A density-matching approach for optimization under uncertainty

Abstract

A monotonic, non-kernel density variant of the density-matching technique for optimization under uncertainty is developed. The approach is suited for turbomachinery problems which, by and large, tend to exhibit monotonic variations in the circumferentially and radially mass-averaged quantities-such as pressure ratio, efficiency and capacity-with common aleatory turbomachinery uncertainties. The method is successfully applied to de-sensitize the effect of an uncertainty in rear-seal leakage flows on the fan stage of a modern jet engine. IntroductionAero-engine manufacturers are becoming increasingly reliant on computational predictions of compressor and turbine flows to guide their design decisions. As this trend will continue in the years to come, there will be a growing emphasis on leveraging these methods within optimization routines. In tandem, there will also be a demand for the application of uncertainty quantification (UQ) methods, which seek to compute error bars, probability density functions (pdfs) and probabilities of failure for a given compressor or turbine design. Computational simulations of turbomachinery flows can be used to yield these statistics in an efficient manner. Central to this endeavor is the identification of operational uncertainties and their variations. These may be of the form of operational variations in the tip gap, blade and endwall manufacturing deviations and leakage flows between rotating and stationary components-to name but a few.Finally, to close the loop for redesign, this statistical data needs to be fed to an optimizer in a suitable manner. Here one would like to depart from expensive multi-objective methods and opt for computationally tractable alternatives. A framework that can leverage adjoint-based design sensitivities is also extremely favorable. These perspectives give rise to the main focus area of this research. In this paper, a monotonic variant of a probability density-matching technique for optimization under uncertainty (OUU) is presented. A brief background on density-matching and its advantages over existing robust and reliability-based design optimization techniques is given in Section 2. In Section 3, the mathematics behind monotonic density-matching and the motivation for this technique is presented. Finally, in Section 4, the method is successfully applied to a real turbomachinery problem: fan sub-system design under rear-seal leakage uncertainty. Robust design optimization (RDO)Robust design methodologies seek to optimize the mean performance metric while minimizing its variance [1]. Typically RDO lends itself to a multi-objective optimization strategy where there is a trade-off between these two objectives. This approach yields a Pareto set of feasible design solutions from which the designer will select a single design that offers a reasonable trade-off between the two objectives. By and large, multi-objective genetic algorithms are the optimizer of choice for these problems. Aerospace applications range from the design of compressor blades [2] to en...

Horsetail matching: a flexible approach to optimization under uncertainty

“…Utilizing the entire distribution of a quantity of interest avoids losing information by extracting just the first couple of moments. However, it was noted in the development of density matching (Seshadri et al 2016) that requiring a target in an optimization formulation placed a lot of responsibility on the designer, since if the target is not feasible then density matching performs poorly (discussed further in Section 5.2); so it might seem that requiring a target for horsetail matching restricts the approach. In contrast, the target provides horsetail matching with considerable flexibility.…”

“…Instead, other methods attempt to optimize the entire distribution of the quantity of interest. The most direct application of this philosophy is the recently developed density matching approach presented in Seshadri, Constantine, Iccarino, and Parks (2016), where a distance metric between a design's probability density function (PDF) and a designer-specified target PDF is minimized. Additionally, in Petrone, Iaccarino, and Quagliarella (2011), an approach is presented that minimizes the area between sections of a design's cumulative distribution function (CDF) and an ideal target in a multi-objective formulation.…”

“…This philosophy to robust design is also evident in the development of the CDF-based methods of Petrone et al (2011) and Quagliarella et al (2014), where the value of q for different sections/values of the CDF are minimized or traded-off: these methods avoid stochastically dominated designs. Additionally, only considering mean and variance does not take into account higher-order moments or the tails of the distribution, and so loses information which might be important to a designer; this was one of the motivations behind the density matching approach in Seshadri et al (2016).…”

Multi-fidelity modelling via recursive co-kriging and Gaussian–Markov random fields

“…Design criteria come in different flavours and generally reflect our priorities and objectives in identifying configurations of the design variables x that qualify as optimal. For example, one may pursue to maximize the performance of a system for a wide range of operating conditions [13], identify a design with a probability density function (PDF) that matches a target performance [14], or minimize the risk associated with undesirable realizations [15] (table 1). All concepts introduced in the subsequent sections aim to address single-objective design optimization problems, where the quantity of interest is the output of a scalar function of the inputs.…”

Optimization of Aerospace Structures under Uncertainty using an Iterative Distribution Evolutionary Algorithm

“…Alternative optimization strategies have been proposed to overcome these problems, for instance aggressive design procedures 13 . 14 The difficulty in such an approach, is that, depending on the problem, it is not assured that the desired target is matched 'close' enough or that the computational effort is actually reduced. In this scenario, the necessity arises to develop a method that can assure reliability for an engineering structure whose analysis is time demanding even in the absence of uncertainty.…”