The reliability index is related to the probability of failure (pf) and the reliability (R) of the linearized constraint function as. For each iteration of the outer loop, the reliability constraints must be evaluated, thus increasing the number of function evaluations. This is the traditional reliability-based optimization approach, where FORM limit state analysis is used to estimate the reliability constraints of the optimization problem.
Methods to separate the reliability calculations from the optimization loop are desirable to increase the efficiency of the optimization procedure.
Single Loop Methods
When x tends to the design point x*, the displacement vector si reaches a constant value and thus we get gi*, which is the value of the limit state operation. This method does not perform an iterative reliability evaluation, but instead uses derivatives that are calculated only when the optimizer changes any of the design variable values.
Introduction
Decoupled approach with standard deviations as design parameters
If at this MPP point the value of the limit state is less than zero, then this means that the design solution has not yet reached the target reliability (if for that limit state a positive value indicates the safe region). The MPPs will again need to be updated for the new values of the design variables and subsequent calculations will continue until the converged objective function value is obtained.
Simulation-Based Reliability Assessment
Using Correction Factors in Simulation-based SORA
If βSIM or βFORM are much larger than 2*βTarget, they are considered equal to 2*βTarget. Once the correction factors for each limit state are obtained, the standard normal MPP for each limit state is obtained using Eq.
Examples
In this case, the FORM limit state approach provided a good reliability estimate, so the simulation result is comparable to that of the inverse FORM.
Example 2
All constraints are treated as probabilistic with the target reliability as 90% corresponding to a β value of 1.28. Solution: RBDO is performed with three different initial models (nominal values, lower bound and upper bound). The results, obtained using the inverse FORM method for reliability assessment, are given in Table 3.3.
The computational effort for this method is very high (between 100,000 and 200,000 function evaluations are needed to achieve a converged solution), but the results meet the reliability requirement more accurately. The comparison of the achieved reliability indices for each limit state as obtained from the inverse FORM and the four simulation runs is given below. In Table 3.5, the limit states that do not meet the intended reliability are shown in bold.
It can be observed that the simulation results give the reliability index closer to the target.
Summary
Overview
Variance of the Objective Function
If the target mean is also to be minimized, then we only need to minimize fα2. Similarly, if the mean of the objective function is to be maximized, we only need to maximize the lower percentile value (ie fα) of the objective function. A weighted three-level numerical integration is performed on the first two moments of the objective function, thereby providing an estimate of its mean and variance.
In this method, the function is first evaluated using the mean values of the design variables. The weighted sum of these three function values gives us an estimate of the mean value of the objective function. The objective function for robust design will be the weighted sum of the mean and variance values obtained in the above equations.
To get close to the actual variances of the target, sensitivities will have to be derived using a more global technique. In the third method, the variance is calculated using function and sensitivity evaluations at three points to include a more accurate behavior of the function with respect to the mean value. It turns out that the first-order method provides a good estimate of the variance for a linear function and is the most computationally efficient.
Multi-objective Optimization
Example 1
The robust design process for the problem was carried out using three different initial design points viz. the first-order method was used to evaluate the variance, since the objective function in this problem is linear. For the percentile formulation, the α2 percentile value to be minimized is taken as the 90th percentile for illustration.
The number of function evaluations for different starting points was different, so a range is given in the table above. By comparing the results in table 4.2, it can be seen that both formulations give the same solution. This is because for a linear objective function with constant standard deviations, the robust design problem collapses to an RBDO problem.
When standard deviations are considered as design parameters, they reach their lower bounds for this problem. Again, the computational efficiency of the single-loop method is higher than that of the uncoupled approach. Some convergence problems were observed in some initial designs in the case of the single-loop method.
Example 2
A range of values for the number of feature evaluations is provided, as different initial designs lead to different numbers of feature evaluations. It can be observed that both methods (i.e. percentile and weighted sum) result in the same solution. It was also found that the computational effort for the percentile formulation is greater than the weighted sum method.
In the weighted sum approach, when the weight is kept unreasonably low, there are difficulties in the convergence of some variables when using the single loop method. Next, the standard deviations are treated as deterministic design parameters and their bounds are given in Table 4.5. The results for the robust design optimization including standard deviations as design variables are shown in Table 4.6.
For all cases, the single loop was found to be better in computational efficiency than the decoupled method. For some initial designs and very low weight values, the decoupled loop was able to converge to a lower optimal solution than single loop, but used more function evaluations.
Summary
Introduction
Importance of the problem
Background
Problem setup
Now that the objective and constraints are listed, the optimization problem is written as Minimize (Total Damage Probability). Here NTF, NTE and NCF are the neck damage coefficients for different load cases. The relevant design and other random variables and their distributions are given in Table 5.2 and Table 5.3.
Solution
Preliminary Analysis
The improved RBDO methods proposed in Section 3 and the robust design methods discussed in Section 4 were used to solve the problem. First, the results obtained for the RBDO techniques will be shown, followed by the robust design.
Reliability based design optimization
Using the resulting solution as the mean of the design variables, 1000 samples are generated and the corresponding HIC and chest G values are calculated. We see that quite a few samples for the first solution and two samples for the third solution have a negative HIC value. This may be because most of the design variables approached one of the boundaries and the response surfaces were not accurate in this region.
Therefore, the original limits are narrowed by a value of 2σ to obtain the new design limits, as shown in Table 5.7. It was found that a target reliability of 97% could not be met as the design space was becoming too limited. Therefore, the target reliability was reduced to 90% and the corresponding reliability index was reduced to 1.28 from the previous value of 1.96.
The new results after performing sequential optimization and reliability analysis are given in Table 5.8. The graphical plots for the solutions obtained in Table 5.8 do not show any negative values for HIC.
Robust Design
The variance in the above formulation is calculated using the first-order method discussed in the previous chapter. For the percentage formulation, optimization was performed within the original limits given in Table 5.5. The value of 90 percent of the total probability of injury is the objective being minimized.
Again, the standard deviations are constants and the target reliability for limit states is set at 97%. Next, the above procedure is repeated for the design limits given in Table 5.7, keeping the standard deviations constant. Also, the design space seems to become very limited with the inclusion of the second target, leading to unmanageable results.
Summary
Conclusions
Future Work
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