Document Sample

Proceedings of DETC’01 ASME 2001 Design Engineering Technical Conferences and Computers and Information in Engineering Conference Pittsburgh, Pennsylvania, September 9-12, 2001 DETC2001/DAC-21044 HYBRID ANALYSIS METHOD FOR RELIABILITY-BASED DESIGN OPTIMIZATION Kyung K. Choi Byeng D. Youn Center for Computer-Aided Design and Center for Computer-Aided Design and Department of Mechanical Engineering Department of Mechanical Engineering College of Engineering College of Engineering The University of Iowa The University of Iowa Iowa City, IA 52242, USA Iowa City, IA 52242, USA kkchoi@ccad.uiowa.edu ybd@ccad.uiowa.edu ABSTRACT fX ( x) Joint Probability Density Function (JPDF) of the random Reliability-Based Design Optimization (RBDO) involves parameter evaluation of probabilistic constraints, which can be done in two Φ(•) Standard normal Cumulative Distribution Function different ways, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). It has been reported in the (CDF) Φ(•) literature that RIA yields instability for some problems but PMA is FG (•) CDF of the performance function G(X) robust and efficient in identifying a probabilistic failure mode in the βs Safety reliability index RBDO process. However, several examples of numerical tests of β s,FORM First order approximation of safety reliability index β s PMA have also shown instability and inefficiency in the RBDO process if the Advanced Mean Value (AMV) method, which is a βt Target reliability index numerical tool for probabilistic constraint evaluation in PMA, is G (X) Performance function; the design is considered “fail” if used, since it behaves poorly for a concave performance function, G(X) < 0 even though it is effective for a convex performance function. Gp Probabilistic performance measure To overcome difficulties of the AMV method, the Conjugate Mean Value (CMV) method is proposed in this paper for the concave u* ( U ) =0 G Most Probable Failure Point (MPFP) in first-order performance function in PMA. However, since the CMV method reliability analysis exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type u* = βt β Most Probable Point (MPP) in first-order inverse of the performance function is identified, either the AMV method or reliability analysis the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is u* (A)MV MPP using (advanced) mean value method in PMA referred to as the Hybrid Mean Value (HMV) method. u* MPP using conjugate mean value method in PMA CMV The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO u* HMV MPP using hybrid mean value method in PMA process. It is shown that PMA with a spherical equality constraint is n Normalized steepest descent direction of performance easier to solve than RIA with a complicated equality constraint in function estimating the probabilistic constraint in the RBDO process. ∆Gabs , ∆Grel Absolute and relative changes in performance measure NOMENCLATURE ς Criteria for the type of performance function X Random parameter; X = [X1, X2,…, Xn]T L ( X) Crack initiation fatigue life x Realization of X; x = [x1, x2,…, xn]T Lt Target crack initiation fatigue life U Independent standard normal random parameter u Realization of U; u = [u1, u2,…, un]T INTRODUCTION µ Mean of random parameter X A commonly used design optimization methodology for d Design parameter; d = [d1, d2,…, dn]T engineering systems comprises deterministic modeling and simulation-based design optimization. However, the existence of d L , dU Lower and upper bounds of design parameter d uncertainties in physical quantities such as manufacturing tolerances, P (•) Probability function material properties, and loads requires a reliability-based approach to 1 Copyright © 2001 by ASME design optimization [1,2]. Given the increased computational where the cost can be any function of the design vector capabilities developed during the last few years, fundamental issues d = [diT ] = µ( X) , X = [ X i ]T (i = 1, 2,L, n) is the random vector, and relating to the inclusion of quantitative estimation of uncertainty have been recently addressed. Techniques have been explored which the probabilistic constraints are described by the performance incorporate uncertainty during design optimization at an affordable function Gi subject to uncertainty X, their probabilistic models, and computational cost. their prescribed confidence level β t . There has been a recent development in the Reliability-Based The statistical description of the failure of the performance Design Optimization (RBDO) incorporating probabilistic constraints function Gi(X) is characterized by the Cumulative Distribution that can be evaluated using two different approaches, the Reliability Function (CDF) FGi (0) as Index Approach (RIA) and the Performance Measure Approach P (Gi ( X) ≤ 0) = FGi (0) ≤ Φ (− β t ) (2) (PMA) [3,4]. The evaluation of a probabilistic constraint in the RBDO model is an essential step and thus the probabilistic constraint where the CDF is described as in the RBDO model must be computationally stable and affordable so FGi (0) = ∫G ( X)≤0L ∫ f X (x)dx1K dxn , i = 1, 2,L, np (3) i that the RBDO process can be effective. It has been shown that PMA is equivalent to RIA in prescribing the probabilistic constraint [3]. In Eq. (3) f X (x) is the Joint Probability Density Function (JPDF) of However, these approaches are not equivalent in computational all random parameters. The evaluation of Eq. (3) requires reliability robustness in evaluating probabilistic constraints in the RBDO analysis where the multiple integration is involved as shown in Eq. process. That is, RIA may demonstrate instability whereas PMA is (3). Some approximate probability integration methods have been stable in evaluating a probabilistic constraint [3]. However, several developed to provide efficient solutions [1], such as the First-Order examples of numerical tests of the PMA show inefficiency and Reliability Method (FORM) or the asymptotic Second-Order instability in the assessment of a probabilistic constraint during the Reliability Method (SORM) with a rotationally invariant measure as RBDO process as the result of an ineffective numerical method, i.e., the reliability [1,2]. The FORM often provides adequate accuracy the Advanced Mean Value (AMV) method [5,6]. In general, the [1,2] and is widely used for RBDO applications. In FORM, AMV method exhibits divergence or slow rate of convergence in reliability analysis requires a transformation T [15,16] from the addressing a concave performance function, although it is good for a original random parameter X to the independent and standard normal convex performance function. random parameter U. The performance function G ( X) in X-space With respect to a concave performance function, numerical can then be mapped onto G(T(X)) ≡ G(U) in U-space. instability as well as inefficiency in PMA using the AMV method As described in Section 1, the probabilistic constraint in Eq. (2) highlights the need for a stable and efficient computational algorithm can be further expressed in two different ways through inverse that utilizes a conjugate direction, namely, the Conjugate Mean Value transformations [3] as: (CMV) method. However, the CMV method is computationally more expensive than the AMV method for a convex performance β si = (−Φ −1( FGi (0))) ≥ β t (4) function. Consequently, the Hybrid Mean Value (HMV) method is − G pi = FGi1(Φ (− β t )) ≥ 0 (5) proposed in this paper to adaptively select either the AMV method or the CMV method once the performance function type is identified. where β si and G pi are respectively called the safety reliability It has been noted in Refs. 3 and 4 that the efficiency of RIA and index and the probabilistic performance measure for the ith PMA to assess the probabilistic constraint depends on activeness of probabilistic constraint. Equation (4) is employed to describe the the probabilistic constraint. The previous research, however, has not probabilistic constraint in Eq. (1) using the reliability index, i.e., the been dealt with the HMV method proposed in this paper. Hence, a so-called Reliability Index Approach (RIA). Similarly, Eq. (5) can comparative study between RIA and PMA from an efficiency and replace the probabilistic constraint in Eq. (1) with the performance robustness perspective, with respect to probabilistic constraint measure, which is referred to as the Performance Measure Approach evaluation in the RBDO process, is presented in this paper. It is (PMA). shown that the conventional reliability analysis model in RIA causes ineffectiveness in the RBDO process, while the inverse reliability First-Order Reliability Analysis in RIA analysis model in PMA provides an efficient and robust RBDO In RIA, the first-order safety reliability index β s,FORM is process using the proposed HMV method. Popular numerical methods for RIA are the HL-RF method obtained using the FORM by formulating as an optimization problem [7,8], Modified HL-RF [8], and Two-Point Approximation (TPA) with one equality constraint in U-space, which is defined as a limit [9,10]. For PMA, the AMV [5,6] is a popular numerical method. In state function: this paper, the proposed HMV method will is used to show efficiency minimize U and robustness in probabilistic constraint assessment for PMA. (6) subject to G (U ) = 0 GENERAL DEFINITION OF RBDO MODEL where the optimum point on the failure surface is called the Most In the system parameter design, the RBDO model [11-14] can Probable Failure Point (MPFP) u* (U ) =0 G and thus be generally defined as Minimize Cost(d) β s,FORM = u* (U ) = 0 . G subject to P(Gi ( X) ≤ 0) − Φ (− β t ) ≤ 0, i = 1, 2,L, np (1) Either MPFP search algorithms specifically developed for the first-order reliability analysis or general optimization algorithms [17] d ≤ d ≤ d , d∈R L U n can be used to solve Eq. (6). In this paper, the HL-RF method is employed to perform reliability analyses in RIA due to its simplicity and efficiency. 2 Copyright © 2001 by ASME kinds of difficulties can be overcome by using both the current and First-Order Reliability Analysis in PMA previous MPP information as applied in the proposed Conjugate Reliability analysis in PMA can be formulated as the inverse of Mean Value (CMV) method. The new search direction is obtained by reliability analysis in RIA. The first-order probabilistic performance combining n(u ( k − 2) ) , n(u ( k −1) ) , and n(u ( k ) ) with an equal CMV CMV CMV measure G p,FORM is obtained from a nonlinear optimization weight, such that it is directed towards the diagonal of the three problem [3] in U-space defined as consecutive steepest descent directions. That is, minimize G (U) (7) (1) (2) u (0) = 0, u CMV = u (1) , uCMV = u (2) , CMV AMV AMV subject to U = βt ( k −1) ( k − 2) n(u ( k ) ) + n(u CMV ) + n(u CMV ) (11) where the optimum point on a target reliability surface is identified as u ( k +1) = β t CMV for k ≥ 2 CMV ( k −1) ( k − 2) the Most Probable Point (MPP) u* = βt with a prescribed reliability n(u ( k ) ) + n(u CMV ) + n(u CMV ) β CMV where β t = u* = βt , which will be called MPP in the paper. Unlike RIA, β ∇U G (u( k ) ) n(u( k ) ) = − CMV . (12) only the direction vector u* = βt β u* = β t β needs to be determined by CMV ∇U G (u( k ) ) CMV exploring the spherical equality constraint U = β t . Consequently, the conjugate steepest descent direction General optimization algorithms can be employed to solve the significantly improves the rate of convergence, as well as the optimization problem in Eq. (7). However, the AMV method is well stability, compared to the AMV method for the concave performance suited for PMA [5,6] due to its simplicity and efficiency. function. However, as will be seen in the next section, the proposed CMV method is inefficient for the convex function. HYBRID RELIABILITY ANALYSIS METHOD FOR PMA It was found that, although the Advanced Mean Value (AMV) Example 1: Convex Performance Function method behaves well for a convex performance function, it exhibits A convex function is given as numerical shortcomings, such as slow convergence or even G ( X) = − exp( X1 − 7) − X 2 + 10 (13) divergence, when applied to a concave performance function. To where X represents the independent random variables with overcome these difficulties, the Conjugate Mean Value (CMV) X i ~ N (6.0,0.8), i = 1, 2 and the target reliability index is set to method is proposed in this paper. However, even though the CMV method always converges, it is inefficient for the convex function. β t = 3.0 . Consequently, the Hybrid Mean Value (HMV) method is proposed in As shown in Fig. 1, the constraint in Eq. (7) is always satisfied this paper to attain both stability and efficiency in the MPP search and the performance function around the MPP is convex with respect algorithm in PMA. to the origin of U-space. The AMV method demonstrates good convergence behavior for the convex function since the steepest Advanced Mean Value (AMV) Method descent direction n(u( k ) ) of the response gradually approaches to AMV Formulation of the first-order AMV method begins with the the MPP, as shown in Fig. 1(a). In Table 1, the convergence rate of Mean Value (MV) method, defined as the AMV method is faster than that of the CMV method for the ∇ G(µ ) ∇ G (0) u* = β t n(0) where n(0) = − X MV =− U (8) convex function because the conjugate steepest descent direction ∇ X G(µ ) ∇U G (0) tends to reduce the rate of convergence for the convex function. That is, to minimize the performance function G (U ) (i.e., the cost Thus, for the convex performance function, the AMV method function in Eq. (7)), the normalized steepest descent direction n(0) performs better than the CMV method. is defined at the mean value. The AMV method iteratively updates Table 1. MPP History for Convex Performance Function the direction vector of the steepest descent method at the probable AMV CMV point u ( k ) AMV initially obtained using the MV method. Thus, the Iter. X1 X2 G X1 X2 G AMV method can be formulated as 1 6.829 8.252 0.905 6.829 8.252 0.905 u (1) = u* , u ( k +1) = β t n(u ( k ) ) AMV MV AMV AMV (9) 2 7.546 7.835 0.438 7.546 7.835 0.438 where 3 8.077 7.203 -0.991 7.839 7.542 0.144 ∇U G (u ( k ) ) n(u ( k ) ) AMV =− AMV (10) 4 8.272 6.774 -0.341 8.043 7.260 -0.097 ∇U G (u ( k ) ) AMV 5 8.311 6.648 -0.357 8.165 7.035 -0.242 As will be shown, this method exhibits instability and inefficiency in 6 8.317 6.625 -0.358 8.234 6.877 -0.312 solving a concave function since this method updates the direction 7 8.272 6.775 -0.341 using only the current MPP. … … Conjugate Mean Value (CMV) Method 11 8.310 6.651 -0.357 When applied for a concave function, the AMV method tends to 12 8.317 6.625 -0.358 be slow in the rate of convergence and/or divergent due to a lack of Converged Converged updated information during the iterative reliability analysis. These 3 Copyright © 2001 by ASME ( a ) AMV Method ( a ) AMV Method ( b ) CMV Method ( b ) CMV Method Figure 1. MPP Search for Convex Performance Function Figure 2. MPP Search for Concave Performance Function 1 Example 2: Concave Performance Function 1 Consider the concave performance function Table 2. MPP History for Concave Performance Function 1 G ( X) = [exp(0.8 X1 − 1.2) + exp(0.7 X 2 − 0.6) − 5]/10 (14) AMV CMV Iter. where X represents an independent random vector with X1 X2 G X1 X2 G X1 ~ N (4.0,0.8) and X 2 ~ N (5.0,0.8) and the target reliability 1 2.989 2.823 0.225 2.989 2.823 0.225 index is set to β t = 3.0 . 2 2.348 3.259 0.234 2.348 3.259 0.234 As shown in Fig. 2, the performance function around the MPP is 3 3.073 2.786 0.238 2.687 2.990 0.204 concave with respect to the origin of U-space. The AMV method 4 2.268 3.338 0.253 2.680 2.996 0.204 applied to the concave response diverges as a result of the oscillation 5 3.162 2.751 0.255 observed in Fig. 2(a). As shown in Table 2, after 34th iteration, 6 2.190 3.424 0.277 oscillation occurs in first-order reliability analysis due to the cyclic behavior of the steepest descent directions, i.e., … … n(u( k ) ) = n(u( k − 2) ) and n(u( k +1) ) = n(u( k −1) ) . 34 1.981 3.703 0.380 AMV AMV AMV AMV This example 35 3.464 2.661 0.335 shows that, unlike the convex function, the AMV method does not … … converge for the concave function. As presented in Table 2, the CMV method applied to the PMA is stable when handling the 999 1.981 3.703 0.380 concave function by using the conjugate steepest direction. 1000 3.464 2.661 0.335 Diverged Converged Example 3: Concave Performance Function 2 A different situation using another concave function is presented Although the AMV method has converged in this case, it G ( X) = 0.3 X1 X 2 − X 2 + 0.8 X1 + 1 2 (15) requires substantially more iterations than the CMV method. Similar where X represents the independent random variables with to Example 2, the slow rate of convergence is the result of oscillating X1 ~ N (1.3,0.55) and X 2 ~ N (1.0,0.55) and the target reliability behavior of reliability iterations when using the AMV method. Based on the previous examples, it can be concluded that the of β t = 3.0 is used. AMV method either diverges or performs poorly compared to the 4 Copyright © 2001 by ASME CMV method, for the concave performance function. Thus, a Hybrid Mean Value (HMV) Method desirable approach is to select either the AMV or CMV methods once To select an appropriate MPP search method, the type of the type of performance function has been determined to achieve the performance function must be first identified. In this paper, the most efficient and robust evaluation of probabilistic constraint, as function type criteria is proposed by employing the steepest descent discussed in the following section. directions at the three consecutive iterations as follows ς ( k +1) = (n( k +1) − n( k ) ) ⋅ (n( k ) − n( k −1) ) sign(ς ( k +1) ) > 0 : Convex type at u( k +1) w.r.t. design d HMV (16) ≤ 0 : Concave type at u( k +1) HMV w.r.t. design d ( k +1) where ς is the criterion for the performance function type at the k+1 th step and n ( k ) is the steepest descent direction for a performance function at the MPP u( k ) at the kth iteration. Once the HMV performance function type is defined, one of two numerical algorithms, AMV and CMV, is adaptively selected for the MPP search. The proposed numerical procedure is therefore denoted as the Hybrid Mean Value (HMV) method, and is summarized as: ( a ) AMV Method Step 1. Set the iteration counter k=0. Select the convergence parameter ε . Compute the steepest descent direction of the performance function in U-space where ∇U G (u (0) ) n(u (0) ) = − HMV HMV ∇U G (u (0) ) HMV where u (0) = 0 (origin in U -space) HMV Step 2. If the performance function type is convex or k < 3 , calculate the MPP using the AMV method (note that Step 2 of AMV method is the same as that of HMV method when k < 3 ) as u( k +1) = β t n(u( k ) ) HMV HMV If the performance function is concave and k ≥ 3 , compute the MPP using the CMV method as n(u( k ) ) + n(u( k −1) ) + n(u ( k − 2) ) ( b ) CMV Method u( k +1) = β t HMV HMV HMV n(u( k ) ) + n(u( k −1) ) + n(u ( k − 2) ) HMV HMV HMV HMV Figure 3. MPP Search for Concave Performance Function 2 ∇U G (u( k ) ) Table 3. MPP History for Concave Performance Function 2 where n(u( k ) ) = − HMV HMV ∇U G (u( k ) ) HMV AMV CMV Iter. X1 X2 G X1 X2 G Step 3. Calculate the performance G (u( k +1) ) and the reliability HMV 1 -0.275 1.491 -0.678 -0.275 1.491 -0.678 index β ( k +1) at the new MPP u( k +1) . Check to see if HMV 2 0.487 2.436 -0.873 0.487 2.436 -0.873 3 4 -0.105 0.368 1.864 2.362 -0.997 -0.959 0.016 0.232 2.036 2.257 -1.023 -1.036 ( max β ( k +1) − β t , ∆Grel+1) , ∆Gabs 1) ≤ ε (k (k + ) where G (u ( k +1) ) − G (u ( k ) ) 5 -0.035 1.969 -1.000 0.119 2.152 -1.048 ∆Grel+1) = (k HMV HMV and 6 0.303 2.315 -1.009 0.174 2.206 -1.047 G (u ( k +1) ) HMV 7 0.009 2.028 -1.020 0.146 2.180 -1.048 (k + ∆Gabs 1) = G (u ( k +1) ) − G (u ( k ) ) HMV HMV 8 0.260 2.281 -1.027 0.160 2.193 -1.048 9 0.041 2.067 -1.033 0.153 2.186 -1.048 If the convergence criteria hold, then stop. Otherwise, go to 10 0.230 2.256 -1.036 0.157 2.190 -1.048 Step 4. 11 0.064 2.094 -1.039 0.155 2.188 -1.048 Step 4. Compute the gradient ∇U G (u ( k +1) ) of the performance HMV … … function and check the criteria ς ( k +1) for performance 23 0.124 2.158 -1.048 function type. Set k = k + 1 and return to Step 2. 24 0.155 2.188 -1.048 Converged Converged 5 Copyright © 2001 by ASME Example 4: Reliability Analysis of Analytical Examples The numerical algorithm proposed in Section 3.3 was applied to 1236 Intersection 1 the previous three examples. For the first example, the proposed b1, b2 Intersection 2 Intersection 4 Torsion 12 b3, b4 Intersection 3 b7, b8 numerical algorithm identifies ς ( k ) as positive, hence the AMV Bar b5, b6 method was then used to search for the MPP and required 6 iterations. For the second and third examples, the values of ς ( k ) were identified as negative and the CMV method was utilized for the MPP search. In conjunction with the numerical algorithm presented x'3 in Section 3.3, the HMV method performed quite well for any type of x'2 performance function. Center of the Roadwheel x'1 Example 5: Reliability Analysis of Durability Model ( b ) Finite Element Model of Roadarm Model A roadarm from a military tracked vehicle shown in Fig. 4 is employed to demonstrate the effectiveness of the HMV method for a Figure 5. Geometry and Finite Element Model for Roadarm Model large-scale problem. Reliability analysis for this example involves the crack initiation fatigue life performance measure. A 17-body Table 4. Critical Nodes for Crack Initiation Fatigue Life dynamics model is created to drive the tracked vehicle on the Aberdeen Proving Ground 4 (APG4) at a constant speed of 20 miles Node ID Life [Load Cycle] Life [Year] per hour forward (positive X2) [13,18]. A 20-second dynamic 885 .9998E+07 6.34 simulation is performed with a maximum integration time step of 889 .1134E+08 7.19 0.05-second using the dynamic analysis package DADS [19]. Three hundred and ten 20-node isoparametric finite elements, 990 .5618E+08 35.63 STIF95, and four beam elements, STIF4, of ANSYS are used for the 994 .6204E+08 39.35 roadarm finite element model shown in Fig. 5. The roadarm is made of S4340 steel with material properties of Young’s modulus E=3.0×107 psi and Poisson’s ratio ν=0.3. Finite element analysis is performed to obtain the Stress Influence Coefficient (SIC) of the roadarm using ANSYS by applying 18 quasi-static loads. To compute the multiaxial crack initiation life of the roadarm, the equivalent von Mises strain approach [20] is employed. The fatigue life contour in Fig. 6 shows critical nodes and the shortest life is listed in Table 4. The computation for fatigue life prediction and for design sensitivity require, respectively, 6950 and 6496 CPU seconds (for 812×8 design parameters) on an HP 9000/782 workstation. Figure 6. Contour for Crack Initiation Fatigue Life The random variables and their statistical properties for the crack initiation life prediction are listed in Table 5. Eight tolerance random parameters characterize four cross sectional shapes of the roadarm. The contour of a cross sectional shape consists of four Roadarm straight lines and four cubic curves, as shown in Fig. 7. Side ′ variations ( x1 -direction) of the cross sectional shapes are defined as Figure 4. Military Tracked Vehicle the random parameters b1, b3, b5, and b7 for intersections 1 to 4, 20 in. ′ respectively, and vertical variations ( x3 -direction) of the cross x' 3 sectional shapes are defined using the remaining four random x'2 variables. For reliability analysis, a failure function is defined as L ( X) G ( X) = −1 (17) Intersection 1 Intersection 3 Lt Intersection 2 Intersection 4 where L ( X) is the number of service blocks to initiate crack at x'2 node 885 and Lt is the number of target service blocks to initiate x'1 crack in the structural component. The number of blocks at node 885 for the current design is 9.998E+6 (20 seconds per block), which ( a ) Geometry of Roadarm Model constitutes the shortest life of the component. The target crack 6 Copyright © 2001 by ASME initiation fatigue life is set as 0.1 years (i.e., 1.577E+5 cycles) to the efficiency in reliability analysis. Rather, it is found that PMA illustrate the concave performance function. with the spherical equality constraint is easier to solve than RIA with a complicated constraint. In other words, it is easier to minimize a Table 5. Definition of Random Variables for Crack Initiation Fatigue complex cost function subject to a simple constraint function than to Life Prediction minimize a simple cost function subject to a complicate constraint Random Distribution function. Mean Value Std. Dev. Variables Type Tolerance b1 1.8776 0.094 Normal Tolerance b2 3.0934 0.155 Normal Tolerance b3 1.8581 0.093 Normal Tolerance b4 3.0091 0.150 Normal Tolerance b5 2.5178 0.126 Normal Tolerance b6 2.9237 0.146 Normal Tolerance b7 4.7926 0.246 Normal Tolerance b8 2.8385 0.142 Normal bi, i = 1,3,5,7 Cubic Curves Straight bi, i = 2,4,6,8 Lines x' 3 ( a ) MPFP Search Space in RIA x'1 Cross Sectional Shape Design Parameters: b1, b3, b5, b7 Design Parameters: b2, b4, b6, b8 Figure 7. Definition of Random Parameters in Roadarm Model The conventional AMV and proposed HMV method are used to calculate the reliability of the crack initiation life. Beginning at the mean point, the HMV method has converged to MPP at x = [1.872, 3.093, 1.708, 2.830, 2.218, 2.755, 4.758, 2.836]T with a target reliability index β t = 3.325 , as obtained from RIA. In contrast, the AMV method has diverged due to oscillation. Consistent with the previous concave function examples, the HMV method has converged while the AMV method has diverged. Table 6. MPP Search History in Roadarm Durability Model ( b ) MPP Search Space in PMA AMV HMV Iter. Figure 8. MP(F)P Search Spaces G(X) β G(X) β ς 0 62.404 0.0 62.404 0.0 N.A. Figure 8(a) illustrates the MPFP search space in RIA over the 1 0.014 3.325 0.014 3.325 N.A. design parameter space, where the first-order safety reliability 2 0.004 3.325 0.004 3.325 N.A. 3 -0.001 3.325 0.001 3.325 -0.0038 indices in Eq. (4) are β s j ,FORM = β j (d k ) = T(x* j =0 ) , j=1, 2. G 4 0.002 3.325 0.000 3.325 -0.0042 Reliability analysis in RIA is carried out by determining the 5 -0.001 3.325 minimum distance between the mean value design point and MPFP 6 0.002 3.325 on the failure surface G j ( X ) = 0, j = 1, 2 . The MPP search space in 7 -0.001 3.325 PMA is illustrated in Fig. 8(b), where the probabilistic performance … … … measures in Eq. (5) are G p j , FORM = G j ( x* j = βt ) , j = 1, 2. β 19 -0.001 3.325 20 0.002 3.325 Reliability analysis in PMA is performed by determining the Diverged Converged minimum performance value on the explicit sphere of the target reliability β j (d) = β t , j = 1, 2 . RIA VS. PMA IN RELIABILITY ANALYSIS Comparing Figs. 8(a) and (b), the MPP search space in PMA is It has been reported [3] that the size of the search space in a smaller than the MPFP search space in RIA if the constraint at the reliability analysis could affect the efficiency of the MP(F)P search. mean value design point is largely inactive or largely violated with However, based on numerical examples in this paper, it has been the large negative reliability index, such as the first probabilistic found that sizes of the MP(F)P search spaces may not be crucial to constraint. Thus, the MPP search in PMA, with the easier 7 Copyright © 2001 by ASME optimization problem in Eq. (7), might be better than RIA in terms of reliability analysis, the comparative study between the conventional efficiency and robustness. On the other hand, the MPFP search space RIA and the enhanced PMA with the HMV method is extended to in RIA is smaller if the constraint at the mean value design point is RBDO of a bracket problem in this section. near actives or lightly violated, such as the second probabilistic constraint in Fig. 8. In this case, although RIA has a smaller MPFP Table 8. Reliability Analysis for Lt = 10 years search space, the optimization problem in Eq. (6) is not easier to RIA (HL-RF) PMA (AMV) PMA (HMV) solve than that of PMA. As a result, a comparison of the efficiency Iter. in RIA and PMA is not clear regarding efficiency and, as such, will G β G β G β G β be examined closely in this section. In this study, the HL-RF method 0 -.366 0.0 -.366 .0 -.366 0.0 -.366 0.0 is used for RIA and both the proposed HMV and conventional AMV 1 .115 -.442 -.002 -.356 -.001 -.356 -.985 3.0 methods are used for PMA. 2 .010 -.364 .001 -.356 .001 -.356 -.985 3.0 For RIA vs. PMA in the reliability analysis, the roadarm 3 -.001 -.356 -.001 -.356 .000 -.356 durability model used in Example 5 will be demonstrated in Example 4 -.001 -.356 .001 -.356 6 and 7. 5 .000 -.356 -.001 -.356 6 .001 -.356 Example 6: RIA with a Larger Search Space than PMA (Such … … … as G1 in Fig. 8) 19 -.001 -.356 In Table 7, a target crack initiation fatigue life of Lt = 300 year 20 .001 -.356 is specified so that the MPFP search space in RIA becomes larger Converged Diverged Converged Converged than the MPP search space in PMA on the infeasible region, as represented by the largely violate deterministic constraint at the mean Example 8: Bracket Problem in RBDO Model value design point. At the second iteration, RIA has diverged - the Figure 9 shows design parameterization and stress analysis life at the first MPFP becomes infinite (1.0E+20 load cycle or result of a bracket at the initial design. A total of 12 design 6.34E+13 years) and all design sensitivities become zero, which lead parameters are selected to define the inner and outer boundary shapes to failure of RIA. In contrast, PMA does not have numerical of the bracket model while maintaining symmetry. Design difficulty in reliability analysis within the prescribed MPP search parameterization is performed by selecting the control points of the space. This example shows that PMA using the HMV method is parametric curves. The bracket is modeled as a plane stress problem better than RIA in terms of stability. using 769 nodes, 214 elements, and 552 DOF with the thickness of 1.0 cm. The boundary condition is imposed to fix two lower holes. Table 7. Reliability Analysis for Lt = 300 year Using FEM, stress analysis required 18.23 sec., while DSA required RIA (HL-RF) PMA (AMV, HMV) 35.44/12=2.95 sec. per design variable. The bracket is made of steel Iteration G β G β G β with E = 207 GPa, ν = 0.3, and the yield stress of σ=400 MPa. Probabilistic constraints are defined on two critical regions using the 0 -0.9789 0.0 -0.979 0.0 -0.979 0.0 von Mises stress as shown in Fig. 9(b). Random parameters are 1 2.11E+11 -35.46 -0.012 -3.182 -0.999 3.0 defined in Table 9 and SQP optimizer is used with a target reliability 2 0.000 -3.182 -0.999 3.0 index of β=3.0 in the RBDO model. 3 0.000 -3.182 Diverged Converged Converged d9 Gp1,Gp2 15000 N d8 Gp3, Gp4 Example 7: RIA with a Smaller Search Space than PMA (Such as G2 in Fig. 8) d7 d6 In Table 8, the target crack initiation fatigue life Lt is specified d12 as 10 years so that the MPFP search space in RIA is smaller than the MPP search space in PMA on the infeasible region, as represented by d5 the slightly violate deterministic constraint at the mean value design d2 point. In this case, RIA searches for the MPFP in the smaller search d1 d4 space than PMA with β t = 3.0 . However, PMA is more efficient d3 than RIA, since the PMA optimization problem in Eq. (7) is easier to d11 solve. Note that the HMV method demonstrates superiority over the d10 AMV method, which has diverged, as shown in Table 8. Based on the examples presented in this section, it can be ( a ) Design Parameterization ( b ) Stress Contour at Initial Design concluded that PMA is superior to RIA, regardless of sizes of the MP(F)P search spaces. Consequently, it is recommended to use PMA Figure 9. Initial Bracket Design with the spherical equality constraint in reliability analysis and not RIA with the complicate constraint for all cases. Figure 10 shows several design iterations throughout the RBDO process. At the optimum design, the overall area is substantially RBDO USING PMA WITH HMV METHOD reduced at the inner boundary and slightly at the outer boundary. As described in Section 2.1, the probabilistic constraints in the Figure 11 (a) shows the stress contour at the MPP of the initial design RBDO model can be evaluated by two different reliability analyses: where all probabilistic constraints are largely inactive. Figure 11 (b) RIA and PMA. Based on the results of previous sections in shows the stress contour at the MPP of the optimum design. 8 Copyright © 2001 by ASME Table 9. Random Variables in Bracket Model model, the RIA fails to converge in reliability analysis, whereas PMA Lower Mean Upper successfully obtains an optimal design for the bracket model. In Random Std. Distrib. addition, PMA with the HMV method performs better than with the Design Value Design Variable Dev. Type conventional AMV method in terms of numerical efficiency (195 Bound (Design) Bound 1 0.800 1.006 3.000 0.2 Normal analyses vs. 295 analyses). 2 1.600 3.004 3.500 0.2 Normal 3 0.000 0.000 1.500 0.2 Normal 4 4.470 6.388 7.000 0.2 Normal 5 3.850 4.139 4.500 0.2 Normal 6 2.690 3.332 3.800 0.2 Normal 7 13.030 13.32 14.000 0.2 Normal 8 1.850 2.493 2.800 0.2 Normal 9 15.550 15.84 16.500 0.2 Normal 10 2.500 3.509 3.800 0.2 Normal 11 0.000 0.000 1.200 0.2 Normal 12 6.000 7.776 14.000 0.2 Normal ( a ) Stress At Initial Design ( b ) Stress At Optimum Design Figure 11. Analysis Results Comparison 1.1 1.0 0.9 Cost 0.8 0.7 ( a ) Initial Design ( b ) 1ST RBDO Iteration 0.6 0.5 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ( a ) Volume History 0.4 0.2 0 ( c ) 4TH RBDO Iteration ( d ) 7TH RBDO Iteration -0.2 -0.4 -0.6 Gp1 Gp2 -0.8 Gp3 Gp4 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ( b ) Probabilistic Constraint History 5 4 ( e ) Optimum Design D1 D2 D3 D4 3 D5 D6 D7 D8 D9 D10 D11 D12 2 Figure 10. Shape Design History in RBDO Process 1 0 Design histories are shown in Fig. 12. The area of the -1 reliability-based optimum design is reduced by 47% of the original -2 area. The first probabilistic constraint becomes active while other -3 probabilistic constraints inactive at the optimum design with 99.9% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 reliability as shown in Fig.12 (b). The significantly changed shape design parameters are 12th, 1st, and 2nd parameters. In Table 10, the PMA with both HMV and AMV methods is compared to RIA in ( c ) Design Parameter History terms of computational efficiency and robustness. As in the roadarm Figure 12. Design Optimization with 99.9 % Reliability 9 Copyright © 2001 by ASME Table 10. Computational Efficiency and Robustness in RIA and PMA 5. Wu, Y.T., Millwater, H.R., and Cruse, T.A., 1990, “Advanced PMA RIA Probabilistic Structural Analysis Method for Implicit Opt. HMV AMV HL-RF Performance Functions,” AIAA Journal, Vol. 28, No. 9, pp. Iter. Line Anal Line Anal Line Anal 1663-1669. Search ysis Search ysis Search ysis 6. Wu Y.T., 1994, “Computational Methods for Efficient Structural Reliability and Reliability Sensitivity Analysis,” AIAA Journal, 0 1 5 1 5 1 40 Vol. 32, No. 8, pp. 1717-1723. 1 1 7 1 7 3 120 7. Hasofer, A.M. and Lind, N.C., 1974, “Exact and Invariant 2 2 14 2 14 1 N.A. Second-Moment Code Format, Journal of Engineering 3 1 5 1 5 Mechanics Division ASCE, 100(EMI), pp. 111-121. 4 3 25 3 46 8. Liu, P.L. and Kiureghian, A.D., 1991, “Optimization 5 2 16 2 16 Algorithms For Structural Reliability,” Structural Safety, Vol. 9, 6 1 10 1 12 pp. 161-177. 7 1 10 1 16 9. Wang, L.P. and Grandhi, R.V., 1994, “Efficient Safety Index 8 1 9 1 17 Calculation For Structural Reliability Analysis,” Computers & 9 1 10 1 8 Structures, Vol. 52, No. 1, pp. 103-111. 10 1 14 1 8 10. Wang, L.P. and Grandhi, R.V., 1996, “Safety Index Calculation 11 1 16 1 9 Using Intervening Variables For Structural Reliability,” 12 1 9 1 18 Computers & Structures, Vol. 59, No. 6, pp. 1139-1148. 13 2 25 1 19 11. Enevoldsen, I. and Sorensen, J.D., 1994, “Reliability-Based 14 1 10 1 19 Optimization In Structural Engineering,” Structural Safety, Vol. 15, pp. 169-196. 15 1 10 2 25 12. Wu, Y.-T. and Wang, W., 1996, “A New Method for Efficient 16 3 51 Reliability-Based Design Optimization,” Probabilistic Opti Failure to Mechanics & Structural Reliability: Proceedings of the 7th 21 195 24 295 mum Converge Special Conference, pp 274-277. 13. Yu, X., Choi, K.K., and Chang, K.H., 1997, “A Mixed Design Approach for Probabilistic Structural Durability,” Journal of CONCLUSIONS Structural Optimization, Vol. 14, No. 2-3, pp. 81-90. Advances in the RBDO are made by developing the HMV 14. Grandhi, R.V. and Wang, L.P., 1998, “Reliability-Based method for the PMA in this paper. It has been shown that PMA with Structural Optimization Using Improved Two-Point Adaptive a spherical equality constraint is easier to solve than RIA with a Nonlinear Approximations,” Finite Elements in Analysis and complicate constraint in reliability analysis. However, it has been Design, Vol. 29, pp. 35-48. found that the conventional MPP search algorithm, the AMV method, 15. Rackwitz, R. and Fiessler, B., 1978, “Structural Reliability exhibits numerical instability and inefficiency for the concave Under Combined Random Load Sequences,” Computers & performance function. Therefore, the HMV method is proposed for Structures, Vol. 9, pp. 489-494. effective evaluation of probabilistic constraints in the RBDO process 16. Hohenbichler, M. and Rackwitz, R., 1981, “Nonnormal in order to take advantages of PMA. Based on numerical efficiency Dependent Vectors in Structural Reliability,” Journal of the and robustness in reliability analysis, the HMV method is very Engineering Mechanics Division ASCE, 107(6), 1227-1238 effective numerical tool for estimating probabilistic constraints in the 17. Arora, J.S., 1989, Introduction to Optimum Design, McGraw- RBDO process. The comparison study between RIA and PMA has Hill, New York, NY. been extended to the RBDO problem, demonstrating that the PMA 18. Yu, X., Chang, K.H. and Choi, K.K., 1998, “Probabilistic using HMV method provides the best result in the RBDO process. Structural Durability Prediction,” AIAA Journal, Vol. 36, No. 4, pp. 628-637. ACKNOWLEDGMENTS 19. CADSI Inc., DADS User’s Manual, Rev. 7.5, Oakdale, IA, Research is partially supported by the Automotive Research 1994. Center sponsored by the U.S. Army TARDEC. 20. DRAW, Durability and Reliability Analysis Workspace, Center for Computer-Aided Design, College of Engineering, The REFERENCES University of Iowa, Iowa City, IA, 1994. 1. Madsen, H.O., Krenk, S., and Lind, N.C., 1986, Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ. 2. Palle, T.C. and Michael J. B., 1982, Structural Reliability Theory and Its Applications, Springer-Verlag, Berlin, Heidelberg. 3. Tu, J. and Choi, K.K., 1999, “A New Study on Reliability-Based Design Optimization,” Journal of Mechanical Design, ASME, Vol. 121, No. 4, 1999, pp. 557-564. 4. Tu, J., Choi, K.K., and Park, Y.H., 2001, “Design Potential Method for Robust System Parameter Design,” to appear in AIAA Journal. 10 Copyright © 2001 by ASME

DOCUMENT INFO

Shared By:

Tags:

Stats:

views: | 33 |

posted: | 2/29/2012 |

language: | English |

pages: | 10 |

OTHER DOCS BY kunfyeekun

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.