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					                                                                                             Proceedings of DETC’01
                                  ASME 2001 Design Engineering Technical Conferences and Computers and Information
                                                                                          in Engineering Conference
                                                                      Pittsburgh, Pennsylvania, September 9-12, 2001


                               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
      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

                                                                          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

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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)
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
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 .
      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)
                                                                                                    (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
β t = u* = βt , which will be called MPP in the paper. Unlike RIA,
                                                                                                           ∇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 ) )
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.
     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
   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
                                                                                                 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
                                                                                          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
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
                                                                        (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
                                                                       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) ) = −
                                                                                                          ∇U G (u (0) )

                                                                                     where u (0) = 0 (origin in U -space)
                                                                       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
    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 ) ) = −
                                                                                                                   ∇U G (u( k ) )
                   AMV                         CMV
             X1      X2        G        X1       X2       G            Step 3. Calculate the performance G (u( k +1) ) and the reliability
    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
    2      0.487   2.436     -0.873   0.487    2.436    -0.873
                                                                                     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) )
    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
   …                 …                                                              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

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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
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
                         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
 3                                                                                                                     ( a ) MPFP Search Space in RIA

                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.
           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.
          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
                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
     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

                                                                                     0.9                                Cost

             ( a ) Initial Design ( b ) 1ST RBDO Iteration


                                                                                            0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

                                                                                                    ( a ) Volume History


       ( c ) 4TH RBDO Iteration       ( d ) 7TH RBDO Iteration                       -0.2


                                                                                                                      Gp1      Gp2
                                                                                     -0.8                             Gp3      Gp4
                                                                                            0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

                                                                                            ( b ) Probabilistic Constraint History
                        ( e ) Optimum Design                                                             D1     D2      D3      D4
                                                                                       3                 D5     D6      D7      D8
                                                                                                         D9     D10     D11     D12
          Figure 10. Shape Design History in RBDO Process
     Design histories are shown in Fig. 12. The area of the
reliability-based optimum design is reduced by 47% of the original
area. The first probabilistic constraint becomes active while other
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
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                                                                           10                                      Copyright © 2001 by ASME

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