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Ranking Based Model-Form Uncertainty Quantification for a Multi-fidelity Design Approach J. Umakant$1 Defence Research and Development Laboratory, Hyderabad-500058, India K. Sudhakar¶ , P.M. Mujumdar¶ IIT Bombay, Mumbai-40076, India and C. Raghavendra Rao# University of Hyderabad-500058, India Computer simulation based design processes are being extensively used in preliminary design phase of complex aerospace vehicles like scramjet powered hypersonic vehicles. Analysis tools of varying fidelity are generally used to assess the system performance metrics. Often there is a constraint on the number of simulations using high fidelity analysis tools to predict the performance metrics, due to attendant computational demands. A low fidelity analysis tool complemented with an uncertainty model enables the designer to make better design decisions in such situations. A novel approach based on ranks is proposed in this study to aggregate high fidelity information in a cost effective manner. Based on this information, a cumulative distribution function for the difference between high fidelity and low fidelity responses is constructed. The approach is demonstrated initially for a synthetic example and subsequently for quantification of uncertainty in a typical disciplinary metric for hypersonic vehicle design. Nomenclature CDF = cumulative distribution function CFD = computational fluid dynamics DACE = design and analysis of computer experiments HFA = high fidelity analysis H_cr = cruise altitude LFA = low fidelity analysis ma = mass flow capture of air M = Mach number 1, 2,1 = forebody compression angles = angle of attack I. Introduction D esign of any system involves trade-off among various options and selecting one that meets the requirements best. In general, it may be desirable to assess the various options using a model that accurately evaluates the performance metrics of the system. In a computer based design framework these models are generally, computationally expensive functions or HFA models. However, such functions have been usually used to examine or analyze a particular design in great detail rather than to evaluate large number of designs. This is mainly due to practical difficulties like unrealistic cycle time and issues related to integration of several HFA models as may be needed in the design of a complex multidisciplinary system. Hence the designer is confronted with the challenge of taking decisions in an environment wherein uncertainty is ever present. It is important here to distinguish between variability and uncertainty. $1 Scientist, Aerodynamics Division, also Research Student at IIT, Bombay ¶ Professor, Department of Aerospace Engineering , CASDE # Reader, Department of Mathematics and Statistics 1 American Institute of Aeronautics and Astronautics Variability [1] is inherent randomness in the system. The use of probability theory to represent variability is well established. Uncertainty is defined as “a potential deficiency in any phase or activity of the modeling process that is due to lack of knowledge”. Uncertainty may also arise when there is a scarcity of high fidelity information. This is true when new classes of systems like scramjet powered hypersonic vehicles are being developed and no historical database exists. Probabilistic approaches to handle uncertainty in lieu of replacing the expensive high fidelity analysis with low/medium fidelity analysis and their application in design scenarios have been demonstrated recently. Quantification of uncertainty using a Bayesian approach to update the uncertainty model was proposed in [2] in the context of an aerospace design. In [3], the probability density function was discretized for various confidence levels and a response surface model was created to achieve aircraft design that is robust in performance with respect to stability and control disciplines. In [4] the engineering methods for various participating disciplines in Reusable Launch Vehicle design, is used together with a multiplier coefficient that is characterized by assuming a probability distribution. A probabilistic design approach for hypersonic vehicle has been also demonstrated in [5] by the authors of this papers. In this study, the effect of fidelity uncertainty in a disciplinary metric namely mass flow capture of air was propagated onto a system metric namely thrust deliverable and a design that maximized the system metric was sought through formal optimization. Still, there are several issues that need to be addressed. Basically the development of a non-deterministic design process involves three major steps, namely: (a) Based on the information available, construct an input uncertainty model. (b) Propagate the effect of disciplinary uncertainty onto system performance metrics. (c) Assess the system performance and take design decision under uncertainty. In most of the studies, discussed above, the focus has been on the last two steps. An uncertainty model was assumed based on the disciplinary expert‟s recommendation regarding the prediction accuracy of the low fidelity analysis or based on evaluating the analysis tool with respect to similar applications. For example, if the disciplinary expert declared that the low fidelity analysis was accurate within 10%, then a normal distribution with mean zero and standard deviation 33% is assumed. Based on four high fidelity observations for a disciplinary metric, namely mass flow capture of air, a Weibull distribution was constructed by the authors of this paper in [5]. These four observations correspond to arbitrary points in the design space. Therefore it is required to ascertain the validity of the assumed probability distribution if another set of observations were used. The present study seeks to bridge this gap by proposing a novel method to aggregate high fidelity information in a systematic manner and construct an empirical probability model to complement the use of low/medium fidelity analysis tool for application in a multidisciplinary design environment. II. Rank Transformation Approach This section describes rank transformation approach to model uncertainty in the estimation of a function. It is assumed that the function is computationally expensive; restricting its frequent use as may be demanded while searching the design space during optimization. Hence a low fidelity model that is computationally light is used, albeit with less accuracy. Uncertainty therefore arises in the estimation of the function. The concept of ranks is introduced and a method that exploits this concept is proposed for uncertainty modeling. A. Concept of ranks Ranking refers to the process of ordering a sample (say of size „N‟) with respect to a system performance metric. For minimization problems, the observation with the least value receives the highest rank (rank N) while the observation with the maximum value is given the lowest rank (rank 1). Rank based approaches have been mainly used in agricultural research and property valuation research. Ranking procedures are one approach in multiple decision theory [6] where a simple loss function (zero-one) is used and the risk is an incorrect decision. Dell and Clutter [7] show that using relatively cheap methods, like “judgment ordering”, where it is difficult or expensive to obtain the characteristic of interest, may still increase the efficiency of using the sample mean as an estimator for the population mean. Cronan et.al.,[8] report that for small sample sizes, the rank regression technique produces a model with better estimates of residential property value as compared to the model based on multiple regression analysis technique. However to the best of the author‟s knowledge there has been no study in the area of uncertainty modeling based on rank 2 American Institute of Aeronautics and Astronautics transformation approach. As discussed in the previous section, creating a sample for the high fidelity information constitutes the initial step in process of uncertainty modeling. Rather than trying to model the uncertainty for the entire design support, a novel method is proposed to sequentially aggregate a high fidelity sample from the regions where the expensive function is potentially attractive. Design points with low response value receive higher ranks while those with higher response values receive lower ranks. A stopping criterion is used to limit the sample size to a reasonable number. Based on this information, residue is estimated as the difference between the high fidelity response value and the corresponding low fidelity response value. A probabilistic model for the residues is then constructed. Uncertainty model for the estimation of expensive function is now defined as the low fidelity model complemented with probabilistic model of the residue. It may be noted that in the context of optimization (for minimization), the inaccuracy of the model in the regions where the function is relatively higher is not of much interest. Rank transformation of the response enables to introduce the preferential characteristics in the uncertainty model. B. Notations Let Z = F(X), denote a high fidelity model that describes the system behavior. The low fidelity model is represented by z = f(X). X D Rn, denotes the input vector for both the models. D is the design space in „n‟ dimensional real space Rn. F(.) is the high fidelity or expensive function and Z is the high fidelity response. f(.) is the low fidelity function and z is the low fidelity response. X typically describes the parameterization of the system while Z or z describes a performance metric of the system. Then for one dimensional space R the following notations are adopted. D=[a b] for design support /space X1,……XK,…… for design points in D Z1,……ZK,……. for high fidelity responses corresponding to the design points z1,……zK ,…….. for low fidelity responses corresponding to the design points SX = {X1,………XN, XN+1,………XM} for set of design points SR = {Z1,……… ZN, zN+1,………zM } for set of high fidelity and low fidelity responses corresponding to the points at SX ri for rank of the ith response in SR Pi for empirical probability at X = Xi in SX Ci for empirical cumulative probability at X = Xi in SX ei for residue Zi – zi at the point X = Xi X0 =a < X1 < ….<Xp-1 < Xp = b for partition of D with size p f(D) = [fL fU] for low fidelity response space for D fL =z0 < z1 < ….<zp-1 < zp= fU for partition of f(D) with size p SL = { (X,z) | X D , z = f(X) } for low fidelity system state descriptor SH = { (X,Z) | X D , Z = f(X) }for high fidelity system state descriptor SA = { (X,Z) | (X,Z) SH and Z ≤ z1 }for system state descriptor in attractive zone nA for cardinality of SA indicating number of observations of the sample in attractive zone. The expensive function is estimated using the model, Z ~ z + U(Z), where U(Z) is a probability distribution that represents the uncertainty in the estimation of Z when a low fidelity model is used in lieu of the high fidelity model. C. Rank Transformation Approach The various steps involved in the rank transformation approach are now discussed. 1. Selection of Initial Sample Assume that to begin with there are „K‟ (K≥2) high fidelity responses available. These responses correspond to arbitrarily selected points in the design space. Alternately expert opinion can be solicited as to where to perform the initial expensive function evaluations. A typical table consisting of three initial points is shown in table1a. 2. Augmentation of the sample with low fidelity responses Low fidelity responses are now used to augment the sample. These responses are evaluated at design points resulting from uniform gridding of the design support. Augmenting is done to encourage global representation of the function and the design space, while ranking. In case of multimodal functions this helps to avoid gathering high fidelity information that is restricted to a local valley. However, caution must 3 American Institute of Aeronautics and Astronautics be exercised such that at any point of time the number of low fidelity observations is not very significantly higher as compared to the number of high fidelity observations. Otherwise the trend of the function behavior as predicted from the sample will be dominated by the low fidelity model. Table1b shows the data after addition of six low fidelity observations. It may be argued that low fidelity responses may be added sequentially. However, adding one low fidelity response at a time may not encourage global representation in the sample, especially in the initial stages when the available number of responses is small. 3. Rank Transformation of responses The high fidelity responses and low fidelity responses are combined to form the dataset SR and the elements of the set are sorted in a descending order. Rank transformation is now applied to the responses by assigning ranks to them in a serially increasing manner, starting from one to the value of the maximum rank. The maximum rank has a value equal to M, the total number of high fidelity and low fidelity responses. Thus the response with the minimum value receives the maximum rank while the response with maximum value receives a rank of one. If a response value occurs more than once, then same rank is assigned to all its occurrences. Typical rank transformation of the responses is given in the fourth column of table1c. 4. Mapping of the ranks onto design support The ranks for the responses are now mapped to their corresponding design points in SX. Thus the design points are now given an additional attribute, namely rank that defines its preference for selection. 5. Computation of the empirical PDF and CDF of the design support The design points together with their associated ranks are now sorted in an ascending order with respect to their respective values. The empirical probability density of the i th point is then computed as the ratio of its rank to the summation of ranks of all the points, and is given as below: ri Pi M r i 1 i The lower bound and upper bound of the design support are assigned respectively zero and one as their empirical probability densities. Since the points are sorted in ascending order, the empirical cumulative density of the ith point in the table is the ratio of sum of ranks of all the points above it to the total summation of ranks and is denoted as below: i r j 1 j 0.5 Ci M r i 1 i The factor 0.5 is used for continuity correction to connect the first and last points of the table respectively with the lower bound and upper bound of the design support. Piece-wise linear interpolation yields the cumulative density at any other point in the design space. Empirical PDF and CDF for the design support are thus constructed. The fifth and sixth columns of table 1.1c depict the probability and cumulative probability of the design points. 6. Selection of new point for HFA evaluation The CDF of the design support can now be used to choose a new point where the expensive function can be evaluated. This is accomplished by applying a random inverse transformation on the CDF. Points where the response is likely to be relatively lower have greater probability of getting selected than the points where the function responses are likely to be higher. However, allowing some chance for unfavorable points to also get selected enables the algorithm to scan the whole design support. This feature is useful when the expensive function exhibits multimodal characteristics. Thus there are now „K+1‟ high fidelity observations. An updated table is shown at table1d. The steps of augmenting the sample with low fidelity observations, ranking the collection of the responses, mapping the responses onto the design support are repeated to update the CDF of the design 4 American Institute of Aeronautics and Astronautics support. The updated distribution is used for selection of the next design point for high fidelity evaluation. This process is repeated till the desired number of high fidelity responses is aggregated. Alternatively a heuristic stopping criteria described in the next section can be used. 7. Stopping Criterion A heuristic criterion is used to decide when to stop the process of collecting high fidelity information. The CDF for the residues and the design support are examined for after every update and if the variation between the successive trials is negligible, the process is terminated. Thus „N+K‟ high fidelity observations when the process is terminated after „N‟ trials are performed. D. Limitations of the rank transformation and means to circumvent them The rank transformation procedure described in the previous section has the following limitations: i) The procedure may be sensitive to the initial sample and consequently influence the selection of subsequent high fidelity responses ii) Since the inverse transformation for selecting new design point is random, there is a possibility of clustering of the design points. The first limitation can be largely circumvented by starting with an initial sample that spans the entire design support and the range of the response. A strategy based on stratification is adopted to implement this. The second limitation is addressed by defining a minimum distance criterion. 1. Stratification Algorithm Stratification refers to classifying or coding the design space D and f(D) into „p‟ number of levels or strata sij with the following property: Xs = i , iff Xi-1 < X ≤ Xi and zs = i , iff zi-1 < z ≤ zi and A typical system state descriptor SL = (X, z) is coded as (Xs, zs). Each level can be interpreted as an isocontour. Our purpose is to choose the initial points such that there is a representation of all the strata in x and z, at least once. In other words, p points are chosen such that each point is representative of one strata of x and z. This problem can be formulated as an assignment problem (or an integer programming problem), in Boolean space. However, in the present study this is implemented as described below. The entire design support is uniformly gridded into large number of points. Low fidelity tool is now used for evaluating the response at each of the grid points. Typical representation for 15 grid points is shown at table2a.The grid points and the corresponding response values are then coded or stratified according to the definition above. Thus a stratification table of coded values for the design points and their respective response values is set up. Table2b illustrates this process for three stratification levels. A consequence of the stratification is that the points falling in the same strata lose their distinct identity, resulting in duplicate points that do not possess any additional information. For further analysis, such points are not retained in the stratification table. Table2c highlights the data after deletion of such points. A frequency table indicating the number of occurrences of xs and zs is then set up and a typical result is shown in table 2d. Strata level corresponding to the maximum number of occurrences either in x s or zs is identified and a point from the set of coded values corresponding to this strata level is selected. Any tie that occurs is resolved randomly. Other points having the same stratification identity either in support space or in the response space are then removed. For example, table 2d shows that the maximum of occurrences is three for z s. So we choose to select the point at serial number 6 in the table2c. The corresponding strata level for xs is also three. Thus all the entries in table 2c having strata level three either in x s or zs are removed from the table and the updated stratification table is illustrated in table 2e. The process is repeated for the specified number of strata levels, resulting in „p‟ distinct coded points with their corresponding response codes also being distinct. An illustration of the updated table after selecting the second point is shown in table 2f. Table 2g summarizes the selected strata levels for the support points and the corresponding response. The selected stratified levels are then mapped to the physical domain and the points are randomly selected from the respective strata. At these initial points we perform the expensive function evaluations. For example, the entry at serial number 3 in table 2g corresponds to a design point in strata level 2 with its response in strata level 1. In the physical domain, the entries at serial number 6 through 10 of table 2a correspond to these strata levels and any one of these points is randomly selected. 5 American Institute of Aeronautics and Astronautics 2. Minimum Distance Criterion Random inverse transformation of the cumulative distribution function to select the new design points can sometimes result in the new point to lie within the neighborhood of existing points. To avoid this, the L1 distance of the new point with respect to the existing points is calculated and the point is retained only if the norm is greater than a specified tolerance bound; otherwise another point is selected from the distribution. It may be noted that this check is performed before the expensive function is evaluated. E. Uncertainty Modeling After aggregating the high fidelity sample, an empirical cumulative distribution needs to be constructed. Residue, defined as the difference between the high fidelity response and the corresponding low fidelity response. Diagnosis of the trajectory of the residues with respect to the low fidelity response enables to infer whether or not the residues are correlated with low fidelity response. In case the trajectory exhibits a random path, it is inferred that the differences are random. On the other hand, a systematic variation in the trajectory exemplifies a correlation between the low fidelity and the high fidelity responses. 1. CDF for the Residue The residues, denoted by „e‟, for the N+K high fidelity responses available are sorted in ascending order according to their values. Homogenous distribution for the residues is assumed for computing the probability distribution. The cumulative density for the ith residue is defined as Ci e i 0.5 i 1,2,, N K NK The lower bound of the residues is defined as the minimum value of the observed residues decremented by one unit. Similarly the upper bound of residue is defined as the maximum value of the observed residue incremented by one unit. Cumulative densities of zero and one are assigned respectively to the lower and upper bounds. A fit for this data will give a smooth representation of empirical CDF. Linear interpolation may also be used to estimate the cumulative density of for any intermediate values of residue. This type of distribution is also referred to as non-parametric distribution. 2. Restricted CDF of residue for attractive zone It may be recalled that using the low fidelity model as a guiding tool, the range of the response was stratified into three contour levels. The zone defined by the first contour level is designated as the attractive zone, since the response value corresponding to the first level is smaller as compared to that for the second and third levels. In the context of minimization, we are interested only in those high fidelity observations that are contained in the first contour level. Hence, the N+K responses are screened to retain only those responses that are within the attractive zone. The cumulative distribution function of the residue for the attractive zone is constructed in a similar manner as described in the previous section. It may be noted that the uncertainty bounds of the residue will now be typically lower than that obtained in the previous section and hence the estimate of the expensive function is not unnecessarily conservative. This is consistent with the philosophy to create an uncertainty model that is more appropriate in the regions of interest, rather than trying to characterize the uncertainty for the entire design support. F. Validation of Algorithm In the above procedure, the new design point for high fidelity estimation has been chosen randomly using inverse transformation of the cumulative distribution function of the design support. Hence the repeatability or robustness of the algorithm is verified by conducting large number of simulations and examining the results. Verification is based on the following criteria: i) The probability of number of observations in the attractive zone should be non-trivial with at least 95% confidence. ii) The empirical cumulative distribution of the design support and residue are examined to check if the same type of distribution is obtained in most of the simulations. iii) Coefficient of variation, defined as the percentage ratio of standard deviation to mean, is examined for the lower and upper bounds of the residue. The algorithm can be considered to be stable [9] if the coefficient of variation is less than 33%. 6 American Institute of Aeronautics and Astronautics III. Uncertainty Quantification for Synthetic Example Results to illustrate the performance of above algorithm are presented initially for a synthetic univariate bimodal function. The low fidelity model is represented by a quadratic function. Three levels of stratification were adopted. Low fidelity tool was used to partition the expected range of the response into three contour levels. The first level has the lowest contour value. Attractive zone for the response then corresponds to first contour level. The details regarding the functions are given in tables 3a .Figure 1a shows the high fidelity and low fidelity function contours. It may be noted that the function has a global minima at X=1.05 and another local minima at X=3.75. The symbols in this figure represent the selected points. The initial points selected by stratification are distinguished by the symbol „*‟. It can be observed that the algorithm has selected design points near both the valleys present in the high fidelity function and three points out of the sample size of thirteen are in the attractive zone. However, only two points would have been in the attractive zone if the selection was based on simple grid of the design support. It may be noted that the algorithm has successfully selected points in the favorable regions even though the low fidelity model is unimodal whereas the expensive function is bimodal. The CDF for the design support is shown in figure 1b. It can be noticed that the probability of selecting a point from the unfavorable region ie., X greater than 4 is less than 0.02 while it is only 0.2 for the other unfavorable region ie.,X less than 1. The CDF of the residue based on the total sample of high fidelity information is depicted in figure 1c and the corresponding prediction of the expensive function using this model is shown in figure 1d. It can be seen that the uncertainty has been characterized over most of the design support. Figure 1e shows the restricted CDF based on restricting the sample to the observations within the attractive zone. This enabled better characterization of uncertainty in the expensive function in favorable regions than in other regions and the result is visualized in figure 1f. Results of Monte-Carlo simulations are shown in figures 2a-2d. The non-trivial probability of number of observations in the attractive zone is 7.7% with 97% confidence. The CDF for the design support and residue exhibits similar characteristics in most of the simulations. The coefficient of variation for the bounds is tabulated in table 3b. It can be noticed that the lower bounds of the residue have coefficient of variation less than 33% while for the upper bounds it is greater than 33%. This may be mainly due to the fact that the low fidelity model is unimodal whereas the high fidelity is bimodal. Still, the algorithm has characterized the uncertainty better in the attractive zone. The effect of number of sample size on the CDF for residue is shown in figure 3. Trial 1 refers to the sample with the initial three points selected based on stratification algorithm. From the results it can be observed that the CDF of the residues exhibits similar characteristics for trial numbers more than ten. Hence we may limit the size of the sample to ten. In the next section the method is applied to quantify uncertainty for a typical disciplinary metric for hypersonic vehicle. IV. Uncertainty Quantification of a typical disciplinary metric in Hypersonic Vehicle Design Scramjet powered hypersonic vehicle typically exhibit a highly integrated airframe-engine consideration. A generic representation of such a vehicle is shown in figure 4a. Mass flow capture of air is a critical disciplinary performance metric needed as input in the design of intake of hypersonic vehicle. The design of the forebody largely dictates the mass flow capture of air of the configuration and the intake entry conditions. At the same time the forebody has a strong influence on the body aerodynamics and also affects the sizing of the vehicle. The flow past the forebody is dominated by the presence of strong shocks and viscous phenomena. Thus, a CFD code simulating viscous phenomena is ideally needed for calculating the disciplinary metrics. The computational demands of such codes allow only few numbers of high fidelity simulations in the preliminary design environment. Instead, a low or medium fidelity model that is computationally light like an oblique shock theory based model may be preferred for rapid estimation of mass flow capture of air. However the inaccuracy in the estimation of the performance metric will affect the entire downstream design of the propulsion flowpath. Thus, it is important to quantify the uncertainty in the estimation of mass flow of air in a systematic manner. In the present study, the forebody is assumed to have three compression ramps with body width of 0.8m. The compression angles for all the three ramps are allowed to vary between 0 and 6. The intake entry dimensions are 0.240m x 0.500m. The free-stream conditions are as follows: M∞ = 6.5, H_cruise = 32.5 km and = 4. Rank transformation approach described in the previous section is used to construct the uncertainty model. In this case it is desired to maximize the mass flow capture of air. The high fidelity model is a CFD based DACE model while the low 7 American Institute of Aeronautics and Astronautics fidelity model is based on oblique shock theory. The details of both the models, developed as part of this research, are given in [5]. Figure 4b shows the CDF of the residue. Recall that the residue is the difference between the high fidelity response and the corresponding low fidelity response. In this application, it was observed that the high fidelity responses were always lower as compared to the low fidelity responses. If the bias is large, then there may not be any points in the attractive zone. To avoid this, a constant has been included in the low fidelity model. This constant is defined as Z - z ; where Z denotes the mean of the initial high fidelity responses obtained by stratification and z denotes the mean of the corresponding low fidelity responses. Using the lower bound and upper bound of the CDF defined in figure 4b, the disciplinary metric is computed for the entire support. For a given value of mass flow capture of air based on the low fidelity model, the corresponding tolerance bounds are shown in figure 4c. It can be noticed that the method has been able to characterize the uncertainty in the estimation of ma over the entire support. The refinement in the CDF based on attractive zone is shown in figure4d. The corresponding uncertainty bounds in the estimation of ma are shown in figure 4e. It can be observed that the width of the tolerance bounds is now less conservative in the regions where the response values of ma are higher. This uncertainty model can now complement the low fidelity model. It may be noted that ten high fidelity simulations have been used to develop the uncertainty model. Developing a surrogate model for the same performance metric required twenty four high fidelity simulations [5]. V. Conclusions A rank based approach has been demonstrated to enable quantification of uncertainty in disciplinary performance metrics when a low fidelity analysis tool is used instead of a computationally expensive analysis tool. Rather than relying on subjective opinions regarding the accuracy of the low fidelity analysis tools, the proposed method seeks to aggregate limited number of high fidelity information in a sequential manner. Based on this information, an empirical cumulative distribution function for the residue is constructed. Results for synthetic example demonstrate the validity of the method. The approach has been subsequently used to develop a cumulative distribution function to represent the uncertainty in the estimation of a typical disciplinary metric in the design of hypersonic vehicle. The method can be extended to develop similar models for other performance metrics. The models can be incorporated in the system synthesis design tool and used in a multidisciplinary design optimization environment for hypersonic vehicle. References [1] Oberkampf,W.L., Bibliography for Verification and Validation in Computational Simulation. SAND 98-2041. [2] Mantis,George C., Quantification and Propagation of Disciplinary Uncetainty via Bayesian Statistics, PhD thesis, Georgia Institute of Technology, 2002 [3] DeLaurentis, Daniel A. A Probabilistic Approach to Aircraft Design Emphasizing Stability and Control Uncertainties, Ph D thesis, Georgia Institute of Technology, 1998 [4] Charania, A. C., Bradford, John E., Olds, John R., Graham, Matthew., “ System level uncertainty assessment for collaborative RLV Design,” 2002 JANNNAF 38th Combustion subcommittee/ 26th Airbreathing Propulsion subcommittee/ 20th Propulsion Systems Hazards subcommittee / 2nd Modeling and Simulation subcommittee Joint Meeting, 8-12 April / Destin, Florida [5] Umakant, J., Sudhakar, K., Mujumdar, P.M., Panneerselvam, S., “ Configuration Design of Air- Breathing Hypersonic Vehicle considering Fidelity Uncertainty,” AIAA-2004-4543, 10th AIAA / ISSMO Multi-disciplinary Analysis and Optimization Conference, 30th August-1st September, 2004. Albany, New York [6] Gupta, S. S., Huang, D.Y. Multiple Statistical Decision Theory. Springer-Verlag, New York, 1981 [7] Dell,T.R. and Clutterr,J.L. Biometrics, 28, 545-553, 1972 [8] Cronan, Timothy P., Epley, Donald R., Perry, Larry G., “The Use of Rank Transformation and Multiple Regression Analysis in Estimating Residential Property Values with a Small Sample,” The Journal of Real Estate Research Vol.1No.1 Fall 1986 [9] Agarwal, B.L., Basic Statistics. Third Edition, New Age International (P) Limited Publishers, 1996. 8 American Institute of Aeronautics and Astronautics 35 1 30 ___ high fidelity Cumulative Probability _ _ _ low fidelity 0.8 Response 25 0.6 20 0.4 15 0.2 10 5 0 0 1 2 3 4 5 0 1 2 3 4 5 Design Variable, X Design Variable, X Fig.1a. High Fidelity and Low fidelity Fig.1b. Cumulative Distribution Function contours with selected design points of Design Support 1 35 30 ___expensive function Cumulative Probability 0.8 _ _ _ lower and upper bounds Response 25 0.6 20 0.4 15 10 0.2 5 ___expensive function 0 _ _ _ lower and upper bounds 0 -6 -4 -2 0 2 4 6 0 1 2 3 4 5 Residue Design Variable, X Fig.1c. Cumulative Distribution Function Fig.1d. Uncertainty in estimation of of Residue Expensive Function 1 35 ___expensive function Cumulative Probability 30 0.8 _ _ _ lower and upper bounds 25 Response 0.6 20 0.4 15 10 0.2 5 0 0 -5 -4 -3 -2 -1 0 1 0 1 2 3 4 5 Residue Design Variable, X Fig.1e. Restricted Cumulative Distribution Fig.1f. Uncertainty in estimation of Function of Residue Expensive Function using restricted CDF 9 American Institute of Aeronautics and Astronautics 40 1 Number of Simulations Cumulative Probability 0.8 30 0.6 20 0.4 10 0.2 0 0 1 2 3 4 5 6 0 1 2 3 4 5 Number of responses in attractive zone Design Variable, X Fig.2a. Histogram of number of hits in Fig.2b. Cumulative Distribution Function the attractive zone of design support for various simulations 1 1 Cumulative Probability 0.8 0.8 Cumulative Probability 0.6 0.6 0.4 0.4 0.2 0.2 0 -10 -5 0 5 10 15 0 -6 -4 -2 0 2 Residue Residue Fig.2c. Cumulative Distribution Function Fig.2 d. Restricted Cumulative Distribution of Residue for various simulations Function of Residue for various simulations 10 American Institute of Aeronautics and Astronautics 1 1 Trial 1 Trial 5 Cumulative Probability Cumulative Probability 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -60 -40 -20 0 20 -60 -40 -20 0 20 Residue Residue 1 1 Trial 10 Trial 15 Cumulative Probability Cumulative Probability 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -60 -40 -20 0 20 -60 -40 -20 0 20 Residue Residue 1 1 Trial 20 Trial 50 Cumulative Probability Cumulative Probability 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 -60 -40 -20 0 20 0 -60 -40 -20 0 20 40 Residue Residue Fig.3 History of Cumulative Distribution Function of Residue 11 American Institute of Aeronautics and Astronautics y x n-pl z x w-cant 1 2 3 L mid Lab Fig.4a. Generic Hypersonic Vehicle Configuration 1 0.8 Cumulative Probability 0.6 0.4 0.2 0 -4 -3 -2 -1 0 1 2 Residue, ma (kg/s) Fig.4b. Cumulative distribution function for mass flow capture of air 12 American Institute of Aeronautics and Astronautics 20 ___ upper bound ( top line) ____ lower bound ( bottom line) ____ lfa response – 3.2 Mass flow capture of air, ma (kg/s) 15 ____ HF response ( thick line ) 10 5 0 6 8 10 12 14 16 18 Mass flow capture of air, ma (kg/s) Fig.4c Uncertainty in the estimation of mass flow capture of air 1 0.8 Cumulative Probability 0.6 0.4 0.2 0 -1.5 -1 -0.5 0 0.5 1 1.5 Residue, ma (kg/s) Fig.4d Restricted Cumulative distribution function for mass flow capture of air 13 American Institute of Aeronautics and Astronautics 20 ___ upper bound ( top line) ____ lower bound ( bottom line) Mass flow capture of air, ma (kg/s) ____ lfa response – 3.2 15 ____ HF response ( thick line ) 10 5 0 6 8 10 12 14 16 18 Mass flow capture of air, ma (kg/s) Fig.4e Uncertainty in the estimation of mass flow capture of air using restricted CDF S.No. SX SR S.No. X Z 1 1.21 46.09 1 1.21 46.09 2 6.92 55.69 2 6.92 55.69 3 9.07 88.22 3 9.07 88.22 4 0.50 65.20 5 2.50 22.96 6 4.50 39.60 Tab 1a: Typical Sample of 7 6.50 72.58 High Fidelity Responses 8 8.50 119.41 9 10.50 179.49 Tab1b. Typical Augmented data 14 American Institute of Aeronautics and Astronautics S.No. SX SR Rank Probability Cumulative Density Probability 4 0.50 65.20 5 0.11 0.10 1 1.21 46.09 7 0.16 0.26 5 2.50 22.96 9 0.20 0.46 6 4.50 39.60 8 0.18 0.63 7 6.50 72.58 4 0.09 0.72 2 6.92 55.69 6 0.13 0.86 8 8.50 119.41 2 0.04 0.90 3 9.07 88.22 3 0.07 0.97 9 10.50 179.49 1 0.02 0.99 Tab1c. Typical data showing rank transformation of responses and cumulative probabilities of design points S.No. X Z 1 1.21 46.09 2 6.92 55.69 3 9.07 88.22 4 5.10 36.61 Tab1d. Typical High Fidelity sample after selecting one point from CDF of design space S.No. zs xs S.No. zs xs --------------------------- --------------------------- 1 2 1 S.No. zs xs 1 17.60 0.00 ------------------ 2 11.76 0.35 2 1 1 3 1 1 1 1 1 3 9.86 0.71 2 1 2 4 10.16 1.07 4 1 1 5 1 1 3 1 3 5 11.29 1.42 4 2 1 6 12.29 1.78 6 1 2 7 1 2 5 2 3 7 12.56 2.14 6 3 3 8 11.94 2.50 8 1 2 9 10.61 2.85 9 1 2 10 9.16 3.21 10 1 2 11 1 3 Tab.2c Typical Coded data 11 8.58 3.57 12 1 3 after deleting duplicate 12 10.25 3.92 13 2 3 points 13 15.92 4.28 14 27.75 4.64 14 3 3 15 48.28 5.00 15 3 3 Tab.2a. Typical data in Tab. 2b Typical Coded physical domain for data in after stratification stratification 15 American Institute of Aeronautics and Astronautics S.No. zs xs zs xs -------------------- --------------------------- 1 1 1 Level 1 3 2 2 1 2 Level 2 2 1 3 2 1 Level 3 1 3 Tab.2e Typical Coded Tab.2d Frequency table of data after selecting first coded data point S.No. zs xs S.No. zs xs ------------------- -------------------- 1 1 2 1 3 3 ` 2 2 1 Tab.2f Typical Coded 3 1 2 data after selecting second point Tab.2g Summary of Coded data selected for three levels Function Design Space Attractive Zone High Fidelity Z X 0.5 X 2 X 4 X 3.25 10 D 0 5 Z 8.3 Low Fidelity z 1.8785 * X 2 7.8888 * X 15.7313 D 0 5 Tab3a. Univariate Bimodal Function Residue Restricted Residue Lower Bound Upper Bound Lower Bound Upper Bound Mean -3.70 5.70 -3.11 -0.59 Standard 0.49 2.35 0.50 0.36 Deviation Coefficient of 13.20 41.20 16.00 60.00 Variation , % Tab. 3b. Metrics from Monte-Carlo Simulations of Univariate Bimodal Function 16 American Institute of Aeronautics and Astronautics

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upper bound, Cumulative Probability, Cumulative Distribution Function, hypersonic vehicle, uncertainty model, low fidelity, high fidelity, design support, design points, mass flow

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posted: | 12/27/2010 |

language: | English |

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