This thesis proposes formulations and algorithms for design optimization under both aleatory (i.e. natural or physical variability) and epistemic uncertainty (i.e. imprecise probability information), from the perspective of system robustness and reliability. This thesis proposes formulations and algorithms for design optimization of multidisciplinary systems under both aleatory uncertainty (i.e. natural or physical variability) and epistemic uncertainty (i.e. imprecise probability information), from the perspective of system robustness and reliability.
Introduction
This dissertation focuses specifically on epistemic uncertainty arising from imprecise probabilistic information (especially sparse point data and interval data) on random variables. The following sections present an overview of existing methods in the literature for uncertainty quantification, propagation, and design optimization under uncertainty.
Uncertainty quantification with sparse point data
In some cases, random variables with sparse data can be modeled using convex uncertainty models (Ben-Haim and Elishakoff, 1990). Several of these current methods for uncertainty propagation under data and distribution uncertainty can be computationally expensive.
Uncertainty quantification with interval data 1 Sources of Interval data 1 Sources of Interval data
Following is a brief discussion of some of the popular uncertainty theories discussed in the workshop above, and of interval data in general. One of the disadvantages of current approaches is the need for nested analysis in the presence of interval variables.
Uncertainty quantification considering correlations
However, this multivariate Johnson distribution cannot match the sample correlation matrix of the original data set if some of the marginal distributions have large skewness. Most of the existing methods using statistical correlation in uncertainty analysis have only been developed in the context of aleatory uncertainty in the input random variables (eg Noh et al, 2009).
Design optimization with epistemic uncertainty
A few studies on robust design optimization are reported in the literature to deal with epistemic uncertainty resulting from lack of information. Therefore, there is a need for efficient robust design optimization and RBDO methods that handle both aleatory and epistemic uncertainty.
Uncertainty quantification in multidisciplinary systems
An efficient methodology for multidisciplinary uncertainty propagation with both aleatoric and epistemic uncertainty operating within a probabilistic framework of uncertainty representation awaits development. In this treatise, we extend this notion of a decoupled formulation and propose probabilistic methods for multidisciplinary reliability analysis under both aleatory and epistemic uncertainty.
Multidisciplinary design optimization with epistemic uncertainty
Therefore, an efficient method for multidisciplinary robust design optimization under both aleatory and epistemic uncertainty awaits development. First, the epistemic uncertainty arising from a lack of knowledge about the distribution type of the random variables is considered.
Proposed Methods for Uncertainty Representation
Fitting the Johnson Distribution to Point Data
These methods include the method of moments (which requires the first four moments of the data), percentile matching (using four points and solving a system of nonlinear equations for the distribution parameters), least squares estimation (minimizing the sum of squared errors in percentage values of the probability distribution), and minimizing the error rate of the Johnson distribution when compared to the empirical CDF. Once the parameters of the distribution of the experimental data have been estimated, regenerating the random variable x that follows this distribution is easy.
Statistical Uncertainty Quantification via Jackknife
The assumption that the distribution parameters also have the Johnson distribution allows for the possibility of a non-normal distribution for the distribution parameters. Remove observation i from the original set of observations. Estimate the Johnson distribution parameters based on the remaining N-1 points.
Proposed Methods for Uncertainty Propagation
FORM, Inverse FORM, and Sensitivity Analysis
Optimization-based approaches to obtain confidence limits in reliability estimation are described in detail in the next section. This sensitivity vector shows the relative contribution of each of the random variables to the variance in the limit state function.
Optimization-Based Confidence Intervals for CDF and Reliability Estimates In this chapter, uncertainty analysis is carried out using the probabilistic
It should be noted that, given the presence of epistemic uncertainty, the probability of failure itself is uncertain due to the uncertainty in the distributions of the basic random variables. Therefore, the solution of the problem gives confidence limits for the probability of failure with a confidence level of 1 - α/2 equal to Φ(-|βT|). This vector gives an indication of the sensitivity of the failure probability to the uncertainty in each distribution parameter.
Numerical Illustration
Because the variability of the mission parameters is described by sparse point data, uncertainty arises about the distribution parameters of Mach and AoA. This is intuitive given the larger gradients of the response surface in the AoA direction and the broader distribution of AoA. The CDFs of the worst and best distributions of Mach and AoA are shown in Figures 7 and 8, respectively.
Conclusion
In this chapter, we present efficient algorithms based on continuous optimization to find bounds on second and higher moments of interval data. Using the moment bounds computed using the proposed approach, we fit a family of Johnson distributions to the interval data. Third, using moment bounds, we fit a family of Johnson distributions to the interval data.
Estimating Bounds on Moments for Interval Data
The lower bound of the mean for a single interval is therefore the lower bound of the interval itself. The upper bound of the mean for a single interval is therefore the upper bound of the interval. Similarly, the above upper bound maximization at the second instant yields a PMF of 0.5 at both endpoints of the single interval.
Fitting Johnson Distributions to Interval Data
Note that the objective function of the above optimization problem may require scaling as the moments may be of vastly different sizes. Note that the minimization yields the leftmost bound of the family of distributions for each α. The value of the objective function, xα, can be found by applying the Johnson transformation (see Eq. 1 in Chapter III) to a standard normal variable corresponding to the given α.
Numerical Examples
The moments of the family of Johnson distributions fall within the previously calculated moment limits; The moments of the lognormal variable (X), m1dist and m2dist are calculated in terms of the corresponding normal variable moments (μY, σY) as follows. The family of Johnson distributions thus fitted can be used as the probabilistic representation of the interval data.
Uncertainty Propagation using Probabilistic Analysis
Sampling-based methodology for uncertainty propagation
Generate a family of CDFs for each of the input variables described by single or multiple interval data using the procedure described in Section 2. Expand each of the CDFs from the input family of CDFs through the system response equation using any probabilistic uncertainty propagation method (eg , FORM, SORM or MCS). Using the procedure described in Section 2, generate a family of n CDFs for each of the distribution parameters of the input variables described by one or more interval data.
Optimization-based methods for uncertainty propagation
Once the distribution parameters are obtained, any probabilistic uncertainty propagation method (eg FORM or MCS) can be used to construct approximate bounds on the CDF of the system response. Probabilistic uncertainty propagation method (eg FORM or MCS) for each of the two sets of distribution parameter values. Solving these nested formulations provides the realization of distribution parameters that minimize or maximize the expectation of the system's response.
Numerical Examples
Challenge Problem A: Algebraic problem set
The input variables a and b are assumed to be independent of each other and both a and b are positive real numbers. The task for each group problem is to quantify the uncertainty in y given information about a and b. Multiple interval types are classified as i) consonant collection of intervals (intervals are correlated), ii) stable collection of intervals (no overlap between intervals) and iii) arbitrary collection of intervals (no assumption of overlap or relationship between intervals) .
Challenge Problem B: ODE Problem
Problems 2, 3, and 5 are further divided into three subproblems based on the multi-interval nature. In the prescribed set of problems, the parameter m is given by a triangular probability distribution defined on the interval [mmin,mmax] by mmod mode. The parameter c is described by several intervals, and ω is given by a triangular probability distribution defined on the interval [ωmin,ωmax] with the mode ω mod.
Numerical Results
- Challenge Problem A Problem A-1
- Challenge Problem B
Several sample cumulative density functions from the family of Johnson distributions for input variable b are shown in Figure 7. Several sample cumulative density functions from the family of triangular distributions for input variable k are shown in Figure 15. Several sample cumulative density functions from the family of triangular distributions for input variable ω are shown in figure 16.
Conclusion
We formulate optimization problems of deriving the bounds of the cumulative probability distribution of the system response using correlations between input variables that are described by interval data. This chapter develops a multivariate input model for random variables with marginal Johnson distributions based on the Nataf transformation. Furthermore, for interval data, the correlations between the input variables are unknown, and there are very few computationally efficient random and statistical uncertainty propagation methods that account for the correlations between the interval variables.
Input modeling with Johnson distributions using correlations
Then, instead of using the integral equation in Eq. 1), we calculate the reduced correlation coefficient ρ0,ij with a numerical technique based on optimization. Generate correlated standard normal variations (Z) from the joint PDF given in Eq. 3) with the reduced correlation matrix [C']. Generate correlated standard normal variations (Z) from the joint PDF given in Eq. 3) with the reduced correlation matrix [C'].
Proposed Methodology for uncertainty propagation under uncertain correlations correlations
We propose the following algorithm to quantify the uncertainty of distribution parameters and correlation coefficients. We now have four marginal distributions for the distribution parameters of the underlying random variables. Calculate the correlation coefficients ρij for the distribution parameters from the set of distribution parameters obtained in step 1.
Example Problems
Figure 2 shows that the correlations between the basic random variables have a significant impact on the system response distributions, especially on the tail of the distributions. Various methods have been reported in the literature to estimate the mean and standard deviation of the performance function. The performance of the robustness-based design can be determined from the mean and variance of the performance function.
Proposed methodology
Therefore, in this chapter, we propose a general formulation for robust design that considers some of the epistemic variables as non-design variables, which leads to a conservative design under epistemic uncertainty. The proposed methods are illustrated for the conceptual level design process of a two-stage-to-orbit (TSTO) vehicle, where random input distributions are described by sparse point and/or interval data. In the proposed methodology, we use the variance as a measure of the performance function change to achieve objective stability, the feasible region reduction method to achieve feasibility stability, a first-order Taylor series expansion to estimate the mean and performance variance. function and a weighted sum method for aggregating multiple objectives.
Robustness-based design optimization under data uncertainty
- Robustness-based design with sparse point data
- Determination of optimal sample size for sparse point data
In design optimization (Eq. 7)), the mean values of the design variables (either aleatory or epistemic) are controlled by given design bounds. However, since the mean values of unplanned variables cannot be controlled in design optimization, the proposed robustness-based design optimization methodology considers the uncertainty about the mean values of such epistemic variables by optimizing in Eq. For unplanned epistemic variables described by interval data, the constraints on the decision variables in Eq. 8) are carried out by estimating the limits of averages using the methods described in Chapter IV.
Example Problem
- Robustness-based design optimization with sparse point data
It is seen in Figure 2 that the solutions become more conservative (i.e., the mean and standard deviation of GW take on higher values) as we add uncertainty to the design problem. As the collection of more data reduces data uncertainty, the solutions become less sensitive (i.e. the standard deviation of GW takes on lower value) to the variations of the input random variables as the sample size (n) increases. In this case, we get the most conservative robust design, that is, the mean and the standard deviation of GW take the maximum of all possible values.
RBDO for single discipline systems
- Deterministic design optimization
- RBDO under epistemic uncertainty
Note that the first constraint (i.e., the reliability constraint) in Eq. 6) is necessary to ensure that the optimization is driven by all non-design epistemic variables, because sometimes the objective function is not a function of all non-design epistemic variables. It is assumed that only sparse point data are available for some design variables, as well as for epistemic non-design variables. In the design optimization (Eq. 5)), the average values of the design variables (aleatoric or epistemic) are determined by the given design boundaries.
Numerical Example Shaft-Gear Assembly Shaft-Gear Assembly
