AN INVESTIGATION OF NONLINEARITY OF RELIABILITY-BASED DESIGN OPTIMIZATION APPROACHES. β Target reliability index. t β s Safety reliability index; s

Transcription

1 Proceedings of DETC 0 ASME 00 Design Engineering Technical Conferences and Computers and Information in Engineering Conference Montreal, CANADA, September 9-October, 00 DETC00/DAC-XXXXX AN INVESTIGATION OF NONLINEARITY OF RELIABILITY-BASED DESIGN OPTIMIZATION APPROACHES Kyung K. Choi Center for Computer-Aided Design and Department of Mechanical & Industrial Engineering College of Engineering The niversity of Iowa Iowa City, IA 54, SA Kkchoi@ccad.uiowa.edu Byeng D. Youn Center for Computer-Aided Design and Department of Mechanical & Industrial Engineering College of Engineering The niversity of Iowa Iowa City, IA 54, SA ybd@ccad.uiowa.edu ABSTRACT Deterministic optimum designs that are obtained without consideration of uncertainty could lead to unreliable designs, which call for a reliability approach to design optimization, using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space (X-space) and reliability analysis in an independent and standard normal random space (-space). This process requires numerous nonlinear mapping between X- and - spaces for a various probability distributions. Therefore, the nonlinearity of RBDO problem will depend on the type of distribution of random parameters, since a transformation between X- and -spaces introduces additional nonlinearity to reliability-based performance measures evaluated during the RBDO process. Evaluation of probabilistic constraints in RBDO can be carried out in two different ways: the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity of RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of highly nonlinear transformations involved. However, PMA is rather independent of probability distributions because of little involvement of the nonlinear transformation. NOMENCLATRE X Random parameter; X = [X 1, X,, X n ] T x Realization of X; x = [x 1, x,, x n ] T Independent and standard normal random parameter u Realization of ; u = [u 1, u,, u n ] T µ Mean of random parameters X; µ = [µ 1, µ,, µ n ] T d Design parameter; d = [d 1, d,, d n ] T L d, d Lower and upper bound of design parameter d P () Probability function f X ( x ) Joint probability density function of the random parameters Φ () Standard normal Cumulative Distribution Function (CDF) G( X ) Design performance function; considered as fail if G(X)<0 F G( ) CDF of the performance function G(X); F G (g)=p(g(x)<g) F 1 G () Inverse of CDF of G(X) β Target reliability index t β s Safety reliability index; s s,form β = Φ 1 (P f )= Φ 1 (F G (g)) β First order approximation of safety reliability index β s G p Probabilistic performance measure; G p = G ( ) = 0 F 1 ( Φ ( β )) u Most Probable Point (MPP) in first-order reliability β = β t analysis u MPP in first-order inverse reliability analysis u MV MPP using mean value method in PMA u AMV MPP using advanced mean value method in PMA u CMV MPP using conjugate mean value method in PMA u HMV MPP using hybrid mean value method in PMA n Normalized steepest descent direction ς Criteria for the type of performance function T Transformation between X- and -spaces 1. INTRODCTION Due to an increasingly global, competitive market, optimum designs are pushed to the limit of system failure boundaries by using multidisciplinary design optimization techniques, leaving very little or no room for tolerances in modeling and simulation uncertainties and/or for manufacturing imperfections. Consequently, deterministic optimum designs that are obtained without taking uncertainty into account could lead to unreliable designs. RBDO can be carried out by evaluating probabilistic constraints in two alternative ways: the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). Reliability measures in G t 1 Copyright 00 by ASME

2 RIA or performance measures in PMA describe probabilistic constraints in the RBDO model. It has been reported that PMA is inherently robust and far more effective when the probabilistic constraint is either very feasible or very infeasible [1], since it is easier to minimize a nonlinear cost function subject to a simple constraint function with known distance (i.e., the reliability index) than it is to minimize a simple cost function subject to a nonlinear constraint function. The RBDO process iteratively carries out two optimizations: design optimization in an original random space (Xspace) and reliability analysis in an independent and standard normal random space (-space) [,3]. Efficiency and robustness of the RBDO process depends on nonlinearity of two optimization models (reliability analysis model and design model) in RBDO. It is presented in this paper that RBDO nonlinearity is strongly related to distribution types of random parameters, since transformations between X- and -spaces affect nonlinearity of performance measures evaluated during the RBDO process. For the RBDO process, random variables that characterize physical quantities under uncertainties must be modeled as probability distributions. Through the nonlinear mappings between X- and -spaces for given probability distributions [4,5], nonlinearity of constraints in two optimization models is changed in the RBDO formulations. Therefore, different reliability analysis approaches employed in RIA and PMA result in different nonlinearity of the RBDO formulations. It is found that small nonlinearity is introduced in PMA, as significant nonlinearity is introduced in RIA. Some example problems, including a largescale model, demonstrate the efficiency and robustness of PMA for a diverse class of probability distributions. A popular numerical method for RIA is the HL-RF method [6,7], since it requires no line search in reliability analysis. It has been reported that for PMA the Hybrid Mean-Value (HMV) method [1] is very efficient and robust, which adaptively utilizes the Conjugate Mean-Value (CMV) method [1] and the Advanced Mean- Value (AMV) method [8,9] in probabilistic constraint assessment during the RBDO process.. RELIABILITY-BASED DESIGN OPTIMIZATION.1 General Definition of the RBDO Model In system parameter design, the RBDO model [1,10-13] can generally be defined as minimize Cost( d) subject to PG ( ( X) 0) Φ( β ) 0, i= 1,,, np i t i L ndv nrv d d d, d R and X R d = µ ( X) is the design vector, X is the random vector, and the probabilistic constraints are described by the performance function Gi(X), their probability distributions, and their prescribed confidence level β. t i The statistical description of the failure of the performance function G i (X) is characterized by the Cumulative Distribution Function (CDF) (0) as F Gi the CDF is defined as In Eq. (3) (1) PG ( ( X ) 0) = F (0) Φ( β ti ) () F Gi f X ( x) i G i (0) f ( x) dx = X Gi ( X) 0 (3) is the Joint Probability Density Function (JPDF) of all random parameters. The evaluation of probabilistic constraints in Eq. () requires a reliability analysis the multiple integration of Eq. (3) is involved. Some approximate probability integration methods have been developed to provide efficient solutions [,3], such as the First-Order Reliability Method (FORM) and the asymptotic Second-Order Reliability Method (SORM) with a rotationally invariant measure as the reliability []. FORM often provides adequate accuracy [1-3] and is widely used for RBDO applications. In FORM, reliability analysis requires a transformation T [4,5] from the original random parameter X to the independent and standard normal random parameter. That is, the performance function G( X) in X-space is mapped onto G(T(X)) G() in -space. For the RBDO formulation, the probabilistic constraint in Eq. () can be further expressed through inverse transformations in two alternative ways [1,14,15]: s i si 1 β = ( Φ ( F (0))) β (4) Gi ti G = F 1 ( Φ ( β )) 0 (5) pi Gi ti β and are called the safety reliability index and the G pi probabilistic performance measure for the i th probabilistic constraint, respectively. Equation (4) is employed to describe the probabilistic constraint in Eq. (1) using the reliability index, i.e., the so-called Reliability Index Approach (RIA). Similarly, Eq. (5) can replace the probabilistic constraint in Eq. (1) with the performance measure, which is referred to as the Performance Measure Approach (PMA) [1,14].. First-Order Reliability Analysis in RIA In RIA, the first-order safety reliability index β s,form is obtained using FORM by formulating an optimization problem with an equality constraint in -space, which is the failure surface, as minimize (6) subject to G ( ) = 0 The minimum point on the failure surface is called the Most Probable Point (MPP) u G ( ) = 0 and the reliability index is defined by β s,form = G( ) = 0 u. For the solution of Eq. (6), either an MPP search algorithm that has been specifically developed for the first-order reliability analysis or a general optimization algorithm [7] can be used. Due to its simplicity and efficiency, the HL-RF method is a popular choice for reliability analysis in RIA..3 First-Order Reliability Analysis in PMA The reliability analysis in PMA is formulated as the inverse of the reliability analysis in RIA. The first-order probabilistic performance measure is obtained from the optimization G p,form problem [1,14,15] in -space, minimize G( ) subject to = β The minimum point on the target reliability surface is called MPP β = β t u with the prescribed reliability t β t β = β t (7) = u and the probabilistic performance measure is defined by p,form ( β = βt G = G u ). nlike RIA, only the direction vector Copyright 00 by ASME

3 β = βt β= βt u u needs to be determined by taking advantage of the spherical equality constraint = βt to find MPP u β = β t. General optimization algorithms can be employed to solve the optimization problem in Eq. (7). Although the AMV method is well suited for PMA [7-9] due to its simplicity and efficiency, it may show instability or inefficiency for concave performance measures..4 Reliability Analysis Methods for RIA and PMA The HL-RF method is a popular choice for RIA, and HMV method, which adaptively combines the AMV and CMV methods, is used for PMA in this paper. HL-RF Method in RIA As shown in Eq. (6), the reliability analysis in RIA is to find the minimum distance G = 0 from the origin of the standard normal space to the failure surface G ( ) = 0 HL-RF method is n ( ). The iterative algorithm of the ( k+ 1) G( u ) = + u u n n n G( u ) G( u ) = G( u ) (8) is the steepest descent direction of the performance function G( ) at u find a direction with the shortest distance to the failure surface, and the second term on the right side of Eq. (8) is a correction term to reach G( ). HMV Method In PMA a. Advanced Mean-Value (AMV) Method Formulation of the first-order AMV method begins with the Mean Value (MV) method, defined as XG( µ ) G( 0) umv = βtn( 0) n( 0) = = (9) G( µ ) G( 0) That is, to minimize the cost function G( ) in Eq. (7), the normalized steepest descent direction nµ ( ) is obtained at the meanvalue point (i.e., the origin in -space). The AMV method iteratively updates the steepest descent direction at the probable AMV point u starting from. The iterative algorithm of the AMV method is u MV (0) (1) ( k+ 1) AMV =, AMV = MV, AMV = β t ( AMV u 0 u u u n u G( uamv AMV = G uamv nu ( ) X ) ( ) ) (10) (11) As presented in Ref. 1, this method exhibits instability and inefficiency in solving a concave performance function, since it only updates the direction using the current MPP. b. Conjugate Mean-Value (CMV) Method When applied to a concave performance function, the AMV method tends to converge very slowly or divergence due to lack of updated information during the iterative optimization for reliability analysis. This kind of difficulty can be overcome by using both the current and previous MPP search information as applied in the proposed Conjugate Mean Value (CMV) method [1]. The new search direction is obtained by combining nu ( ), nu ( ), and nu ( CMV ) ( k ) CMV ( k 1) CMV with an equal weight, such that it is directed towards the diagonal of the three consecutive steepest descent directions. That is, (0) (1) (1) () () CMV = CMV = AMV CMV = AMV u 0, u u, u u, u ( k 1) ( k ) ( k + 1) nucmv + nucmv + nucmv CMV βt ( k 1) ( k ) nucmv + nucmv + nucmv ( ) ( ) ( ) = for k ( ) ( ) ( ) G ucmv CMV = G ucmv nu ( ) ( ) ( ) (1) (13) Consequently, the conjugate steepest descent direction significantly improves the rate of convergence as well as the stability, as compared to the AMV method for the concave performance function. However, the CMV method was found inefficient for the convex performance function [1]. Thus, it is desirable to combine the AMV and CMV methods for the MPP search to treat both the convex and concave performance functions. c. Hybrid Mean-Value (HMV) Method To select an appropriate MPP search method, the type of performance function must first be identified. The function type can be determined by employing steepest descent directions at the three consecutive iterations, as follows: ( k+ 1) ( k+ 1) ( k 1) ς = ( n n ) ( n n ) sign( ς ) 0 : Convex type at w.r.t. design d 0 : Concave type at w.r.t. design d ( k+ 1) ( k+ 1) > uhmv ( k + 1) uhmv (14) ( k 1) ς + is the criterion for the performance function type at the ( n k) k+1 th step and is the steepest descent direction for a performance function at the MPP u at the k th iteration. Once the performance HMV function type is defined, one of two numerical algorithms, AMV and CMV, is adaptively selected for the MPP search. This proposed MPP search method is the Hybrid Mean Value (HMV) method. 3. NONLINEARITY OF RELIABILITY-BASED DESIGN OPTIMIZATION For a successful RBDO, the uncertainties on random parameters must be modeled with correct types of probability distributions. sing these distribution models, the RBDO process interactively executes two optimization problems: a design optimization in an original random space (X-space) and reliability analyses of performance measures in an independent and standard normal random space (-space), which requires the nonlinear mappings for the input probability distributions. Due to these nonlinear mappings, the nonlinearity of two optimization (reliability analyses and design optimization), which depends upon the distribution types of random parameters, affects efficiency and robustness of the RBDO process. In this paper RBDO nonlinearity is investigated for two different RBDO approaches: RIA and PMA. 3.1 Transformation of Random Parameters As described above, the RBDO process is performed on two different random spaces: the original random parameter space (Xspace) for design optimization and the independent and standard normal space (-space) for reliability analysis. During the RBDO process, a transformation between X- and -spaces at a design point must be carried out to estimate the probabilistic constraint. The 3 Copyright 00 by ASME

4 transformation between two different random spaces at the design point k d is defined as 1 k k X= T ( ) or = T( X) d = µ ( X) (15) and requires the probability distributions of random parameters, such as Joint Probability Density Function (JPDF) of the random vector X. Probability distributions can be developed in numerous ways to represent the uncertainties in physical quantities using suitable distribution types and their parameters. Most transformations required in RBDO, except normal distribution, are highly nonlinear. This paper employs five most commonly used types of probability distributions (normal, lognormal, Weibill, Gumbel, and uniform), which are characterized by their Probability Density Functions (PDFs), their parameters, and their transformations, as presented in Appendix Nonlinearity of RBDO: RIA and PMA The RBDO model constitutes two optimization problems: design optimization with constraints in Eqs. (1), (4), or (5) and the reliability analyses in Eqs (6) or (7), as summarized in Table 1. Two different approaches for RBDO, RIA and PMA, involve different nonlinear transformation, as shown in Table. In general, efficiency of nonlinear optimization significantly depends on the complexity of constraints in the optimization problem. Accordingly, the study of nonlinearity of RBDO approaches is carried out by observing nonlinearity of constraints involving a transformation in two optimization problems: the reliability analysis and design optimization, as shown in Table. Table 1. Two Optimization Problems in the RBDO RIA PMA Design Optimization Minimize Cost Subject to -b s (X)+b t 0 Minimize Cost Subject to -G p (X) 0 Reliability Analysis Minimize b() Subject to G()=0 Minimize G() Subject to b()=b t Nonlinearity of RIA in the RBDO Process a. Reliability Analysis in RIA The performance failure surface G(X)=0 is originally described in X-space; however, the constraint of the reliability analysis in RIA is the performance failure G()=0 in -space, as shown in Table 1. The performance measure G(X) is itself a nonlinear response that requires FEA or other complex engineering analyses. As shown in Table, the failure surface G()=G(T(X))=0 in -space requires a nonlinear transformation for given probability distributions, which may introduce additional nonlinearity to the original failure surface G(X)=0. Furthermore, the reliability analysis in RIA may fail to have a solution whenever the failure surface G()=0 in -space is outside the infinite probability integration domain β = =. It will be demonstrated in Section 4 that RIA failed to converge for probability distributions with bound (e.g., uniform) and extreme type distributions (e.g., Gumbel), in which the infinite integration domain may not include a failure surface G(X)=0. b. Design Optimization in RIA The reliability index β s = u G( ) = 0, β s is used as the RIA probabilistic constraint in design optimization, is a simple, n- dimensional quadratic function in -space. However, the RIA probabilistic constraint in design optimization must be inversely transformed into X-space to perform a design optimization, as shown in Table. Therefore, this inverse transformation introduces additional nonlinearity to the reliability index constraint in the design optimization problem for all probability distributions except for the normal distribution, which requires a linear transformation. Table. Nonlinearity of Probabilistic Constraints in RBDO X-Space -Space RIA b(x)=b(t -1 ()), Nonlinear G()=G(T(X)), Nonlinear inverse transformation of b() transformation of G(X) PMA G(X), Original performance b(), Quadratic function measure with respect to Nonlinearity of PMA in the RBDO Process a. Reliability Analysis in PMA The constraint of reliability analysis in PMA is expressed as a target reliability constraint = βt in -space, β t is the target reliability index. In contrast to RIA, the constraint of reliability analysis in PMA is a simple n-dimensional quadratic function without any nonlinear transformation, as shown in Table. b. Design Optimization in PMA As shown in Table 1, the PMA probabilistic constraint in the RBDO model is expressed using the probabilistic performance β = βt function Gp = G p( x ), x β = β is the MPP on the target t reliability surface. Therefore, the probabilistic constraint in PMA is nothing but the original performance function G( X) evaluated at β = β t x that does not involve any nonlinear transformation. In summary for RIA, constraints in two optimization problems may become highly nonlinear through the nonlinear transformation between X- and -spaces, as for PMA, constraints in these two optimization models do not involve the nonlinear transformation. On the other hand, the cost function of the reliability analysis in RIA does not involve the nonlinear transformation of probability distributions; as, in the inverse reliability analysis in PMA, the cost function involves this nonlinear transformation. As a result, it is expected that the RBDO process using RIA, with non-normal distributions, shows inefficiency and instability because of significant RBDO nonlinearity. On the other hand, the RBDO process using PMA involves less nonlinear transformation for non-normal distributions of random parameters. Furthermore, it has been noted in Ref. 1 that RIA in the RBDO problem is less efficient and robust than PMA, even for a simple problem with normal distribution, due to the desirable optimization problem formulating for reliability analysis in PMA. Thus, PMA is better than RIA regardless of distribution types of random parameters in terms of efficiency and robustness of the RBDO process. In Section 4 some example problems demonstrate the nature of RBDO nonlinearity. 4. EXAMPLES OF RELIABILITY-BASED DESIGN OPTIMIZATION 4.1 Mathematical Example Consider the following mathematical model of RBDO with design variables d = [ µ ( X ), µ ( X )] T. The design optimization problem is defined as 1 4 Copyright 00 by ASME

5 minimize Cost( d) = d + d 1 subject to PG ( ( X) 0) Φ( β ), i= 1, 3 1 X X1 X G ( ) = /0 1 i t i 0 d 10 & 0 d 10 1 X 1 1 G ( ) = ( X + X 5) 30 + ( X X 1) X 1 G ( ) = 80/( X + 8X + 5) 1 (16) (17) and β =.0, i = 1,,3. The initial design is d = [5.0,5.0] T with a ti standard deviation of σ=0.6. Five different types of random distributions are used: normal, lognormal, Weibull, Gumbel, and uniform. The Sequential Quadratic Programming (SQP) optimization algorithm is used for design optimization, and the maximum number of iterations in reliability analysis is set to 0. The efficiency and robustness of the RBDO process for five different probability distributions are shown in Tables 3-6. The total number (0) of function evaluations is used to measure the efficiency of the RBDO process. The results in Tables 3 and 4 are consistent with RBDO nonlinearity, since the total number of analyses increase with nonnormal distributions, i.e., 53 for normal, 68 for lognormal, and 171 for Weibull distributions. The probabilistic constraint function βs( X ) in RIA can be obtained from a circle as shown in Fig. 1 for normal distribution, but become significantly nonlinear for lognormal and Weibull distributions, as shown in Figs. 1(b) and (c). With the Weibull distribution, reliability analysis becomes more nonlinear with the constraint G()=G(T(X))=0 and significantly more function evaluations are required in reliability analysis as shown in Table 3. Moreover, using FORM in RIA for Gumbel and uniform distributions, both 1 st and nd performances failure surface are located at the infinite probability domain, βs ( d ) =, as shown in Fig. 1(d). Consequently, reliability analysis in RIA failed to converge for these distributions. 0 Table 3. Number of Function Evaluations sing RIA for Mathematical Example β 1 β β 3 Evaluation β 1 β β 3 Evaluation β 1 β β 3 Evaluation Normal No. of Lognormal No. of Weibull No. of (5/5/4) (6/6/4) (6/6/5) (4/4/5) (6/6/6) (13/18/18) (4/4/5) (5/5/7) (1/15/18) (4/4/5) (5/5/7) (1/15/18) (4/5/6) Opt. Act. Act. Inact. 53(17/17/19) Act. Act. Inact. 68(//4) Act. Act. Inact. 171(47/59/65) Table 4. Cost and Design Variable History sing RIA for Mathematical Example Normal Lognormal Weibull Cost d 1 d Cost d 1 d Cost d 1 d Opt Table 5. Number of Function Evaluation sing PMA for Mathematical Example G p1 G p G p3 Evaluation G p1 G p G p3 Evaluation G p1 G p G p3 Evaluation Normal No. of Lognormal No. of Weibull No. of (4/4/3) (3/4/4) (5/4/3) (4/4/4) (3/4/4) (5/4/3) (4/4/4) (3/5/4) (4/4/3) (4/5/4) (3/5/4) (4/4/3) Opt. Act. Act. Inact. 48(16/17/15) Act. Act. Inact. 46(1/18/16) Act. Act. Inact. 46(18/16/1) Gumbel No. of niform No. of G p1 G p G p3 Evaluation G p1 G p G p3 Evaluation (3/3/5) (6/5/3) (4/5/5) (6/8/3) (3/5/5) (6/8/5) (3/5/5) (6/8/5) Opt. Act. Act. Inact. 51(13/18/0) Act. Act. Inact. 69(4/9/16) 5 Copyright 00 by ASME

6 Table 6. Cost and Design Variable History sing PMA for Mathematical Example Normal Lognormal Weibull Gumbel niform Cost d 1 d Cost d 1 d Cost d 1 d Cost d 1 d Cost d 1 d Opt ( a )PMA and RIA (Normal Distributions) ( b ) PMA and RIA (Lognormal Distributions) ( c ) PMA and RIA (Weibull Distributions) ( d ) Failure in RIA (niform Distributions) ( e ) PMA (Gumbel Distributions) ( f ) PMA (niform Distributions) Figure 1. RBDO History of PMA and RIA As shown in Tables 5 and 6, PMA requires nearly the same number of analyses (around 50) regardless of probability distributions, since constraints in PMA do not involve nonlinear transformations for any probability distribution. The uniform distribution, however, requires marginally more analyses than the other distribution in the RBDO process, due to a highly nonlinear transformation of the cost G()=G(T(X)) in the reliability analysis, and more analyses in reliability analysis are required for the uniform distribution. 4. Two-Bar Frame Problem Consider RBDO of a two-bar frame subject to out-of-plane loads, as shown in Fig. [13]. Three design parameters, w = width of the member (in), h = height of the member (in), and t = wall thickness (in) are selected for the cross sectional design of a two-bar frame. The two-bar frame is modeled with E=3.0E+07 psi, ν=0.3, and an allowable stress of σ a =40kpsi. The RBDO problem of the two-bar frame is formulated as minimize C( d) = L(wt + ht 4 t ) subject to PG ( ( X) 0) Φ( β ), i= 1, i t i i σ i τ βt σ i a (18) 1 G = 1 ( + 3 ) and = 3.0, i = 1, (19) The members are subject to both bending and tensional stress, σ i and τ, respectively. The von Mises yield stress constraint is imposed at 6 Copyright 00 by ASME

7 points 1 and. The SQP optimization algorithm is used for design optimization. z y 1 L x 1 z, 3 P=10000 lbs Figure. Two-Bar Frame t L=100 in w h Table 7. Random Parameters for Two-Bar Frame d L d d Std. Distrib. Mixed (Mean) Dev. Type Distrib. w Normal, Lognormal Gumbel h Weibull t Lognormal nlike the mathematical example, RIA fails to converge regardless of the distribution types, since both constraints yield 0 βs( d ) at the initial design stage. For the study of RBDO nonlinearity for various distributions in PMA, five different Table 8. RBDO Design History sing PMA for Two-Bar Frame Normal Lognormal Weibull Cost w h t Cost W h t Cost w h t Opt G p E E E-04 G p 9.55E E E-01 No of Eval Table 8. RBDO Design History sing PMA for Two-Bar Frame (continued) Gumbel niform Mixed Distributions Cost w h t Cost w h t Cost w h t Opt G p1.30e e E-04 G p 9.49E E E-01 No of Eval Copyright 00 by ASME

8 distributions are individually tested for all random variables and, in addition, one mixed combination of distributions is tested, as shown in Table 7. As pointed out in Section 4.1, the uniform distribution requires a highly nonlinear transformation. In PMA, uniform distribution requires a larger number of evaluations than other probability distributions, although the number of optimization iterations is about the same. This is due to the fact that the nonlinearity of the design optimization problem in PMA does not depend on the nonlinear transformation (see Table 1), as the nonlinearity of reliability analysis in PMA depends on the type of probability distribution, since the cost function (performance measure G( ) ) in the inverse reliability analysis (see Table 1) becomes highly nonlinear through the nonlinear transformation. 4.3 Bracket Problem Figure 3 (a) shows the design parameterization and stress analysis result of a bracket at the initial design. A total of 1 design parameters are selected to define the inner and outer boundary shapes of the bracket model while maintaining symmetry. Design parameterization is performed by selecting the control points of the parametric curves. The bracket is modeled as a plane stress problem using 769 nodes, 14 elements, and 55 DOF with the thickness of 1.0 cm. The boundary condition is imposed to fix two lower holes. Stress analysis required 18.3 CP sec., while DSA required 35.44/1=.95 sec. per design variable on an HP 9000/78. The bracket is made of steel with E = 07 GPa, ν = 0.3, and a yield stress of σ=400 MPa. Probabilistic constraints are defined at two critical regions using the von Mises stress, as shown in Fig. 3 (b). The SQP optimizer is used for design optimization. Design and random parameters are defined in Table 9 and a target reliability index of β=3.0 is used in the RBDO model. Note that the 3 rd and 11 th design parameters are treated as deterministic parameters with zero standard deviation. As shown in Table 9, five different distributions are individually tested, and, in addition, one mixed combination of distributions is tested N d 7 d1 d 9 d 8 d d d 1 3 d 11 d6 d 5 d 4 d 10 G p3, G p4 G p1,g p ( a ) Design Parameterization ( b ) Stress Contour at Initial Design Figure 3. Initial Bracket Design Table 9. Design and Random Parameters in Bracket Random d L d d Std. Distrib. Mixed Variable (Mean) Dev. Type Distrib Normal Lognorm Lognorm Normal Weibull Weibull Weibull Weibull Gumbel Gumbel Normal Normal niform Table 10. RBDO Design History sing PMA for Bracket Normal Lognormal Weibull G p1 G p G p3 G p4 Eval. G p1 G p G p3 G p4 Eval. G p1 G p G p3 G p4 Eval Opt Copyright 00 by ASME

9 Table 10. RBDO Design History sing PMA for Bracket (continued) Gumbel niform Mixed G p1 G p G p3 G p4 Eval. G p1 G p G p3 G p4 Eval. G p1 G p G p3 G p4 Eval Opt Table 11. Reliability-Based Optimum Designs of Bracket Distribution d 1 d d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 10 d 11 d 1 Normal 3.000, 1.600, 1.500, 6.678, 4.073,.690, 13.03, 1.850, 15.55, 3.797, 1.00, 1.09 Lognormal.999, 1.600, 1.500, 6.460, 4.499,.690, 13.03, 1.850, 15.55, 3.800, 1.00, 1.07 Weibull 3.000, 1.600, 1.500, 6.856, 4.497,.691, 13.04, 1.850, 15.55, 3.799, 1.00, 1.03 Gumbel.999, 1.600, 1.500, 6.800, 4.498,.690, 13.03, 1.850, 15.55, 3.800, 1.00, 1.09 niform 3.000, 1.603, 1.500, 6.616, 3.875,.69, 13.03, 1.850, 15.55, 3.799, 1.00, 1.07 Mixed.999, 1.600, 1.500, 6.83, 4.496,.690, 13.03, 1.850, 15.55, 3.798, 1.00, Cost (Volume) Normal Weibull niform RBDO Iterations Figure 4. RBDO Cost History of Bracket Lognormal Gumbel Mixed Like the two-bar frame, RIA fails to converge regardless of the distribution types, since some of the constraints yield βs( d ) at the initial design. As illustrated in Table 10, the Gumbel and uniform distributions require larger number of evaluations using PMA, although there is little difference in the total number of optimization iterations. That is, the nonlinearity of design optimization problem in PMA does not depend on the type of probability distribution, as the nonlinearity of reliability analysis depends on the type of probability distribution, since the cost function of the reliability analysis becomes highly nonlinear for 0 non-normal distributions. In Figure 4, the cost history of PMA is shown for different probability distributions. Then, reliability-based optimum designs for various distributions are shown in Table11. At the optimum design, significant differences are found in the 4 th, 5 th, and 1 th design variables for different probability distributions. 5. CONCLSIONS The nonlinearity of RIA and PMA in RBDO is discussed by observing mathematical formulations of two optimization problems of the RBDO process. It has been found that nonlinearity of RBDO formulations depends on the distribution types of the random parameters, since the transformation between X- and -space influences nonlinearity of the optimization constraints. Based on the investigation of nonlinearity in the RBDO problem, the RIA depends on the nonlinear transformation too much to the point it does not yield a good RBDO tool. On the other hand, PMA depends on the nonlinear transformation much less and thus can handle all kinds of distributions without significantly increasing the number of function evaluations. Moreover, RIA fails to converge for distributions with bound (e.g., uniform) and extreme type distributions (e.g., Gumbel). 6. ACKNOWLEDGMENTS Research is partially supported by the Automotive Research Center sponsored by the.s. Army TARDEC. 9 Copyright 00 by ASME

10 7. REFERENCES 1. Choi, K.K. and Youn, B.D., Hybrid Analysis Method for Reliability-Based Design Optimization, 7 th ASME Design Automation Conference, September 9-1, 001, Pittsburgh, PA.. Madsen, H.O., Krenk, S., and Lind, N.C., 1986, Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ. 3. Palle, T.C. and Michael J. B., 198, Structural Reliability Theory and Its Applications, Springer-Verlag, Berlin, Heidelberg. 4. Rackwitz, R. and Fiessler, B., 1978, Structural Reliability nder Combined Random Load Sequences, Computers & Structures, Vol. 9, pp Hohenbichler, M. and Rackwitz, R., 1981, Nonnormal Dependent Vectors in Structural Reliability, Journal of the Engineering Mechanics Division ASCE, 107(6), Hasofer, A.M. and Lind, N.C., 1974, Exact and Invariant Second-Moment Code Format, Journal of Engineering Mechanics Division ASCE, 100(EMI), pp Liu, P.L. and Kiureghian, A.D., 1991, Optimization Algorithms For Structural Reliability, Structural Safety, Vol. 9, pp Wu, Y.T., Millwater, H.R., and Cruse, T.A., 1990, Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions, AIAA Journal, Vol. 8, No. 9, pp Wu Y.T., 1994, Computational Methods for Efficient Structural Reliability and Reliability Sensitivity Analysis, AIAA Journal, Vol. 3, No. 8, pp Enevoldsen, I. and Sorensen, J.D., 1994, Reliability-Based Optimization In Structural Engineering, Structural Safety, Vol. 15, pp Wu, Y.-T. and Wang, W., 1996, A New Method for Efficient Reliability-Based Design Optimization, Probabilistic Mechanics & Structural Reliability: Proceedings of the 7 th Special Conference, pp Yu, X., Choi, K.K., and Chang, K.H., 1997, A Mixed Design Approach for Probabilistic Structural Durability, Journal of Structural Optimization, Vol. 14, No. -3, pp Grandhi, R.V. and Wang, L.P., 1998, Reliability-Based Structural Optimization sing Improved Two-Point Adaptive Nonlinear Approximations, Finite Elements in Analysis and Design, Vol. 9, pp Tu, J. and Choi, K.K., 1999, A New Study on Reliability-Based Design Optimization, Journal of Mechanical Design, ASME, Vol. 11, No. 4, 1999, pp Tu, J., Choi, K.K., and Park, Y.H., 001, Design Potential Method for Robust System Parameter Design, to appear in AIAA Journal. 16. Arora, J.S., 1989, Introduction to Optimum Design, McGraw- Hill, New York, NY. 17. Yu, X., Chang, K.H. and Choi, K.K., 1998, Probabilistic Structural Durability Prediction, AIAA Journal, Vol. 36, No. 4, pp Satoshi, K. and Dan, M.F., 1994, Hyperspace Division Method For Structural Reliability, Journal of Engineering Mechanics, Vol. 10, No. 11, pp Lin, H. and Khalessi, M., 1993, Identification of The Most Probable Point In Original Space Applications to Structural Reliability, 34 th AIAA/ASME/ASCE/AHS/ASC Conference, La Jolla, CA, Part, pp Khalessi, M.R., Wu, Y.T., and Torng, T.Y., 1991, A New Most- Probable-Point Search Procedure For Efficient Structural Reliability Analysis, Proceedings of the 3 nd Structures, Structural Dynamics, and Materials Conference, AIAA/ASME/ASCE/AHS/ASC, Part, pp Ayyub, B.M. and McCuen, R.H., 1997, Probability, Statistics, & Reliability for Engineers, CRC Press, New York, NY.. Dai, S.H. and Wang, M.O., 199, Reliability Analysis in Engineering Applications, Van Nostrand Reinhold, New York, NY. 3. W. Chen and C. Yuan, 1999, Probabilistic-based design model for achieving flexibility in design, Journal of Mechanical Design, Vol. 11, pp Appendix 1. PROBABILITY DISTRIBTION AND ITS TRANSFORMATION BETWEEN X- AND -SPACE Normal µ = mean, σ = standard deviation Lognormal σ = + ( σ µ ) Weibull Parameters PDF Transformation µ = mean, σ = standard deviation ln 1, µ = ln( µ ) 0.5σ k > 0, µ = νγ 1+ 1 ( 1 k) ( 1 1 k) σ = ν Γ + Γ + µ = ν α, σ = π 6α Gumbel ( ) niform µ = ( a+ b), σ = ( b a) 1 1 u Φ ( ) = e du. π [( x µ ) σ f( x) = e ], x X = µ + σ πσ [(ln x µ ) σ f( x) = e ], x > 0 X = e µ + σ πxσ k 1 k x k ( x ν ) f( x) = e, x > 0 ν ν ( ) 1 X = ν ln Φ( ) k α( x ν) α( x ν) e = X ν ( ) f( x) αe, x 1 f ( x) =, a x b a b = 1 ln ln ( ) α Φ X = a+ ( b a) Φ ( ) 10 Copyright 00 by ASME