive. Researchers have provided sequential strategies to address this issue, where the deterministic optimization and reliability analysis are decoupled, and the process is performed iteratively until convergence is achieved. These methods, though attractive in terms of obtaining a workable reliable design at considerably reduced computational costs, often lead to premature convergence and therefore yield spurious optimal designs. In this paper, a novel unilevel formulation for RBDO is developed. In the proposed formulation, the lower level optimization (evaluation of reliability constraints in the double-loop formulation) is replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper level optimization. Such a replacement is computationally equivalent to solving the original nested optimization if the lower level optimization problem is solved by numerically satisfying the KKT conditions (which is typically the case). It is shown through the use of test problems that the proposed formulation is numerically robust (stable) and computationally efficient compared to the existing approaches for RBDO.     

Novel decoupling method for time-dependent reliability-based design optimization

Structural and Multidisciplinary Optimization - Time-dependent reliability-based design optimization (RBDO) can provide the optimal design parameter solutions for the time-dependent structure, and...     

An efficient variable screening method for effective surrogate models for reliability-based design optimization

In the reliability-based design optimization (RBDO) process, surrogate models are frequently used to reduce the number of simulations because analysis of a simulation model takes a great deal of computational time. On the other hand, to obtain accurate surrogate models, we have to limit the dimension of the RBDO problem and thus mitigate the curse of dimensionality. Therefore, it is desirable to develop an efficient and effective variable screening method for reduction of the dimension of the RBDO problem. In this paper, requirements of the variable screening method for deterministic design optimization (DDO) and RBDO are compared, and it is found that output variance is critical for identifying important variables in the RBDO process. An efficient approximation method based on the univariate dimension reduction method (DRM) is proposed to calculate output variance efficiently. For variable screening, the variables that induce larger output variances are selected as important variables. To determine important variables, hypothesis testing is used in this paper so that possible errors are contained in a user-specified error level. Also, an appropriate number of samples is proposed for calculating the output variance. Moreover, a quadratic interpolation method is studied in detail to calculate output variance efficiently. Using numerical examples, performance of the proposed method is verified. It is shown that the proposed method finds important variables efficiently and effectively     

Conservative reliability-based design optimization method with insufficient input data

Reliability analysis and reliability-based design optimization (RBDO) require an exact input probabilistic model to obtain accurate probability of failure (PoF) and RBDO optimum design. However, often only limited input data is available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the PoF to be uncertain. Therefore, it is necessary to consider the PoF to follow a probability distribution. In this paper, the probability of the PoF is obtained with consecutive conditional probabilities of input distribution types and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to calculate the probability of the PoF. The probability of the PoF at a user-specified target PoF is defined as the conservativeness level of the PoF. The conservativeness level, in addition to the target PoF, will be used as a probabilistic constraint in an RBDO process to obtain a conservative optimum design, for limited input data. Thus, the design sensitivity of the conservativeness level is derived to support an efficient optimization process. Using numerical examples, it is demonstrated that the conservativeness level should be involved in RBDO when input data is limited. The accuracy and efficiency of the proposed design sensitivity method is verified. Finally, conservative RBDO optimum designs are obtained using the developed methods for limited input data problems.     

An efficient method for reliability-based design optimization of nonlinear inelastic steel space frames

This paper proposes an effective numerical procedure for reliability-based design optimization (RBDO) of nonlinear inelastic steel frames by integrating a harmony search technique (HS) for optimization and a robust method for failure probability analysis. The practical advanced analysis using the beam-column approach is used for capturing the nonlinear inelastic behaviors of frames, while a detail implement of HS for discrete optimization of steel frames is introduced. The failure probability of structures is evaluated by using the combination of the improved Latin Hypercube (IHS) and a new effective importance sampling (EIS). The efficiency and accuracy of the proposed procedure are demonstrated through three mathematical examples and five steel frames. The results obtained in this paper prove that the proposed procedure is computationally efficient and can be applied in practical design. Furthermore, it is shown that the use of nonlinear inelastic analysis in the optimization of steel frames yields more realistic results.     

An adaptive hybrid single-loop method for reliability-based design optimization using iterative control strategy

Single-loop approach (SLA) is one of the most promising methods for solving linear and weakly nonlinear reliability-based design optimization (RBDO) problems. However, since SLA locates the current approximate most probable point (MPP) by using the gradient information of the previous one to reduce the computational cost, it may lead to inaccuracy when the nonlinearity of probabilistic constraints becomes relatively high. To overcome this limitation, a new adaptive hybrid single-loop method (AH-SLM) that can automatically choose to search for the approximate MPP or accurate MPP is proposed in this paper. Moreover, to get the accurate MPP more efficiently and alleviate the oscillation in the search process, an iterative control strategy (ICS) with two iterative control criteria is developed. In each iterative step, the KKT-condition of performance measure approach (PMA) is introduced to check the validity of the approximate MPP. If the approximate MPP is infeasible, ICS will be further carried out to search for the accurate MPP. The two iterative control criteria are used to update the oscillation control step length, then ICS can converge fast for both weakly and highly nonlinear performance functions. Besides, numerical examples are presented to verify the efficiency and robustness of our proposed method.     

An efficient procedure for structural reliability-based robust design optimization

This paper focuses on the development of an optimization tool with the aim to obtain robust and reliable designs in short computational time. The robustness measures considered here are the expected value and standard deviation of the performance function involved in the optimization problem. When using these robustness measures combined, the search of optimal design appears as a robust multiobjective optimization (RMO) problem. Reliable design addresses uncertainties to restrict the structural probability of failure. The mathematical formulation for the reliability based robust design optimization (RBRDO) problem is obtained by adding a reliability based constraint into the RMO problem. As both, statistics calculations and the reliability analysis could be very costly, approximation technique based on reduced-order modeling (ROM) is also incorporated in our procedure. The selected ROM is the proper orthogonal decomposition (POD) method, with the aim to produce fast outputs considering structural non-linear behavior. Moreover, to obtain RBRDO designs with reduced CPU time we propose others developments to be added in the integrated tool. They are: Probabilistic Collocation Method (PCM) to evaluate the statistics of the structural responses and, also, an approximated reliability constraints procedure based on the Performance Measure Approach (PMA) for reliability constraint assessment. Finally, Normal-Boundary Intersection (NBI) or Normalized Normal-Constraint (NNC) multiobjective optimization techniques are employed to obtain fast and even spread Pareto robust designs. To illustrate the application of the proposed tool, optimization studies are conducted for a linear (benchmark) and nonlinear trusses problems. The nonlinear example consider different loads level, exploring the material plasticity. The integrated tool prove to be very effective reducing the computational time by up to five orders of magnitude, when compared to the solutions obtained via classical standard approaches.     

A new method for reliability analysis and reliability-based design optimization

Structural and Multidisciplinary Optimization - We present a novel method for reliability-based design optimization, which is based on the approximation of the safe region in the random space by a...     

A new efficient decoupled reliability-based design optimization method with quantiles

Structural and Multidisciplinary Optimization - The problems of reliability-based design optimization (RBDO) can generally be solved by double-loop methods, single-loop methods or decoupled...     

A self-adaptive modified chaos control method for reliability-based design optimization

In the reliability-based design optimization (RBDO), the Advanced mean value (AMV) method sometimes yields unstable results such as chaotic and periodic solutions for highly nonlinear probabilistic constraints. The chaos control (CC), modified chaos control (MCC) and adaptive chaos control (ACC) methods are more robust than the AMV but inefficient for some moderately nonlinear performance functions. In this paper, a self-adaptive modified chaos control (SMCC) method is developed based on a dynamical control factor to improve the efficiency of MCC for reliability analysis and RBDO. The self-adaptive control factor is dynamically computed based on the new and previous results. The efficiency and robustness of the proposed SMCC are compared with the AMV, CC, MCC and ACC methods using several nonlinear structural/mathematical performance functions and RBDO problems. The results illustrate that the SMCC is more efficient than CC, MCC, and ACC methods, and also more robust than AMV method for highly nonlinear problems.     

An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.     

A derivative-free trust-region algorithm for reliability-based optimization

Structural and Multidisciplinary Optimization - In this note, we present a derivative-free trust-region (TR) algorithm for reliability based optimization (RBO) problems. The proposed algorithm...     

Improved hybrid method as a robust tool for reliability-based design optimization

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.     

A probabilistic design system for reliability-based design optimization

A probabilistic design system for reliability-based design optimization problems called ADAPRES_NET is presented in this paper. ADAPRES_NET includes two main features, one of which is the use of an adaptive response surface method by which the probabilistic constraints are replaced with response functions, the other a distributed computing environment by which the computational applications are distributed on a network. The proposed system is presented with an example in which the well-known mechanical part, the connecting rod, is selected. Finally, the evaluation of the probabilistic constraints is also compared with that of the classical reliability methods, and the results indicate the benefit of using ADAPRES_NET.     

Multiobjective reliability-based optimization for crashworthy structures coupled with metal forming process

Structural and Multidisciplinary Optimization - Crashworthiness design for manufacturing of thin-walled structures remains a main challenge in vehicle industry. Conventionally, there have been two...