Robust design under mixed aleatory/epistemic uncertainties using gradients and surrogates

Abstract

In this paper, mixed aleatory/epistemic uncertainties in a robust design problem are propagated via the use of box-constrained optimizations and surrogate models. The assumption is that the uncertain input parameters can be divided into a set only containing aleatory uncertainties and a set with only epistemic uncertainties. Uncertainties due to the epistemic inputs can then be propagated via a box-constrained optimization approach, while the uncertainties due to aleatory inputs can be propagated via sampling. A statistics-of-intervals approach is used in which the box-constrained optimization results are treated as a random variable and multiple optimizations need to be performed to quantify the aleatory uncertainties via sampling. A Kriging surrogate is employed to model the variation of the optimization results with respect to the aleatory variables enabling exhaustive Monte-Carlo sampling to determine the desired statistics for each robust design iteration. This approach is applied to the robust design of a transonic NACA 0012 airfoil where shape design variables are assumed to have epistemic uncertainties and the angle of attack and Mach number are considered to have aleatory uncertainties. The very good scalability of the framework in the number of epistemic variables is demonstrated as well.

Horsetail Matching for Optimization Under Probabilistic, Interval and Mixed Uncertainties

“…The probabilistic uncertainties represent uncertainty in the operating conditions of the flying wing: the Mach number is uniformly distributed from 0.65 to 0.75, and the actual angle of attack is distributed uniformly over a 4 degree range centered on the nominal angle of attack which is denoted by γ nom and given by design variable x 4 . The interval uncertainty represents uncertainty in the material properties: the failure stress is in the interval [15,25] MPa. The uncertainties are detailed in Table 2 Table 2: Uncertainties for the flying wing design problem.…”

“…Finally we compare horsetail matching to the method commonly suggested in the literature for optimization under mixed uncertainties: using a weighted sum of averages and intervals of statistical moments of the CDFs that make up the horsetail plot. [25][26][27][28] The following weighted sum of three objectives is optimized:…”

“…24 Examples of robust optimization under mixed uncertainties to date almost exclusively use a weighted sum combination of average statistical moments and intervals of statistical moments. [25][26][27][28] For example,…”

Extending Horsetail Matching for Optimization Under Probabilistic, Interval, and Mixed Uncertainties

“…This poses the question of how to use this mixed uncertainty information within an optimization. Existing approaches for design using optimization under mixed uncertainties are few: most examples to date use a weighted sum combination of average statistical moments and intervals of statistical moments [9][10][11][12]. However, such a formulation does not necessarily represent the goal of OUU (this is discussed further in Section II:B), and so in this work we develop an approach to optimization under mixed probabilistic and interval uncertainties that overcomes some of the limitations of the current state-of-the-art.…”

Generalized information reuse for optimization under uncertainty with non‐sample average estimators

“…In OUU, the effects of uncertain inputs on the system must be propagated at every optimization iteration. Various methods have been developed to achieve this propagation at low computational expense, including (but not limited to) quadrature‐based integration, stochastic expansions, and surrogate models . While effective in many scenarios, especially when the problem is smooth, these methods suffer from the “curse of dimensionality” whereby their cost increases exponentially with the number of uncertain inputs.…”