Tushar Goel

Multiple-Surrogate Approach to Helicopter Rotor Blade Vibration Reduction

The advantages of using multiple surrogates for approximation andreduction of helicopter vibration are studied. Multiple approximation methods, including a weighted-average approach, are considered so that pitfalls associated with only using a single best surrogate for the rotor blade vibration-reduction problem are avoided. A vibration objective function corresponding to a flight condition in which blade–vortex interaction causes high levels of vibration is considered. The design variables consist of cross-sectional dimensions of the structural member of the blade and nonstructural masses. The optimized designs are compared with a baseline design resembling a Messerschmitt–Bolkow–Blohm BO-105 blade. The results indicate that at relatively little additional cost compared with optimizing a single surrogate, multiple surrogates can be used to locate various reduced-vibration designs that wouldbeoverlookedifonly asingleapproximationmethodwasemployed,andthemostaccurate surrogatemaynot lead to the best design.

Surrogate Model-Based Optimization Framework: A Case Study in Aerospace Design

Summary. Surrogate-based optimization has proven very useful for novel or exploratory design tasks because it offers a global view of the characteristics of the design space, and it enables one to refine the design of experiments, conduct sensitivity analyses, characterize tradeoffs between multiple objectives, and, if necessary, help modify the design space. In this article, a framework is presented for design optimization on problems that involve two or more objectives which may be conflicting in nature. The applicability of the framework is demonstrated using a case study in space propulsion: a response surface-based multi-objective optimization of a radial turbine for an expander cycle-type liquid rocket engine. The surrogate model is combined with a genetic algorithm-based Pareto front construction and can be effective in supporting global sensitivity evaluations. In this case study, due to the lack of established experiences in adopting radial turbines for space propulsion, much of the original design space, generated based on intuitive ideas from the designer, violated established design constraints. Response surfaces were successfully used to define previously unknown feasible design space boundaries. Once a feasible design space was identified, the optimization framework was followed, which led to the construction of the Pareto front using genetic algorithms. The optimization framework was effectively utilized to achieve a substantial performance improvement and to reveal important physics in the design.

Response surface approximation of pareto optimal front in multi-objective optimization

A systematic approach is presented to approximate the Pareto optimal front (POF) by a response surface approximation. The data for the POF is obtained by multi-objective evolutionary algorithm. Improvements to address drift in the POF are also presented. The approximated POF can help visualize and quantify trade-offs among objectives to select compromise designs. The bounds of this approximate POF are obtained using multiple convex-hulls. The proposed approach is applied to study trade-offs among objectives of a rocket injector design problem where performance and life objectives compete. The POF is approximated using a quintic polynomial. The compromise region quantifies trade-offs among objectives. 2006 Elsevier B.V. All rights reserved.

Performance Estimate and Simultaneous Application of Multiple Surrogates

A typical approach in surrogate-based modeling is to assess the performance of alternative surrogate models and select the model that performs the best. In this paper, we extend the utility of an ensemble of surrogates to: i) identify regions of high uncertainties at locations where predictions of surrogates widely differ, and ii) provide a more robust approximation approach. We explore the possibility of using the best surrogate or a weighted average surrogate model instead of individual surrogate models. The weights associated with each surrogate model are determined based on the errors in surrogates. We demonstrate the advantages of an ensemble of surrogates using analytical problems and an engineering problem of radial turbine design for space launch vehicle. We show that for a single problem the choice of the surrogate can be substantially influenced by the design of experiments.

Application of a Weighted Average Surrogate Approach to Helicopter Rotor Blade Vibration Reduction

The advantages of employing multiple approximation methods and the effectiveness of weighted average surrogate modeling for approximation and reduction of helicopter vibrations is studied. Multiple surrogates, including the weighted average approach, are considered so that the need to identify the “best” approximation method for the rotor vibration reduction problem is eliminated. Various approximation methods are used to generate a vibration objective function corresponding to a flight condition in which bladevortex interaction causes high levels of vibration. The design variables consist of the crosssectional dimensions of the structural member of the blade and non-structural masses. The optimized designs are compared with a baseline design resembling an MBB BO-105 blade. The results indicate that (a) multiple surrogates can be used to locate low vibration designs which would be overlooked if only a single approximation method was employed, and (b) that the weighted average approach protects against the worst individual surrogate, while performing as well as the best individual surrogate. Furthermore, the surrogates were used in a global sensitivity analysis to identify the most significant design variables for the vibration reduction problem.

Global Sensitivity and Trade-Off Analyses for Multi-Objective Liquid Rocket Injector Design

This paper presents sensitivity and trade-off analyses for the design of a gaseous injector used for liquid rocket propulsion whose geometry (hydrogen flow angle, hydrogen and oxygen flow areas and oxygen post tip thickness) is sought to optimize performance (combustion length) and life/survivability (temperatures at three different locations). The analyses are conducted using data available from surrogate models of the design objectives, and the multi-objective optima (Pareto optimal front, POF) generated with the aid of multi-objective genetic algorithm and local search method. The trade-offs among objectives are estimated in different clusters (identified through a hierarchical clustering algorithm) along the POF. Sensitivity analyses are conducted using a variance-based non-parametric approach over the whole design space as well as on the clusters at the POF. The former analysis help identify the contribution of the design variables to the objective variability. The latter analysis highlights: the variability of the design variables and the objectives within a cluster of the POF (using box plots), and the relationship between the design variables and the objectives (using partial correlation coefficients). Additionally, the trade-offs between selected pairs of objectives are also analyzed.

Interaction between Grid and Design Space Refinement for Bluff Body-Facilitated Mixing

The interaction between grid refinement and features in design space is investigated for time dependent Navier-Stokes flows over a bluff body. A mixing measure and total pressure loss are calculated for a range of geometries and several grid refinements. In particular, 52 geometries are analyzed and a response surface surrogate-based outlier analysis revealed that as the grid is refined, three of these configurations stand out with off-trend mixing indices. Both grid refinement and multiple surrogate modeling exercises reveal that very high mixing indices are found in a very small island in design space. This important design region manifests itself only when the grid resolution is adequate. The high value of the mixing index is due to interaction between viscous flows and abrupt geometric variations of the bluff body . based on selected number of original CFD solutions 1-11 . In addition to offering a low-cost alternative for evaluating designs, surrogate models also offer advantages associated with the fact that they require a large number of designs to be evaluated together. Besides obvious advantages in terms of parallel computation, this can also reveal cases with significantly different behavior than others. Histogram and outlier analysis can identify and examine designs exhibiting unusual departure from the overall trend. The outliers can occur due to incomplete convergence or high errors due to inappropriate computational set-up such as grid distributions or boundary conditions. In that case, outlier analysis helps find and possibly correct these problems. However, outlier may also represent designs where the physical behavior changes significantly. In this study, the mixing problem associated with bluff body flows is investigated. Mixing is a process with many practical applications, including propulsion and power generation, homogenization of multiple materials and/or species, and various heat exchange and geophysical processes. In man-made devices, �

Error measures for noise-free surrogate approximations

Error measures are important for assessing uncertainty in surrogate predictions. We use a suite of test problems to appraise a few error estimations for polynomial response surface and kriging. In addition, we study the performance of cross-validation error measures that can be used with any surrogate. We use a large number of experimental designs to obtain the variability of error estimates with respect to experimental designs for each problem. We find that the (actual) errors for polynomial response surfaces are less sensitive to the choice of experimental designs than kriging errors. This is attributed to the variability in the maximum likelihood estimates of the kriging parameters. We find that no single error measure outperforms other measures on all test problems. Computationally expensive integrated local error measures (standard error for polynomials and mean square error for kriging) estimate the actual root mean square error very well. The distribution-free cross-validation error characterized actual errors reasonably well. While estimated root mean square error for polynomial response surface is a good estimate of the actual errors, process variance for kriging is not. We explore methods of simultaneously using multiple error measures and demonstrate that the geometric means of combinations of multiple error measures improve the assessment of the actual errors compared to the individual error measures.

Pointwise RMS bias error estimates for design of experiments

Response surface approximations offer an effective way to solve complex problems. However limitations on computational expense pose restrictions to generate ample data and a simple model is typically used for approximation leaving the possibility of errors due to insufficient model known as bias errors. This paper presents a method to estimate pointwise RMS bias errors in response surface models. Prior to generation of data, RMS bias error estimates can be used to construct design of experiments (DOEs) to minimize the maximal RMS bias errors or compare different DOEs. It is demonstrated that for high dimensional design spaces, central composite design gives a reasonable trade-off between variance and bias errors under conventional assumption of quadratic model and cubic true function. The results indicate that compared to bounds on bias error, RMS estimates are less affected by increase in the dimensionality of the problem. Latin Hypercube Sampling (LHS), though very popular, may yield large unsampled regions which can be successfully detected by implementing an alternate criterion like maximum standard error or maximum RMS bias error in design space. We further demonstrate that poor LHS designs can be eliminated by using more than one DOE. With the help of a non-polynomial example problem, it is demonstrated that the proposed method is correctly able to identify the regions of high errors even when assumed true model is not accurate.

An Assessment of Geometry-based Convergence Metrics for Multi-Objective Evolutionary Algorithms

,Four geometric properties of the archive of non-dominated solutions are used to investigate convergence metrics for multi-objective optimization problems. The metrics are (a) average distances between consecutive non-dominated solutions, (b) spread of the nondominated set, (c) uniformity of the non-dominated set and (d) hypervolume of the nondominated set with respect to a reference point. To demonstrate and compare the practical utility of the metrics, they are applied to two analytical and two crashworthiness optimization problems. It is concluded that the hypervolume change criterion holds great promise because of its inherent stability and monotonicity. Some of the other criteria, especially the spread, are less predictable to a degree depending on the nature of the Pareto Front.