Reliability-based Design Optimization

Reliability-based Design Optimization (RBDO) is an optimization method that can be used to provide optimum designs in the presence of uncertainty.

A two-phase approach is implemented in OptiStruct to solve the RBDO problems efficiently, while preserving accurate reliability assessment of the final design. The design variables, constraints, and objective are tested for reliability based on user-defined reliability requirements.

Three types of variables, including Deterministic design variables, Random design variables (also known as random control factors) and Random parameters (also known as noise factors) can be selected in a RBDO problem. Design Constraints and Objectives can be specified as Deterministic (mean) or Percentile values.

Implementation

The original problem of reliability-based design optimization (RBDO) is mathematically a nested two-level structure that is computationally time consuming for real engineering problems.

To overcome the computational difficulties, many formulations have been proposed in the literature. These include SORA (sequential optimization and reliability assessment) that decouples the nested problems. SLA (single loop approach) further improves efficiency in that reliability analysis becomes an integrated part of the optimization problem. However, even SLA method can become computationally high expensive for real engineering problems involving many reliability constraints. An enhanced version of SLA has now been implemented where the first phase is based on approximation at nominal design point. After convergence of first iterative phase is reached the process transitions to a second phase where approximation of reliability constraints is carried out at their respective minimum performance target point (MPTP). The first phase of the proposed method is as efficient as deterministic optimization. The accuracy of reliability assessment is ensured in the second phase. Thus, the RBDO problems can be solved more efficiently while preserving accurate reliability assessment of the final design. Examples show that the proposed two-phase approach consumes much less finite element analyses while achieving equal solution quality. ¹

The two-phase single loop approach is introduced to efficiently solve the RBDO problem. Reliability analysis is not fully carried out in one optimization iteration. Instead, reliability analysis evolves together with the objective optimization process. When optimization reaches convergence, reliability analysis also reaches convergence and reasonably accurate reliability analysis result is achieved.

Note: The reliability analysis result from intermediate iterations may not be accurate.

Variables

The following design variables and parameters can be used to define the structural design space in OptiStruct:

Random Design Variables

Random design variables are defined via the RAND continuation lines on the DESVAR Bulk Data Entry. Various random distribution types can be selected and their parameters are defined accordingly. In an RBDO process, during reliability and/or robustness analysis, the design should satisfy optimality based on the specified distributions. Additionally, loading can be selected as a design variable using the DVPREL1 and DVPREL2 Bulk Data Entries.

Random Parameters

The definition of Random parameters is similar to that of Random Design Variables, using RANP definition. However, the important difference is that, while the mean values of random variables are changed to improve the design, the mean values of random parameters remain constant. For example, typically sheet metal thickness can be a random variable, due to fabrication variance, while the Young's modulus of a material would typically be a random parameter, if its variance is accounted for.

Deterministic Design Variables

The deterministic design variables are the regular design variables used in an OptiStruct optimization run.

Note: Due to the deviation of the random distribution, the design region should be defined carefully. For example, if a design variable value is intended to be positive, then its lower bound should not be defined lower than

n * δ

; where,

δ

is the standard deviation of the variable; n is a constant multiplier (a value of n=6 is recommended).

Objective

The following design objective types are available in OptiStruct:

Percentile Value (RBDO)

The minimum or maximum of the percentile based objective function can be defined on the DESOBJ Subcase Information Entry. The MINP/MAXP options and the PROB argument can be used to define the required parameters.

Figure 1. Percentile Value Based Objective Depicted

The percentile value based objective is defined as:(1)

\min [P_{r} (f (x))]

or(2)

\max [P_{r} (f (x))]

Where, $f (x)$ is the objective function, and $r$ is the probability level (for example, 95%). The right and left percentile values are available. MINP minimizes the right percentile value and MAXP maximizes the left percentile value.

Deterministic (mean) Value

The deterministic value based objective is the regular objective used in an OptiStruct optimization run. The mean value based objective is defined as follows:(3)

\min [f (x)]

or(4)

\max [f (x)]

Where, $f (x)$ is the objective function.

Constraints

The following design constraint types are available in OptiStruct:

Percentile Value (RBDO)

The probability of one constraint satisfying its bounds should not be less than the predefined reliability value. The reliability value is defined via the PROB field on the DCONSTR Bulk Data Entry.

The reliability-based constraints are defined as:(5)

P (c (x) \leq U B) \geq r

(6)

P (c (x) \geq L B) \geq r

Where, $c (x)$ is the constraint value, UB is the upper bound of the constraint, LB is the lower bound of the constraint, and $r$ is the probability level (for example, 95%).

For the $P (c (x) \leq U B) \geq r$ constraint, the right percentile value of $L B$ is forced to be less than or equal to the upper bound UB. For the $P (c (x) \geq L B) \geq r$ constraint, the left percentile value of $c (x)$ is forced to be greater than or equal to the lower bound LB.

Deterministic (mean) Value

The deterministic value based constraint is the regular constraint definition used in an OptiStruct optimization run. The mean value based constraint is defined as:(7)

c (x) \leq U B

(8)

c (x) \geq L B

Where, $c (x)$ is the constraint value, UB is the upper bound of the constraint, and LB is the lower bound of the constraint.

Example

Problem with mean of random variables as design variables (Yi et al. 2008).

Considering the following mathematical model with two random design variables:(9)

\min C (v_{X}) = v_{X_{1}} + v_{X_{2}}

(10)

s . t . P_{f} (G_{i} (X) \leq 0) \leq Φ (- β_{i}^{t})

Where,

i = 1, 2, 3

(11)

0 \leq v_{X_{j}} \leq 10

Where, $j = 1, 2$ .

Where,

v_{X_{j}}

are the mean values of random variables

X

and

$G_{1} (X) = X_{1}^{2} X_{2} / 20 - 1$
$G_{2} (X) = {(X_{1} + X_{2} - 5)}^{2} / 30 + {(X_{1} - X_{2} - 12)}^{2} / 120 - 1$
$G_{3} (X) = 80 / (X_{1}^{2} + 8 X_{2} + 5) - 1$

and

β_{i}^{t} = 2.0

(corresponds to 2.28% failure probability).

X_{1}

and

X_{2}

follow five different types of random distributions with standard deviation 0.6: normal, lognormal, Weibull, Gumbel, and uniform. The initial design is:(12)

v_{X}^{0} = (5.0, 5.0)

The results of this example are summarized in Table 1. For the two-phase approach, in the first phase one function evaluation at each iteration can generate all the needed response values (performance functions are all evaluated at the same point). So the total number of function evaluations is much less compared to evaluations during second phase. The values in the bottom row of each column, such as 7.268 (3.609, 3.659), are the final objective function and optimal design.

It is clear that the two-phase approach implemented in OptiStruct is much more efficient than SAP. In this example, the distribution types of the random variables have very little influence on the computational efficiency of the two-phase approach while SAP performs quite differently for the different distribution types. The two-phase approach consumes the same computational effort for all the five distribution types except Gumbel, in which number of function evaluations is slightly less than others. For Gumbel distribution, SAP consumes more than 1.6 times of the computational effort of that with Normal distribution.

It should be noted that, for uniform distribution, the result of SAP, i.e. 6.869 (3.521,3.348), violates the constraints slightly when assessed by Monte Carlo Simulation. The two-phase approach reaches a feasible design 7.106 (3.597,3.509). However, the two-phase approach has the same issue in Weibull distribution. The final design 7.549 (3.682,3.866) is only close to the exact optimum, with slight constraint violations. This precision issue is introduced by the high non-linearity of the transformation between X- and U-spaces in probability constraint evaluation, and the approximation error of the first-order reliability method. This is a general weakness of single-loop approach based on first-order reliability method.

Table 1. Number of Function Evaluations and the Optimum in Example 1
Distribution Type		SAP (Yi et al. 2008)	Two-Phase Approach
Normal	Number of function evaluations	54	Phase 1: 7 Phase 2: 15 Total: 22
Normal	Objective (Variables)	7.268 (3.609,3.659)	7.268 (3.609,3.659)
Lognormal	Number of function evaluations	54	Phase 1: 7 Phase 2: 15 Total: 22
Lognormal	Objective (Variables)	7.055 (3.556, 3.499)	7.114 (3.571,3.544)
Weibull	Number of function evaluations	42	Phase 1: 7 Phase 2: 15 Total: 22
Weibull	Objective (Variables)	7.513 (3.668,3.845)	7.549 (3.682,3.866)
Gumbel	Number of function evaluations	90	Phase 1: 7 Phase 2: 12 Total: 19
Gumbel	Objective (Variables)	6.836 (3.491,3.345)	6.817 (3.497,3.320)
Uniform	Number of function evaluations	66	Phase 1: 7 Phase 2: 15 Total: 22
Uniform	Objective (Variables)	6.869 (3.521,3.348)	7.106 (3.597,3.509)

The OptiStruct RBDO (the two-phase approach) offers an efficient tool to consider uncertainty involved in design. For most problems this approach yields reasonable solution accuracy. It should be noted that the two-phase approach is derived from the SLA methods and, therefore, inherits the potential shortcomings in terms of solution accuracy of SLA. Insufficient solution accuracy is observed in the study of the above example. Accurate reliability analysis should be carried out if accurate satisfaction of reliability requirements is critical. Also note that for the OptiStruct RBDO approach, reliability analysis is performed only for retained constraints for which sensitivity is available. You can adjust the screening criteria using the DSCREEN Bulk Data Entry, if required.

¹ Zhou, M & Luo, Z (2017). A Two-Phase Approach based on Sequential Approximation for Reliability-based Design Optimization.