Brian Chandler Barr

Efficient Gradient-Based Algorithms for the Construction of Pareto Fronts

The objective of this study is to develop and assess a gradient-based algorithm that efficiently traverses the Pareto front for multi-objective problems. We use high-fidelity, computationally intensive simulation tools (for eg: Computational Fluid Dynamics (CFD) and Finite Element (FE) structural analysis) for function and gradient evaluations. The use of evolutionary algorithms with these high-fidelity simulation tools results in prohibitive computational costs. Hence, in this study we use an alternate gradient-based approach. We first outline an algorithm that can be proven to recover Pareto fronts. The performance of this algorithm is then tested on three academic problems: a convex front with uniform spacing of Pareto points, a convex front with non-uniform spacing and a concave front. The algorithm is shown to be able to retrieve the Pareto front in all three cases hence overcoming a common deficiency in gradient-based methods that use the idea of scalarization. Then the algorithm is applied to a practical problem in concurrent design for aerodynamic and structural performance of an axial turbine blade. For this problem, with 5 design variables, and for 10 points to approximate the front, the computational cost of the gradient-based method was roughly the same as that of a method that builds the front from a sampling approach. However, as the sampling approach involves building a surrogate model to identify the Pareto front, there is the possibility that validation of this predicted front with CFD and FE analysis results in a different location of the “Pareto” points. This can be avoided with the gradient-based method. Additionally, as the number of design variables increases and/or the number of required points on the Pareto front is reduced, the computational cost favors the gradient-based approach.Copyright

Interpretation of Adjoint Solutions for Turbomachinery Flows

While the mathematical derivation of the adjoint equations and their numerical implementation is well established, there is a scant discussion on the understanding of the adjoint solution by itself. As this is a field solution of similar resolution of the flow-field, there is a wealth of data that can be used for design guidance. This paper addressess this specific topic. In particular, we take representative cases from turbomachinery aerodynamic problems and use the adjoint solution to identify the “physical insight” it provides. We aim to tie the adjoint solution to the flow-field which has physical properties. Towards this end, we first look at three problems 1) a fan, 2) a compressor rotor and stator, 3) a low pressure turbine. In all three of them, we focus on changes related to geometry, but one can also realize the changes using other inputs to the flow solver (eg. boundary conditions). We show how the adjoint counter-part of the density, the velocity fields and the turbulence quantities can be used to provide insights into the nature of changes the designer can induce to cause improvement in the performance metric of interest. We also discuss how to use adjoint solutions for problems with constraints to further refine the changes. Finally, we use a problem where it is not immediately apparent what geometry changes need to be used for further evaluation with optimization algorithms. In this problem, we use the adjoint and flow solution on a turbine strut, to determine the kind of end-wall treatments that reduce the loss. These changes are then implemented to show that the loss is reduced by close to 8%.© 2012 ASME

An Assessment of Non-Intrusive Probabilistic Methods for Turbomachinery Problems

The ability to quantify the impact of uncertainty on performance is an important facet of engineering design. Computational Fluid Dynamics (CFD) studies during the design cycle typically utilize estimates of boundary conditions, geometry and model constants, all of which have uncertainty that could lead to variations in the estimated performance of the design. Traditionally, engineering environments have relied on Monte-Carlo (MC) simulations to obtain probabilistic estimates. But MC methods have poor convergence rate leading to prohibitive computational requirements when used in conjunction with medium to high fidelity computational tools. In this study, we will use an alternate probabilistic approach. We assume that the uncertainties in our computational system can be modeled as random variables with known/prescribed distributions, use CFD solvers to estimate the performance measures and then use a psuedo-spectral probabilistic collocation technique to determine regression/interpolation fits. The psuedo-spectral discrete expansion uses the orthogonal polynomials from the Askey-Wiener basis and finds the coefficients of the expansion [1]. We will restrict our attention to problems with one random variable and hence can without ambiguity choose the Gauss quadratures as the optimal choice to obtain statistical data (mean, variance, moments etc.) of the performance measures. The computational frame-work will be first validated against Monte-Carlo simulations to assess convergence of pdfs. It will then be used to assess the variability in compressor blade efficiency and turbine vane loss due to uncertainty in inflow conditions. The results will be used to answer the following questions. Do we need new probabilistic algorithms to quantify the impact of uncertainty? What is the optimal basis for standard performance metrics in turbomachinery? What are the computational and accuracy requirements of this probabilistic approach? Are there alternate (more efficient) techniques? We believe that the answers to the above questions will provide a quantitative basis to assess the usefulness of non-intrusive (and possibly intrusive) probabilistic methods to analyze variability in engineering designs.Copyright