Robust Design in the Presence of Aleatoric and Epistemic Uncertainty

Abstract

This paper proposes strategies for designing a system whose computational model is subject to aleatory and epistemic uncertainty. Aleatory variables, which are caused by randomness in physical parameters, are draws from a possibly unknown distribution; whereas epistemic variables, which are caused by ignorance in the value of fixed parameters, are free to take any value in a bounded set. Chance-constrained formulations enforcing the system requirements at a finite number of realizations of the uncertain parameters are proposed. These formulations trade off a lower objective value against a reduced robustness by eliminating an optimally chosen subset of such realizations. Risk-aware designs are obtained by accounting for the severity of the requirement violations resulting from this elimination process. Furthermore, we propose a computationally efficient design approach in which the training dataset is sequentially updated according to the results of high-fidelity reliability analyses of suboptimal designs. Robustness is evaluated by using Monte Carlo analysis and Robust Scenario Theory, with the latter approach accounting for the infinitely many values that the epistemic variables can take.

Figures (8)

• Figure 1: Pseudo-distributions of $r(\theta,a^{(i)},{\mathcal{E}})$ for $i=1,\ldots 50$ (left) and aleatory space (right) for $\theta_1^\star$ (top) and $\theta_2^\star$ (bottom). Left plots: each epistemic scenario is marked with a distinctive symbol, with those being neglected (if any) being encircled. Right plots: the nominal circle is shown with a dashed black line, and the boundary of the success domain is shown as a solid blue line. The aleatory scenarios are marked with a "$\times$", where the aleatory outliers are marked with "$\circ$" (if any).
• Figure 2: Pseudo-distributions of $r(\theta,a^{(i)},{\mathcal{E}})$ for $i=1,\ldots 50$ (left) and aleatory space (right) for $\theta_3^\star$ (top) and $\theta_4^\star$ (bottom). Previous conventions apply. The success domain corresponding to all $n_e$ epistemic scenarios is shown as a dotted blue line on the right subplots.
• Figure 3: Pseudo-distributions of $r(\theta, a^{(i)},{\mathcal{E}})$ for $i=1,\ldots 50$ (left) and aleatory space (right) for $\theta_5^\star$ (top) and $\theta_6^\star$ (bottom). Previous conventions apply.
• Figure 4: Distributions of $r(\theta, a^{(i)},{\mathcal{E}})$ for $i=1,\ldots n_a$ (left) and aleatory space (right) for $\theta_7^\star$ (top) and $\theta_{8}^\star$ (bottom). Previous conventions apply.
• Figure 5: Pseudo-distributions of $r(\theta_9^\star, a^{(i)},{\mathcal{E}})$ for $i=1,\ldots 50$ (top left), the aleatory space (top right), the distributions of $r$ and of the moment exceedance for $\alpha_e=0$ (bottom left), and the distributions of $h$ for all aleatory scenarios and for the inliers only (bottom right). The two vertical dashed lines in the bottom-right subplot are the corresponding color-coded moments. Previous conventions apply.

Theorems & Definitions (6)

• Definition 1: Support scenarios
• Definition 2: Point violation and risk
• Definition 3: Set violation and risk
• Definition 4: Set-complexity
• Definition 5: Risk bound
• Theorem 1