A Nonlinear Sensitivity Solver for Aerodynami ShapeOptimization and Rapid Design Spa e ExplorationTravis W. Drayna∗GoHypersoni In orporatedDayton, OH 45402Graham V. Candler† and Heath B. Johnson‡Department of Aerospa e Engineering and Me hani sUniversity of Minnesota, Minneapolis, MN 55455A new nonlinear sensitivity solver has been derived and implemented within the US3DComputational Fluid Dynami s ode. The nonlinear sensitivity equation is derived from thedis rete nite-volume formulation of the Navier-Stokes equations. The sensitivity solveruses a steady-state solution of the ow variables as an initial starting point. A hangeto the system is then introdu ed in the form of a grid or ow parameter perturbation.The perturbed system is solved using an impli it method and rapidly onverged until theuxes are properly balan ed. The new solution at the perturbed state is a ompletely validsolution of the same Navier-Stokes equations used by the CFD solver. Be ause of this, thesensitivity solver an a urately predi t highly nonlinear events su h as sho k wave move-ment and sho k boundary layer intera tions. In this work, the nonlinear sensitivity solveris used to e iently al ulate obje tive fun tion gradients within an inlet optimization y- le. Additionally, the sensitivity solver is used as a onvergen e a eleration me hanism torapidly simulate an angle of atta k sweep of a three-dimensional inward-turning s ramjetinlet. I. Introdu tionThe basis for this work began with the goal of optimizing inward-turning s ramjet inlets for the HyCAUSEand DARPA ASET programs.1, 2 During these eorts, streamline-tra ing was used to arve out various inletdesigns from invis id Busemann parent owelds. The performan e of these invis id designs, however, wasfound to be less than ideal for a number of reasons.3 First, the Busemann oweld is oni al and invis idwhile real inlets have vis ous ee ts with strong three-dimensionality. In addition, real inlets are oftentrun ated to mu h shorter lengths than theoreti ally ideal owelds and have nite radius leading edgesadded for thermal and stru tural reasons. These features reate primary oblique sho k waves that are notpresent in the original Busemann oweld. Furthermore, while theoreti al owelds may be ideal for singledesign points, they an be far from ideal when missions require operation over a wide range of ight onditionssu h as Ma h number, dynami pressure, angle of in iden e, and angle of sideslip.∗Senior Engineer, AIAA Asso iate Member†Professor, AIAA Fellow‡Senior Resear h Asso iate, AIAA Senior Member 1 of 14Ameri an Institute of Aeronauti s and Astronauti s

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-26

Copyright © 2011 by Travis W. Drayna. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

In view of these issues, it was on luded that the simulation of s ramjet inlets requires the use of high-delity CFD to properly apture these real-world ee ts. It was also determined that these simulationsare very expensive to perform in terms of omputational resour es and user intera tion. For example, gridrequirements for a typi al s ramjet inlet might be in the ve to ten million ell range with run times rangingfrom two to six hours per simulation. In addition to this, user intera tion time might require four to eighthours for grid generation depending upon the omplexity and s ale of the geometry.Based on our work during both of these programs, we have on luded that the best way to properly designa mission-spe i s ramjet inlet is to perform a full shape-based aerodynami optimization on the geometry.This on lusion has several impli ations. First, the geometry must be parametri ally dened so that theshape of the surfa e an be represented by a set of input parameters. Se ond, a omputational grid mustbe generated that adequately resolves the ow and aptures all of the important geometri features su has surfa e urvature and dis ontinuities. Also, the omputational grid needs to be generated automati allyand robustly during ea h step of the optimization y le. Finally, the simulation of the ow needs to be doneusing CFD solvers that adequately apture and resolve the physi s of the ow.In our pursuit of full inlet optimizations, we dis overed that gradient-based methods work very wellon s ramjet inlets that are operating within the bounds of their operability envelopes (not nearing inletunstart). The use of su h gradient-based methods means that simulations must be performed to omputethe baseline performan e of the design as well as ea h of the parameter sensitivities. This means that ifan inlet is represented by 20 surfa e parameters, then a total of 21 full CFD solutions will be required toevaluate the obje tive fun tion with gradients. In pra ti e, it is not un ommon for an optimization to runthrough ten or more design y les to rea h the optimal design, for a total of 210 full CFD solutions. If weassume that ea h CFD simulation takes four hours to omplete, then the total omputational ost is 840hours or 35 days. At this point, the problem at hand be omes very lear: A method is needed for rapidlyevaluating obje tive fun tion gradients within the CFD-driven gradient-based optimization y le.II. Sensitivity SolverBa kgroundInitially, two methods were onsidered for the parameter sensitivity evaluation: the dire t sensitivity methodand the adjoint sensitivity method.47 While ea h of these methods has been su essfully used for lowspeed ow appli ations, very little information was found on the su essful appli ation of these methodsto the hypersoni ow regime. It was also noted that while the adjoint method shows promise in termsof e ien y, it requires the development of an entirely new solver that is separate from the original CFDsolver. Furthermore, both the dire t sensitivity method and the adjoint sensitivity method require thedevelopment of non-trivial boundary onditions for proper implementation. Also, the adjoint method is noteasily extensible to arbitrary obje tive fun tions. For these reasons, we hose to onsider a new approa h tothe parameter sensitivity problem that begins with the fundamental assumption of a dis rete steady-statesystem.DerivationWe begin by dening some quantities: α is a design parameter whi h denes some aspe t of the geometry,X(α) is the dis rete representation of the geometry and surrounding spa e on a grid in whi h the oordinatepoints are a fun tion of α. The variable U represents the solution ve tor of onserved variables on a given grid,

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and L is the obje tive fun tion of interest. Given these denitions we an observe the following fun tionality:X = X(α)

U = U(X) = U(X(α))

L = L(U,X) = L(U(X(α)), X(α))To derive the dis rete sensitivity equations, we begin with the ompressible Navier-Stokes equations whi h an be written as:∂U

∂t+ ~∇ · F = 0 (1)The spatially-dis retized nite-volume formulation an be written as:

∂Ui

∂t+

1

Vi

nf∑

j=1

~Fj ·~Sj = 0 (2)where Ui is the ve tor of onserved variables, Vi is the volume of ell i, ~Fj is the ux of onserved variablesthrough fa e j bounding the the ell, and ~Sj is the produ t of the fa e area and the fa e normal ve tor. Thetemporal dis retization of the hange in the solution an then be written as:

∂Ui

∂t≈

Un+1i − Un

i

∆t(3)Dening the uxes a ross a fa e as:

Fj = ~Fj ·~Sj (4)and linearizing hanges in the uxes with respe t to hanges in the onserved variables gives:

Fn+1j = F

nj +

∂Fnj

∂Uni

(

Un+1i − Un

i

) (5)We then dene the hange in the solution as:δUn

i = Un+1i − Un

i (6)Substituting (3), (4), and (6) into (2), the dis rete impli it form of the Navier-Stokes equations be omes:δUn +

∆t

V

∑

j

∂Fn

∂UnδUn = −

∆t

V

∑

j

Fn (7)where we have dropped the subs ripts i and j on the volume and fa e terms, respe tively, to redu e lutterin the equation. These subs ripts are assumed where appropriate in the remainder of the derivation.Finally, we dene the ux ve tor Ja obian as:

A =∂F

∂USubstituting the Ja obian into (7) and multiplying both sides by V/∆t, we arrive at the dis rete impli itform of the Navier-Stokes equations as it is solved within a typi al impli it nite-volume CFD solver:V

∆tδUn +

∑

AnδUn = −

∑

Fn (8)3 of 14Ameri an Institute of Aeronauti s and Astronauti s

When Eq. (8) is solved and the steady-state solution is rea hed on the original omputational grid, thesum of the uxes a ross the fa es of ea h ontrol volume is equal to zero su h that:δUn =

∑

Fn = 0 (9)Thus, at steady-state the uxes balan e out exa tly for the CFD solver. However, in the ase that we wish torun the CFD solver for a sensitivity analysis, we want to perturb the original grid by some small in rement,

∆α, of a design parameter and have the new uxes balan e out on the perturbed grid. Thus, we want:(

∑

F

)

α+∆α= 0 (10)Applying a Taylor series expansion to the uxes about the design parameter α gives:

F (α+∆α) = F(α) +∂F

∂α

∣

∣

∣

X∆α+

∂F

∂α

∣

∣

∣

U∆α+O(∆α2) (11)Applying the hain rule to (11), the term held at onstant X be omes:

∂F

∂α

∣

∣

∣

X=

∂F

∂U

∣

∣

∣

X

∂U

∂α= A

∂U

∂α= AΩ (12)where Ω is the sensitivity of the solution to the design parameter α. The term in (11) held at onstant U an be approximated as:

∂F

∂α

∣

∣

∣

U≈

F (U,X(α+∆α)) −F (U,X(α))

∆α(13)Substituting (12) and (13) ba k into (10) and re alling from (9) that at steady-state in the unperturbedsolution, ∑F =

∑

F (U,X(α)) = 0, we obtain:(

∑

F

)

α+∆α=

∑

(AΩ∆α+ F (U,X(α+∆α))) = 0After rearranging the terms we get:∑

AΩ∆α = −

∑

F (U,X(α+∆α)) (14)For a dis rete hange in the design parameter α, the sensitivity an be approximated as:Ω =

∂U

∂α≈

∆U

∆αSubstituting this into (14) we obtain the dis rete form of the linear sensitivity equation:∑

A∆U = −

∑

F (U,X(α+∆α)) (15)whi h has exa tly the same form (without the unsteady terms) as the dis rete Navier-Stokes equations (8)that are implemented in the CFD solver. Thus, the existing CFD solver an easily be modied to run insensitivity mode to solve the sensitivity equation. In addition, the sensitivity equation an be solved withthe same impli it method and boundary onditions as the CFD solver.On e a solution to the sensitivity equations is obtained, we an onstru t the perturbed solution as:U (X(α+∆α)) = U (X(α)) + ∆UWe then evaluate the obje tive fun tion on the new solution as:4 of 14Ameri an Institute of Aeronauti s and Astronauti s

L (U(X(α+∆α)), X(α+∆α))And ompute the gradient dire tly using a simple nite dieren e:∂L

∂α=

1

∆α(L (U(X(α+∆α)), X(α+∆α)) − L (U(X(α)), X(α)))In this way, the sensitivity solver is used to al ulate a linearized approximation of the solution due to anin remental hange in a design parameter. The obje tive fun tion al ulation is still done independently as apost-pro essing step and the gradients are omputed using a simple nite dieren e. However, it was foundthat the linear approximation an result in poor gradient estimates when strong non-linearities are presentin the ow. In fa t, experien e has shown that in ows with strong nonlinearities, the linear approximation an yield sensitivities with the in orre t sign and magnitude.To remedy this, the linearized sensitivity method is used within an outer iteration loop where the ap-proximate solution from the previous step is used as an initial guess for the next step. In pseudo- ode:do n = 1, nit

∑

A∆Un+1 = −

∑

F (Un, X(α+∆α)) (16)Un+1 = Un +∆Un+1

enddoThe dis rete nonlinear sensitivity solver algorithm presented here is superior to linearized sensitivitysolver formulations for a number of reasons. First, the algorithm omputes exa t solutions of the Navier-Stokes equations in luding nonlinearities by balan ing the same uxes as the CFD solver. Be ause of this, thegradients of the obje tive fun tion are mu h more a urate and reliable than those omputed with a linearizedsensitivity solver, whi h in turn leads to more e ient and robust optimizations. Se ond, the formulation istrivial to implement within virtually any impli it nite-volume CFD ode. The exa t implementation requiresalmost no new ode to be written and uses the solver's existing impli it method. Finally, the formulationrequires no new boundary onditions to be implemented and is extensible to virtually any obje tive fun tion.Thus, the nonlinear sensitivity solver is well suited to CFD solvers with e ient impli it methods of solutionsu h as the DPLR method. In addition, it has been found to work surprisingly well as a solution a elerationmethod for ee tively driving general CFD simulations to steady-state.III. CFD SolverIn this work, the nonlinear sensitivity solver algorithm was implemented within the US3D CFD solver.US3D is a highly s alable parallel unstru tured CFD solver developed by the University of Minnesota forthe simulation of hypersoni and re-entry ows. US3D solves a nite-volume formulation of the ompressibleNavier-Stokes equations with extensions to a ount for nite-rate internal energy ex itation and hemi alkineti s.8, 9 Conve tive uxes are omputed using the modied Steger-Warming ux ve tor splitting method.Higher-order a urate spatial uxes are obtained using a onne tivity-based re onstru tion for hexahedral ells and a gradient-based re onstru tion for non-hexahedral ells. In both ases, a variant of the MUSCLs heme is used to prevent overshoots and os illations a ross sho ks and in regions of strong gradients. Cell- entered gradients are omputed using a weighted-least-squares re onstru tion of the primitive variableswhile vis ous uxes are omputed using a deferred- orre tion approa h. Rapid onvergen e to steady stateis obtained through the use of the Data Parallel Line Relaxation (DPLR) method along lines of ells normalto solid walls.10 In the remainder of the domain, the Full Matrix Point Relaxation (FMPR) method is5 of 14Ameri an Institute of Aeronauti s and Astronauti s

used. US3D in orporates the Spalart-Almaras one-equation turbulen e model with the Catris-Aupioux ompressibility orre tion. In general, US3D has the apability to be run fully laminar, fully turbulent, ortripped turbulent. In the latter ase, the trip lo ations are user dened and an represent natural transition orfor ed tripping depending on the model sele ted. For exibility, US3D in orporates a wide range of standardsurfa e boundary onditions in luding atalyti and partially- atalyti walls with and without radiativeequilibrium, wall blowing and su tion, subsoni inow and outow onditions, and slip wall onditions forraried ow appli ations. IV. Optimization MethodologyDAKOTAIn this work, the axisymmetri inlet design optimization was ontrolled by the DAKOTA toolkit. DAKOTA(Design and Analysis Kit for Optimization and Terras ale Appli ations) is a exible and robust softwaresuite that enables the oupling of advan ed simulation odes to a wide range of iterative systems analysismethods.11 These methods in lude gradient-based and non gradient-based optimization methods, surrogateoptimization methods, optimization under un ertainty methods, un ertainty quanti ation methods, andsensitivity and varian e analysis methods. The majority of the numeri al methods available within thistoolkit are widely used within the omputational s ien e and engineering ommunity. Many of these methodsare prepa kaged with DAKOTA and are in luded under the lesser general publi li ense (GNU LGPL).Additional numeri al methods are available as plug-ins but require li enses that must be pur hased fromthird party sour es.A key feature that makes DAKOTA useful for a wide range of engineering problems is the ability totreat the two main omponents, the optimization ontrol module and the simulation driver module, as bla kboxes that run independently of ea h other ex ept for two spe i instan es during the design evaluation.At the beginning of a design evaluation DAKOTA spe ies the parameters for the design and assigns it aunique design identi ation number. These parameters are then written to a simulation driver input le. Assoon as this le be omes available, the simulation driver reads it and begins exe ution of the pro esses laidout within the simulation driver s ript. Upon ompletion of the simulation s ript, the obje tive fun tions,gradients, and/or Hessians required by the optimization method are written to the output le. This le isthen passed ba k to DAKOTA and the y le is repeated until all of the design evaluations and iterations are omplete.The basi workow for the DAKOTA ontroller and simulation driver is shown in Fig. 1. In thisgure, two distin t modules are shown that en apsulate the DAKOTA ontroller and the simulation driver omponents. The dashed lines onne ting the two modules indi ate the ow of data through the input andoutput les. The simulation driver is a user-dened Perl s ript that laun hes everything ne essary to performthe numeri al simulation and evaluation of a given design. In the inlet optimization ases presented here,the simulation driver s ript laun hes ve individual FORTRAN odes: a surfa e generator, a grid generator,a CFD solver, a sensitivity solver, and an obje tive fun tion post-pro essor.V. ResultsInlet OptimizationIn this work, the nonlinear sensitivity solver was in orporated into a gradient-based optimization y le andtested on an axisymmetri Busemann inlet. The inlet geometry shown in Fig. 2 was onstru ted using a ubi Bezier urve as the ba kbone with a blunted leading edge and a transition radius at the shoulder. AFORTRAN ode was written to onstru t the geometry and to generate the single-blo k stru tured mesh6 of 14Ameri an Institute of Aeronauti s and Astronauti s

automati ally during the optimization. The nal mesh had 701 points in the streamwise dire tion and 221points in the spanwise dire tion for a total of 154,000 ells. The boundary layer region of the mesh had awall spa ing of 1.0× 10−6 m with a geometri growth rate of 1.10. The two-dimensional mesh was smoothedusing an ellipti grid smoother and rotated one degree about the axis to reate a three-dimensional volumegrid. Boundary onditions were set automati ally using a US3D pre-pro essing ode.A single-strategy gradient-based optimization method (OPT++ Quasi Newton) was sele ted for thisappli ation. During the two optimization runs, ve design parameters were allowed to vary: L, θ1, θ2, R1,and R2, while the remaining design parameters were held xed: RLE = 0.003 m , RCAP = 0.4 m, CR = 8,RS = 2RT . The freestream onditions used are: M = 8, q = 1000 psf, TW = 800K. All simulations wereperformed using fully turbulent boundary layers beginning at the leading edge of the inlet. The upper andlower bounds for the design parameters along with their initial values are given in Table 1.The baseline CFD simulations of the axisymmetri inlet required 2000 iterations to rea h proper onver-gen e of the density residual and net mass balan e. This translates into approximately 12.1 minutes runningon ve Opteron nodes (dual 2.3GHz Opteron 2356 pro essors per node with Inniband inter onne t). Ea hsubsequent sensitivity solve required approximately 54 se onds to onverge. Thus, a single design y lerequired a total of 16.6 minutes to omplete the al ulation of the obje tive fun tion and its orrespondinggradients. In omparison, one omplete design y le would have taken 72.6 minutes to omplete if CFDwere used in pla e of the sensitivity solver. This represents a speedup fa tor of approximately 4.4 for thisparti ular appli ation.It should be mentioned that the sensitivity solver has several parameters that an ae t the onvergen erates of the solver. The rst of these parameters is the number of iterations for the nonlinear update loop,nit, as given by Eq. 16. This parameter has been found to be somewhat problem-spe i , but for in remental hanges in a design parameter, a value of 50 has been found to work quite well. The number of nonlinearupdate iterations, however, has been found to be tightly oupled to kmax, whi h is the number of relaxationsolves performed by the impli it method. Within US3D, the impli it system of equations is solved with eitherthe Data-Parallel Line-Relaxation (DPLR) method or the Full-Matrix Point-Relaxation (FMPR) method.In ea h ase, some of the terms resulting from the linearization of the governing equations are moved tothe right-hand side of the linear system solve and their inuen e is added via a set of sub-iterations; theparameter kmax determines how many of these sub-iterations are performed at ea h time step or sensitivitysolution step.In addition to these parameters, a blending fun tion was added to linearly blend between the originalgrid oordinates and the perturbed grid oordinates over a dis rete number of steps. The addition of thisblending fun tion was found to make the system onverge more qui kly and robustly, while simultaneouslyallowing larger deformations of the grid to be made. This modi ation makes sense sin e the system beingsolved during the initial iterations is the original grid with the original onverged solution, whi h is simply theoriginal unperturbed system. Then, as the iterations pro eed, the grid is gradually hanged and the systemis solved with a pres ribed number of kmax sweeps. The solver then updates the solution and pro eedsto the next iteration, where an additional grid hange is introdu ed. In this way the system is perturbedat ea h step until some user-dened number of blending iterations, iblend, have been ompleted. For theaxisymmetri inlet optimization, a value of 10 was found to work quite well.Fig. 3 shows the optimization results for the axisymmetri inlet at Ma h 8 using the quasi-Newtonmethod with the sensitivity solver used to al ulate the gradients of the obje tive fun tions. In Fig. 3a, totalpressure e ien y was used as the obje tive fun tion while in Fig. 3b, ompressive e ien y was used. Thesame initial designs were used in both ases. The baseline design was was sele ted be ause it has extremelypoor performan e and is far from the optimal onguration. This poor starting point makes the problem hallenging, and is a stringent test of the optimization methodology and the sensitivity solver. Despite these7 of 14Ameri an Institute of Aeronauti s and Astronauti s

hallenges, the optimization was able to produ e substantial improvements in the obje tive fun tions byadjusting the shape of the inlet geometry. It is interesting to note how qui kly the optimization onverges tothe nal design. This Newton-like onvergen e is the result of having a urate obje tive fun tion gradientsavailable to the quasi-Newton optimization algorithm. Based on the large improvement in the e ien y ofthe inlet and the rate at whi h the obje tive fun tion onverges, it is lear that the gradients al ulatedwith the sensitivity solver are very a urate. If this were not the ase, the optimization would have qui klydiverged and failed.Figs. 4a through 4 show Ma h number, stati pressure, and total pressure ontours for the initialand nal inlet designs and for both total pressure e ien y and ompressive e ien y obje tive fun tionoptimizations. These gures illustrate the large dieren es between the initial and nal inlet ongurations.They also demonstrate that the optimization method works as expe ted by aligning the ree ted sho k wavewith the shoulder of the inlet and by lengthening the inlet to make the ompression more gradual and loserto isentropi . Fig. 5 shows Ma h number ontours for the step-by-step evolution of the inlet during the ompressive e ien y optimization. In this plot, it is interesting to observe how the optimization overshootsthe optimal length in design two and then qui kly moves ba k towards the optimal value as the optimizationpro eeds. This behavior is onsistent with gradient-based optimization methods and indi ates that thegradients omputed by the sensitivity solver are a urate. It should also be noted that be ause CFD wasused to evaluate the designs, the optimization a ounts for blunt leading edges and vis ous boundary layeree ts. This is a substantial improvement over the more ommon approa h of applying empiri al boundarylayer orre tions to invis id inlet designs.Rapid Design Spa e ExplorationOne very promising use of the sensitivity solver is as a onvergen e a eleration me hanism. During thesimulation of typi al internal and external steady-state ow problems, the ow evolves rapidly during aninitial transient phase as sho k waves move around and mass builds up in the ontrol volume. As thesimulation progresses, the major ow transients die down and the total mass in the ontrol volume begins tosettle to a onstant value. At this point, the rate of onvergen e be omes nearly onstant as the CFD solverdrives the residual towards zero. However, if at this point we shift over to the sensitivity solver mode the rateof onvergen e an be improved. This ee t is demonstrated in Fig. 6a whi h plots the density residual andnet mass balan e for a 5.9 million ell three-dimensional inward-turning inlet running at Ma h 5, q = 1000psf, 0 AOA. In this example, the sensitivity solver was a tivated after 1200 iterations with solver settingsnit = 400 and kmax = 8, and was found to drive down the residual at a redu ed ost ompared to the CFDsolver. In addition, the net mass ow rate through the ontrol volume was found to rapidly approa h zero.Another very promising use of the nonlinear sensitivity solver is in ases where a large number of similardesign evaluations are needed, i.e. a design spa e exploration. Here, the design spa e evaluation may berequired for a number of reasons su h as: dire t omparison to an experimental test matrix, al ulationof an aerodynami performan e database, simulation of a ight traje tory, or as the beginning step in adesign optimization study. Traditionally, CFD is used to populate su h a design spa e resulting in enormous omputational osts when applied to realisti geometries su h as s ramjet engines and full ight vehi les. Inthe ase of design optimization studies, the use of CFD may severely limit the number of designs that an beevaluated due to time onstraints and hardware limitations. This an lead to an inadequate understandingof the design spa e, espe ially if strong nonlinearities are present (whi h is often the ase for hypersoni ows). Furthermore, the design optimization algorithm itself may not be able to nd robust designs due tothe sparsely populated design spa e.In an attempt to redu e the total ost of a design spa e exploration, the sensitivity solver was appliedto an angle of atta k (AOA) sweep between 0 6 α 6 5on a generi 5.9 million ell three-dimensional8 of 14Ameri an Institute of Aeronauti s and Astronauti s

inward-turning inlet. In this ase, an initial CFD solution was obtained at the most severe ondition; thatis the largest positive angle of atta k. The sensitivity solver was then applied to ea h subsequent ase in0.5in rements. The solution at angle of atta k α was used as the starting solution for the next angleof atta k, α + 0.5, in a sequential manner. This pro ess was repeated until the entire sweep range was ompleted.The results for the AOA sweep are summarized in Table 2 and Figure 6b. Here, the ase numbers inTable 2 orrespond dire tly to the ase numbers shown in Fig. 6b. The initial CFD simulation required 81.4minutes of wall time to onverge running on 25 Opteron nodes (dual 2.3GHz Opteron 2356 pro essors pernode with Inniband inter onne t). Ea h subsequent ase, however, required 20 minutes or less to ompletedepending on the angle of atta k. The total run-time for the AOA sweep was 239.5 minutes. If CFD hadbeen used for ea h ase, the total time would have been approximately 895 minutes. Thus, in this appli ationthe sensitivity solver was able to produ e a speedup fa tor of approximately 3.8.The redu tion in time from 20 minutes to 10 minutes is due to the lower angle of atta k ases being easierfor the sensitivity solver to onverge. Also, the lower AOA ases allowed the use of fewer nit iterations andmore kmax sweeps to onverge the system. These values were determined manually through a pro ess oftrial and error. If a value of kmax was sele ted that was too high for the given ase, the sensitivity solverwould blow up. If the value of kmax that was sele ted was too low, onvergen e rates would suer. Optimal ombinations of these parameters are the subje t of ongoing investigation. One possible modi ation toenhan e robustness and onvergen e for ases with large parameter hanges would be the in lusion of afun tion to blend between the initial parameter and the nal parameter, similar to what was used for thegrid oordinate blending. VI. Con lusionsA new nonlinear sensitivity solver has been derived and implemented within the US3D ComputationalFluid Dynami s ode. The sensitivity solver algorithm presented in this work is straight-forward to implementin virtually any impli it nite-volume CFD solver, and does not require the derivation of new boundary onditions. The method omputes valid steady-state solutions of the same equations solved by the CFD solverand is therefore able to apture nonlinearities that that may be present in the ow. The sensitivity solverwas su essfully used within an inlet optimization y le to qui kly and a urately al ulate the gradientsof the obje tive fun tion. The a ura y of these gradients allowed the optimization algorithm to a hieveNewton-like onvergen e to the nal design. Extensions of the sensitivity solver to three-dimensional inletsand hypersoni ight vehi les are urrently being pursued. In addition to optimizations, the sensitivity solverdeveloped here has demonstrated the potential to make design spa e exploration far less ostly to perform.In this work, the sensitivity solver was used to rapidly exe ute an AOA sweep on a three-dimensional inward-turning inlet with a speedup fa tor of 3.8 over running full CFD simulations. Further development of thesensitivity solver for design spa e exploration is the subje t of on-going resear h.A knowledgmentsThis work was sponsored by a O e of Naval Resear h STTR under grant number N68335-08-C0328. Theviews and on lusions ontained herein are those of the author and should not be interpreted as ne essarilyrepresenting the o ial poli ies or endorsements, either expressed or implied, of the U.S. Government.

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Referen es1G. Candler, T. Drayna, and L. Ja obsen. Design and Optimization of the ASET Inward-Turning Inlet. Paper, JANNAF,May 2008. 30th Airbreathing Propulsion Sub ommittee Meeting.2G. Candler, T. Drayna, I. Nompelis, M. Ma Lean, and M. S. Holden. HyCAUSE Inward-Turning Inlet Design and CFDModeling. Paper, JANNAF, De ember 2006. 29th Airbreathing Propulsion Sub ommittee Meeting.3T. Drayna, I. Nompelis, and G. V. Candler. Hypersoni Inward Turning Inlets: Design and Optimization. Paper2006-0297, AIAA, June 2006.4A. Taylor III, G. Hou, and V. Korivi. Methodology for Cal ulating Aerodynami Sensitivity Derivatives. AIAA Journal,30(10):24112419, O tober 1992.5G. Hou, A. Taylor III, and V. Korivi. Dis rete Shape Sensitivity Equations for Aerodynami Problems. InternationalJournal for Numeri al Methods in Engineering, 70:22512266, November 1994.6A. Jameson. Aerodynami Shape Optimization Using the Adjoint Method. Le ture notes, Von Karman Institute, February2003.7G. Carpentieri, B. Koren, and M. van Tooren. Adjoint-based Aerodynami Shape Optimization on Unstru tured Meshes.Journal of Computational Physi s, 224(1):267287, February 2007.8I. Nompelis, T. W. Drayna, and G. V. Candler. Development of a Hybrid Unstru tured Impli it Solver for the Simulationof Rea ting Flows Over Complex Geometries. Paper 2004-2227, AIAA, June 2004.9I. Nompelis, T. W. Drayna, and G. V. Candler. A Parallel Unstru tured Impli it Solver for Hypersoni Rea ting FlowSimulation. Paper 2005-4867, AIAA, June 2005.

10M. J. Wright, G. V. Candler, and D. Bose. A Data-Parallel Line-Relaxation Method for the Navier-Stokes Equations.Paper 97-2046CP, AIAA, June 1997.11Bohnho W.J. Dalbey K.R. Eddy J.P. Eldred M.S. Gay D.M. Haskell K. Hough P.D. Adams, B.M. and L.P. Swiler.DAKOTA, A Multilevel Parallel Obje t-Oriented Framework for Design Optimization, Parameter Estimation, Un ertaintyQuanti ation, and Sensitivity Analysis: Version 5.0 User's Manual. Sandia Te hni al Report SAND2010-2183, Sandia NationalLaboratories, De ember 2009.

Objective

Definition

Optimization

ControllerParameters

Grid

Generator

CFD

Solver

Sensitivity

Solver

Objective Function

& Gradients

OptimalNo

Optimization

Strategy

Results

Input Files Output Files

Simulation Driver

Parameter

DefinitionFinal

DesignYes

X(α) U(α)

X(α+∆α) U(α+∆α)

α F

Surface

Generator

Optimization Controller

dF/dα

Figure 1: Optimization ontroller and simulation driver.10 of 14Ameri an Institute of Aeronauti s and Astronauti s

DesignVariable LowerBound UpperBound InitialValue Final ValueF [ηPT ]

Final ValueF [ηB]

L 10 30 11 20.00 20.61

θ1 0.0 10.0 2.0 0.5447 0.0540

θ2 0.0 20.0 10.0 6.894 7.042

R1 0.1 0.6 0.3 0.2842 0.2678

R2 0.1 0.6 0.3 0.2863 0.3272Table 1: Axisymmetri inlet design parameter bounds, initial, and nal values.θ1

R1 R2

θ2

L

RS

RT

RCAP

RLE

Figure 2: Parametri denition of axisymmetri inlet.

Design ID

Nor

mal

ized

Par

amet

er

0 2 4 6 8 100.00

0.20

0.40

0.60

0.80

1.00LengthRadius 2Theta 2Radius 1Theta 1

0.25

0.30

0.35

0.40

0.45

0.50

0.55

Total Pressure Efficiency

Axisymmetric Inlet Optimization

ηPT

(a) Total pressure e ien y optimization.0.55

0.60

0.65

0.70

0.75

0.80

Compressive Efficiency

Axisymmetric Inlet Optimization

ηB

Design ID

Nor

mal

ized

Par

amet

er

0 2 4 6 8 10

0.00

0.20

0.40

0.60

0.80

1.00

Inlet LengthRadius 2Theta 2Radius 1Theta 1

(b) Compressive e ien y optimization.Figure 3: Obje tive fun tion and design parameter evolution.11 of 14Ameri an Institute of Aeronauti s and Astronauti s

(a) Ma h number ontours.

(b) Stati pressure ontours.

( ) Total pressure ontours.Figure 4: Axisymmetri inlet optimization results showing baseline and optimal designs.12 of 14Ameri an Institute of Aeronauti s and Astronauti s

Figure 5: Design evolution of ompressive e ien y optimization.13 of 14Ameri an Institute of Aeronauti s and Astronauti s

Case AOA Solver Iterations kmax Time [min1 5.0 CFD 2200 4 81.42 4.5 SENS 300 8 19.53 4.0 SENS 300 8 20.04 3.5 SENS 300 8 19.65 3.0 SENS 300 8 19.56 2.5 SENS 200 12 15.17 2.0 SENS 200 12 15.18 1.5 SENS 150 16 12.59 1.0 SENS 150 16 12.810 0.5 SENS 100 24 10.211 0 SENS 100 24 10.2Total 235.9Table 2: Inward-turning inlet AOA sweep results.

CPU Time

Den

sity

Res

idua

l

Net

Mas

s[k

g/s]

0 20 40 60 80100

101

102

103

104

105

106

107

108

-20

-15

-10

-5

0

5

10CFD - ResidualCFD - Net MassSENS - ResidualSENS - Net Mass

Convergence History

(a) Convergen e vs. wall time for CFD solver and sensitivity solver. CPU Time

Den

sity

Res

idua

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Net

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s[k

g/s]

0 50 100 150 200100

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-20

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Net Mass

Convergence History

8 1110976Case 1 5432(b) Convergen e vs. wall time for AOA sweep.Figure 6: Appli ation of sensitivity solver to three-dimensional inward-turning inlet.

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