aerodynamic response quantification of complex...
TRANSCRIPT
Aerodynamic Response Quantification of Complex
Hypersonic Configurations using Variable-Fidelity
Surrogate Modeling
James A. Tancred ∗
University of Dayton, Dayton, Ohio, 45469, USA
Markus P. Rumpfkeil †
University of Dayton, Dayton, Ohio, 45469, USA
At hypersonic speeds, the design of a flight-worthy air vehicle to sustain endo-atmosphericflight remains to be a challenge to the designer at the conceptual and preliminary stages.The complex and extreme environment in which such vehicles fly presents physical phe-nomena and engineering challenges which are very difficult to simulate in such a way thatmodel approximation errors do not compromise vehicle integrity and function in flight.This paper seeks to capture the aerodynamic response, more commonly known to the en-gineering community as aero-database generation, of complex hypersonic vehicles usingvariable-fidelity (VF) kriging surrogate models. This approach merges the response of dif-ferent flow prediction simulations, varying in flow approximation accuracy, into a singlesurrogate model. This model can be built for different vehicle aerodynamic responses suchas lift, drag, and pitching moment. The VF kriging approach interpolates the simulation re-sponses of the highest fidelity level while being guided by the trends of any number of lowerfidelity responses. This can accelerate the database generation process dramatically by tak-ing advantage of the lower computational cost of the lower fidelity simulations. Examplesimulations are performed on a generic hypersonic configuration using Euler simulationsas a high-fidelity approximation and the Modified Newtonian method as a low-fidelity ap-proximation. Performance and error of the VF kriging approach is assessed to show itsadvantages for the use in conceptual and preliminary design of hypersonic configurations.
Nomenclature
CD Drag coefficientCL Lift coefficientCM Pitching moment coefficientCp Pressure coefficientD Vector of design variablesIl Total number of training points at the lth fidelity leveli Specific training pointJ Objective functionL Total number of fidelity levelsl Specific fidelity level, l = 1 is the highest fidelityM Mach numberm Total number of design variablesn Body surface normalp0,2 Stagnation pressure behind a normal shock-wavep∞ Free-stream pressureq∞ Free-stream dynamic pressureR Domain (parameter-space) of the objective function JV∞ Free-stream velocityα Angle-of-attack
∗Master’s Candidate, Dept. of Mechanical and Aerospace Engineering, [email protected], Student Member AIAA†Assistant Professor, Dept. of Mech. and Aerospace Engineering, [email protected], Senior Member AIAA
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53rd AIAA Aerospace Sciences Meeting
5-9 January 2015, Kissimmee, Florida
AIAA 2015-1703
Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner.
AIAA SciTech
θ Local surface inclination angleρ∞ Free-stream densityφ Angle between body surface normal and free-stream velocity
I. Introduction and Considerations
Difficulty plagues the design of flight vehicles that dwell in high speed and high Reynolds number flight.In particular, predicting the high non-linearity of the vehicle response to flow phenomena experienced inhypersonic flight is masked by numerous complex flow physics including turbulence, high thermal gradients,and, among numerous others, the perplexing nature of viscous-inviscid interactions1,2 found within the shock-layer. Combined, these phenomena take on a fluid-dynamic mechanism that is challenging to simulate eithernumerically or experimentally on the ground. For the current state-of-the-art, only after successfully reachingtrue flight conditions can a more accurate understanding of hypersonic vehicle aerodynamic responses begained. However, reaching flight conditions can be problematic, to say the least, if predictions by designsimulations are not accurate. A paradox unfolds such that knowledge is required to reach flight, yet flightitself is the key to unlock some of that knowledge. Attempting to accommodate such complexity numericallyor experimentally becomes taxing, requiring heavy supercomputing or incurring very high costs from groundtesting. This is the challenge of hypersonic vehicle design.
The challenge of flight, especially at hyper-velocity, complicates the efforts of the designer seeking aflight-worthy air vehicle. Design itself calls for the exploration of various concepts that yield options andperhaps better alternatives for vehicle and mission requirements. This implies timely searching and devel-opment of concepts in order to meet program schedules and to provide efficient designs. Due to the complexnature of hypersonic flight, accuracy in flight performance predictions is also a paramount requirement ofdesign. However, due to that very complexity, the time it takes to reach accurate predictions can becomeprohibitive. This work attempts to alleviate, to some degree, the time necessary to reach more accurateaerodynamic performance predictions of hypersonic configurations through the use of variable-fidelity (VF)kriging, thereby accelerating the exploration of a tractable vehicle trade space. Emphasizing the use of VFkriging on a generic hypersonic configuration, the vehicle shown in Figure 1 was analyzed using Euler flowequations and the Modified Newtonian method. The vehicle is not optimized for a given flight condition,but rather is proposed merely to penetrate a hypersonic configuration design space.
Figure 1. Generic hypersonic vehicle used to demonstrate variable-fidelity kriging.
This paper begins with an overview of the VF kriging method in Section II. Section III describes the high-and low-fidelity simulation codes, Cart3D3–6 and the Modified Newtonian method,7 respectively, in moredetail. Verification results via grid convergence studies of the analysis codes to ensure proper grid set-upare shown in Section IV. The aero-databases for the complex hypersonic vehicle configuration generated bythe two analysis codes on a Cartesian mesh of equispaced nodes are presented in Section V. Finally, theapplication of VF kriging for the generation of the same hypersonic aero-database is discussed in Section VI.
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II. Variable-Fidelity Kriging Overview
According to Chiles and Delfiner, the term kriging was first used by Georges Matheron in 1963 in honorof the statistical work of Danie Krige within the field of geostatistics.8 A quite appropriate quote also fromChiles and Delfiner regarding a fundamental problem within geostatistics is as follows:8
A central problem in geostatistics is the estimation of a variable of interest over a domain on thebasis of values observed at a limited number of points.
The problem of spatial quantification, a problem about which the field of geostatistics is intimately aware,is ironically the same as that found when quantifying the aerodynamic response of flight vehicles. Ratherthan quantifying geological phenomena, here the nature of how perturbations of a body in fluid flow affectthe dynamics of flight is considered.
In general, kriging is a predictive method in which statistics are used to build a surrogate model (ormetamodel) that passes strictly through function values of sampled points of an underlying function ofinterest.8,9 In other words, given an m-dimensional design variable vector, D, kriging will pass a surfacethrough the function of interest J(Di), i = 1, 2, 3, . . . , I1, where I1 is the total number of training points atthe highest fidelity level. When building a kriging surrogate model, a spatial correlation function is usedwhich depends only on the distance between two training points.9
Kriging also supports the usage of both high- and low-fidelity training points.9–13 The key idea of avariable-fidelity surrogate model is to map the trend of the unknown function underlying the intensivelysampled low-fidelity data to the less intensively sampled high-fidelity data. Some possible combinationsfor low- and high-fidelity data sources are shown in Table 1, and an application to an analytical functionexample where a more accurate surrogate model is constructed by using the trends of a low-fidelity objectivefunction is shown in Figure 2.
Table 1. Some possible combinations of variable-fidelity methods
High-Fidelity Model Low-Fidelity Model
Experimental data CFD results
RANS Euler
Finer mesh CFD results Coarser mesh CFD results
Fully converged solutions Partly converged solutions
Figure 2. Analytical function example for the use of variable-fidelity kriging.
The most popular method for this mapping currently used is a correction-based method.13 The correctionis called bridge function, scaling function or calibration. The correction can be multiplicative, additive orhybrid multiplicative/additive. For instance, a multiplicative bridge function is used to locally scale thelow-fidelity function to approximate the high-fidelity function and is typically a low-order polynomial (ofconstant, linear or second order). An additive bridge function was developed as a global correction andhas become the most popular method for variable-fidelity optimization or for data fusion.13 The VF krigingapproach used in the present work follows that of Yamazaki and Mavriplis9 using an additive bridge functionand incorporates the dynamic sampling method developed by Boopathy and Rumpfkeil.14,15
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III. Analysis Code Descriptions
To demonstrate the VF kriging formulation discussed in Section II, two different fidelity computationalsimulation methods are used for a generic hypersonic vehicle. The first, considered to be the high-fidelitysimulation, is Cart3D.3–6 The second, designated as the low-fidelity simulation, is an implementation of theModified Newtonian local inclination method.7 Each of these methods is described briefly in SubsectionsIII.A and III.B, respectively.
III.A. High-Fidelity Simulations: Cart3D
The high-fidelity simulations performed for this research used Cart3D for the flow simulations. Cart3D isa computational software suite suitable for rapid generation of CFD solutions for the Euler flow equations.Using a robust cell-centered, finite volume5,6 method, Cart3D is capable of producing numerous flow solutionswithout user intervention. For the present hypersonic configuration, a Mach and angle-of-attack domain wasused, given by R ∈ [(M,α) | 6 < M < 15,−5◦ < α < 15◦] with increments of ∆M = 0.3 and ∆α = 0.5degrees. Thus, it requires 1, 271 flow solutions to build this aero-database totaling 166 hours (6.9 days) ofwall-clock time using 32 cores. The flow solutions in the corners of the domain and the corresponding finaladjoint-adapted meshes are shown in Figures 3 and 4, respectively.
M = 6, α = −5 degrees M = 6, α = 15 degrees M = 15, α = −5 degrees
M = 15, α = 15 degrees
Figure 3. Cart3D solution Mach contours in the corners of the M − α domain. (Contour legend omitted.)
Figure 5 shows the convergence history of these four simulations. The “oscillatory” nature of the residualsis due to the adjoint-based mesh adaptation implemented in Cart3D. As the solution progresses, errors in aflow functional (such as lift, drag, or pitching moment) relative to a user-specified tolerance are assessed byCart3D. Here, the pitching moment, CM , is monitored and cells in the mesh with adjoint weighted errorsabove a given tolerance are sub-divided every 100 iterations which is reflected by the “spikes” in the residuals.
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M = 6, α = −5 degrees M = 6, α = 15 degrees M = 15, α = −5 degrees
M = 15, α = 15 degreesFigure 4. Adjoint-based adapted final meshes from Cart3D in the corners of the M −α domain. Note the clustering ofcells in regions where there are large flow gradients, such as the boundaries of the shock structures along the vehicle.
M = 6, α = −5 degrees M = 6, α = 15 degrees
M = 15, α = −5 degrees M = 15, α = 15 degrees
Figure 5. Cart3D convergence histories in the corners of the M − α domain.
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III.B. Low-Fidelity Simulations: Modified Newtonian Method
For low-fidelity simulations, the Modified Newtonian (MN) method7 was used. This method assumes thatflow over a body travels so fast that its deflection is tangential to any point on the body. Hence, Anderson7
refers to this approximation as a local surface inclination method. It is quite useful for very fast, approximatecalculations on arbitrary bodies. A brief description of the method follows.
Given a body subjected to an airflow with surface normal n, flow inclination angle θ between a surfacetangent plane and the free-stream velocity V∞, and angle φ between n and V∞, the following complementaryequation is apparent:
θ =π
2− φ (1)
Further, given a pressure p induced by the incoming airflow acting in the opposite direction of n, the MNmethod assumes that the pressure coefficient, normalized by dynamic pressure q∞ = 1
2ρ∞|V∞|2, that impacts
the body is given by
Cp = κ sin2 θ (2)
where κ is the pressure coefficient resulting from the stagnation pressure behind a normal shock-wave:7
κ =2
γM2∞
[p0,2p∞− 1
]. (3)
The fractionp0,2p∞
is given by the Rayleigh-Pitot tube formula7 as
p0,2p∞
=
[(γ + 1)
2M2∞
4γM2∞ − 2 (γ − 1)
] γγ−1 [
1− γ + 2γM2∞
γ + 1
]. (4)
Tracking each surface normal, the value for φ may be obtained from the dot product of the normal andfree-stream velocity vectors:
cosφ = −n · V∞ (5)
Note that Equation (5) aligns n and V∞ head to tail. If the flow does not directly impact the body, whichis known as shadowed or leeward region, the value of Cp is set to zero. Using Equations (1)-(5), forces andmoments may be obtained about the body in hypersonic flow. The same domain as used for the high-fidelitysolutions in Subsection III.A was simulated16,17 with the MN method just described. The 1, 271 solutionstook approximately 30 minutes of wall-clock time on a single core.
IV. Verification of Analysis Codes
To produce confidence in the simulation approach, grid convergence analyses for the high- and low-fidelitysimulations were performed. Flow conditions for all combinations of the sets M ∈ [6, 10, 15] and α ∈ [5◦, 15◦]were completed with different meshes for both fidelity levels. The values of the lift and drag coefficients weremonitored and for the sake of brevity, where appropriate, the results for the domain points R(M,α) = (6, 5◦)and R(M,α) = (15, 15◦) are shown.
IV.A. Grid Convergence Study of Cart3D
The grid convergence study for Cart3D explored various starting meshes which then underwent between fiveto nine adjoint guided adaptation cycles. The different starting meshes are rather coarse, and increase insize up to roughly 20, 000 volume cells. These meshes, along with their final state after nine adaptations,are shown in Figure 6. By increasing the starting mesh size, the volume grid resolution resulting after thefinal adaptation is also increased. Therefore, having a larger starting mesh is preferable to reach an accuratesolution. However, it is noted that if the starting mesh resolution is too high, Cart3D will not recognize thaterror in the solution exists around the body, and will cluster cells in the off-body region rather than near thebody itself. An initial starting mesh resolution of roughly 20, 000 cells is suggested for proper adaptation
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and convergence of Cart3D about a body of similar shape as that depicted by the present generic hypersonicvehicle.
Scatter plots of CL and CD, reflecting different combinations of starting meshes and adaptation evolutionare shown in Figure 7. Changes in both monitored coefficients begin to slow after roughly seven adaptations,indicating that Cart3D has converged. The change in each force coefficient, as well as the recorded wall-clocktime, from seven to nine adaptations is shown in Table 2. The changes in the coefficients are quite smallacross this range of adaptations. This indicates that seven adaptations is sufficient to reach a convergedsolution with accompanied significant time savings. Thus, seven grid adaptations are going to be used forall runs with Cart3D using an initial starting mesh resolution of roughly 20, 000 cells, which yields aboutone million cells after the last adaptation.
Table 2. Change in Force Coefficients from 7 to 9 Adaptations
∆CL ∆CD Change in Wall-Clock Time (min)
R(M,α) = (6, 5◦) −3.6 · 10−4 7.0 · 10−5 43
R(M,α) = (15, 15◦) −3.0 · 10−4 2.1 · 10−4 40
M = 6, α = 5 degrees M = 15, α = 15 degreesFigure 6. Cart3D grid adaptation evolution from starting mesh (image left) through 9 adaptations (image right).Increasing the cell count of the starting mesh clearly increases the resolution of the volume grid after final adaptation.A starting mesh resolution of roughly 20000 cells is suggested for proper adaptation and convergence.
IV.B. Grid Convergence Study of the Modified Newtonian Method
The Modified Newtonian method was exercised using eight surface meshes of varying cell resolution. Selectedmeshes used in the grid convergence study are shown in Figure 8. To illustrate how well each of thesemeshes facilitate a solution, plots of CL and CD are plotted in Figure 9. After employing roughly 200, 000surface cells, the lift and drag coefficients begin to approach asymptotic values. This demonstrates sufficientresolution of the surface mesh to indicate solution convergence. Therefore, the mesh with 244, 252 cells,pictured in Figure 8, was chosen for use in the full VF kriging simulations. This mesh is indicated by thevertical dashed (magenta) line in Figure 9.
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Figure 7. Scatter plots of CL (left) and CD (right) solutions for different starting meshes that evolve from the fifth toninth adaptation for R(M,α) = (6, 5◦) (top) and R(M,α) = (15, 15◦) (bottom). Adaptations increase from left to right ineach plot.
Ncells: 49,340 Ncells: 117,022 Ncells: 244,252
Ncells: 734,944
Figure 8. Example surface meshes used to exercise the Modified Newtonian method.
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CL, α = 5 degrees CD, α = 5 degrees CL, α = 15 degrees
CD, α = 15 degreesFigure 9. Grid convergence study of CL and CD across the Mach number set M ∈ [6, 10, 15] for the Modified Newtonianmethod using different surface meshes.
V. Aero-databases using Cartesian Mesh
Using the high- and low-fidelity simulations with the grid resolutions described in the previous section on aCartesian mesh of 31×41 = 1, 271 equispaced nodes on the domain R ∈ [(M,α) | 6 < M < 15,−5◦ < α < 15◦]the resulting “actual” aero-databases for the lift and drag coefficients are displayed in Figure 10. The Cart3Dsurfaces are plotted in red (translucent) while those from the MN method are plotted in green (solid). Onecan infer that both the Cart3D and the MN results trend in similar fashions. As expected, more of thephysical flow features are captured by Cart3D which leads to higher non-linear regions in the responses.However, since the trends are similar for both the high- and low-fidelity simulations, the VF kriging methodperforms very well, as shown in the next Section.
Lift Coefficient Drag CoefficientFigure 10. Lift and drag performance of a generic hypersonic vehicle predicted by Cart3D (red, translucent) and theMN Method (green, solid).
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VI. Variable-Fidelity Kriging Results for a Hypersonic Aero-Database
The promise of the VF kriging method is that it has the potential to capture high-fidelity results withreasonable accuracy but with considerably less computational cost. By using the VF kriging formulationdescribed in Section II, the use of many levels of fidelity along with dynamic sampling of kriging trainingpoint locations enables an efficient means of estimating complex responses. To investigate the performanceof VF kriging for the generic hypersonic vehicle considered in this paper, the test matrix with the numberof high-fidelity (HF) and low-fidelity (LF) sample points shown in Table 3 was implemented.
Table 3. Matrix of number of high- and low-fidelity samples used for VF kriging
Num. LF Samples
Num. HF Samples 0 50 200
10 X X X
20 X X X
40 X — —
For each case, the following sequence of analysis was implemented to generate a kriging surface:
1. Obtain the five high-fidelity responses from the corners and the center of the M − α domain
2. Obtain all requested low-fidelity training points through latin hypercube sampling and evaluate theirfunction values
3. Build initial VF kriging surface
4. Continue to obtain high-fidelity training points through dynamic sampling14,15 until all requests aremet; build intermediate kriging surfaces after every five high-fidelity training points are determinedthrough dynamic sampling
The construction of the VF kriging model and required calls to Cart3D and the MN method was orchestratedby a Python script and run on 32 cores.
VI.A. Variable-Fidelity Kriging Qualitative Results
The VF kriging surfaces for CL and CD are shown in Figures 11-12 for 10 and 20 HF training points,respectively. Observing plots of lift or drag coefficients for 10 HF points, there is noticeable improvementin each response surface with the increase of LF samples. Note that, in the plot for CD in Figure 11(d), thekriging surface does not represent drag well because 10 HF samples is insufficient to resolve the curvatureof the drag response. However, as the LF sample points are increased to 200, the kriging model is able tocapture the drag response properly. Lift coefficient, having less curvature across the M − α domain, is aneasier response for the kriging formulation to model. Similar to the drag response, as the LF samples areincreased significantly to 200, the kriging models for CL yield a smoother response surface as well.
Comparisons of the kriging surfaces to the “actual” database are shown in Figures 13 and 14. Note thevery good agreement across all surfaces. However, one can observe that adding 50 LF samples to surfaceswith 20 HF samples (Figures 12(b) and 12(e)) actually reduces the accuracy of the resulting response surfacesbecause the LF samples drive the kriging surface away from the high-fidelity model that is already relativelyaccurate without LF sampling. Increasing the number of low-fidelity samples further to 200 drives thesurfaces back to a more accurate representation. Using 20 HF samples, or more, essentially captures the“actual” databases from Section V.
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(a) (b) (c)
(d) (e) (f)Figure 11. Variable-fidelity kriging surfaces for CL (top) and CD (bottom) using 10 high-fidelity samples (responsevalues omitted). High-fidelity samples are given as large spheres (gray) while low-fidelity samples are shown as smallspheres (yellow): (a) [CL] 0 LF, (b) [CL] 50 LF, (c) [CL] 200 LF, (d) [CD] 0 LF, (e) [CD] 50 LF, (f) [CD] 200 LF
(a) (b) (c)
(d) (e) (f)Figure 12. Variable-fidelity kriging surfaces for CL (top) and CD (bottom) using 20 high-fidelity samples (responsevalues omitted). High-fidelity samples are given as large spheres (gray) while low-fidelity samples are shown as smallspheres (yellow): (a) [CL] 0 LF, (b) [CL] 50 LF, (c) [CL] 200 LF, (d) [CD] 0 LF, (e) [CD] 50 LF, (f) [CD] 200 LF
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(a) (b) (c)
(d) (e)Figure 13. Comparison of CL kriging surfaces to the “actual” database (response values omitted): (a) 10 HF, 200 LF;(b) 20 HF, 200 LF; (c) 20 HF, 0 LF; (d) 40 HF, 0 LF; (e) 1271 HF
(a) (b) (c)
(d) (e)Figure 14. Comparison of CD kriging surfaces to the “actual” database (response values omitted): (a) 10 HF, 200 LF;(b) 20 HF, 200 LF; (c) 20 HF, 0 LF; (d) 40 HF, 0 LF; (e) 1271 HF
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VI.B. Variable-Fidelity Kriging Quantitative Results and Discussion
To give more significance to the qualitative results presented above, the root-mean square error (RMSE)of each kriging surface relative to the “actual” aerodynamic database was monitored. Here, the RMSE isgiven by
RMSE =
√√√√ 1
1271
31∑a=1
41∑b=1
(Jab − yab)2 (6)
where Jab is the value of the high-fidelity response, J , (lift or drag coefficient) at the abth location in theM−α domain, and yab is the corresponding response value predicted by the VF kriging surface. The RMSEhistories for the different cases in Table 3 are shown in Figure 15.
(a) (b)
(c) (d)Figure 15. Evolution of RMSE errors using 10 HF training points (top) and 20 HF training points (bottom) comparedto the use of 40 HF points: (a) [CL] 10 HF, (b) [CD] 10 HF, (c) [CL] 20 HF, (d) [CD] 20 HF
Each of the 10 HF and 20 HF simulations are compared against the 40 HF simulations containing noLF sampling. Observing the RMSE across all figures, adding LF sampling significantly reduces the error ifthere are 10 HF samples or less. One can also observe that the RMSE increases for the cases of 20 HF and50 LF samples (found in Figures 15(c) and 15(d)) as noted previously in Section VI.A. The use of only amoderate amount of LF samples in a model that has already a larger number of high-fidelity samples tendsto corrupt the model, adding error that otherwise would not be included. However, increasing the numberof LF samples to 200 does actually reduce the RMSE to reasonable levels compared to the use of 40 HFsamples in all cases. Once the number of HF samples increases to 20 or above, the benefit of the LF samplingfor the kriging models diminishes. That is, the error in kriging models with no LF sampling reduces to levelsat or below those of the VF kriging models.
Illustrating quantitatively the trends in RMSE across the kriging models studied, RMSE values areshown for CL and CD in Tables 4 and 5, respectively.
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Table 4. RMSE in Lift Coefficient, CL, for different VF kriging models
—RMSE— Num. LF Samples
Num. HF Samples 0 50 200
10 3.33 · 10−3 3.52 · 10−3 2.69 · 10−3
20 5.25 · 10−4 3.32 · 10−3 4.00 · 10−4
40 1.28 · 10−4 — —
Table 5. RMSE in Drag Coefficient, CD, for different VF kriging models
—RMSE— Num. LF Samples
Num. HF Samples 0 50 200
10 7.24 · 10−3 7.91 · 10−4 6.20 · 10−4
20 1.56 · 10−4 4.72 · 10−4 2.17 · 10−4
40 6.51 · 10−5 — —
Driving the RMSE below three orders of magnitude appears to create reasonable approximations tothe “actual” databases as shown in Figures 13 and 14. This can be achieved with larger HF sample sizes(greater than 20) or with lower HF sample sizes (below 20) combined with larger LF sample sizes (200).Once again, one can observe as HF sample sizes trend above 20, the use of LF samples introduces error intothe kriging model, creating a lower accuracy response surface. This is evidenced, for example, in the secondrow of Table 4. The RMSE spikes from 5.25 · 10−4 to 3.32 · 10−3 when 50 LF samples are introduced. Theerror then settles back to 4.00 · 10−4 when 200 LF samples are used. From these numbers, it is clear thatVF kriging is best suited for very low HF sample sizes and very high LF sample sizes.
Interestingly, using low HF sample sizes and very high LF sample sizes is an ideal condition for designpurposes. The VF kriging approach can reduce the number of required HF training points, considered inthe studies of this article, by factors of 127 and 63 for 10 and 20 HF training points, respectively. Thisis a significant finding. To put such factors into perspective, consider the average wall-clock time for asingle Cart3D run in this study. Using the suggested mesh settings from the grid convergence study inSection IV.A, the average runtime on 32 cores was 7.2 minutes. With this average runtime as well as VFkriging for 10 HF samples and 200 LF samples, the wall-clock time to build a single database reduces from6.4 days to 1.2 hours. Generating perhaps more resolved databases only marginally increases this time, asshown in Table 6. This makes VF kriging a very viable option for conceptual hypersonic vehicle databasegeneration. Furthermore, given the reasonable error found in the kriging models relative to the “actual”database, the use of kriging and dynamic sampling also appears to be ideal for preliminary design. Havingthe ability to resolve an underlying aerodynamic response quickly and within a close tolerance is whatpreliminary hypersonic vehicle design requires. Such a demonstration of acceleration and accuracy (relativeto a specified benchmark, in this case a benchmark of Euler solutions) of hypersonic aerodynamic databasegeneration shows promise for both VF kriging and kriging with dynamic sampling.
Table 6. Cart3D Wall-Clock Time for Various Requested HF Training Points (32 Cores, 7.2 min. Mean Runtime)
Num. HF Samples Wall-clock Time
10 1.2 hrs
20 2.4 hrs
40 4.8 hrs
1271 6.4 days
A final remark on the aerodynamic database generation with VF kriging is on the accuracy of resultswith respect to true flight conditions. The results of the kriging surrogate models are only as good as theunderlying assumptions of the highest fidelity analysis code employed and the error inherent to the krigingformulation. In the case of this work, the surrogate models generated are only valid under the assumptions
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of Cart3D (inviscid, three-dimensional Euler solutions). No viscous effects are considered. If more physicalrealism is required by the designer, other analysis tools incorporating viscous effects, such as the Reynolds-Averaged Navier-Stokes (RANS) equations, should be chosen as the highest fidelity level. Keep in mind,however, that this still is not the end-all solution to obtaining a database that is truly representative ofactual flight conditions. Once again, as mentioned at the beginning of this article, computational modelsand experiment both have shortfalls to the prediction of true flight behavior. The positive note here is thatthe designer now has the ability, with accelerated database generation, to use state-of-the-art computationalmodeling at both conceptual and preliminary phases of hypersonic vehicle design through VF kriging.
VII. Conclusion
The use of variable-fidelity kriging was exercised to show the viability of its use for hypersonic vehicledesign. Two levels of fidelity, Euler solutions from Cart3D and impact solutions through a Modified New-tonian method, were employed. A high-fidelity database on a Cartesian mesh of equispaced nodes usingCart3D was generated to quantitatively assess the variable-fidelity approach. Grid convergence studies werealso performed to ensure that each fidelity level was set up properly to achieve grid-converged solutions. Itwas shown that variable-fidelity kriging can significantly reduce the time required to build databases for liftand drag coefficients for a generic hypersonic vehicle in the Mach and angle-of-attack domain. The utilityof variable-fidelity kriging was found to be best when using low high-fidelity sample sizes (in this case lessthan 20 training points) and large low-fidelity sample sizes (200 samples). Reduction in computational costfor the generation of a relatively accurate aerodynamic database exceeded two orders of magnitude for theextreme case (limited high-fidelity information) of 10 high-fidelity training points and 200 low-fidelity sam-ples. This acceleration as well as the accuracy of the resulting surrogate models show promise for the use ofvariable-fidelity kriging and kriging with dynamic sampling during conceptual and preliminary hypersonicvehicle design phases.
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