2. RBDO for single discipline systems
2.3 RBDO under epistemic uncertainty
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( ) ( ) ( )
0 ..
) 3 ( min
=
= Y g t s
Y Y
Y
β T
In Eq. (3), Y denotes all the random variables in uncorrelated standard normal space.
Function gY is transformed functions such that gY
( )
Y g(
T( )
x)
−1
= where T is the transformation function from original space, x, to standard normal space Y. For more details about the implementation of FORM, the reader is referred to Ditlevsen and Madsen (1979), Haldar and Mahadevan (2000), and Nowak and Collins (2000).
In the following section, we develop the methodology for RBDO under epistemic uncertainty for single discipline problems.
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where Zl and Zu are the vectors of lower and upper bounds of the decision variables µz of the inner loop optimization problem.
Note that in this formulation, the outer loop decision variables d may consist of stochastic design variables as well as epistemic design variables. The outer loop optimization is a design optimization problem, where an RBDO is carried out for a fixed set of non-design epistemic variables. The inner loop optimization is the analysis for the non-design epistemic variables, where the optimizer searches among the possible values of the non-design epistemic variables for a conservative solution of the RBDO.
This nested optimization problem can be decoupled and expressed as:
( )
( )
( )
(
g X)
p i kP p t
s
d f d
i z
i f
z d
i , 0 for 1,2,...,
. .
) 5 ( ,
min arg
*
*
*
=
<
≤
=
=
µ µ
( )
( )
( )
( )
u z l
i z
i f
z z
Z Z
k i
p X
g P p t s
d f
i z
≤
≤
=
<
≤
=
= µ
µ µ
µ µ
,..., 2 , 1 for 0
, .
.
) 6 ( ,
max arg
*
*
*
The optimization problems in Eqs. (5) and (6) are solved iteratively until convergence.
Note that the first constraint (i.e., the reliability constraint) in Eq. (6) is required to ensure that the optimization is driven by all non-design epistemic variables, because sometimes the objective function may not be a function of all non-design epistemic variables. In cases when the objective function is the function of all non-design epistemic variables, this constraint is not required.
Since Eq. (5) is solved with a fixed set of non-design epistemic variables, Eq. (5) is equivalent to an RBDO problem under aleatory uncertainty alone. Eq. (6) is referred to
184
as uncertainty analysis for the non-design epistemic variables throughout this dissertation. The RBDO formulations presented above are general and can handle all varieties of design and non-design variables, such as one or more design or non-design variables being deterministic, aleatory or epistemic. Since Eq. (5) is equivalent to traditional RBDO under aleatory uncertainty, it can accommodate both deterministic and aleatory design variables as well as both deterministic and aleatory non-design variables.
Eq. (5) also accommodates epistemic design variables. The propose methodology accommodates non-design epistemic variables by employing a search among the possible values of non-design epistemic variables through the formulation in Eq. (6).
RBDO with sparse data
This section develops a methodology for RBDO with sparse point data, using the formulations in Eqs. (5) and (6). It is assumed that only sparse point data are available for some of the design variables as well as non-design epistemic variables.
When a variable, either design or non-design, is described by sparse point data, there is uncertainty about the mean and variance calculated from the samples. In the design optimization (Eq. (5)), the mean values of the design variables (either aleatory or epistemic) are controlled by the given design bounds. However, since the mean values of the non-design variables cannot be controlled in the design optimization, the proposed RBDO methodology accounts for the uncertainty about mean values of such epistemic variables through the optimization in Eq. (6).
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The constraints on the non-design epistemic variables in Eq. (6) are implemented through the construction of confidence intervals about mean values using Eq. (9) of Chapter VII.
The proposed RBDO methodology accounts for the uncertainty about the variances for all epistemic variables by first estimating confidence bounds on variances and then solving the optimization formulations in Eqs. (5) and (6) using the upper bound variances for the input random variables xi and zi. Solving the optimization formulations in Eqs. (5)-(6) using the upper bound variances for all the epistemic variables ensures that the resulting solution is least sensitive to the variations in the input random variables. The confidence bounds on variances are estimated using Eq. (10) of Chapter VII.
RBDO with interval data
This section develops a methodology for RBDO with interval data, using Eqs. (5) and (6). In this case, the only information available for one or more input random variables is in the form of single interval or multiple interval data.
The methodology for RBDO with interval data is similar to sparse point data as described earlier. However, the estimation of mean values and variances for interval data is not straightforward. For interval data, the moments (e.g., mean and variance) are not single-valued, rather only bounds can be given (see Chapter IV). We have proposed methods to compute the bounds of moments for both single and multiple interval data in Chapter IV. Once the bounds on the mean and variance of interval data are estimated, we use the upper bounds of the variances to solve the formulations of RBDO under epistemic
186
uncertainty in Eqs. (5) and (6). Therefore, the resulting solution becomes least sensitive to the variations in the uncertain variables.
For non-design epistemic variables described by interval data, the constraints on the decision variables in Eq. (6) are implemented through estimating the bounds on the mean values by the methods as described in Chapter IV.
In the following section, the proposed RBDO formulations are illustrated for a Shaft- Gear Assembly.
3. Numerical Example