Pranay Seshadri

A density-matching approach for optimization under uncertainty

Abstract Modern computers enable methods for design optimization that account for uncertainty in the system—so-called optimization under uncertainty (OUU). We propose a metric for OUU that measures the distance between a designer-specified probability density function of the system response (the target ) and the system response’s density function at a given design. We study an OUU formulation that minimizes this distance metric over all designs. We discretize the objective function with numerical quadrature, and we approximate the response density function with a Gaussian kernel density estimate. We offer heuristics for addressing issues that arise in this formulation, and we apply the approach to a CFD-based airfoil shape optimization problem. We qualitatively compare the density-matching approach to a multi-objective robust design optimization to gain insight into the method.

Understanding Micro Air Vehicle Flapping-Wing Aerodynamics Using Force and Flowfield Measurements

Experimental studies were conducted by flapping a rigid rectangular wing with a mechanism that is capable of emulating complex insect wing kinematics, including figure-of-eight motions, in order to explore the fundamental unsteady flow on a flapping wing at micro-air-vehicle-scale Reynolds numbers. Force and moment measurements were obtained from a miniature six-component force transducer installed at the wing root. The wing was flapped in air and vacuum at the same frequency, and wing kinematics, and the resultant forces, were subtracted in order to obtain the pure aerodynamic forces. In the first part of this paper, the forces produced on the wing undergoing single-degree-of-freedom fixed-pitch pure flapping motions (no pitching or out-of-the-plane coning motions) were determined for a variety of pitch angles. The unsteady aerodynamic coefficients measured during these tests were almost six times the steady-state values measured in the wind tunnel. Flow visualization and particle image velocimetry tests...

Leakage Uncertainties in Compressors: The Case of Rotor 37

This paper revisits an old problem of validating computational fluid dynamics simulations with experiments in turbomachinery. The case considered here is NASA rotor 37. Prior computational fluid dynamics studies of this blade have been unable to predict a total pressure deficit at the hub as observed in the experiments. A possible explanation for this discrepancy is a small hub leakage flow emanating fore of the leading edge, between the forward stationary center body and the rotating disk. In this work, a large-scale high-fidelity uncertainty quantification study is carried out to investigate whether this indeed was the case. Computations are carried out on a 4.5-million-cell rotor 37 mesh with a small cavity fore of the leading edge. This cavity has an inlet with three boundary conditions: all assumed to be uncertain. A nonintrusive, orthogonal polynomial-based technique using sparse grids is used to propagate these three uncertainties, namely, leakage mass flow, leakage whirl velocity, and radial flow ...

Optimization of a supersonic airfoil using the multi-objective alliance algorithm

A baseline NACA0012 two-dimensional (2D) airfoil is optimized for supersonic flight conditions using a recently introduced optimization algorithm: the multi-objective alliance algorithm (MOAA). The efficacy of the algorithm is demonstrated through comparisons with NSGA-II for 300, 600 and 1000 function evaluations. Through epsilon/hypervolume indicators and the Mann-Whitney statistical test, we show that MOAA outperforms NSGA-II on this problem.

Effectively Subsampled Quadratures for Least Squares Polynomial Approximations

This paper proposes a new deterministic sampling strategy for constructing polynomial chaos approximations for expensive physics simulation models. The proposed approach, effectively subsampled quadratures involves sparsely subsampling an existing tensor grid using QR column pivoting. For polynomial interpolation using hyperbolic or total order sets, we then solve the following square least squares problem. For polynomial approximation, we use a column pruning heuristic that removes columns based on the highest total orders and then solves the tall least squares problem. While we provide bounds on the condition number of such tall submatrices, it is difficult to ascertain how column pruning effects solution accuracy as this is problem specific. We conclude with numerical experiments on an analytical function and a model piston problem that show the efficacy of our approach compared with randomized subsampling. We also show an example where this method fails.

Aggressive Design Under Uncertainty

This paper presents a novel approach for design under uncertainty—aggressive design. For a given objective, aggressive design seeks to minimize the distance between a designs’ PDF under uncertainty and a given target. This target is the probability distribution of the quantity of interest given uncertainty in the model inputs. The objective of aggressive design is to find a design whose PDF most closely resembles the target PDF. The framework outlined here is designed to be computationally cheaper than robust, multi-objective optimization. To evaluate the efficacy of this approach, detailed numerical experiments are carried out on the the Gaussian and beta families of PDFs—both for smooth and discrete PDFs. Finally the approach is applied to the design of an airfoil under Mach number uncertainty and compared with results obtained using robust design.

Turbomachinery Active Subspace Performance Maps

Turbomachinery active subspace performance maps are two-dimensional (2D) contour plots that illustrate the variation of key flow performance metrics with different blade designs. While such maps are easy to construct for design parameterizations with two variables, in this paper, maps will be generated for a fan blade with twenty-five design variables. Turbomachinery active subspace performance maps combine active subspaces—a new set of ideas for dimension reduction—with fundamental turbomachinery aerodynamics and design spaces. In this paper, contours of (i) cruise efficiency, (ii) cruise pressure ratio (PR), (iii) maximum climb flow capacity, and (iv) sensitivity to manufacturing variations are plotted as objectives for the fan. These maps are then used to infer pedigree design rules: how best to increase fan efficiency; how best to desensitize blade aerodynamics to the impact of manufacturing variations? In the present study, the former required both a reduction in PR and flow capacity—leading to a reduction of the strength of the leading edge bow wave—while the latter required strictly a reduction in flow capacity. While such pedigree rules can be obtained from first principles, in this paper, these rules are derived from the active subspaces. This facilitates a more detailed quantification of the aerodynamic trade-offs. Thus, instead of simply stating that a particular design is more sensitive to manufacturing variations; or that it lies on a hypothetical “efficiency cliff,” this paper seeks to visualize, quantify, and make precise such notions of turbomachinery design. [DOI: 10.1115/1.4038839]

Robust Compressor Blades for Desensitizing Operational Tip Clearance Variations

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Effective-Quadratures (EQ): Polynomials for Computational Engineering Studies

Effective Quadratures is a suite of tools for uncertainty quantification, regression and numerical integration. It is a python research code designed for the development and testing of new polynomial techniques in the areas listed above.

Sensitivity analysis of a coupled hydrodynamic-vegetation model using the effectively subsampled quadratures method (ESQM v5.2)

Coastal hydrodynamics can be greatly affected by the presence of submerged aquatic vegetation. The effect of vegetation has been incorporated into the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system. The vegetation implementation includes the plant-induced three-dimensional drag, in-canopy waveinduced streaming, and the production of turbulent kinetic energy by the presence of vegetation. In this study, we evaluate the sensitivity of the flow and wave dynamics to vegetation parameters using Sobol’ indices and a least squares polynomial approach referred to as the Effective Quadratures method. This method reduces the number of simulations needed for evaluating Sobol’ indices and provides a robust, practical, and efficient approach for the parameter sensitivity analysis. The evaluation of Sobol’ indices shows that kinetic energy, turbulent kinetic energy, and water level changes are affected by plant stem density, height, and, to a lesser degree, diameter. Wave dissipation is mostly dependent on the variation in plant stem density. Performing sensitivity analyses for the vegetation module in COAWST provides guidance to optimize efforts and reduce exploration of parameter space for future observational and modeling work.