AIAA 2004-4007 GLOBAL SENSITIVITY AND TRADE-OFF ANALYSES FOR MULTI-OBJECTIVE LIQUID ROCKET INJECTOR DESIGN

Rajkumar Vaidyanathan(1), Tushar Goel(1), Wei Shyy(1), Raphael T. Haftka(1), Nestor V. Queipo(2) and P. Kevin Tucker(3) University of Florida(1) Gainesville, FL University of Zulia(2) Venezuela NASA Marshall Space Flight Center(3) Huntsville, AL

40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit

11-14 July 2004 / Fort Lauderdale, FL

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

Rajkumar Vaidyanathan*, Tushar Goel§, Wei Shyy¥ and Raphael T. Haftka†

Department of Mechanical and Aerospace Engineering University of Florida, Gainesville, FL 32611.

Nestor V. Queipo‡

Applied Computing Institute, Faculty of Engineering University of Zulia, Venezuela

and

Kevin Tucker#

NASA Marshall Space Flight Center MS/TD64, MSFC, AL 35812

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.

I. Introduction

A critical goal for space propulsion design is to make the device safer, more affordable and more reliable. The design of combustion devices, namely, injectors, chambers and nozzles, will help in meeting these goals. The characteristics of the injector design are a key factor for both performance and thrust chamber environments. The thrust chamber performance is estimated by the rate and the extent to which mixing and resultant combustion occurs. The location of the mixing and resultant combustion determines the injector and thrust chamber thermal environments. These environments include temperatures on the combustor wall, the injector face and, for coaxial injectors, the oxidizer post tip. The difficulty encountered in designing injectors that perform well and have manageable environments is that the factors that promote

∗ Graduate Student, Student Member AIAA §Graduate Student ¥Professor and Department Chair, Fellow AIAA †Distinguished Professor, Fellow AIAA ‡Professor #Aerospace Engineer, Member AIAA Copyright 2004 by authors. Published by the American Institute of Aeronautics and Astronautics, Inc, with permission.

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performance often lead to increased heating of the solid surfaces of the injector and combustor thereby reducing the life expectancy or survivability of these components. Injector design tools have undergone a considerable amount of evolution over time. In the past the tools were largely empirically based1-5. The experimental databases, and thus the tools developed from them are limited in terms of design space, to specific element configurations that have been tested6. In terms of scope, these design tools typically focus on performance, with the life expectancy/survivability defined in the context of the thermal environment. The amount of information available from these tools is usually one-dimensional and not functionally related to the details of the injector design. It is very doubtful that application of these traditional design tools will successfully address the goals of the future propulsion devices. Hence more advanced tools like CFD-based optimization have been developed to address the design of these devices. The application of CFD to injector design has lagged behind other areas such as turbomachinery because the physical models are more complicated for multiphase, turbulent reacting flows. Recently, new numerical models that efficiently account for some of these complex processes 7 and thus improve the solution fidelity have become available. Also, advances in computer speed and progress in parallel processing have begun to reduce the computation time. However, the three-dimensional geometry of multi-element injectors and the complex physical processes inherent in flows that issue from them create major obstacles in the validation process. The high pressure and temperature environments typical of injector flows create significant difficulty in obtaining experimental data of satisfactory quality to validate and guide further development of computational models. It has long been known that small changes in injector geometry can have significant impact on performance8, as well as on environmental variables such as combustion chamber wall and injector face temperatures and heat fluxes. Vaidyanathan et al.9 in their work studied the injector design based on different trade-offs among the objectives and drawn conclusion on design trends. In a companion study, Goel et al.10 presented a systematic approach to approximate the multi-objective optima (Pareto optimal front, POF) using the data obtained from surrogate models of the design objectives. The current paper focuses on sensitivity and trade-off analyses for the design of a gaseous injector for liquid rocket propulsion. The data for the analyses is obtained from surrogate models of the design objectives, and the multi-objective optima (Pareto optimal front, POF) generated by Goel et al.10 with the aid of multi-objective genetic algorithm and a local search method (∈-constraint strategy). The regions of the POF that represent different trade-offs among the objectives are obtained through a hierarchical clustering algorithm. The information from the global sensitivity analysis can be useful in reducing the number of variables to be included in the construction of the surrogate models . For example, Knill et al.11 have used linear aerodynamics to identify the important terms in the polynomia l-based RSA which were then used for creating the surrogate model from Euler analyses. This reduced considerably the number of points needed for the Euler design of experiments. The current study is broadly divided into two parts. Firstly a sensitivity study is carried over the whole design space. The contribution of the design variables to the objectives variability is calculated using a variance-based non-parametric approach and correlations between objectives are investigated. Secondly, sensitivity analyses are conducted on clusters along the POF. Box plots are used to highlight the variability of the design variables and the objectives within a cluster. Additionally, the linear relationships between the design variables and the objectives are explored with the aid of partial correlation coefficients. These measures can be used to tune the design variables in a chosen cluster so as to improve on the objectives as per design requirements.

II. Injector Model Liquid rocket propulsion injector elements can be categorized into two basic types based on propellant mixing. The first type is an impinging element (Figure 1a) where mixing occurs by direct impingement of the propellant streams at an acute angle. This enhances mixing by head-on interaction between the oxidizer and fuel8. The second type of injector consists of non-impinging elements where the propellant streams flow in parallel, typically in coaxial fashion (Figure 1b). Here, mixing is accomplished through a shear-mixing processes6. From a design standpoint, both element types have appealing as well as undesirable characteristics. For instance, if the impinging element has a fuel-oxidizer-fuel (F-O-F) arrangement (Figure 1a), the mixing occurs rapidly, which can yield high performance. However, since the combustion zone is close to the injector face, the potential for high levels of injector face heating must be considered since it adversely affects the life/survivability of the injector. If the non-impinging element is assumed to be a shear coaxial ele ment, mixing across the shear layer is relatively slow8, requiring longer chambers to allow complete combustion. However, since the combustion zone spreads over a longer axial distance, the injector face is generally exposed to less severe thermal environments which results in longer life/survivability.

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The above-mentioned characteristics of injector types can be combined to develop new designs. For example, it has been noted8 that performance improvement in the shear coaxial element can be realized by thinning the oxidizer post wall and directing the fuel toward the oxidizer stream rather than parallel to it. The first modification causes the shear coaxial element to take on some aspects of the F-O-F impinging element. These notions lead to the injector element (Figure 2) that has been developed by The Boeing Company (U. S. Patent 6253539). The injector design used for the current study is based on the injector shown in Figure 2. The design variables chosen for this study are the angle, α, at which H2 is directed toward the oxidizer, the change in H2 flow area from the baseline, ∆HA, the change in O2 flow area from the baseline, ∆OA, and the oxidizer post tip thickness, OPTT (Figure 3a). The fuel and oxidizer flow rates are held constant. The ranges of the design variables are shown in Figure 3a. The H2 flow angle varies between 0° and α°, the H2 flow area varies between 0% and 25% of the baseline H2 flow area ((1+0.25) × baseline H2 flow area), the O2 flow area varies between 0% and 40% of the baseline O2 flow area ((1-0.40) × baseline O2 flow area) and the O2 post tip thickness varies between x inches to 2x inches. The design variables are normalized between zero and one based on their minimum and maximum values to avoid scalability issues. Similarly, the objectives are also normalized between zero and one based on the minimum and maximum values of the CFD solutions in the whole design space. The four objectives chosen for the injector design are TTmax, the maximum temperature on the oxygen post tip, TFmax, the maximum temperature on the injector face, TW4, the adiabatic wall temperature three inches from injector face and Xcc, the x-location on the injector centerline where combustion is 99 % complete (Figure 3b). Of the four objectives, the temperatures control the life/survivability of the injector and the combustion length, Xcc, controls the performance. Minimizing the temperatures increases the life/survivability and minimizing Xcc improves the performance.

III. Approach Vaidyanathan et al.9 in their work have generated the surrogate models of the 4 objectives based on the CFD results for the designs obtained using design of experiment (DOE)12, 13 techniques. The numerical procedure and the response surface methodology (RSM)12 used in their study are described in subsections A and B, respectively. In the current work, the surrogate models shown in subsection B are used to conduct sensitivity and trade-off analyses. The approaches used to estimate the sensitivity indices, namely the global indices14 and the partial correlation coefficients15 are given in subsections C. Section D describes the details of the box plot15 used to visualize the variability of design variables and objectives. The multi-objective genetic algorithm (MOGA)16 and local search method (∈-constraint strategy)17 along with the response surfaces approximations (RSA) used to obtain the POF are explained in subsection E. Finally in subsection F, a hierarchical clustering algorithm18 used to identify regions along the POF is explained. The variability of the design variables and objectives in these regions are analyzed using box plots and partial correlation coefficients. A. Flow Solver A pressure-based, finite difference, Navier-Stokes solver, FDNS500-CVS19-21, is used in this study. The Navier-Stokes equations, the two-equation turbulence model, and kinetic equations are solved. Convection terms are discretized using either a second order upwind, third order upwind or a central difference scheme, with adaptively added second order and fourth order dissipation terms. For the viscous and source terms, second order central differencing scheme is used. First order upwind scheme is used for scalar quantities, like turbulence kinetic energy and species mass fractions, to ensure positive values. Steady state is assumed and an implicit Euler time marching scheme is used for computational efficiency. The chemical species continuity equations represent the H2 - O2 chemistry. It is represented with the aid of a 7-species and 9-reaction set19-21. The simulation domain and the boundary conditions used in all the CFD cases are shown in Figure 4. Because of the very large aspect ratio, both the injector and chamber have been shortened (at the cross hatched areas) in Figure 4 for clarity. Both fuel and oxidizer flow in through the west boundary where the mass flow rate is fixed for both streams. The nozzle exit at the east boundary is modeled by an outlet boundary condition. The south boundary is modeled with the symmetry condition. All walls (both sides of the oxygen post, the outside of the fuel annulus, the outside chamber wall, and the faceplate) are modeled with the no-slip adiabatic wall boundary condition. B. Response Surface Methodology The response surface methodology (RSM)12 consists of performing a series of experiments or numerical analyses, for a prescribed set of design points, and to construct a RSA of the calculated quantity over the design space. Design of experiment (DOE)12,13 is used for the selection of the design space as it helps reduce the effect of noise on the fitted response surface.

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The response surface approximations are constructed by standard least-squares regression with a quadratic/cubic polynomial using JMP, a statistical analysis software15. The quality of fit between different surfaces is evaluated by comparing different error measures10, 12. A measure of the variability in an objective accounted by its surrogate model is given by the coefficient of multiple determination11, R2, given as

2 1 E

yy

SSR

SS= − (1)

where SSE is the sum of squares of errors (difference between the predicted value and the observations) and SSyy is the sum of squares of the difference between the mean and the observations. For a good fit, R2 should be close to one. Vaidyanathan et al.9 used orthogonal array (OA)22 to generate 54 design points for the single element rocket injector based on the 4 design variables, of which 14 were selected for testing the RSAs. These 14 design points were selected using a k-fold cross validation technique23. Of the 40 design points used two were unacceptable because the corresponding CFD solutions exhibited unsteady behaviors. Quadratic polynomials were generated for TFmax, TW4 and Xcc. A reduced cubic polynomial was fit for TTmax. For TTmax, all 52 design points were used for the RSA and the testing was done using the press statistic measure12. The normalized design variables and objectives are used for the RSAs. The obtained RSAs are shown below. TFmax = 0.692 + 0.477(α) – 0.687(∆HA) – 0.080(∆OA) – 0.0650(OPTT) – 0.167(α)2 – 0.0129(∆HA)(α) + 0.0796(∆HA)2 – 0.0634(∆OA)(α) – 0.0257(∆OA)(∆HA) + 0.0877(∆OA)2 – 0.0521(OPTT)(α) + 0.00156(OPTT)(∆HA) + 0.00198(OPTT)(∆OA) + 0.0184(OPTT)2 (2) TW4 = 0.758 + 0.358(α) – 0.807(∆HA) + 0.0925(∆OA) – 0.0468(OPTT) – 0.172(α)2 + 0.0106(∆HA)(α) + 0.0697(∆HA)2 – 0.146(∆OA)(α) – 0.0416(∆OA)(∆HA) + 0.102(∆OA)2 – 0.0694(OPTT)(α) – 0.00503(OPTT)(∆HA) + 0.0151(OPTT)(∆OA) + 0.0173(OPTT)2 (3) TTmax = 0.370 – 0.205(α) + 0.0307(∆HA) + 0.108(∆OA) + 1.019(OPTT) – 0.135(α)2 + 0.0141(∆HA)(α) + 0.0998(∆HA)2 + 0.208(∆OA)(α) – 0.0301(∆OA)(∆HA) – 0.226(∆OA)2 + 0.353(OPTT)(α) – 0.0497(OPTT)(∆OA) – 0.423(OPTT)2 + 0.202(∆HA)(α)2 – 0.281(∆OA)(α)2 – 0.342(∆HA)2(α) – 0.245(∆HA)2(∆OA) + 0.281(∆OA)2(∆HA) – 0.184(OPTT)2(α) + 0.281(∆HA)( α)(∆OA) (4) Xcc = 0.153 - 0.322(α) + 0.396(∆HA) + 0.424(∆OA) + 0.0226(OPTT) + 0.175(α)2 + 0.0185(∆HA)(α) – 0.0701(∆HA)2 – 0.251(∆OA)(α) + 0.179(∆OA)(∆HA) + 0.0150(∆OA)2 + 0.0134(OPTT)(α) + 0.0296(OPTT)(∆HA) + 0.0752(OPTT)(∆OA) + 0.0192(OPTT)2 (5) Vaidyanthan et al.9 have compared different error measures to show that the generated RSAs are accurate enough. Additionally, the R2

nonlinear listed in Table 2 for the 4 objectives have values close to 1 for all the objectives suggestion a good fit. C. Sensitivity Analysis Unlike local sensitivity where the partial derivatives are used to locally estimate the sensitivity of an objective to a specific design variable, global sensitivity allows the study of overall model behavior14. Sobol14 has proposed a variance-based non-parametric approach to estimate the global sensitivity using Monte Carlo methods. To explain the concept, the surrogate model of the performance measure, Xcc as a function of the normalized design variables, (α, ∆HA, ∆OA, OPTT) is used. To calculate the total sensitivity to any design variable, say α, the design variable set is divided into two complementary subsets, α and Z (Z = ∆HA, ∆OA, OPTT). The purpose of using these subsets is to isolate the influence of α on Xcc variability from the influence of the remaining design variables included in Z. The total sensitivity index for α is then defined as

total totalS D Dα α= (6)

where

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,total

ZD D Dα α α= + (7)

Dα is the partial variance associated with α and Dα,Z is the sum of all partial variances associated with any combination of the remaining variables representing the interactions between α and Z. Similarly the partial variance for Z can be defined as DZ. Therefore the total variability of the performance measure can be written as

,Z ZD D D Dα α= + + (8)

In this study, the multidimensional integrals for computing the partial variances can be evaluated analytically. However, the Monte Carlo approach which is applicable under rather general conditions (e.g., any model, design under uncertainty) has been adopted. The designs for the current approach are selected without any preference of one design over the other from the uniformly distributed design space of unit sides. Hence the design variables are uncorrelated which is essential for the effective implementation of the method proposed by Sobol14. The variance estimates can be obtained using the following expressions:

( ) 01

1 ,N

cc jjX Z f

Nα

=→∑ (9)

where f0 defines the mean of Xcc.

( ) ( ) 20

1

1 , ,N

cc cc jjX Z X Z D f

N αα α=

′ → +∑ (10)

( )2 20

1

1 ,N

cc jjX Z D f

Nα

=→ +∑ (11)

( ) ( ) 20

1

1, ,

N

cc cc Zj j

X Z X Z D fN

α α=

′ → +∑ (12)

where the terms (α, Z) and (α’, Z’) represent random sample designs. Equations (10), (11) and (12) give the estimates for Dα, D and DZ, respectively and Dα,Z can be calculated using Equation (8). Once these estimates are known the main factor and total sensitivity indices of the objectives variability with respect to a given design variable can be obtained using Equation (6). The influence of a design variable on an objective variability without accounting for any of its interactions with other variables is called a main factor index and given as

S D Dα α= (13) For the current study, to estimate the global sensitivity indices 107 pairs of random designs were necessary to achieve a convergence to the third significant digit in the index values. Note that: each pair of random samples requires three different objective function evaluations (eg. Xcc(α, Z), Xcc(α, Z’) and Xcc(α’, Z)), the mean and the total variance of an objective need to be estimated only once, and only two Monte Carlo integrals per design variable are necessary to compute the main factor and total sensitivity indices. The sensitivity indices are computed using a Matlab24 code which is validated using a well-known benchmark problem25. These indices will be used to understand the influence of design variables of the injector on the variability of life/survivability and performance related objectives. The design variables at different regions along the POF share some similar features. Therefore, the design variables are correlated and this correlation has to be accounted for before estimating the influence of a design variable on an objective. For this purpose, a partial correlation coefficient15 was calculated. A partial correlation coefficient15 gives a measure of the linear relationship of a design variable, say α, with an objective, say Xcc, after the influence of other variables have been filtered out. Linear fits are obtained for α and Xcc as a function of Z (Z = ∆HA, ∆OA, OPTT) and the residuals measured as the difference between the data used for fitting and the predicted

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value of the fit are estimated. A partial correlation coefficient is the correlation between these two residuals , r1 and r2 and is given by

( )( )1 2

1 1 2 2corr

r r

E r r r rR

σ σ

− − = (14)

D. Visualization Using Box Plot Box plot15 is a visualization tool which can help understand the variability in the design variables and objectives within the clusters of the POF. It can also assist identifying the design variable that could be fixed when analyzing a given cluster. A schematic of a box plot with the y-axis representing the range of the normalized design variable is shown in Figure 5. Figure 5 shows that 25%, 50% and 75% of the data lie below the lower hinge, median and upper hinge, respectively. The difference between the upper and lower hinges is known as the “H-spread”. The inner fence is located 1 step beyond the hinges which is equal to 1.5 times the H-spread. The upper adjacent value is identified as the largest value below the upper inner fence. The lower inner fence and lower adjacent value can be similarly determined. These box plots will be used to visualize the variability along the POF. The spread between the upper and lower adjacent values gives the range of variation of a variable in a given set. Any variable lying beyond this spread is a potential outlier of the set. Small range gives a tight bound on that particular variable. Sometimes the box plot is known to collapse to a single point which suggests that the variable should be fixed at that value. E. Multi-Objective Genetic Algorithm (MOGA) for Pareto Optimal Front MOGA has been widely used to address multi-objective optimization problem. Before describing the algorithm, it is essential to understand the concept of dominated and non-dominated solutions. In a multi-objective optimization scenario, when the objectives are conflicting in nature, many representative optimal solutions can be obtained. All these solutions comprise a set called the Pareto-optimal set26. These solutions are such that no improvement is possible in any objective without sacrificing at least one of the remaining objectives. Hence these solutions are non-dominated. On the other hand if for a given solution there are other solutions where improvement in any objective is possible without sacrificing the remaining objectives, that solution is said to be dominated. The function space of all the non-dominated solutions in the Pareto-optimal set is termed the Pareto-optimal front (POF)26. When there are two objectives, the POF is a curve. When there are three objectives, the POF is represented by a surface. If there are more than three objectives, it is represented by hyper-surfaces. MOGA works on the principle of survival of the fittest. Genetic operators like reproduction, crossover and mutation are employed for finding the optimal solution. The multi-objective injector design optimization problem has been solved by Goel at al.10. In this study, NSGA -II16 is used as the MOGA for finding the Pareto-optimal solutions. The algorithm of NSGA -II can be described as:

1. Randomly initialize population (injector designs) of size N. 2. Compute objective functions for each design. 3. Rank the population using non-domination criteria. 4. Compute crowding distance (This distance finds the relative closeness of the solution with other solutions in the

objective space.) 5. Employ genetic operators – selection, crossover & mutation – Create new population. 6. Evaluate objective functions for the new population. 7. Combine the two populations, rank them and compute the crowding distance. 8. Select N best individuals. 9. Go to step 3 and repeat till termination criteria is reached, which in the current study is chosen to be the number of

generations. It is shown in a number of studies that using a combination of GA and local search (also known as hybrid GA), helps achieve faster convergence to the global Pareto optimal solution set17, 27-30. The posteriori hybrid method17 used in this study assumes the set of solutions generated by GA simulations as a starting point for the local search. Most of the local search methods are very efficient only for single objective optimization problems. Hence, a ∈-constraint strategy17 is used to reduce the dimensionality of the current problem. Sequential quadratic programming, available in Matlab24, serves as the local search optimizer. This gives a set of solutions from which the dominated and duplicate solutions are removed to obtain the global Pareto optimal solution set.

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The multi-objective optimization problem is solved using the NSGA -II algorithm10 and further refined using the ∈–constraint local search strategy coded in Matlab24. In this strategy, first, one of the objectives is treated as the objective function and rest of the objectives are treated as equality constraints . The constraint value is set equal to the corresponding objective value as found by NSGA -II simulation. The corresponding design variable vector is used as initial guess. This procedure is repeated for all individuals (designs) in the population. This gives a set of Pareto optimal solutions, referred to as Set 1 in this study. Next, a different objective is chosen as the objective and the remaining objective functions are treated as constraints. For this problem, the obtained Pareto optimal set is referred to as Set 2. Similarly, that many Pareto optimal set are obtained as the number of objectives. Now all the obtained Pareto optimal solutions (Sets 1, 2, …) and the original NSGA -II set are combined. To find the true Pareto optimal front, a non-domination check is carried out on this set of solutions and duplicates are removed. F. Hierarchical Clustering Method The POF can be divided into a number of clusters using a hierarchical clustering algorithm18 to assist the designer in selecting the optimal solution of choice. The clustering algorithm can be summarized as:

1. Start by assuming all the solutions as individual clusters. 2. Find the mean of each cluster. 3. Find the distance between clusters. 4. Merge closest clusters. 5. Go to step 2 till the number of clusters is equal to K. 6. Find the member of each cluster closest to the mean of the cluster. This is the representative element.

IV. Results and Discussion

It has been observed by Vaidyanathan et al.9 that the four objectives, based on the designs obtained during single objective minimization, largely fall into two groups: (TFmax, TW4) and (TTmax, Xcc). However, there are differences between them. The change in oxygen flow area, ∆OA, affects the two objectives in each group differently. The observation also indicates that the three life/survivability -related objectives require compromises among design variables. Minimizing TFmax and TW4 leads to a design with α equal to zero (shear coaxial element), maximum fuel flow area and the thickest post tip. This design also yields moderate to poor performance due to the slow mixing across the shear layer. Minimizing TTmax and Xcc results in an impinging-like design with α equal to one. It also has the minimum fuel flow area and the thinnest post tip thickness. This design performs well, but has very high wall and injector face temperatures. In the present effort, the sensitivity of the objective variability to the design variables, trade-offs between selected objectives and the POFs are systematically investigated.

A. Global Analyses Table 1 summarizes the results of this study and lists the essential and non-essential design variables with respect to individual objective variability. A design variable is considered essential if it is responsible for at least 5% of the objective variability. Figure 6 shows the percentage of main factor (Si) contribution of different design variables to individual objectives. The variability of TFmax is largely influenced by ∆HA and moderately by α (Figure 6a). The effect of the other design variables is marginal suggesting that they are non-essential and in principle could be fixed. The variability of TW4 is considerably influenced only by ∆HA (Figure 6b). TTmax is influenced considerably by OPTT and marginally by α (Figure 6c). For Xcc, ∆HA, ∆OA and α have considerable influence (Figure 6d). The total sensitivity indices (Si

total) were computed and compared with the main factors (Si) and it was found that the contributions of the cross-interactions among the design variables to the objectives variability were negligible. In the current injector design study, for example, the design variables ∆OA and OPTT can be fixed at their mean value (0.5) for objective TFmax as this does not result in significant differences in the prediction. The mean error (difference between predictions) throughout the design space between modified surrogate models (obtained by fixing the non-essential design variables at their mean values), and the actual surrogate models listed in Equations (2)-(5), are given in Table 1. The error for TW4 is about 12% suggesting that the cut-off for the non-essential variables may have to be lowered to capture additional features of the original model. The mean error in the modified RSA of the remaining objectives is about 6%, suggesting that they capture the original RSA reasonably well. This information can be used for dimensionality reduction and therefore, to ease the search for optimum designs. But since the number of design variables is small the remaining studies are carried out without fixing the non-essential variables. From an engineer’s viewpoint and interest, this study

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provides an insight into the physics of a design problem by highlighting the design features that govern the individual objectives. Linear surrogate models for the four objectives are constructed as a function of the four design variables. The corresponding coefficients are shown in Table 2. The magnitude of the coefficients agrees well with the essential and non-essential nature of the design variables for each objective. Additionally, the R2 values for the linear RSA are compared with those of the RSAs generated by Vaidyanathan et al.9 (R2

nonlinear). Comparing R2linear with R2

nonlinear shows that most of the variability is accounted for by the linear model and the additional terms in the nonlinear model give marginal improvement. A correlation analysis was then carried out to observe the interactions between objectives. A total of 14641 (114) design points were generated over the complete design space by varying one variable at a time by a constant value. The objectives were calculated using the RSAs. The correlation matrix Cdes and corresponding p-values are computed using Matlab28. The correlation matrix, Cdes shows that there is a strong correlation between objectives TFmax and TW4, as the corresponding coefficient is very close to 1. P-values and 95% confidence intervals for the correlation coefficients also establish the statistical significance of the results. Low P-values (<< 0.05) confirms the significance of the correlation results. This finding is in agreement with the observations made by Vaidyanathan et al.9. TW4 is excluded and the optimization problem is formulated with the remaining three objectives as given by Equations (2), (4) and (5).

Cdes =

max cc 4 max

max

cc

4

max

TF X TW TT

TF 1.00 -0.773 0.947 -0.272

X -0.773 1.00 -0.583 0.263

TW 0.947 -0.583 1.00 -0.193

TT -0.272 0.263 -0.193 1.00

B. Pareto Front Analyses The trade-offs between objectives and sensitivity analyses are carried out at the POF. The Pareto fronts of (TFmax, Xcc) and (TFmax, TTmax) are of interest as these objectives conflict one another. Figure 7a shows the relation between TFmax and Xcc. The O2 post tip temperature, TTmax is ignored. It can be seen that the POF in this case is linear over a large region. A small increase in the value of TFmax (≈ 10%) reduces the combustion length, Xcc, by nearly 50%. Figure 7b shows the relation between TFmax and TTmax. The combustion length, Xcc is ignored. It is obvious that the POF is non-convex. It is also seen that a small drop in the value of the face temperature ( ≈ 10%) can reduce the tip temperature TTmax by nearly 60%. Hence at a small cost of TFmax both Xcc and TTmax improves considerably. Following the trade-off studies, the three objectives (TFmax, Xcc and TTmax) optimization problem is solved using the NSGA -II algorithm10 and the ∈–constraint local search strategy as described earlier. A ll the Pareto optimal solutions obtained using this approach and the original NSGA -II set are combined. A non-domination check on this set of 400 solutions yields 254 non-dominated solutions. After removing the duplicates, there are 249 solutions, which are on the POF. These solutions are shown in Figure 8. This solution set is the global Pareto optimal solution set. The hierarchical clustering algorithm18 is used to divide the obtained POF into 9 clusters for sensitivity and trade-off analyses . Values of the design variables and objectives for these designs are shown in Table 3. Graphically, the solutions are shown on the POF in Figure 8. It is clear from Figure 8 that solutions are selected uniformly over the design space. In Figure 9, the box plots of the design variables for clusters 1, 3, 6 and 9 are shown. The box plots for the objectives are shown in Figure 10. These plots highlight the variability of the design variables and objectives in each cluster. For cluster 1 it is seen that the value of a is fixed at 0 (shear coaxial injector) (Figure 9a) and ∆HA is fixed at 1 (Figure 9b). This suggests that in this cluster, the designs are sensitive to a and ∆HA both of which reach their extreme values. The remaining two design variables ∆OA (Figure 9c) and OPTT (Figure 9d), vary over a range. It is observed that TFmax is minimized (Figure 10a) where as Xcc and TTmax lie near their maximum and have little variability. Hence the designs in cluster 1 tend to minimize TFmax and represent shear coaxial injector designs. The design variables ∆OA and OPTT do not influence TFmax but affect the remaining objectives, Xcc and TTmax. Partial correlation coefficients are estimated to obtain

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the relationship between these design variables and objectives (Table 4). It is noticed that as ∆OA increases, Xcc increases (Rcorr = 1.00) and TTmax decreases (Rcorr = -0.638). As OPTT decreases both Xcc and TTmax decrease (Rcorr = 1.00 for both). Similar observations can be made for clusters 3, 6 and 9. Figures 10a-c show that with increase in TFmax, from cluster 1 to cluster 9, Xcc and TTmax decrease (based on the median of the box plots). This highlights the trade-off between the objectives. Cluster 9 provides information about the opposing trend as to what was observed in cluster 1. As objectives Xcc and TTmax are minimized (Figures 10b, 10c) an impinging injector design is obtained (α ~ 1, Figure 9a) with TFmax exhibiting high values (Figure 10a). The ∆HA is near minimum (Figure 9b) contrary to the design in cluster 1 where high ∆HA minimized TFmax. The ∆OA has considerable variation which suggests that the variability of objectives, Xcc and TTmax are not largely affected by this design variable. Figure 9d shows that Xcc and TTmax are minimized for the minimum value of OPTT. Table 4 gives the partial correlation coefficients for the set of design variables and the objectives in each cluster which shows considerable variation. The partial correlation coefficients for the combinations left out are effectively zero.

V. Summary and Conclusion Different tools have been used to conduct sensitivity and trade-off analyses of a liquid rocket injector design. Global analyses have been conducted to observe the influence of design variables on the objective variability and estimate the correlation between objectives. A POF, generated using a RSA-MOGA coupled approach, is used to conduct trade-off studies between objectives. A 3-objective POF has been divided into 9 clusters and box plots have been used to observe the variability of the designs in each cluster. Additionally, partial correlation coefficients have been calculated to derive a relationship between the objectives and design variables in each cluster. Such analyses provide the designer with enough information to filter out designs based on his specific needs. Some observations based on the tools used, their performance, and the different design aspects of the injector are highlighted below. • The global sensitivity indices (main factor and total sensitivity indices) help identify which design variable is essential

for objective variability. Additionally, point out the effect of cross interactions among the design variables on the objectives variability.

• The variability of TFmax depends on a and ∆HA, TW4 on ∆HA, TTmax on a and OPTT and Xcc on a, ∆HA and ∆OA. The rest of the design variables for each objective can, in principle, be fixed, as their contribution to the objective variability is marginal. It is noticed that the mean errors between the modified RSAs and the original RSAs are about 6-12%.

• It is observed that TFmax and TW4 are strongly correlated and hence TW4 is excluded from the multi-objective optimization study.

• The conflicting nature of the objectives (TFmax, TTmax) and (TFmax, Xcc) is observed by examining the corresponding POFs. For a small cost in TFmax(~10%) , both Xcc and TTmax (~50%) can be decreased considerably.

• The POF obtained for the 3-objective study (TFmax, TTmax and Xcc) gives an idea of the various comp romises among the different objectives. A hierarchical clustering approach helps divide the POF into regions of similar objective goals.

• Box plots and partial correlation coefficients can assist the design selection along the POF. Box plots identify the variability of design variables and objectives in each cluster. Partial correlation coefficient identifies linear trends between the design variables and objectives.

• To minimize TFmax it is essential to have a shear coaxial injector (α = 0) and maximum change in baseline H2 flow area. To minimize Xcc and TTmax, an impinging injector (α > 0) is required with minimum change in baseline H2 flow area.

This study has offered in-depth information about the different design aspects pertaining to the design variables and objectives. We have been able to identify the importance of different design variables for individual objectives and also notice the influence of the objectives in each other. The POF of the 3-objective study gives different choices for the designer to look into but the key observation is that for a marginal cost in the injector face temperature, TFmax, the performance, Xcc and O2 post tip temperature, TTmax can be considerably reduced.

Acknowledgment The present work has been partially supported by the NASA University Research and Technology Institute (URETI) program and NASA Marshall Space Flight Center.

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References 1. Rupe, J. H., “An Experimental Correlation of the Nonreactive Properties of Injection Schemes and Combustion Effects in a Liquid Rocket Engine,” NASA TR 32-255, 1965. 2. Pieper, J. L., “Oxygen/Hydrocarbon Injector Characterization,” PL-TR 91-3029, October 1991. 3. Nurick, J. H., “DROPMIX - A PC Based Program for Rocket Engine Injector Design,” JANNAF Propulsion Conference, Cheyenne, Wyoming, 1990. 4. Dickerson, R., Tate, K., Nurick, W., “Correlation of Spray Injector Parameters with Rocket Engine Performance,” AFRPL-TR-68-11, January 1968. 5. Pavli, A. L., “Design and Evaluation of High Performance Rocket Engine Injectors for Use with Hydrocarbon Fuels,” NASA TM 79319, 1979. 6. Calhoon, D., F. Ito, J. I. and Kors, D. L., “Investigation of Gaseous propellant Combustion and Associated Injector-Chamber Design Guidelines,” Aerojet liquid rocket company, NASA Cr-121234, Contract NAS3-13379, July 1973. 7. Cheng, G.C., and Farmer, R.C., "CFD Spray Combustion Model for Liquid Rocket Engine Injector Analyses," AIAA Paper 2002-0785, 40th AIAA Aerospace Sciences Meeting & Exhibit, Jan. 14-17, 2002. 8. Gill, D. S. and Nurick, W. H., “Liquid Rocket Engine Injectors,” NASA SP-8089, NASA Lewis Research Center, March, 1976. 9. Vaidyanathan, R., Tucker, P.K., Papila, N., Shyy, W., “CFD-Based Optimization for a Single Element Rocket Injector,” 41st Aerospace Sciences Meeting & Exhibit, Paper No. 2003-0296, January 2003 (Accepted for publication in Journal of Power and Propulsion).

10. Goel, T., Vaidyanathan, R., Haftka, R. T., Queipo, N., Shyy, W. and Tucker, K., “Response surface approximation of global Pareto optimal front in multi-objective optimization of simulation-based models,” 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany NY, 2004.

11. Knill, D.L., Giunta, A.A., Baker, C.A., Grossman, B., Mason, W.H., Haftka, R.T. and Watson, L.T., “Response Surface Methods Combining Linear and Euler Aerodynamics for Supersonic Transport Design,” Journal of Aircraft, 36(1), pp. 75-86, 1999.

12. Myers, R. H., and Montgomery, D. C., Response Surface Methodology – Process and Product Optimization Using Designed Experiment, John Wiley & Sons, 1995.

13. JMP Design of Experiments, Version 5, Copyright 2002, SAS Institute Inc., Cary, NC, USA. 14. Sobol I M, 1993, “Sensitivity Estimates for Nonlinear Mathematical Models,” Mathematical Modeling and Computational Experiment, New York, NY, John Wiley & Sons, Vol. 1, pp 407-414.

15. JMPTM, The Statistical Discovery SoftwareTM, Version 5, Copyright 1989-2002, SAS Institute Inc., Cary, NC, USA. 16. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T., "A Fast and Elitist Multi-objective Genetic Algorithm for Multi-objective Optimization: NSGA -II", Proceedings of the parallel problem solving from Nature VI conference, Paris, pp. 849 – 858, 2000.

17. Deb, K., and Goel, T., "A hybrid multi-objective evolutionary approach to engineering shape design." Proceedings of Evolutionary Multi-criterion Optimization Conference, Zurich, pp. 385 – 399, 2001.

18. Jain, A. K. and Dubes, R.C., Algorithms for Clustering Data, Prentice Hall College, New Jersey, 1988. 19. Chen, Y. S., “Compressible and Incompressible Flow Computation with a Pressure-Based Method,” AIAA89-0286, AIAA 28th Aerospace Sciences Meeting, January 9-12, 1989.

20. Wang, T. S. and Chen, Y. S., “A United Navier-Stokes Flowfield and performance Analysis of Liquid Rocket Engines,” AIAA 90-2494, AIAA 26th Joint Propulsion Conference, July 16-18, 1990.

21. Chen, Y. S. and Farmer, R. C., “CFD Analysis of Baffle Flame Stabilization,” AIAA 91-1967, AIAA 27th Joint Propulsion Conference, June 24-26, 1991.

22. Owen, A., "Orthogonal Arrays for: Computer Experiments, Integration and Visualization," Statistica Sinica, Vol. 2, No.2, pp. 439-452, 1992.

23. Mullin, M. and Sukthankar, R., "Complete Cross-Validation for Nearest Neighbor Classifiers," 17th International Conference on Machine Learning (ICML) , Stanford, California, 2000.

24. "MATLAB®}, The Language of Technical computing, Version 6.5 Release 13. © 1984-2002, The MathWorks, Inc. 25. McKay, M. D., “Nonparametric Variant based Methods of Assessing Uncertainty Importance,” Reliability and System Safety, 57, pp.267-279, March, 1997.

26. Miettinen, K. M., Nonlinear Multiobjective Optimization . Kluwer, Boston, 1999. 27. Deb, K., and Goel, T., "Multi-objective Evolutionary Algorithms for Engineering Shape Design", Evolutionary optimization (Eds. R. Sarker, M. Mohammadin, X. Yao), Kluwer, pp. 147 – 176, 2000.

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29. Goel, T. and Deb, T., "Hybrid methods for multi-objective evolutionary algorithms", SEAL'02, Proceedings of the 4th Asia-Pacific Conference on Simulated Evolution And Learning Computational Intelligence for the E-Age November 18 – 22, Singapore, pp. 188-192, 2002.

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Design Variables Mean Error (%)

Objective

α ∆HA ∆OA OPTT

TFmax √ √ × × 5.2

TW4 × √ × × 11.5

Life/Survivability objectives

TTmax √ × × √ 6.7

Performance objective

Xcc √ √ √ × 6.1

Table 1: List of essential (√) and non-essential (× ) variables for each objective and the mean errors between the modified RSA and original RSA. An essential variable accounts for at least 5% of the objective’s variability.

α ∆HA ∆OA OPTT Intercept R2linear R2

nonlinear

TFmax 0.237 -0.634 -0.033 -0.066 0.733 0.990 0.999

TW4 0.0719 -0.775 0.114 -0.052 0.826 0.986 0.999

TTmax -0.222 0.108 -0.063 0.662 0.367 0.918 0.994

Xcc -0.234 -0.402 0.433 0.108 0.138 0.958 0.997 Table 2: Coefficient associated with the different terms in the linear RSA and comparison of R2 values of linear RSA with that of nonlinear RSA.

Cluster α ∆HA ∆OA OPTT TFmax Xcc TTmax 1 0.00 1.00 0.842 0.712 0.0231 1.09 0.88 2 0.00 1.00 0.356 0.587 0.0276 0.749 0.890 3 0.0000 1.00 0.442 0.0144 0.0541 0.750 0.452 4 0.0939 1.00 0.00 0.0146 0.126 0.453 0.466 5 0.668 1.00 0.732 0.00 0.259 0.681 0.229 6 0.600 0.670 0.00 0.00 0.489 0.264 0.226 7 0.295 0.108 0.00 0.354 0.719 0.129 0.641 8 0.314 0.0656 0.00 0.0554 0.776 0.0969 0.357 9 1.00 0.0140 0.680 0.00 0.935 0.138 -0.0432

Table 3: Objective function and design variables of nine (9) representative designs from the Pareto optimal solution set.

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Cluster Design Variable TFmax Xcc TTmax

∆OA - 1.00 -0.638 1 OPTT - 1.00 0.991

3 ∆OA - 1.00 - α 0.982 -0.735 -0.983 6

∆HA -0.999 0.994 -0.729 α 0.877 -0.203 -0.769

∆HA -0.992 0.983 0.816 9

∆OA -0.977 0.997 -0.940 Table 4: Partial correlation coefficients (Rcorr) of design variables vs. objectives for different clusters along the POF.

(a) (b) Figure 1: Schematic of the GO2/GH2 impinging and coaxial injector elements. (a) F-O-F impinging element, (b) coaxial element.

Figure 2: Schematic of Boeing injector element (U. S. Patent 6253539)

do

df

GO2

GH2

GH2

s

αdo

df

GO2

GH2

GH2

s

α

GO2

GH2

GH2

do dfi dfoGO2

GH2

GH2

do dfi dfoGO2

GH2

GH2

do dfi dfo

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(a)

(b)

Figure 3 (a): Design variables and their range, (b): Objective functions.

O2 Flow Area (∆OA=0--40%)

H2 Flow Area (∆HA=0-25%)

O2 Post Tip Thickness (OPTT=0.01-0.02 in.)

H2 Flow Angle (α=0o-20o)

H2

O2

H2

O2 Flow Area (∆OA=0--40%)

H2 Flow Area (∆HA=0-25%)

O2 Post Tip Thickness (OPTT=0.01-0.02 in.)

H2 Flow Angle (α=0o-20o)

H2

O2

H2

H2 Flow Angle (α= 0o-αo)

O2 Post Tip Thickness (OPTT=x-2x in)

Injector detail

X cc

TT max

TF max TW 4

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Figure 4: Simulation domain and boundary conditions.

Figure 5: Schematic of a box plot

0

1

Inner f ence Upper Adjacent Value

Upper hinge

Median Lower hinge

O utside value

Lower Adjacent Value

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(a) (b)

(c) (d)

Figure 6: Main factor (Si,) influence on objective variability, (a) TFmax, (b) TW4, (c) TTmax and (d) Xcc.

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(a) (b)

Figure 7: Pareto optimal front: (a) TFmaxvs Xcc, (b) TFmax vs TTmax

Figure 8: Pareto optimal solution set and nine (9) representative solutions from the same set. The circles identify the

representative of a specific cluster.

TFmax vs TTmax

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

TFmax

TT

max

TFmax vs Xcc

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

TFmax

Xcc

Xcc

TFmax

TTmax

3

6

9

1

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(a) (b)

(c) (d) Figure 9: Box plots for the design variables in cluster 1, 3, 6 and 9. (a) α, (b) ∆ HA, (c) ∆OA and (d) OPTT.

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(a) (b)

(c) Figure 10: Box plots for the objectives in cluster 1, 3, 6 and 9. (a) TFmax, (b) Xcc and (c) TTmax.