arXiv subjects

Pranay Seshadri

Publications and source records attributed to Pranay Seshadri.

29 records · Page 2Linked to original sources

arXiv2019

Supporting Multi-point Fan Design with Dimension Reduction

Motivated by the idea of turbomachinery active subspace performance maps, this paper studies dimension reduction in turbomachinery 3D CFD simulations. First, we show that these subspaces exist across different blades---under the same parametrization---largely independent of their Mach number or Reynolds number. This is demonstrated via a numerical study on three different blades. Then, in an attempt to reduce the computational cost of identifying a suitable dimension reducing subspace, we examine statistical sufficient dimension reduction methods, including sliced inverse regression, sliced average variance estimation, principal Hessian directions and contour regression. Unsatisfied by these results, we evaluate a new idea based on polynomial variable projection---a non-linear least squares problem. Our results using polynomial variable projection clearly demonstrate that one can accurately identify dimension reducing subspaces for turbomachinery functionals at a fraction of the cost associated with prior methods. We apply these subspaces to the problem of comparing design configurations across different flight points on a working line of a fan blade. We demonstrate how designs that offer a healthy compromise between performance at cruise and sea-level conditions can be easily found by visually inspecting their subspaces.

stat.AP↗

arXiv2019

Spatial Flow-Field Approximation Using Few Thermodynamic Measurements Part II: Uncertainty Assessments

In this second part of our two-part paper, we provide a detailed, frequentist framework for propagating uncertainties within our multivariate linear least squares model. This permits us to quantify the impact of uncertainties in thermodynamic measurements---arising from calibrations and the data acquisition system---and the correlations therein, along with uncertainties in probe positions. We show how the former has a much larger effect (relatively) than uncertainties in probe placement. We use this non-deterministic framework to demonstrate why the well-worn metric for assessing spatial sampling uncertainty falls short of providing an accurate characterization of the effect of a few spatial measurements. In other words, it does not accurately describe the uncertainty associated with sampling a non-uniform pattern with a few circumferentially scattered rakes. To this end, we argue that our data-centric framework can offer a more rigorous characterization of this uncertainty. Our paper proposes two new uncertainty metrics: one for characterizing spatial sampling uncertainty and another for capturing the impact of measurement imprecision in individual probes. These metrics are rigorously derived in our paper and their ease in computation permits them to be widely adopted by the turbomachinery community for carrying out uncertainty assessments.

stat.AP↗

arXiv2019

Spatial Flow-Field Approximation Using Few Thermodynamic Measurements Part I: Formulation and Area Averaging

Our investigation raises an important question that is of relevance to the wider turbomachinery community: how do we estimate the spatial average of a flow quantity given finite (and sparse) measurements? This paper seeks to advance efforts to answer this question rigorously. In this paper, we develop a regularized multivariate linear regression framework for studying engine temperature measurements. As part of this investigation, we study the temperature measurements obtained from the same axial plane across five different engines yielding a total of 82 data-sets. The five different engines have similar architectures and therefore similar temperature spatial harmonics are expected. Our problem is to estimate the spatial field in engine temperature given a few measurements obtained from thermocouples positioned on a set of rakes. Our motivation for doing so is to understand key engine temperature modes that cannot be captured in a rig or in computational simulations, as the cause of these modes may not be replicated in these simpler environments. To this end, we develop a multivariate linear least squares model with Tikhonov regularization to estimate the 2D temperature spatial field. Our model uses a Fourier expansion in the circumferential direction and a quadratic polynomial expansion in the radial direction. One important component of our modeling framework is the selection of model parameters, i.e. the harmonics in the circumferential direction. A training-testing paradigm is proposed and applied to quantify the harmonics.

stat.AP↗

arXiv2019

Dimension Reduction via Gaussian Ridge Functions

Ridge functions have recently emerged as a powerful set of ideas for subspace-based dimension reduction. In this paper we begin by drawing parallels between ridge subspaces, sufficient dimension reduction and active subspaces, contrasting between techniques rooted in statistical regression and those rooted in approximation theory. This sets the stage for our new algorithm that approximates what we call a Gaussian ridge function---the posterior mean of a Gaussian process on a dimension-reducing subspace---suitable for both regression and approximation problems. To compute this subspace we develop an iterative algorithm that optimizes over the Stiefel manifold to compute the subspace, followed by an optimization of the hyperparameters of the Gaussian process. We demonstrate the utility of the algorithm on two analytical functions, where we obtain near exact ridge recovery, and a turbomachinery case study, where we compare the efficacy of our approach with three well-known sufficient dimension reduction methods: SIR, SAVE, CR. The comparisons motivate the use of the posterior variance as a heuristic for identifying the suitability of a dimension-reducing subspace.

stat.ME↗

arXiv2018

Numerical Evaluation of Entropy Generation in Isolated Airfoils and Wells Turbines

In recent years, a number of authors have studied entropy generation in Wells turbines. This is potentially a very interesting topic, as it can provide important insights into the irreversibilities of the system, as well as a methodology for identifying, and possibly minimizing, the main sources of loss. Unfortunately, the approach used in these studies contains some crude simplifications that lead to a severe underestimation of entropy generation and, more importantly, to misleading conclusions. This paper contains a re-examination of the mechanisms for entropy generation in fluid flow, with a particular emphasis on RANS equations. An appropriate methodology for estimating entropy generation in isolated airfoils and Wells turbines is presented. Results are verified for different flow conditions, and a comparison with theoretical values is presented.

physics.flu-dyn↗

arXiv2018

Quadrature Strategies for Constructing Polynomial Approximations

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies.

math.NA↗

arXiv2017

Kronecker Product Least Squares

In this rather brief note we present and discuss techniques for solving Kronecker matrix product least squares problems. Our main contribution is an iterative approach that uses the efficient Kronecker matrix-vector multiplication strategy (Fernandes et al. 1998) with a conjugate gradient solver. Numerical results contrast this approach---in terms of running times and accuracy---against the direct approach.

math.NA↗

arXiv2017

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.

math.NA↗

arXiv2016

An Aerodynamic Analysis of a Robustly Redesigned Modern Aero-Engine Fan

This paper documents results from a recent computational study aimed at de-sensitizing fan stage aerodynamics---in a modern, high bypass ratio aero-engine---to the effects of rear-seal leakage flows. These flows are the result of seal erosion between a rotor and stator disk in an engine, and deterioration over the life of an engine. The density-matching technique for optimization under uncertainty was applied to this problem. This involved RANS and adjoint flow solves of a full fan stage carried out at two different leakage conditions. Here a detailed analysis of the fan stage aerodynamics is carried out to determine why exactly the new design is more insensitive to the effects of leakage flows. Specifically, it is shown that this insensitivity is attributed to three main factors: a slight rearward shift in loading, and thus a reduction in incidence; a reduction in the cross-passage pressure gradient; and a re-acceleration of the flow towards the trailing edge, which prevented any corner separation.

physics.flu-dyn↗

arXiv2015

Density-Matching for Turbomachinery Optimization Under Uncertainty

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.

stat.AP↗

arXiv2015

A density-matching approach for optimization under uncertainty

Modern computers enable methods for design optimization that account for uncertainty in the system---so-called optimization under uncertainty. We propose a metric for OUU that measures the distance between a designer-specified probability density function of the system response the target and 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 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.

math.OC↗