Rajkumar Vaidyanathan

Response Surface and Neural Network Techniques for Rocket Engine Injector Optimization

Theresponsesurfacemethodologyforrocketengineinjectordesignoptimizationforwhichonlymodestamounts of data may exist is examined. Two main aspects are emphasized: relative performance of quadratic and cubic polynomial response surfaces and enhancement of the e delity of the response surface via neural networks. A data set of 45 design points from a semi-empirical model for a shear coaxial injector element using gaseous oxygen and gaseous hydrogen propellants is used to formulate response surfaces using quadratic and cubic polynomials. This original data set is also employed to train a two-layered radial basis neural network (RBNN). The trained network is then used to generate additional data to augment the original information available to characterize the design space. Quadratic and cubic polynomials are again used to generate response surfaces for this RBNN-enhanced data set. The response surfaces resulting from both the original and RBNN-enhanced data sets are compared for accuracy. Whereas the cubic e t is superior to the quadratic e t for each data set, the RBNN-enhanced data set is capable of improving the accuracy of the response surface if noticeable errors from polynomial curve e ts are encountered. Furthermore, the RBNN-enhanced data set yields more consistent selections of optimal designs between cubic and quadratic polynomials. The techniques developed can be directly applied to injectordesign and optimization for rocket propulsion.

Sensitivity Evaluation of a Transport-Based Turbulent Cavitation Model

A sensitivity analysis is done for turbulent cavitating flows using a pressure -based Navier-Stokes solver coupled with a phase volume fraction transport model and non-equilibrium k-e turbulence closure. Four modeling parameters are adopted for evaluation, namely, Ce1 and Ce2, which directly influences the production and dissipation of turbulence kinetic energy, and Cdest and Cprod, which regulate the evaporation and condensation of the phases. Response surface methodology along with design of experiments is used for the sensitivity studies. The difference between the computational and experimental results is used to judge the model fidelity. Under non-cavitating conditions, the best selections of Ce1 and Ce2, exhibit a linear combination with multiple optima. Using this information, cavitating flows around an axi -symmetric geometry with a hemispherical fore-body and the NACA66(MOD) airfoil are assessed. Analysis of the cavitating model shows that the favorable combinations of Cdest and Cprod, are inversely proportional to each other for the geometries considered. A set of cavitation numbers is selected for each of the geometries to demonstrate the predictive capability of the present modeling approach for attached, turbulent cavitating flows.

Neural Network and Response Surface Methodology for Rocket Engine Component Optimization

The goal of this work is to compare the performance of response surface methodology (RSM) and two types of neural networks (NN) to aid preliminary design of two rocket engine components. A data set of 45 training points and 20 test points, obtained from a semi-empirical model based on three design variables, is used for a shear coaxial injector element. Data for supersonic turbine design is based on six design variables, 76 training data and 18 test data obtained from simplified aerodynamic analysis. Several RS and NN are first constructed using the training data. The test data are then employed to select the best RS or NN. Quadratic and cubic response surfaces, radial basis neural network (RBNN) and back-propagation neural network (BPNN) are compared. Twolayered RBNN are generated using two different training algorithms, namely, solverbe and solverb. A two-layered BPNN is generated with Tan-Sigmoid transfer function. Various issues related to the training of the neural networks are addressed, including number of neurons, error goals, spread constants, and the accuracy of different models in representing the design space. A search for the optimum design is carried out using a standard, gradient-based optimization algorithm over the response surfaces represented by the polynomials and trained neural networks. Usually a cubic polynomial performs better than the quadratic polynomial but exceptions have been noticed. Among the NN choices, the RBNN designed using solverb yields more consistent performance for both engine components considered. The training of RBNN is easier as it requires linear regression. This coupled with the consistency in performance promise the possibility of it being used as an optimization strategy for engineering design problems.

Computational-Fluid-Dynamics-Based Design Optimization for Single-Element Rocket Injector

A computational-fluid-dynamics-based design optimization approach, utilizing the response surface method, has been proposed for a single-element rocket injector. The overall goal of the effort is to demonstrate the integration of a set of computational and optimization tools to enable the injector designer to objectively determine the trades between performance and life during the design cycle. Using design of experiment techniques, 54 cases are selected, and computational solutions based on the Navier‐Stokes equations, finite rate chemistry, and the k‐e turbulence closure are obtained. The response surface methodology is employed as the optimization tool. Four independent design variables are selected, namely, H2 flow angle, H2 and O2 flow areas with fixed flow rates, and O2 posttip thickness. Design optimization is guided by four design objectives. The maximum temperature on the injector element oxidizer posttip, the maximum temperature on the injector face, and a combustion chamber wall temperature are chosen as life indicators. The length of the combustion zone is selected as an indicator of mixing and performance. In the context of this effort, the design optimization tools performed efficiently and reliably. In addition to establishing optimum designs by varying emphasis on the individual objectives, better insight into the interplay between design variables and their impact on the design objectives is gained. The need to include environmental design objectives early in the design phase is clearly established.

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.

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.

An optimization-based approach to injector element design

An injector optimization methodology, method i, is used to investigate optimal design points for gaseous oxygen/gaseous hydrogen (GO2/GH2) injector elements. A swirl coaxial element and an unlike impinging element (a fuel-oxidizer-fuel triplet) are used to facilitate the study. The elements are optimized in terms of design variables such as fuel pressure drop, APf, oxidizer pressure drop, deltaP(sub f), combustor length, L(sub comb), and full cone swirl angle, theta, (for the swirl element) or impingement half-angle, alpha, (for the impinging element) at a given mixture ratio and chamber pressure. Dependent variables such as energy release efficiency, ERE, wall heat flux, Q(sub w), injector heat flux, Q(sub inj), relative combustor weight, W(sub rel), and relative injector cost, C(sub rel), are calculated and then correlated with the design variables. An empirical design methodology is used to generate these responses for both element types. Method i is then used to generate response surfaces for each dependent variable for both types of elements. Desirability functions based on dependent variable constraints are created and used to facilitate development of composite response surfaces representing the five dependent variables in terms of the input variables. Three examples illustrating the utility and flexibility of method i are discussed in detail for each element type. First, joint response surfaces are constructed by sequentially adding dependent variables. Optimum designs are identified after addition of each variable and the effect each variable has on the element design is illustrated. This stepwise demonstration also highlights the importance of including variables such as weight and cost early in the design process. Secondly, using the composite response surface that includes all five dependent variables, unequal weights are assigned to emphasize certain variables relative to others. Here, method i is used to enable objective trade studies on design issues such as component life and thrust to weight ratio. Finally, combining results from both elements to simulate a trade study, thrust-to-weight trends are illustrated and examined in detail.

CFD-Based Design Optimization for Single Element Rocket Injector

To develop future Reusable Launch Vehicle concepts, we have conducted design optimization for a single element rocket injector, with overall goals of improving reliability and performance while reducing cost. Computational solutions based on the Navier-Stokes equations, finite rate chemistry, and the k-E turbulence closure are generated with design of experiment techniques, and the response surface method is employed as the optimization tool. The design considerations are guided by four design objectives motivated by the consideration in both performance and life, namely, the maximum temperature on the oxidizer post tip, the maximum temperature on the injector face, the adiabatic wall temperature, and the length of the combustion zone. Four design variables are selected, namely, H2 flow angle, H2 and O2 flow areas with fixed flow rates, and O2 post tip thickness. In addition to establishing optimum designs by varying emphasis on the individual objectives, better insight into the interplay between design variables and their impact on the design objectives is gained. The investigation indicates that improvement in performance or life comes at the cost of the other. Best compromise is obtained when improvements in both performance and life are given equal importance.