A99-24608 AIAA-99-1210

VARIABLE COMPLEXITY STRUCTURAL RELIABILITY ANALYSIS FOR EFFICIENT RELIABILITY- BASED DESIGN OPTIMIZATION

Patrick N. Koch’ and Srinivas Kodiyalam+ Engineous Software Inc.

1800 Perimeter Park West, Suite 275 Morrisville. North Carolina 27560. USA

ABSTRACT The computational expense of structural reliability analysis, in terms of the number of function evaluations necessary to accurately estimate the probability of failure, has made reliability-based structural optimization impractical for realistic structural design problems. More efficient reliability analysis methods often lead to a sacrifice in accuracy and unacceptable solutions. In this paper we present and demonstrate a variable complexity approach to structural reliability analysis for more efficient reliability-based design optimization. In this approach, the desirable characteristics of both the traditional first order reliability methods (accuracy) and mean value first order methods (efficiency) are exploited to increase the efficiency of identifying reliable su-uctural designs through reliability-based design optimization. Three example problems are presented, the design of a three- bar truss, the classical tension-compression spring design problem, and the design of a composite tube under internal pressure, modeled using MSCfNastran. Results for these problems show a 40-70% reduction in the number of function evaluations when using the variable complexity approach as compared to using the traditional fust order reliability method within reliability-based design optimization.

1 INTRODUCTION Methods for structural reliability analysis have been developed in recent years to incorporate uncertainties associated with geometrical and material properties, loading and boundary conditions, and operational environment into structural analysis and design.“5 These uncertainties are incorporated through the definition of random variables and probabilistic distribution functions, to assess the probability of failure of a structural component or system. Design variables can then be adjusted to reduce the probability of failure (and thus increase the reliability) to an acceptable level.

* Member AIAA. koch@engineous.com + Senior Member AIAA. srinivas@engineous.com

Copyright Q 1999 by Engineous So2ware Inc. Published by the Institute of Aeronautics and Astronautics. Inc. with permission. 66

Although reliability analysis methods can be significantly more effective than the traditional -safety factor methods, which often lead to conservative, over- designed structures, reliability methods require significantly larger computational effort and analysis time, particularly for large, complex problems employing intensive computer analyses. Monte Carlo based methods, while straightforward to implement and highly accurate, require on the order of thousands of analyses or failure function evaluations to assess the probability of failure. Due to this extreme computational expense, Thoft-Christensen and Baker’ recommend that “as a general rule.. .Monte-Carlo methods should be avoided it at all possible.” The mean value first-order, second moment (MVFO) method of reliability analysis’ is much more efficient, requiring onIy one time failure function and sensitivity evaluations (number of analyses necessary is thus a function of the number of random variables). However MVFO is suited only for linear failure functions and normally distributed random variables, and experience has been shown MVFO to be inaccurate.4

More recently first order reliability methods (FORM) have been developed based on a reliability index defined by Hasofer and Lind’ to identify the “most probable (failure) point”. While FORM is more accurate than MVFO, FORM is significantly less efficient, requiring the solution of an optimization problem to &am the reliabilit index and evaluate the probability of failure. Thus the number of problem- specific analyses for a structural design problem with FORM is not only a function of the number of random variables, but also a function of the number of potential failure modes being evaluated and the number of iterations necessary for convergence. Efforts have been made to increase the efficiency of FORM through use of approximations,8,9 which improve convergence and perform better for nonlinear failure functions, but these approaches still generally require three or more times more failure function evaluations than MVFO.

The efficiency of reliability analysis methods Jecomes even more crucial when these methods are

couple with techniques for structural optimization. Reliability-based design (=D) optimization

techniques are implemented by augmenting a deterministic optimization model through the defmition and incorporation of probabilistic constraints. A formulation for probabilistic constraints using the Hasofer-Lind fust order, second moment criteria’ is presented in Ref. 10. These probabilistic constraints seek to ensure the achievement of a desired level of reliability, or conversely a specified acceptable failure probability. During each RBD optimization iteration, however, in addition to the analyses necessary for optimization, reliability analyses are necessary to evaluate the probability of failure associated with each potential failure mode. Thus a computationally expensive deterministic optimization problem can become unmanageable when converted into a RBD optimization problem.

Due to its low computational expense, MVFO was initially used for reliability analysis within RBD optimization. ‘“y” Because of the inaccuracies of MVFO, however, other authors have implemented FORM for RBD optimization.9~‘2*‘3 To reduce the number of reliability analyses necessary during structural optimization, Choi et al. presented a mixed design approach in which a deterministic optimization is first performed, followed by a FORM analysis of the deterministic optimum, and FORM RBD optimization if necessary to improve the reliability of the structure.‘3 Even when starting from a deterministic solution, however, RBD optimization with FORM remains computationally expensive; for the example presented in Ref. 13, for only two RBD iterations ten FORM analyses are performed, requiring 120 cpu hours.

Given the problem of computational expensive of RBD optimization, the focus in this paper is on increasing the efficiency of RBD optimization by exploiting the desirable properties of both FORM (accuracy) and MVFO (efficiency) reliability analyses. A review of both of these methods is provided in the next section. A variable complexity reliability-based design optimization approach is then presented in Section 3. In Section 4, the details and results of three example problems are presented. A discussion of results and current investigations is provided in closing in Section 5.

2 REVIEW OF STRUCTURAL RELIABILITY ANALYSIS METHODS

The goal in structural reliability analysis is to assess the probability of failure (and conversely the reliability) of a structural component or system, given performance variation caused by random (uncertain) variables. The probability of failure for a structural reliability problem can be expressed as:

p, = JJ...Jfx(X,,XZ Y..., x&x [II

where X = [Xl,,X2, . . . . XnIT is the vector of random variables and n is the number of random variables, g(X) is the failure function, and fx(X1, X2, . . ., Xn) is the corresponding joint probability density function of the random variables X. The failure surface, g(X) = 0, separates the domain into a safe region, g(X)>O, and a failure region, g(X)<0 (see Figure 1). However, the failure surface is often an implicit function of the random vector X, and even if the form of the function is know-n, the multi-dimension numerical integration of Eqn. 1 over the failure region is extremely difficult and computationally expensive. To overcome these difficulties, various reliability methods have been proposed. Summaries of two such methods, FORM and MVFO, are given in this section.

2.1 First-Order Reliability Method (FORM) The fust-order reliability method (FORM) is shown pictorially in Figure 1. Specific details for FORM can be found in Refs. 4 and 5; an overview of this method is given here.

Figure 1 First-Order Reliability Method47579

FORM takes advantage of the desirable properties of the standard normal probability distribution. Hasofer and Lind’ defined the reliability index as the shortest distance from the origin of the standard normal space (U-space) to a point on the failure surface. Mathematically, determining the reliability index is a -minimization problem with one equality constraint:

p = rn$lUl

s.t. g(X) = g(T-I(U)) = g(U) = 0 PI

where a transformation T is introduced to map the original random vector X (in X-space) to the standard,

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uncorrelated normal vector -using U=T(X). The solution of the minimization problem in U-space, U*, is called the Most Probable Point (MPP). If the failure function g(U) is linear in terms of the normally disuibuted random variables Ui, the failure probability is calculated as:

p, = W-P) [31 where @ is the standard normal distribution function. If the failure function is nonlinear or the random variables are not normally distributed, a good approximation can still be obtained using Eqn. 3, provided that the principal curvatures of the failure surface at the MPP are not too large in magnitude.

The algorithm traditionally used to solve for the MPP is the Hasofer-Lind-Rackwitz-Fiessler @IL-RF’) method, a special scheme which does not require line search during optimization. The HL-RF method was originally derived from the Kuhn-Tucker necessary condition by Hasofer and Lind.’ Rackwitz and Fiessler’ suggested that non-normally distributed random variables can be transformed to equivalent normal variables for use in the Hasofer-Lind analysis.

The HL-RF method requires the fust-order sensitivity of failure functions with respect to random variables. Numerical results show that the HL-RF method generally converges rapidly. However, convergence is not guaranteed for all circumstances. When the curvature of the failure surface becomes large, the HL-RF method requires a large number of iterations to converge and can fail to converge.

2.2 Mean Value, First Order (MVFO) Method The mean value, first order (MVFO) reliability ana& method5,7.%‘o.1 1 utilizes the Taylor’s series expansion of failure functions g(X) at the mean values of the random variables, l-r x ,:

The mean and standard deviation of g(X) are then calculated as follows:

Pg = d&x> [51

[61

where B, is the standard deviation of X. The mean- value reliability index can then be calculated as follows:

P= g Px II

=clp c( 1 n ag - 2(oxi)2

=g [71

i=l axi

Given the reliability index from Eqn. [i’], the probability of failure can then again be calculated as Pf =a(-PI (Eqn. [31).

The major advantage of the MVFO method is efficiency; it requires only one time failure function and sensitivity evaluations. The MVFO reliability index, however, is only accurate for linear failure functions with normally distributed random variables. In most other situations, the most probable point is not on the failure surface and thus the mean-value reliability index is not accurate.

2.3 Reliability-Based Design Optimization Each of the reliability analysis methods reviewed % this section can be incorporated within an optimization formulation to facilitate reliability-based design (RBD) optimization. A deterministic optimization problem is converted to a RBD optimization problem by converting the necessary deterministic constraints (failure or limit state functions) to probabilistic constraints. The probabilistic RBD optimization problem is then formulated as:

Find the set of design variables. X (deterministic, d and random, r)

that Minimizes or Maximizes. F(X)

Subject to. P,~P(gj(X)~O)SPju,j= l,...,n X’IXIX

where, F(X) is the objective function, P, is the probability of failure for the limit state function gj, and n is the number of probabihstic constraints. Pi” is the required upper bound for the probability of failure of the jth constraint. Alternately, the reliability for each limit state function can be calculated as R, =l- Pa, where R, is referred to as a reliability constraint; the constraints are then set as R, 2 R,“, where I$” is the required lower bound on the reliability of the jth constraint. Upper and lower bounds are specified on the design variables.

During the RBD optimization, the reliability analysis (using FORM or MVFO, for example) is performed each time a function evaluation (design analysis) is necessary -- for the design point analysis as well as for sensitivity analyses (for each gradient analysis). The reliability analysis in turn requires

multiple executions of the design analysis. As a result, the RRD optimization is very much computationally expensive for problems that involve costly simulations.

3 VARIABLE COMPLEXITY RELIABILITY- BASED DESIGN OPTIMIZATION

A variable complexity approach is presented and demonstrated in this paper for increased efficiency in RBD optimization. This approach, termed variable complexi~ reliability-based design (VCRBD), is illustrated in Figure 2. Given a starting point or initial design for optimization, a deterministic optimization is first performed; thus the mixed design approach proposed in Ref. 13 is implemented here within the VCRBD approach. An identified deterministic optimum is likely not a sufficiently reliable solution since one or more constraints are likely active. In the absence of a good baseline design or previous solution, however, the deterministic optimization will push the design towards feasible, desirable solutions; performing a deterministic optimization first will prevent unnecessary and expensive reliability analyses for infeasible and otherwise undesirable design points. The focus with the following RBD optimization, then, is to move the deterministic solution away from the active constraints to achieve a reliable structural design.

After the deterministic optimization, RBD optimization is performed multiple times using a variable complexity approach to reliability analysis. With this variable complexity approach, reliabilities calculated using the efficient MVFO method are adjusted to match the more accurate reliabilities calculated using FORM using the following adjustment factors, AF,:

R F0W R MVFO;

1101

where R,,, is the reliability for probabilistic constraint i calculated using FORM and RmFo, is the that calculated using MVFO. Adjustment factors are calculated for each probabilistic constraint since the degree of nonlinearity may be different for each, which will affect the factor calculation. Here the conservative approach to system reliability analysis is taken, as suggested in Ref. 1, where the reliability is evaluated for each probabilistic constraint separately. The minimum reliability for all probabilistic constraints is taken as the system reliability. During RED optimization, then, the MVFO method is employed for efficiency and the calculated reliabilities are multiplied by the adjustment factors determined prior to initiating each RBD optimization.

Deterministic

( Adjustme? Factors 1

Analysis L

1 VCREID Loop: 1

1 No 1

Figure 2 Variable Complexity Reliability-Based Design (VCRBD) Optimization Flowchart

Before the fast RBD optimization, the adjustment factors are initialized to 1.0 for each probabilistic constraint. The first RBD optimization is then performed using the MVFO method for reliability analysis and the reliabilities are not adjusted (multiplied by 1.0). The reason the reliabilities are not adjusted for this first loop is because this fust RBD optimization represents the largest design point shift, from deterministic optimum to a more reliable region of the design space. Testing has shown that adjustment factors calculated based on the deterministic optimum are highly inaccurate, and thus the extra reliability

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analyses (particularly the FORM analyses) are not justified. In addition, for some problems the MVFO reliability analysis is sufficient-without adjustment and the extra FORM analysis can be avoided.

After each RBD optimization, FORM and MVFO reliability analyses are performed on the RBD solution. If convergence between the FORM reliabilities and the adjusted mean value reliabilities is achieved, VCRBD is terminated. The following convergence criterion is used for this check:

IboRMi - AFi*RmroJ I E ill1 where E is a sufficiently small number; E is set to 0.00001 for the analyses in H&paper.

If convergence is not achieved, and the current VCRBD cycle number is less than the maximum allowable cycles, the adjustment factors are updated based on the current FORM and MVFO analyses, and RBD optimization is repeated with the new adjustment factors. The maximum number of cycles for VCRBD is set at six, based on the difference in number of analyses required for FORM-RBD and MVFO-RBD for the class of problems investigated here.

RBD optimization using the MVFO analysis method is significantly more efficient than with FORM. The belief in implementing the VCREID approach of Figure 2 is that performing multiple RBD optimizations with the adjusted MVFO method is still more efficient than a single RBD optimization with FORM, and is equally accurate in the reliability analysis (to the precision of the convergence criterion). This hypothesis is tested and verified in the next section.

4 EXAMPLE PROBLEMS AND RESULTS The VCRBD approach of Figure 2 .is tested here using three structural design problems: a three-bar truss (Sections 4. I), a tension-compression spring (Section 4.2) and a composite tube under internal pressure (Section 4.3). For all three examples, results obtained using VCRBD are compared with those obtained with a single REID optimization using FORM.

It should be noted here that the focus in these example problems is the comparison of VCRBD and FORM-RBD optimization results, particularly the required number of function evaluations for each, and not the efficiency of optimization. For all three example problems, although analytic gradients could have been used to increase the efficiency of optimization, finite differencing was used for all gradient calculations. Since the reliability analyses are performed for each gradient calculation in addition to the design point analyses, the number of function evaluations for RBD optimization is high. Also,

approximation concepts are not incorporated here for optimization or for reliability analysis,

4.1 Example 1: Three-Bar Truss The fust test problem is the classical example of a symmetric three-bar truss, shown in Figure 2; this example is taken from Ref. 11. The three-bar truss is to be designed for minimum volume to support a force of 40,000 lbs., applied at 45 degrees at node 4 in Figure 2. The structure must be designed to support failure modes in stress, in each of the members, and deflection. The allowable stress for members 1 and 3 is 5000 psi and for member 2 is 20,000 psi, and the allowable displacement in the x and y directions is 0.005 inches. The design variables are the cross-sectional areas of members 1 and 2 (Area,=Area,). The lower bound on the member cross-sectional areas is taken as 0.1 in’, The deterministic design optimization model is:

Minimize: Vohme

F=l(2&A, +A*) [I21

Subject to:

x displacement

JzPl 2 5 0.005 A,E

y displacement

JzP,l

(A, +&A,)E <- 0.005

[13]

[I41

member I stress I~

(A, +‘&A,) ‘5000 -1 WI ;

member 2 stress

member 3 stress

-EL <5000 1 A, - r.171

variable bounds

A,,A, 20.1 [181

where 1 = 10 in., P = 40,000 lbs., 8 = 45”, P, = Pcos 8 and P, = Psi&, and Young’s Modulus (E) is 1.0 x 10’ psi.

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node 3

Figure 2 Three-bar truss configuration for Example 1

The starting point for optimization is taken as A, = A, = 1.0 in2. The deterministic optimum is A, = A, = 7.95 in2 and A, = 0.1 in2, which gives a volume of 225.87 id. For the reliability-based design optimization model for this example, the Young’s Modulus (E) and allowable displacements in the x and y direction (6x,, and 6y,,,) are taken as random variables, each with a coefficient of variation of 10%. The x and y displacement constraints (Equations 13 and 14) then become probabilistic constraints, for which a minimum reliability level of 0.90 or 90% is set. For the deterministic solution, the system reliability is calculated to be only 48% (based on normally distributed random variables; both displacement constraints are active). Two probabilistic distribution cases are investigated with this example, one in which the random variables are normally distributed, and a

second in which Gumbel exueme value distributions are used. Since the MVFO reliability analysis method is strictly suited only for normally distributed random variables, using Gumbel distributions in Case 2 allows more extensive testing of the mean value adjustment in the VCRBD approach.

The RBD optimization results for the three-bar truss example problem are presented in Table 1. As can be seen, for both random variable distribution cases, the reliability of the three-bar truss is increased from 48% with the deterministic solution to the desired 90%. In both cases, this reliability level is achieved by increasing the cross-sectional area of member 1 (and 3) from 7.950 in2 for the deterministic solution to 9.67 in2 for Case 1 and roughly 9.6 in2 for Case 2; the cross- sectional area of member 2 does not change from the lower bound value of 0.100 for the deterministic solution. With the increase the A, and the system reliability, the truss volume is increased from 225.87 in3 for the deterministic solution to roughly 275 in3 for Case 1 and 272 in’ for Case 2.

In implementing the VCRBD approach for Case 1 of this example, 38% fewer function evaluations are executed when compared to using FORM within the RBD optimization (223 analyses compared to 357). For Case 2, with Gumbel distributions, the savings are even greater, with 71% fewer evaluations with VCRBD. These resuIts suggest that the MVFO method can be successfully employed with non-normally distributed random variables if the MVFO reliability values are adjusted based on FORM reliability calculations within the VCRBD approach.

Table 1 Reliability-based design optimization solutions for Example 1, three-bar truss problem

Distribution Type: a Case 1: Normal Case 2: Gumbel Optimizati

Initial Design

Final Design

1/RBD Method: Deterministic FORM-RBD 4 (in’) 1.0 1.0 A2 ma> 1 1.0 II 1.0 A, W) II 7.950 9.675 Al Volume I

0.100 u

0.100 225.87 274.66

Reliability 1 . ’ (x displacement)

48.24% 90.02%

Analysis Savings

1.0 1.0 9.674 I 9.585 0.100

I 0.100

274.62 272.10

VCRBD 1.0 1.0

9.602 0.100

272.59

90.29%

92.54%

175 n

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4.2 Example 2: Tension-Compression Spring The second example employed here to test the VCGD approach is the classical tension-compression spring problem. The spring is to be designed for minimum mass subject to constraints on minimum deflection, shear stress, surge frequency, and limits on the outside diameter and on the design variables. The wire diameter, d, coil diameter, D,and the number of coils, n, as shown in Figure 3, have been considered as normally distributed random variables, and their mean values are considered as the design variables, With random variables also as design variables, RBD optimization times are increased since the random variable values (means as well as standard deviations for fixed coefficients of variation) are updated with each design point move. Thisexample is formulated to test the VCRBD~ approach for this more expensive type of structural R.BD problem.

The deterministic optimization problem for this example is stated as follows:

Minimize: Mass Cost Function

Subject to:

deflection

F = (n + 2)Dd’ P91

g,=l- D’n 7.1875x10’d4

50 iP1

shear stress

4DZ - dD 1 gZ= 12566xlO’(Dd’-d’) 5.108x103d2

-150 PII

surgefrequency

g, =l- 140.45d < o

D2n - size

D+d --110 g4= 1.5

PI

v31

variable bounds

d kO.O5,D2:0_10,n21 v41

For the objective function calculation, Eqn. 19, the constant rt2p/4 (where p is the material density) has been removed from the mass calculation, since it simply scales the objective, and thus the objective is a cost function related to the mass. Additjonal details for

this problem formulation can be found in Ref. 14. The constraints g,-g, (Eqns. 20-23) become’ probabilistic constraints for the reliability-based design problem, with reliability levels set to 95% for each. The coefficients of variation for d, D, and n are 0.5, 5.0, and 10% respectively.

Figure 3 Tension-Compression Spring, Example 2

The solutions obtained for this problem are given in Table 2. The starting point for all optimizations (deterministic as well as RBD) is the variable lower bounds: d=O.O5 in., D=O.lO in., and n=l.OO. The deterministic solution is d=O.O5 15 in., D=0.35 1 in., and n=11.633, which gives an objective value of 0.0127. With two active constraints (deflection, g,, and shear stress, gJ, the reliability of this solution is 49%. The desired reliability levels of 95% are obtained with both FORM-based REID optimization and VCRBD by increasing the wire diameter, coil diameter, and number of coils by roughly lo%, 20%, and 8% respectively, which increases the objective by nearly 50%. However, the VCRE3D approach arrives at the reliable solution with half of the function evaluations required by FORM-RBD optimization.

As shown in Table 2, the number of fimction evaluations executed for the RESD optimization for both approaches is very high. This is a result of all three design variables being random variables. With fixed coefficients of variation and the mean values changing with the current design point, the standard deviation is also recalculated for each move. Changing the standard deviation affects both the reliability calculations and the probabilistic constraint gradient calculations, For

this problem since the design variables are increased to improve the reliability, the mean of the random variables is increased, and thus the standard deviation of the random variables is increased also (coefficient of variation = standard deviation / mean). Increasing the standard deviations has the effect of reducing the reliability at the current design point (or lessening the effect of the current move based on the previous gradients). This shifting target significantly increases the RBD optimization iterations and thus the number of function evaluations.

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Table 2 Reliability-based design optimization solutions for Example 2, tension-compression spring problem

% Analvsis Savings

4.3 Example 3: Composite Tube under Internal Pressure

The third example problem presented to demonstrate the advantages of the VCRBD approach is a composites ply laynp optimization problem taken from the MSCMastran solution 200 example problems (d200cOl.dat).” In this example a composite tube under an internal pressure load of 100 psi is modeled using shell elements; this model is shown in Figure 4. The composite tube is to be designed for minimum weight subject to constraints on the individual ply stresses and the design variables. The ply thickness, TPLY, and two ply orientation angles, THETA1 and THETA2, are given as the design variables.

Figure 4 Composite Tube under Internal Pressure, Nastran model for Example 3

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The deterministic optimization problem for this example is stated as follows:

. . . Mmumze: Weight

Subject to:

ply direct stresses

0.001 I PlyStressFailureCriteriai I 0.9

variable bounds ~251

0.00001 I TPLY I 0.10 WI

-90.0 I THETA, 5 90.0 ~271

where i = 1 to 8 for the eight plies, and j = 1,2 for the two ply orientation angles. Here all plies have equal thickness and the ply stresses are measured through a failure criteria index using the Hill failure theory. Additional details for this example problem formulation can be found in Ref. 15.

Given a starting point of TPLY=O.Ol in., THETA1=85.0”, and THETA2=60.0”, the deterministic optimization solution for this example given by MSUNastran (Version 70.5) is TPLY=O.O00546 in., THETAl= 47.796”, and THETA2=5.877”, which gives a weight of 0.0873 lbs. (see Table 3). The reliability of this solution is calculated using the FORM analysis to be 50%.

Table 3 Reliability-based design optimization solutions for Example 3, composite tube problem

OptimizationKBD Method: H Deterministic

Initial Design

Final Design

TPLY THETA 1

(in.) 0.01 (“) 85.5

THETA2 TPLY

60.0 iI.

THETA 1 THETA2 Weight (lbs.) Reliability 2 (Ply 1 Stress - upper) Reliability 4 (Ply 2 Stress - upper) Reliability 6 (Ply 3 Stress - upper) Reliability 8 (Ply 4 Stress - upper) Reliability 10 (Ply 5 Stress - upper) Reliability 12 (Ply 6 Stress - upper) Reliability 14 (Ply 7 Stress - upper)

47.796 5.877

0.08732 5 1.73% 52.51% 62.19% 61.82% 61.45% 61.64% 5 1.60%

Reliability 16 (Ply 8 Stress - upper) # Function Gals # VCBD loops % Analysis Savings

49.96% ---- ---- ----

For the reliability-based design optimization problem, the modulus of elasticity in the longitudinal direction, El, and the modulus of elasticity in the lateral direction, E2, are taken as normally distributed random variables. The mean values are set at 1.0701 x 10’ psi and 5.4375 x lo5 psi for El and E2 respectively (the material for all plies is Kevlar 49 epoxy), and the coefficient of variation is set at 5.0% for both. The 16 ply stress constraints (lower and upper bounds for each of the eight plies) become the probabilistic, reliability constraints.

The solutions obtained for this problem using FORM-REID optimization and VCRBD are presented in Table 3 along with the MSC/Nastran deterministic solution. Only reliabilities calculated for the upper bound ply stress constraints are reported in this table since the lower bounds of these constraints are not active (reliabilities with respect to the ply stress lower bound constraints are all 100%). The structural reliability is increased with both approaches from the 50% reliability of the deterministic solution to the desired 95% reliability by slightly increasing the ply thickness, TPLY, and slightly decreasing the ply angles, THETA1 and THETA2. As a result, the weight of the composite tube is increased by roughly 4%.

Again, the VCRBD approach requires significantly fewer function evaluations than the FORM-RBD optimization approach, nearly 70% fewer for this

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FORM-IXBD

0.000546 47.796 5.877

0.000567 - 47.085 5.833

0.09061 95.22% 95.44% 97.74% 97.99% 98.00% 97.57% 95.20% 94.81%

VCRBD

0.000546 47.796 5.877

0.000565 47.063 5.837

0.09033 95.07% - 95.29% 95.48% 95.94% 95.28% 95.75% 95.08% 94.65%

---- 67%

problem. The large number of function evaluations for this problem is due primarily to the number of constraints. The FORM reliability analysis is performed for each constraint individually; thus the number of function evaluations increases with the number of constraints. For this problem, with the FORM-RBD optimization approach only 5 1 of the total function evaluations (roughly 1%) are due to optimization; the rest (4368 evaluations!) are required for FORM reliability analysis. With the VCRBD approach, 858 function evaluations (nearly 60%) are due to the multiple optimizations, while 603 are executed for reliability analysis.

5 DISCUSSION AND CLOSING REMARKS In this paper a variable compkxity approach to reliability-based structural design optimization is introduced and tested. This approach is compared to RBD optimization using the traditional first order reliability method (FORM) for reliability analysis. Results obtained for three test problems demonstrate the increased efficiency of the VCRBD approach over FORM-RBD optimization. With the VCRBD approach, multiple optimizations using the more efficient mean value, first order (MVFO) reliability analysis method, adjusted to match FORM calculated reliabilities, has been shown to identify equivalent reliable structural designs while requiring 40-70%

fewer function evaluations than conducting the RBD optimization using FORM for reliability analysis.

The number of function evaluations for these example structural design problems, however, remains larger than desirable. With the FORM approach the high number of function evaluations is associated with the optimization problem necessary to calculate the reliability of each probabilistic constraint individually. With the VCRBD approach, the number of function evaluations is associated with the level of inaccuracy of the MVFO method for a given problem, and thus the number of optimization loops necessary to correct the reliabilities to match those calculated using FORM. However, it should be reiterated here that for these three example problems, no attempt was made to increase the efficiency of the optimization. Although analytical gradients could have been used for all three problems, fmite differencing was employed, which increased the number of function evaluations due to both the optimization and the reliability analysis. Also, no approximation concepts were employed for either the optimization evaluations or the reliability analyses; approximations could be employed to reduce the number of evaluations for both of these aspects of RBD optimization.

Currently the authors are investigating alternate reliability analysis methods to replace the expensive FORM analysis within the VCRBD approach, as well as alternate approaches to RBD optimization, to be implemented in iSIGHT (the flagship product of Engineous Software Inc.). The single loop RBD optimization method presented in Ref. 16 has also shown to require significantly fewer function evaluations for RBD optimization than FORM-RBD optimization. The accuracy of this method is currently being investigated and compared to that of the approaches tested in this paper.

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