A decoupled approach for non-probabilistic reliability-based design optimization
Introduction
There are many uncertainties stemming from practical engineering, such as geometric dimensions, material properties and external loads. All these inherent uncertain factors may lead to large variations of the structural properties and even failure. Thus, the non-deterministic structural optimization models, including reliability-based design optimization (RBDO) and robust design optimization (RDO), exhibit some attraction in both of theoretical research and practical applications [1], [2], [3], [4], [5]. Based on classical probability theory, RBDO and RDO approaches have been extensively studied in the methodology and applications, and the precise information on the probabilistic distribution of random variables should be provided for both RBDO and RDO [6], [7]. Otherwise, the optimal design of RBDO or RDO may generate unacceptable error for practical applications [8], [9]. Until now, the quantification of uncertain parameters still is a challenge for complex system, which seriously hinders the application of uncertain design optimization based on the probability theory [10].
Therefore, the non-probabilistic reliability-based design optimization (NRBDO) models, such as interval set model [11], [12], [13] and convex model [14], [15], [16], are suggested to deal with the non-deterministic structural analysis and optimization problems with respect to limited uncertain information. The purpose of convex model is to provide an effective tool for optimal design with uncertain-but-bounded parameters, which pays an extremely important role in non-probabilistic reliability analysis. In 1990s, Ben-Haim and Elishakoff [17], [18], [19] first introduced the concept of non-probabilistic reliability through convex model theory, which plays a well alternative role for RBDO when only a limit of information is available for uncertain factors. Then, Qiu and Elishakoff [20], Elishakoff et al. [21] suggested the interval set model for truss structures optimization with uncertain-but-bounded parameters. Majumder and Rao [22] developed an interval-based multi-objective optimization approach for design of aircraft wing structures. Although interval set model performs well for NRBDO, it suffers from the problem of interval extension. To this end, the parameterization method [23], [24] was introduced. In general, NRBDO is effective for uncertain optimization when only limited experiment samples are applied for uncertain variables.
Similar to the RBDO, NRBDO convex model also is achieved by a nested, double loop optimization model, where the non-probabilistic constraints are calculated repeatedly at each iterative step of the deterministic optimization. Evidently, it will result in a daunting computational effort. Thus, how to reduce the number of function evaluations is crucial for NRBDO. Early attempts have been made by e.g. Lombardi and Haftka [25], who used the anti-optimization technique to alleviate the computational burden. Kang et al. [26] applied the concept of performance measure approach (PMA) of RBDO and introduced the concerned performance approach (CPA) to improve the efficiency of NRBDO approaches. Then, the linearization-based approach [27] and sequential approximate programming [28] are further constructed to deal with the hybrid models with probabilistic and non-probabilistic variables efficiently. Generally, the expensive computational cost has become an important problem as the development of NRBDO convex model, which hinders its applications seriously.
In order to improve the efficiency of NRBDO, the metamodels are also employed for NRBDO to substitute the actual objective and performance functions. Jiang et al. [29] applied the latin hypercube design (LHD) and response surface method for nonlinear interval model. Li et al. [30] combined the LHD and Kriging model to deal with the multi-objective optimal problem with uncertain-but-bounded problems. Khodaparast et al. [31] also suggested an iterative procedure based on the Kriging model for interval model. Recently, Yang et al. [32] applied the efficient global optimization for hybrid reliability model with probabilistic and non-probabilistic variables.
Decoupled approach has been deemed as one of the most promising strategy for solving RBDO problems in terms of accuracy, efficiency and robustness [1], [33], [34], [35]. Until now, a series of non-probabilistic convex models have been developed, such as interval set, single-ellipse and multi-ellipse convex models, and the selection of these models should be determined by the experimental data [13]. However, there are few studies on developing the decoupled approach for these NRBDO convex models [27], [28]. Thus, a universal decoupled strategy is urgently required to solve the associated NRBDO problems.
In this paper, a mathematic definition is given to distinguish the types of the NRBDO convex models (including interval set model, single-ellipse model and multi-ellipse model). Then, a new sequential optimization approach (SOA) is proposed based on CPA to reduce the number of function calls of NRBDO convex models, which performs the non-probabilistic reliability analysis and deterministic optimization sequentially. Moreover, a feasibility-checking criterion using the approximate or actual concerned performance value is constructed to identify the active or inactive constraint during the iterative process, and then the non-probabilistic constraints can be evaluated efficiently.
The outline of this paper is presented as follows: Section 2 introduces the basic concept of the non-probabilistic reliability-based design optimization. Then, the proposed method is described in detail in Section 3. Using a mathematical example, a ten truss structure, a welded beam and a stiffened shell design, the performance of the proposed method is demonstrated in Section 4. Finally, the concluding remarks are drawn in Section 5.
Section snippets
Non-probabilistic reliability-based design optimization
In this section, a review of NRBDO is presented. The formulation of NRBDO is formulated aswhere C represents the objective function. d is the nd-dimensional deterministic design variable vector, x and p are nx-dimensional uncertain design variable vector and np-dimensional uncertain variable vector, respectively. is the nominal value of x. is the radius of x, and is assumed unchanged during the iterative process. The denotes the
Sequential optimization approach for non-probabilistic reliability-based design optimization
NRBDO model in Eq. (3) with CPA is still composed by a nested double loop structure, thus it also suffers from the unbearable computational cost. In this section, the sequential optimization approach (SOA) is proposed to improve the efficiency of the NRBDO approach, which is inspired by sequential optimization and reliability assessment (SORA) of probabilistic optimization [34].
Numerical examples
In order to verify the efficiency and accuracy of the proposed method, four examples are tested and compared to NRIA and CPA. The results of all NRBDO approaches are confirmed by SQP at the optimum. The computations are conducted on a personal computer with the following specifications: Intel(R) Core(TM) i7-6700 CPU @ 3.40 GHz 3.41 GHz, 16.0 GB of RAM, 64-bit Windows 10 operating system.
Conclusion
It is well known that there are many NRBDO convex models, and all these models are composed by a nested double loop structure. Thus, how to enhance their efficiency is crucial. In this paper, a general decoupled strategy is proposed, which improves the efficiency significantly for interval set, single-ellipse and multi-ellipse convex models. The feasibility of the proposed SOA is also proved. It is shown that the SOA can be applied for other NRBDO problems that the optimal model of
Acknowledgments
The research is supported by the National Natural Science Foundation of China (Nos. 11072073, 51478086 and 11502063) and the Central Universities of China (No. JZ2016HGBZ0751). The authors also thank Dr. Hao Hu for their comments and discussion.
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