ASME 2001 Design Engineering Technical Conferences and Computers and Information
in Engineering Conference
Pittsburgh, Pennsylvania, September 9-12, 2001
DETC2001/DAC-21044
HYBRID ANALYSIS METHOD FOR RELIABILITY-BASED DESIGN OPTIMIZATION
Kyung K. Choi
Center for Computer-Aided Design and Department of Mechanical Engineering
College of Engineering The University of Iowa Iowa City, IA 52242, USA
Byeng D. Youn
Center for Computer-Aided Design and Department of Mechanical Engineering
College of Engineering The University of Iowa Iowa City, IA 52242, USA
ABSTRACT
Reliability-Based Design Optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the RBDO process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the Advanced Mean Value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function.
To overcome difficulties of the AMV method, the Conjugate Mean Value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the Hybrid Mean Value (HMV) method.
The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process.
NOMENCLATURE
X Random parameter; X = [X1, X2,…, Xn]T x Realization of X; x = [x1, x2,…, xn]
T
U Independent standard normal random parameter
u Realization of U; u = [u1, u2,…, un]T
µ Mean of random parameter X
d Design parameter; d = [d1, d2,…, dn]T
,
L U
d d Lower and upper bounds of design parameter d
( )
P• Probability function
( )
fX x Joint Probability Density Function (JPDF) of the random parameter
( )
Φ • Standard normal Cumulative Distribution Function
(CDF) ( )Φ • ( )
G
F • CDF of the performance function G(X) s
β Safety reliability index
,FORM
s
β First order approximation of safety reliability index βs t
β Target reliability index
( )
G X Performance function; the design is considered “fail” if G(X) < 0
p
G Probabilistic performance measure
* ( ) 0
GU=
u Most Probable Failure Point (MPFP) in first-order
reliability analysis *
t
β β=
u Most Probable Point (MPP) in first-order inverse
reliability analysis *
(A)MV
u MPP using (advanced) mean value method in PMA
* CMV
u MPP using conjugate mean value method in PMA
* HMV
u MPP using hybrid mean value method in PMA
n Normalized steepest descent direction of performance
function abs, rel
G G
∆ ∆ Absolute and relative changes in performance measure ς Criteria for the type of performance function
( )
L X Crack initiation fatigue life t
L Target crack initiation fatigue life
INTRODUCTION
A commonly used design optimization methodology for engineering systems comprises deterministic modeling and simulation-based design optimization. However, the existence of uncertainties in physical quantities such as manufacturing tolerances, material properties, and loads requires a reliability-based approach to
[ i] (T 1, 2, , ) design optimization [1,2]. Given the increased computational
capabilities developed during the last few years, fundamental issues relating to the inclusion of quantitative estimation of uncertainty have been recently addressed. Techniques have been explored which incorporate uncertainty during design optimization at an affordable computational cost.
There has been a recent development in the Reliability-Based Design Optimization (RBDO) incorporating probabilistic constraints that can be evaluated using two different approaches, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA) [3,4]. The evaluation of a probabilistic constraint in the RBDO model is an essential step and thus the probabilistic constraint in the RBDO model must be computationally stable and affordable so that the RBDO process can be effective. It has been shown that PMA is equivalent to RIA in prescribing the probabilistic constraint [3]. However, these approaches are not equivalent in computational robustness in evaluating probabilistic constraints in the RBDO process. That is, RIA may demonstrate instability whereas PMA is stable in evaluating a probabilistic constraint [3]. However, several examples of numerical tests of the PMA show inefficiency and instability in the assessment of a probabilistic constraint during the RBDO process as the result of an ineffective numerical method, i.e., the Advanced Mean Value (AMV) method [5,6]. In general, the AMV method exhibits divergence or slow rate of convergence in addressing a concave performance function, although it is good for a convex performance function.
With respect to a concave performance function, numerical instability as well as inefficiency in PMA using the AMV method highlights the need for a stable and efficient computational algorithm that utilizes a conjugate direction, namely, the Conjugate Mean Value (CMV) method. However, the CMV method is computationally more expensive than the AMV method for a convex performance function. Consequently, the Hybrid Mean Value (HMV) method is proposed in this paper to adaptively select either the AMV method or the CMV method once the performance function type is identified.
It has been noted in Refs. 3 and 4 that the efficiency of RIA and PMA to assess the probabilistic constraint depends on activeness of the probabilistic constraint. The previous research, however, has not been dealt with the HMV method proposed in this paper. Hence, a comparative study between RIA and PMA from an efficiency and robustness perspective, with respect to probabilistic constraint evaluation in the RBDO process, is presented in this paper. It is shown that the conventional reliability analysis model in RIA causes ineffectiveness in the RBDO process, while the inverse reliability analysis model in PMA provides an efficient and robust RBDO process using the proposed HMV method.
Popular numerical methods for RIA are the HL-RF method [7,8], Modified HL-RF [8], and Two-Point Approximation (TPA) [9,10]. For PMA, the AMV [5,6] is a popular numerical method. In this paper, the proposed HMV method will is used to show efficiency and robustness in probabilistic constraint assessment for PMA.
GENERAL DEFINITION OF RBDO MODEL
In the system parameter design, the RBDO model [11-14] can be generally defined as Minimize Cost( ) subject to ( ( ) 0) ( ) 0, 1, 2, , , i t L U n P G i np R β ≤ − Φ − ≤ = ≤ ≤ ∈ d X d d d d (1)
where the cost can be any function of the design vector ,
[diT] ( )
= =
d µ X X= X i= n is the random vector, and
the probabilistic constraints are described by the performance function Gi subject to uncertainty X, their probabilistic models, and
their prescribed confidence level βt.
The statistical description of the failure of the performance function Gi(X) is characterized by the Cumulative Distribution
Function (CDF) (0) as i G F ( ( ) 0) (0) ( ) i i G P G X ≤ =F ≤ Φ −βt (2)
where the CDF is described as 1 ( ) 0 (0) ( ) , 1, 2, , i i G G n F =
∫
≤∫
fX dx dx i np X x … = (3)In Eq. (3) is the Joint Probability Density Function (JPDF) of all random parameters. The evaluation of Eq. (3) requires reliability analysis where the multiple integration is involved as shown in Eq. (3). Some approximate probability integration methods have been developed to provide efficient solutions [1], such as the First-Order Reliability Method (FORM) or the asymptotic Second-Order Reliability Method (SORM) with a rotationally invariant measure as the reliability [1,2]. The FORM often provides adequate accuracy [1,2] and is widely used for RBDO applications. In FORM, reliability analysis requires a transformation T [15,16] from the original random parameter X to the independent and standard normal random parameter U. The performance function in X-space can then be mapped onto G(T(X)) ≡ G(U) in U-space.
( ) fX x
( ) G X
As described in Section 1, the probabilistic constraint in Eq. (2) can be further expressed in two different ways through inverse transformations [3] as: 1 ( ( (0))) i i s FG t β = −Φ− ≥β (4) 1 ( ( )) 0 i i p G t G =F− Φ −β ≥ (5) where i s
β and are respectively called the safety reliability index and the probabilistic performance measure for the i
i
p
G
th
probabilistic constraint. 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).
First-Order Reliability Analysis in RIA
In RIA, the first-order safety reliability index βs,FORM is obtained using the FORM by formulating as an optimization problem with one equality constraint in U-space, which is defined as a limit state function:
minimize
subject to G( )=0
U
U (6)
where the optimum point on the failure surface is called the Most
Probable Failure Point (MPFP) u*G( ) 0U= and thus
* ,FORM ( ) 0
s G
β = u U = .
Either MPFP search algorithms specifically developed for the first-order reliability analysis or general optimization algorithms [17] 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.
First-Order Reliability Analysis in PMA
Reliability analysis in PMA can be formulated as the inverse of reliability analysis in RIA. The first-order probabilistic performance
measure is obtained from a nonlinear optimization
problem [3] in U-space defined as ,FORM p G minimize ( ) subject to t G β = U U (7)
where the optimum point on a target reliability surface is identified as the Most Probable Point (MPP) *
t
β β=
u with a prescribed reliability *
t
t β β
β = u = , which will be called MPP in the paper. Unlike RIA, only the direction vector * *
t t
β β= β β=
u u needs to be determined by
exploring the spherical equality constraint U =βt.
General optimization algorithms can be employed to solve the optimization problem in Eq. (7). However, the AMV method is well suited for PMA [5,6] due to its simplicity and efficiency.
HYBRID RELIABILITY ANALYSIS METHOD FOR PMA
It was found that, although the Advanced Mean Value (AMV) method behaves well for a convex performance function, it exhibits numerical shortcomings, such as slow convergence or even divergence, when applied to a concave performance function. To overcome these difficulties, the Conjugate Mean Value (CMV) method is proposed in this paper. However, even though the CMV method always converges, it is inefficient for the convex function. Consequently, the Hybrid Mean Value (HMV) method is proposed in this paper to attain both stability and efficiency in the MPP search algorithm in PMA.
Advanced Mean Value (AMV) Method
Formulation of the first-order AMV method begins with the Mean Value (MV) method, defined as
* MV ( ) ( ) ( ) where ( ) ( ) ( ) X U t X U G G G µ β µ ∇ ∇ = = − = − ∇ ∇ 0 u n 0 n 0 0 G ) k (8) That is, to minimize the performance function (i.e., the cost function in Eq. (7)), the normalized steepest descent direction is defined at the mean value. The AMV method iteratively updates the direction vector of the steepest descent method at the probable point initially obtained using the MV method. Thus, the AMV method can be formulated as
( ) G U ( ) n 0 ( ) AMV k u (1) * ( 1) ( ) MV
AMV , AMV ( AMV
k t β + = = u u u n u (9) where ( ) ( ) AMV AMV ( ) AMV ( ( ) ( ) k U k k U G G ∇ = − ∇ u n u u ) (10) As will be shown, this method exhibits instability and inefficiency in
solving a concave function since this method updates the direction using only the current MPP.
Conjugate Mean Value (CMV) Method
When applied for a concave function, the AMV method tends to be slow in the rate of convergence and/or divergent due to a lack of updated information during the iterative reliability analysis. These
kinds of difficulties can be overcome by using both the current and previous MPP information as applied in the proposed Conjugate Mean Value (CMV) method. The new search direction is obtained by
combining , , and with an equal
weight, such that it is directed towards the diagonal of the three consecutive steepest descent directions. That is,
( 2) CMV ( k− ) n u n u( (CMVk−1)) n u( ( )CMVk ) (0) (1) (1) (2) (2) CMV CMV AMV CMV AMV ( ) ( 1) ( 2) ( 1) CMV CMV CMV CMV ( ) ( 1) ( 2) CMV CMV CMV , , , ( ) ( ) ( ) for 2 ( ) ( ) ( ) k k k k t k k k k β − − + − − = = = + + = ≥ + + u 0 u u u u n u n u n u u n u n u n u (11) where ( ) ( ) CMV CMV ( ) CMV ( ) ( ) ( ) k U k k U G G ∇ = − ∇ u n u u . (12) Consequently, the conjugate steepest descent direction
significantly improves the rate of convergence, as well as the stability, compared to the AMV method for the concave performance function. However, as will be seen in the next section, the proposed CMV method is inefficient for the convex function.
Example 1: Convex Performance Function A convex function is given as
1 2
( ) exp( 7) 10
G X = − X − −X + (13)
whereX represents the independent random variables with ~ (6.0, 0.8), 1, 2
i
X N i= and the target reliability index is set to
3.0 t
β = .
As shown in Fig. 1, the constraint in Eq. (7) is always satisfied and the performance function around the MPP is convex with respect to the origin of U-space. The AMV method demonstrates good convergence behavior for the convex function since the steepest descent direction of the response gradually approaches to the MPP, as shown in Fig. 1(a). In Table 1, the convergence rate of the AMV method is faster than that of the CMV method for the convex function because the conjugate steepest descent direction tends to reduce the rate of convergence for the convex function. Thus, for the convex performance function, the AMV method performs better than the CMV method.
( ) AMV ( k
n u )
Table 1. MPP History for Convex Performance Function
AMV CMV Iter. X1 X2 G X1 X2 G 1 6.829 8.252 0.905 6.829 8.252 0.905 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 4 8.272 6.774 -0.341 8.043 7.260 -0.097 5 8.311 6.648 -0.357 8.165 7.035 -0.242 6 8.317 6.625 -0.358 8.234 6.877 -0.312 7 8.272 6.775 -0.341 … … 11 8.310 6.651 -0.357 12 8.317 6.625 -0.358 Converged Converged
( a ) AMV Method
( b ) CMV Method
Figure 1. MPP Search for Convex Performance Function Example 2: Concave Performance Function 1
Consider the concave performance function
1 2
( ) [exp(0.8 1.2) exp(0.7 0.6) 5]/10
G X = X − + X − − (14)
where represents an independent random vector with
and and the target reliability
index is set to X 1~ (4.0, 0.8) X N X2~N(5.0, 0.8) 3.0 t β = .
As shown in Fig. 2, the performance function around the MPP is concave with respect to the origin of U-space. The AMV method applied to the concave response diverges as a result of the oscillation observed in Fig. 2(a). As shown in Table 2, after 34th iteration, oscillation occurs in first-order reliability analysis due to the cyclic behavior of the steepest descent directions, i.e.,
and . This example
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 concave function by using the conjugate steepest direction.
( ) ( 2)
AMV AMV
( k )= ( k−
n u n u ) n u( (AMVk+1))=n u( (AMVk−1))
Example 3: Concave Performance Function 2
A different situation using another concave function is presented 2
1 2 2 1
( ) 0.3 0.8 1
G X = X X −X + X + (15)
where represents the independent random variables with
and and the target reliability
of X 1~ (1.3, 0.55) X N X2~N(1.0, 0.55) 3.0 t β = is used. ( a ) AMV Method ( b ) CMV Method
Figure 2. MPP Search for Concave Performance Function 1 Table 2. MPP History for Concave Performance Function 1
AMV CMV Iter. X1 X2 G X1 X2 G 1 2.989 2.823 0.225 2.989 2.823 0.225 2 2.348 3.259 0.234 2.348 3.259 0.234 3 3.073 2.786 0.238 2.687 2.990 0.204 4 2.268 3.338 0.253 2.680 2.996 0.204 5 3.162 2.751 0.255 6 2.190 3.424 0.277 … … 34 1.981 3.703 0.380 35 3.464 2.661 0.335 … … 999 1.981 3.703 0.380 1000 3.464 2.661 0.335 Diverged Converged
Although the AMV method has converged in this case, it requires substantially more iterations than the CMV method. Similar to Example 2, the slow rate of convergence is the result of oscillating behavior of reliability iterations when using the AMV method.
Based on the previous examples, it can be concluded that the AMV method either diverges or performs poorly compared to the
CMV method, for the concave performance function. Thus, a desirable approach is to select either the AMV or CMV methods once the type of performance function has been determined to achieve the most efficient and robust evaluation of probabilistic constraint, as discussed in the following section.
( a ) AMV Method
( b ) CMV Method
Figure 3. MPP Search for Concave Performance Function 2 Table 3. MPP History for Concave Performance Function 2
AMV CMV Iter. X1 X2 G X1 X2 G 1 -0.275 1.491 -0.678 -0.275 1.491 -0.678 2 0.487 2.436 -0.873 0.487 2.436 -0.873 3 -0.105 1.864 -0.997 0.016 2.036 -1.023 4 0.368 2.362 -0.959 0.232 2.257 -1.036 5 -0.035 1.969 -1.000 0.119 2.152 -1.048 6 0.303 2.315 -1.009 0.174 2.206 -1.047 7 0.009 2.028 -1.020 0.146 2.180 -1.048 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 10 0.230 2.256 -1.036 0.157 2.190 -1.048 11 0.064 2.094 -1.039 0.155 2.188 -1.048 … … 23 0.124 2.158 -1.048 24 0.155 2.188 -1.048 Converged Converged
Hybrid Mean Value (HMV) Method
To select an appropriate MPP search method, the type of performance function must be first identified. In this paper, the function type criteria is proposed by employing the steepest descent directions at the three consecutive iterations as follows
( 1) ( 1) ( ) ( ) ( 1) ( 1) ( 1) HMV ( 1) HMV ( ) ( )
sign( ) 0 : Convex type at w.r.t. design 0 : Concave type at w.r.t. design
k k k k k k k k ς ς + + − + + + = − ⋅ − > ≤ n n n n u d u d (16)
where ς(k+1) is the criterion for the performance function type at the k+1th step and is the steepest descent direction for a performance function at the MPP at the k
( )k
n
( ) HMV
k
u th 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: 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
(0) (0) HMV HMV (0) HMV (0) HMV ( ) ( ) ( )
where (origin in -space)
U U G G U ∇ = − ∇ = u n u u u 0
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
( 1) ( )
HMVk+ =βt ( HMVk )
u n u
If the performance function is concave and , compute the MPP using the CMV method as
3 k≥ ( ) ( 1) ( 2) ( 1) HMV HMV HMV HMV ( ) ( 1) ( 2) HMV HMV HMV ( ) ( ) ( ( ) ( ) ( k k k k t k k k β − − + − − + + = + + n u n u n u u n u n u n u ) ) where ( ) ( ) HMV HMV ( ) HMV ( ) ( ) ( ) k k U k U G G ∇ = − ∇ u n u u
Step 3. Calculate the performance and the reliability index ( 1) HMV ( k Gu + ) (k 1)
β + at the new MPP u(HMVk+1). Check to see if
(
( 1) (k 1) (k 1))
rel abs max β k+ −βt , ∆G + , ∆G + ≤ε where ( 1) ( ) ( 1) HMV HMV rel ( 1) HMV ( ) ( ( ) k k k k G G G G + + + − ∆ = u u u ) and ( 1) ( 1) ( ) HMV HMV abs ( ) ( ) k k k G + G + G ∆ = u − uIf the convergence criteria hold, then stop. Otherwise, go to Step 4.
Step 4. Compute the gradient of the performance
function and check the criteria ( 1) HMV ( k UG + ∇ u ) (k 1) ς + for performance function type. Set k= +k 1 and return to Step 2.
Example 4: Reliability Analysis of Analytical Examples
The numerical algorithm proposed in Section 3.3 was applied to the previous three examples. For the first example, the proposed numerical algorithm identifies ς( )k as positive, hence the AMV 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 in Section 3.3, the HMV method performed quite well for any type of performance function.
Example 5: Reliability Analysis of Durability Model
A roadarm from a military tracked vehicle shown in Fig. 4 is employed to demonstrate the effectiveness of the HMV method for a large-scale problem. Reliability analysis for this example involves the crack initiation fatigue life performance measure. A 17-body dynamics model is created to drive the tracked vehicle on the Aberdeen Proving Ground 4 (APG4) at a constant speed of 20 miles per hour forward (positive X2) [13,18]. A 20-second dynamic
simulation is performed with a maximum integration time step of 0.05-second using the dynamic analysis package DADS [19].
Three hundred and ten 20-node isoparametric finite elements, STIF95, and four beam elements, STIF4, of ANSYS are used for the 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 4. Military Tracked Vehicle
Intersection 1 Intersection 2 Intersection 3 Intersection 4 20 in. 1 x' 3 x' 2 x' 2 x'
( a ) Geometry of Roadarm Model
1236 12 Torsion Bar Center of the Roadwheel 1 x' 3 x' 2 x' Intersection 1 b1, b2 Intersection 2 b3, b4 Intersection 3 b5, b6 Intersection 4 b7, b8
( b ) Finite Element Model of Roadarm Model
Figure 5. Geometry and Finite Element Model for Roadarm Model Table 4. Critical Nodes for Crack Initiation Fatigue Life
Node ID Life [Load Cycle] Life [Year]
885 .9998E+07 6.34 889 .1134E+08 7.19 990 .5618E+08 35.63 994 .6204E+08 39.35
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 straight lines and four cubic curves, as shown in Fig. 7. Side variations (x1′-direction) of the cross sectional shapes are defined as the random parameters b1, b3, b5, and b7 for intersections 1 to 4, respectively, and vertical variations (x3′-direction) of the cross sectional shapes are defined using the remaining four random variables.
Roadarm
For reliability analysis, a failure function is defined as ( ) ( ) 1 t L G L = X − X (17)
where is the number of service blocks to initiate crack at node 885 and
( ) L X
t
L is the number of target service blocks to initiate 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 constitutes the shortest life of the component. The target crack
initiation fatigue life is set as 0.1 years (i.e., 1.577E+5 cycles) to illustrate the concave performance function.
Table 5. Definition of Random Variables for Crack Initiation Fatigue Life Prediction
Random
Variables Mean Value Std. Dev.
Distribution 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 1 x' Design Parameters: b1, b3, b5, b7 Design Parameters: b2, b4, b6, b8 Cross Sectional Shape
Straight Lines Cubic Curves 3 x' bi, i = 1,3,5,7 bi, i = 2,4,6,8
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
with a target reliability index
[1.872, 3.093, 1.708, 2.830, 2.218, 2.755, 4.758, 2.836]T
=
x
3.325 t
β = , 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
AMV HMV Iter. G(X) β G(X) β ς 0 62.404 0.0 62.404 0.0 N.A. 1 0.014 3.325 0.014 3.325 N.A. 2 0.004 3.325 0.004 3.325 N.A. 3 -0.001 3.325 0.001 3.325 -0.0038 4 0.002 3.325 0.000 3.325 -0.0042 5 -0.001 3.325 6 0.002 3.325 7 -0.001 3.325 … … … 19 -0.001 3.325 20 0.002 3.325 Diverged Converged
RIA VS. PMA IN RELIABILITY ANALYSIS
It has been reported [3] that the size of the search space in a reliability analysis could affect the efficiency of the MP(F)P search. However, based on numerical examples in this paper, it has been found that sizes of the MP(F)P search spaces may not be crucial to
the efficiency in reliability analysis. Rather, it is found that PMA with the spherical equality constraint is easier to solve than RIA with a complicated constraint. In other words, it is easier to minimize a complex cost function subject to a simple constraint function than to minimize a simple cost function subject to a complicate constraint function.
( a ) MPFP Search Space in RIA
( b ) MPP Search Space in PMA Figure 8. MP(F)P Search Spaces
Figure 8(a) illustrates the MPFP search space in RIA over the design parameter space, where the first-order safety reliability
indices in Eq. (4) are ,FORM ( )
j k s j β =β d ( * 0 j G = = T x ) t , j=1, 2. Reliability analysis in RIA is carried out by determining the minimum distance between the mean value design point and MPFP
on the failure surface . The MPP search space in
PMA is illustrated in Fig. 8(b), where the probabilistic performance measures in Eq. (5) are
( ) 0, j G X = j=1, 2 * , ( ) j j p FORM j G =G xβ β= , j = 1, 2.
Reliability analysis in PMA is performed by determining the minimum performance value on the explicit sphere of the target reliability ( )βj d =βt, 1, 2j= .
Comparing Figs. 8(a) and (b), the MPP search space in PMA is smaller than the MPFP search space in RIA if the constraint at the mean value design point is largely inactive or largely violated with the large negative reliability index, such as the first probabilistic constraint. Thus, the MPP search in PMA, with the easier
optimization problem in Eq. (7), might be better than RIA in terms of efficiency and robustness. On the other hand, the MPFP search space in RIA is smaller if the constraint at the mean value design point is near actives or lightly violated, such as the second probabilistic constraint in Fig. 8. In this case, although RIA has a smaller MPFP search space, the optimization problem in Eq. (6) is not easier to solve than that of PMA. As a result, a comparison of the efficiency in RIA and PMA is not clear regarding efficiency and, as such, will be examined closely in this section. In this study, the HL-RF method is used for RIA and both the proposed HMV and conventional AMV methods are used for PMA.
For RIA vs. PMA in the reliability analysis, the roadarm durability model used in Example 5 will be demonstrated in Example 6 and 7.
Example 6: RIA with a Larger Search Space than PMA (Such as G 1 in Fig. 8)
In Table 7, a target crack initiation fatigue life of Lt=300 year is specified so that the MPFP search space in RIA becomes larger than the MPP search space in PMA on the infeasible region, as represented by the largely violate deterministic constraint at the mean value design point. At the second iteration, RIA has diverged - the life at the first MPFP becomes infinite (1.0E+20 load cycle or 6.34E+13 years) and all design sensitivities become zero, which lead to failure of RIA. In contrast, PMA does not have numerical difficulty in reliability analysis within the prescribed MPP search space. This example shows that PMA using the HMV method is better than RIA in terms of stability.
Table 7. Reliability Analysis for Lt=300 year
RIA (HL-RF) PMA (AMV, HMV)
Iteration G β G β G β 0 -0.9789 0.0 -0.979 0.0 -0.979 0.0 1 2.11E+11 -35.46 -0.012 -3.182 -0.999 3.0 2 0.000 -3.182 -0.999 3.0 3 0.000 -3.182
Diverged Converged Converged
Example 7: RIA with a Smaller Search Space than PMA (Such as G 2 in Fig. 8)
In Table 8, the target crack initiation fatigue life Lt is specified 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 the slightly violate deterministic constraint at the mean value design point. In this case, RIA searches for the MPFP in the smaller search space than PMA with βt=3.0. However, PMA is more efficient than RIA, since the PMA optimization problem in Eq. (7) is easier to solve. Note that the HMV method demonstrates superiority over the AMV method, which has diverged, as shown in Table 8.
Based on the examples presented in this section, it can be concluded that PMA is superior to RIA, regardless of sizes of the MP(F)P search spaces. Consequently, it is recommended to use PMA with the spherical equality constraint in reliability analysis and not RIA with the complicate constraint for all cases.
RBDO USING PMA WITH HMV METHOD
As described in Section 2.1, the probabilistic constraints in the RBDO model can be evaluated by two different reliability analyses: RIA and PMA. Based on the results of previous sections in
reliability analysis, the comparative study between the conventional RIA and the enhanced PMA with the HMV method is extended to RBDO of a bracket problem in this section.
Table 8. Reliability Analysis for Lt=10 years
RIA (HL-RF) PMA (AMV) PMA (HMV)
Iter. G β G β G β G β 0 -.366 0.0 -.366 .0 -.366 0.0 -.366 0.0 1 .115 -.442 -.002 -.356 -.001 -.356 -.985 3.0 2 .010 -.364 .001 -.356 .001 -.356 -.985 3.0 3 -.001 -.356 -.001 -.356 .000 -.356 4 -.001 -.356 .001 -.356 5 .000 -.356 -.001 -.356 6 .001 -.356 … … … 19 -.001 -.356 20 .001 -.356
Converged Diverged Converged Converged
Example 8: Bracket Problem in RBDO Model
Figure 9 shows design parameterization and stress analysis result of a bracket at the initial design. A total of 12 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, 214 elements, and 552 DOF with the thickness of 1.0 cm. The boundary condition is imposed to fix two lower holes. Using FEM, stress analysis required 18.23 sec., while DSA required 35.44/12=2.95 sec. per design variable. The bracket is made of steel 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 von Mises stress as shown in Fig. 9(b). Random parameters are defined in Table 9 and SQP optimizer is used with a target reliability index of β=3.0 in the RBDO model.
Gp1,Gp2
d9
( a ) Design Parameterization ( b ) Stress Contour at Initial Design Figure 9. Initial Bracket Design
Figure 10 shows several design iterations throughout the RBDO process. At the optimum design, the overall area is substantially reduced at the inner boundary and slightly at the outer boundary. Figure 11 (a) shows the stress contour at the MPP of the initial design where all probabilistic constraints are largely inactive. Figure 11 (b) shows the stress contour at the MPP of the optimum design.
d8 Gp3, Gp4 15000 N d7 d 6 d12 d5 d2 d 4 d1 d10 d11 d3
Table 9. Random Variables in Bracket Model Random Variable Lower Design Bound Mean Value (Design) Upper Design Bound Std. Dev. Distrib. Type 1 0.800 1.006 3.000 0.2 Normal 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 ) Initial Design ( b ) 1ST RBDO Iteration
( c ) 4TH RBDO Iteration ( d ) 7TH RBDO Iteration
( e ) Optimum Design
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% 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 terms of computational efficiency and robustness. As in the roadarm
model, the RIA fails to converge in reliability analysis, whereas PMA successfully obtains an optimal design for the bracket model. In addition, PMA with the HMV method performs better than with the conventional AMV method in terms of numerical efficiency (195 analyses vs. 295 analyses).
( a ) Stress At Initial Design ( b ) Stress At Optimum Design Figure 11. Analysis Results Comparison
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Cost ( a ) Volume History -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Gp1 Gp2 Gp3 Gp4
( b ) Probabilistic Constraint History
-3 -2 -1 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12
( c ) Design Parameter History
Table 10. Computational Efficiency and Robustness in RIA and PMA PMA RIA HMV AMV HL-RF Opt. Iter. Line Search Anal ysis Line Search Anal ysis Line Search Anal ysis 0 1 5 1 5 1 40 1 1 7 1 7 3 120 2 2 14 2 14 1 N.A. 3 1 5 1 5 4 3 25 3 46 5 2 16 2 16 6 1 10 1 12 7 1 10 1 16 8 1 9 1 17 9 1 10 1 8 10 1 14 1 8 11 1 16 1 9 12 1 9 1 18 13 2 25 1 19 14 1 10 1 19 15 1 10 2 25 16 3 51 Opti mum 21 195 24 295 Failure to Converge CONCLUSIONS
Advances in the RBDO are made by developing the HMV method for the PMA in this paper. It has been shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicate constraint in reliability analysis. However, it has been found that the conventional MPP search algorithm, the AMV method, exhibits numerical instability and inefficiency for the concave performance function. Therefore, the HMV method is proposed for effective evaluation of probabilistic constraints in the RBDO process in order to take advantages of PMA. Based on numerical efficiency and robustness in reliability analysis, the HMV method is very effective numerical tool for estimating probabilistic constraints in the RBDO process. The comparison study between RIA and PMA has been extended to the RBDO problem, demonstrating that the PMA using HMV method provides the best result in the RBDO process.
ACKNOWLEDGMENTS
Research is partially supported by the Automotive Research Center sponsored by the U.S. Army TARDEC.
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