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Statistical Methods for Rapid Aerothermal Analysisand

Design Technology .........

Douglas DePriest,

Carolyn Morgan

: ;_ Hampton Un_hrers]_

Hampton, VA 23668

January 31, 2002

Abstract

The cost and safety goals for NASA's next generation of reusable launch vehicle (RLV)

will require that rapid high-fidelity aerothermodynamic design tools be used early in the

design cycle. To meet these requirements, it is desirable to establish statistical models

that quantify and improve the accuracy, extend the applicability, and enable combined

analyses using existing prediction tools. The research work was focused on establishing

the suitable mathematical/statistical models for these purposes. It is anticipated that the

resuIfing models can be incorporated into a software tool to provide rapid, variable-

fidelity, aerothermal environments to predict heating along an arbitrary trajectory. This

work will support development of an integrated design tool to perform automated thermal

protection system (TPS) sizing and material selection.

Introduction

Recent design experience with NASA's X-37 has demonstrated the need for considering

higher fidelity aerodynamic heating early in the design cycle. In the case of X-37, the

vehicle shape was optimized for aerodynamic performance and resulted in severe

aerodynamic heating that forced costly redesign of the nose and wing surfaces and

lowered flight margins. The availability of higher fidelity aerothermal analysis earlier in

the design cycle could have prevented these problems.

Development of this technology impacts NASA's goal of reduced cost by enabling faster

and more optimized design cycles. Utilizing higher fidelity analyses earlier in the design

will avoid the delay and expense of late design changes. Optimizing the thermal-

structural and TPS design in the process will minimize vehicle weight leading to lower

launch cost. The technology described is applicable across all next generation

systems/architectures for any vehicle configuration and includes metallic and ceramicTPS and hot structures.

There will be two phases to this effort and they are model development and model

validation. The initial phase will be to explore and/or develop the

statistical/mathematical methods that can be used to transform the point wise aeroheating

predictions of current tools to yield complete aerothermal environments through a

trajectory corridor. The approach is intended to identify statistical/math models that

best characterize and/or model a set of sphere stagnation data. Once several acceptable

models have been identified, they will be tested in the second phase to see if they are able

to predict heating values within the normal trajectory of corridor values.

2

Description of Sphere Stagnation Data Sets

There were two worksheets for analysis; one containing the full trajectory space (1269

measurements) and the second with measurements only within the entry corridor (138

measurements). For these data sets, a sphere shape configuration is used and the

measurements are taken at the stagnation point on the sphere. For simplicity, a sphere of

one foot radius is assumed. Generally, as the sphere radius decreases, the heat ratemeasurements will increase.

For each case, eleven (11) variables are labeled on the worksheets. The first three

variables (altitude, velocity, and wall temperature) are considered the basic independent

variables. The other variables are derived from these basic three variables. For example,

the density, pressure, and temperature are direct functions of altitude. The Mach number,

dynamic pressure, Reynolds number, and energy variables are combinations of the 3

basic variables, e.g. dynamic pressure equals density*velocity*velocity. The responsevariable of interest is heat rate.

The heating rate values in the spreadsheet were generated using MINIVER-tape

calculations. The datasets contain values for velocity and altitude but no angle-of-attack

values because of the sphere assumption.

Analysis Methods

Our goal is to begin with some rudimentary analysis on these type datasets to explore the

behavior and relationships, correlations, summary statistics, graphics including contour

plots and 3-D plots, robust and residual analyses. Several regression models were

investigated including multiple linear regression, polynomial regression, step-wise, best

subset, quadratic and cubic regression models as well as some nonlinear regression

models. Regression analysis allows one to model the relationship between a response

variable and one or more predictor variables. One of the useful features of a regression

model is that it can be used to predict or estimate a future response value based on a

given set of values of the predictor variables.

Regression analysis results usually include the following: regression equation, predictor

table, summary statistics, ANOVA Table, list of unusual observations, contour plots, 3-D

plots and residual plots. To appropriately use the t-test, F-test and associated confidence

intervals, the data are assumed to meet Certain .........conditions. These include (1) the residuals

(error component) are assumed to be normally distributed, (2) variation is constant

(homoscedasity) and (3) independence. The study of unusual patterns in the residuals

through residual analysis may indicate underlying weaknesses in the model.

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Data Exploration

Table 1 in the Appendix lists the 138 data cases used to develop a predictive

model relating the response variable, heat rate, to the 10 predictor variables shown. Due

to the large quantity of data, the full trajectory data set will note be included in the

appendix. It should be noted that unless otherwise indicated tables and figures will appear

in the Appendix. The summary statistics for the independent and response variables areprovided in Tables 2 and 2a.

Prior to the model building activities, graphical methods were used to help

identify any underlying relationships between the variables being studied. Matrix plots

of cross-graphs of the variables are provided in Tables 3 and 3a. The plots highlight

• Quadratic relationship between heat rate and altitude, temp

• Strong positive linear relationship between mach and velocity, mach and altitude,

velocity and altitude

• Exponential relationship between heat rate and Reynolds number.

On the other hand, Tables 4 and 4a consist of the Correlation Matrix that specifies the

Pearson correlation and corresponding p-values. Given the inherent relationships

between the independent or predictor variables, a principal components analysis was

performed to help define a set of orthogonal variables so that the first principal

component accounts for the largest possible amount of the total dispersion in the data, the

......second principal component accounts for the second largest possible amount of the total

dispersion in the data, etc. This would be beneficial in helping to identify a subset list of

candidate predictor variables for the analysis. Tables 5a and 5b show the results of the

_alysis using the correlation matrix of the predictor variables as input to the principal

components analysis.

Another method that was used to identify a subset list of candidate predictor variables is

best subset selection. Table 6 shows the results of this analysis for the corridor data set.

Best subset selection identified altitude, velocity, mach, dynamic pressure and Reynolds

as the top five prediction variables.

Classification and Regression Tree (CART) based models are exploratory techniquesfor uncovering structure in data that are used for:

• developing prediction rules that can be rapidly evaluated

• screening variables

• assessingthe adequacy of linear modeis .....

• summarizing large data sets for both classification and regression problems.

Tables 7a and 7b show the results of the CART _alysis and Figure 1 shows the resulting

CART tree. CART selected velocity, mach, altitude, energy and dynamic pressure as the

primary prediction variables. The tree indicates that for those data cases with velocity

measurements]ess than 13750, the predicted Values for heat rate_e generally below

thirty. Furthermore, if values ofmach are below 7.7, then the predicted heat rate is

approximately 3.909.

The classical regression methods are often used to obtain models for prediction. The

challenge is the development of the best mathematical expression to describe in somesense the behavior of a random variable of interest as a function of one or more

independent or predictor variables. The classical regression techniques however make

several strong assumptions about the underlying data, and the data can fail to satisfy these

assumptions in several different ways as indicated the Analysis Methods section.

In the case where there are one or more outliers in the data or the data may not be fitted

well by any straight line, robust regression methods come into play. These methods

minimize the effect of the outliers and can be useful in helping to identify the outliers inthe data.

Scatterplot smoothers are useful tools for fitting arbitrary smooth functions to a scatter

plot of data points. The smoother summarizes the trend of the response as function of the

predictor variables. All of the above analysis methods are used to explore the given datasets.

Results

Several approaches were used in the exploration and identification of

statistical/mathematical models for the given data sets including the classical multiple

regression, classification and regression trees (CART), and other advanced analysis

methods. One of the interesting results concerns a comparison of some initial multiple

regression model types using only the independent predictor variables for both full andcorridor data sets. These results are summarized below in Table A that includes Data,

Coefficient of Determination R 2, Adjusted R 2, fit standard error S, F-statistics and the

model type.

Table A: Comparison of Model Types for Full and Corridor Data using Three (3)

Independent Predictor Variables

Data R 2 Ad i R 2 Std Error F-statistic Model Tv_e

Full 48.3% 48.2% 196.5 394.15 Linear

Corridor 85.2% 84.9% 8.72 257.21 Linear

Full 84.4% 84.4% ....... _!08.0 1140.45 Quadratic

Corridor 97.6% 97.5% 3.54 895.68 Quadratic

Full 85.6% 85.5% 103.9 937.21 Cubic, Interact

Corridor 99.7% 99.6% 1.34 4751.89 Cubic, Interact

................................... ..-w ..................... ......-.. ....................................

- :-:- i.:̧

In reviewing Table A, an obvious conclusion is that the regression models appear to be

more appropriate for the corridor data than the full trajectory data for all model types.=

5

For everymodeltypecomparisonin TableA, therearehigherR2andadjustedR2valuesandlowerstandarderrorvaluesfor thecorridordataset. A morein-depthanalysiswasconductedon thecorridordatasetasit morerealisticallysimulatespossibleentrytrajectoryandheatingratesof anexperimentalspacevehicle. The structuredapproachthatwasusedfor model identificationconsistsof thefollowing: (1) analyzethesummarystatisticsresultsfor errorsandconsistency,(2) conductpreliminaryanalysisusingonlythe independentpredictorvariables,(3) usegraphicalmethodsto identify underlyingrelationships,(4) employmethods(BestSubsetSelection,PrincipalComponents,etc..)thataid in identifying mostlikely additionalpredictorvariablesand(5) specifyamodelusing classicalregression,classificationandregressiontrees(CART) andotheradvancedstatisticalmethods.Theresultsaresummarizedin TableB belowthatincludesrank, (R2),adjustedR2,fit standarderror(S),F-statisticsandthemodeltype. Additional analysisandresultsonthenumber1,number2 andnumber6 rankedmodelsareprovidedin theAppendixin Tables8 to 16and Figures2 to 12. Regressionmodelranked6 hasallpredictorvariablesexceptaltitude,velocity andTwall.

Table B: Identificationof ModelTypesfor theCorridorDataSet

Rank R 2 Adi R 2 Std Errgr F-,statistic Model

1 99.8 99.7 1.130 4140.87 Quad&Interacts (5V)

2 99.7 99.6 1.348 4751.89 Cubic, Interact. (3V)

3 97.6% 97.5% 3.54 895.68 Quad& Inter (3V)4 96.77 .... 4.1533 483.15 CART

5 99.4 99.3 1.851 1665.11 Cubic WO Indep (7V)

6 99.4 99.4 1.724 3843.80 Linear (6V)

7 99.1% 99.0% 2.209 2798.68 Linear 1 (5V)

8 85.2% 84.9% 8.72 257.21 Linear (3V)

Conclusions

A number of methods were considered in thi_s analysis including classical multiple

regression, polynomial regression, classification regression trees (CART), principal

components, correlationmatrix and resicl_! analysis. Several graphical methods were

_iSed in model development and assessing model adequacy. In addition, several

• tec_iques were used to screen/identify underlying relationships. For example the matrix

plot in Tables 3 and 3a suggested the inclusion of quadratic and interaction terms in our• modelsl Whereas, principal components and best subset selection were used to screen

and identify the main predictor variables. All of these methods helped to guide us in the

..... selection of the predictor variables ino_ models. Using all of the above methods,

Several promising candidate models have been developed that may be used to predict the

response variable, heat rate for the entry corridor data set. In the next phase of our

research, validation of the adequacy of our models and other advanced methods will be

explored.

6

References

1. Bowles, J.V. and Henline, W.D., "Development of Aerothermodynamic Environments

Database for the Integrated Design of the X-33 Prototype Flight Test Vehicle",AIAA 98-0870.

2. Montgomery, Douglas C.. (2001). Design and Analysis of Experiments, 5 th edition.John Wiley & Sons, New York, NY

. Meyers, R. H. and D. C. Montgomery (1995). Response Surface Methodology."

Process and Product Optimization Using Designed Experiments. John Wiley & Sons,New York, NY.

. Thompson, R.A., "Automated TPS/Structural Sizing Method", (Personal

communication), September 1999.

5. Wurster, K.E.; Riley, C.J.; and Zoby, E.V., "Engineering Aerothermal Analysis forX34 _e_al Protection System", Journal of Spacecraft and Rockets, Vol. 36, 1999,

pp. 216-227.

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(5V Corridor Data)

The regression equation is

Heat Rate, BTU/ft2/s = - 815 - 2.71 Altitude, kft

+ 0.0521 Velocity, ft/sec + 3.59 Temp, deg R

- 1.45 Dyn. Pres. ib/ft2 +0.000648 Reynolds per

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Velocity

Temp,

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0.0006481

0.007743

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0.00005516

0.0002828

0.00000222

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0.4786 -5.66 0.000

0.005991 8.69 0.000

0.5892 6.09 0.000

0.3884 -3.74 0.000

0.0001496 4.33 0.000

0.001294 5.98 0.000

0.0005719 -5.49 0.000

0.00000000 -5.65 0.000

0.00001424 -7.12 0.000

0.00000845 -6.21 0.000

0.00000750 7.36 0.000

0.0006029 0.47 0.640

0.00000035 6.42 0.000

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Table 14: Regression Analysis Without Alt., Vel &Twall

_. i _ m

The zeg=eisAon equation _s

Haat D_te,

Mach

ib/ft2

i . i . ._ mm --

BTU/ft2/S _ 241 - 0.0266 Mach*3 + 1.68 Mach*2 - 33.2

-0.000030 DynPress*3 + 0.0133 DynPress*2 - 2.99 Dyn. Pros

-0.000000 Energy*2 +0.000117 Energy, ft3/soc

- 43.5 Pressure, ib/ft2 +26874067 Density, slugs/ft3

- 0.0589 Temp, deg R +0.000453 Reynolds per ft

Prodictor

Constant

Mach*3

Math*2

Math

DynPzess

DynPress

Dyn, Pre

EnergY2

Energy,

Pressure

Densi_,

Te_p,

Reynolds

Coaf SE Coof T P

241.19 61.32 3,93 0.000

-0.026629 0.004115 -6.47 0.000

1.6772 0.2470 6.79 0.000

-33.223 4.995 -6.65 0.000

-0.00002972 0.00000566 -5,25 0.000

0.013323 0.002475 5.38 0.000

-2.9908 0.5574 -5.37 0.000

-0,00000000 0.00000000 -3.2.94 0.000

0.00011696 0,00001058 11,06 0.000

-43.51 23.52 -1.85 0.067

26874067 16016537 1.68 0.096

-0.05891 0.08058 -0,73 0.466

0,0004529 0.0001233 3,67 0.000

S = 1.851 R-Sq = 99.4% R-Sq(adj) = 99.3%

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