Hybrid Analysis Method for Reliability-based Design Optimization

Reliability-Based Design Optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the RBDO process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the Advanced Mean Value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function. To overcome difficulties of the AMV method, the Conjugate Mean Value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the Hybrid Mean Value (HMV) method. The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process. NOMENCLATURE X Random parameter; X = [X1, X2,..., Xn] x Realization of X; x = [x1, x2,..., xn] U Independent standard normal random parameter u Realization of U; u = [u1, u2,..., un] T μ Mean of random parameter X d Design parameter; d = [d1, d2,..., dn] , L U d d Lower and upper bounds of design parameter d ( ) P • Probability function ( ) fX x Joint Probability Density Function (JPDF) of the random parameter ( ) Φ • Standard normal Cumulative Distribution Function (CDF) ( ) Φ • ( ) G F • CDF of the performance function G(X) s β Safety reliability index ,FORM s β First order approximation of safety reliability index s β t β Target reliability index ( ) G X Performance function; the design is considered “fail” if G(X) < 0 p G Probabilistic performance measure * ( ) 0 G = U u Most Probable Failure Point (MPFP) in first-order reliability analysis * t β β = u Most Probable Point (MPP) in first-order inverse reliability analysis * (A)MV u MPP using (advanced) mean value method in PMA * CMV u MPP using conjugate mean value method in PMA * HMV u MPP using hybrid mean value method in PMA n Normalized steepest descent direction of performance function abs rel , G G ∆ ∆ Absolute and relative changes in performance measure ς Criteria for the type of performance function ( ) L X Crack initiation fatigue life t L Target crack initiation fatigue life INTRODUCTION A commonly used design optimization methodology for engineering systems comprises deterministic modeling and simulation-based design optimization. However, the existence of uncertainties in physical quantities such as manufacturing tolerances, material properties, and loads requires a reliability-based approach to Black and Decker Best Paper Award in the 2001 ASME Design Automation Conference 2 Copyright © 2001 by ASME [ ] ( 1, 2, , ) T i design optimization [1,2]. Given the increased computational capabilities developed during the last few years, fundamental issues relating to the inclusion of quantitative estimation of uncertainty have been recently addressed. Techniques have been explored which incorporate uncertainty during design optimization at an affordable computational cost. There has been a recent development in the Reliability-Based Design Optimization (RBDO) incorporating probabilistic constraints that can be evaluated using two different approaches, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA) [3,4]. The evaluation of a probabilistic constraint in the RBDO model is an essential step and thus the probabilistic constraint in the RBDO model must be computationally stable and affordable so that the RBDO process can be effective. It has been shown that PMA is equivalent to RIA in prescribing the probabilistic constraint [3]. However, these approaches are not equivalent in computational robustness in evaluating probabilistic constraints in the RBDO process. That is, RIA may demonstrate instability whereas PMA is stable in evaluating a probabilistic constraint [3]. However, several examples of numerical tests of the PMA show inefficiency and instability in the assessment of a probabilistic constraint during the RBDO process as the result of an ineffective numerical method, i.e., the Advanced Mean Value (AMV) method [5,6]. In general, the AMV method exhibits divergence or slow rate of convergence in addressing a concave performance function, although it is good for a convex performance function. With respect to a concave performance function, numerical instability as well as inefficiency in PMA using the AMV method highlights the need for a stable and efficient computational algorithm that utilizes a conjugate direction, namely, the Conjugate Mean Value (CMV) method. However, the CMV method is computationally more expensive than the AMV method for a convex performance function. Consequently, the Hybrid Mean Value (HMV) method is proposed in this paper to adaptively select either the AMV method or the CMV method once the performance function type is identified. It has been noted in Refs. 3 and 4 that the efficiency of RIA and PMA to assess the probabilistic constraint depends on activeness of the probabilistic constraint. The previous research, however, has not been dealt with the HMV method proposed in this paper. Hence, a comparative study between RIA and PMA from an efficiency and robustness perspective, with respect to probabilistic constraint evaluation in the RBDO process, is presented in this paper. It is shown that the conventional reliability analysis model in RIA causes ineffectiveness in the RBDO process, while the inverse reliability analysis model in PMA provides an efficient and robust RBDO process using the proposed HMV method. Popular numerical methods for RIA are the HL-RF method [7,8], Modified HL-RF [8], and Two-Point Approximation (TPA) [9,10]. For PMA, the AMV [5,6] is a popular numerical method. In this paper, the proposed HMV method will is used to show efficiency and robustness in probabilistic constraint assessment for PMA. GENERAL DEFINITION OF RBDO MODEL In the system parameter design, the RBDO model [11-14] can be generally defined as Minimize Cost( ) subject to ( ( ) 0) ( ) 0, 1,2, , , i t L U n P G i np