Time Dependent Adjoint-based Optimization for Coupled Aeroelastic Problems

Transcription

1 ime Dependent Adjoint-based Optimization for Coupled Aeroelasti Problems Asitav Mishra Karthik Mani Dimitri Mavriplis Jay Sitaraman Department of Mehanial Engineering,University of Wyoming, Laramie, WY A formulation for sensitivity analysis of fully oupled time-dependent aeroelasti problems is given in this paper. Both forward sensitivity and adjoint sensitivity formulations are derived that orrespond to analogues of the non-linear aeroelasti analysis problem. Both sensitivity analysis formulations make use of the same iterative disiplinary solution tehniques used for analysis, and make use of an analogous oupling strategy. he information passed between fluid and strutural solvers is dimensionally equivalent in all ases, enabling the use of the same data strutures for analysis, forward and adjoint problems. Sensitivities from both forward and adjoint formulations for the fully oupled aeroelasti problem are verified using the omplex step method and agreement to mahine preision is demonstrated. he fully oupled adjoint formulation is then used to perform rotor blade design optimization on a Hart2 rotor in hover while onstraining the time-integrated thrust oeffiient to the baseline value. he optimized rotor ahieves 2% redued torque with a penalty of 1% redution of thrust. I. Introdution In the reent past, the use of adjoint equations has beome a popular approah for solving aerodynami design optimization problems based on omputational fluid dynamis (CFD). 1 6 Adjoint equations are a very powerful tool in the sense that they allow the omputation of sensitivity derivatives of an objetive funtion to a set of given inputs at a ost whih is essentially independent of the number of inputs. his is in ontrast to the brute-fore finite-differene method, where eah input or design variable has to be perturbed individually to obtain a orresponding effet on the output. his is a tedious and ostly proess whih is of little use when there are a large number of design variables or inputs. Another major shortoming of the finite-differene method is that it suffers from step-size limitations whih affet the auray of the omputed gradients. While the use of adjoint equations is now fairly well established in steady-state shape optimization, only reently have inroads been made into extending them to unsteady flow problems. Unsteady disrete adjoint-based shape optimization was initially demonstrated in the ontext of two-dimensional problems by Mani and Mavriplis 7 and also by Rumpfkeil and Zingg. 8 Preliminary demonstration of the method s feasibility in three-dimensional problems was done by Mavriplis. 9 Full implementation in a general sense and appliation to large sale problems involving heliopter rotors was then arried out by Nielsen et.al. in the NASA FUN3D ode. 1, 11 Sine engineering optimization is an inherently multidisiplinary endeavor, the next logial step involves extending adjoint methods to multidisiplinary simulations and using the obtained sensitivities for driving multidisiplinary optimizations. In the ontext of fixed and espeially rotory wing airraft, aeroelasti oupling effets an be very important and must be onsidered in the ontext of a suessful optimization strategy. he oupling of omputational fluid dynamis (CFD) and omputational strutural dynamis (CSD) and the use of sensitivity analysis on suh a system has been addressed in the past primarily from a steady-state standpoint. Until now, relatively little work has been done addressing unsteady aeroelasti optimization problems, mainly due to omplexities in the linearization of oupled time-dependent systems. In previous work, we have derived the fully oupled adjoint Postdotoral Researh Assoiate; amishra3@uwyo.edu Assoiate Researh Sientist; kmani@uwyo.edu Professor; mavripl@uwyo.edu Assistant Professor; jsitaram@uwyo.edu 1 of 23

2 problem for a two-dimensional aeroelasti airfoil problem and demonstrated the use of adjoint-derived sensitivities for performing time-dependent aeroelasti optimization inluding flutter suppression. 12 he objetive of the urrent work is to extend the previously developed time-dependent aerodynami adjoint-based optimization apability 13 to inlude fully oupled aero-strutural dynamis effets for three dimensional flexible rotor design. he strutural deformations are obtained using a Hodges-Dowell type beam finite-element based solver with adjoint formulation for omputing funtional sensitivities to aerodynami and strutural design variables. he CFD solver used is a 3-D Navier-Stokes Unstrutured (NSU3D) finite-volume ode, whih has previously been extended to inlude a steady 14 and unsteady 15 disrete adjoint apability. he following setions desribe the formulation and validation of eah disiplinary omponent of the ombined aero-strutural analysis and sensitivity problems, as well as the oupling of these omponents to form the oupled aero-strutural optimization apability. II. Aerodynami Analysis and Sensitivity Formulation II.A. Flow Solver Analysis Formulation he base flow solver used in this work is the NSU3D unstrutured mesh Reynolds-averaged Navier-Stokes solver. NSU3D has been widely validated for steady-state and time-dependent flows and ontains a disrete tangent and adjoint sensitivity apability whih has been demonstrated previously for optimization of steady-state and timedependent flow problems. As suh, only a onise desription of these formulations will be given in this paper, with additional details available in previous referenes he flow solver is based on the onservative form of the Navier-Stokes equations whih may be written as: U(x, t) t + F(U) = (1) For moving mesh problems these are written in arbitrary Lagrangian-Eulerian (ALE) form as: V U t + [F(U) ẋu] ndb = (2) db(t) Here V refers to the area of the ontrol volume, ẋ is the vetor of mesh fae or edge veloities, and n is the unit normal of the fae or edge. he state vetor U onsists of the onserved variables and the artesian flux vetor F = (F x,f y,f z ) ontains both invisid and visous fluxes. he equations are losed with the perfet gas equation of state and the Spalart-Allmaras turbulent eddy visosity model 16 for all ases presented in this work. he solver uses a vertex-entered median dual ontrol volume formulation that is seond-order aurate, where the invisid flux integral S around a losed ontrol volume is disretized as: n edge S = [F(U) ẋu] ndb = db(t) i=1 F e i (V ei,u,n ei )B ei (3) where B e is the fae area, V e is the normal fae veloity, n e is the unit normal of the fae, and Fe is the normal flux aross the fae. he normal flux aross the fae is omputed using the seond-order aurate matrix dissipation sheme 17 as the sum of a entral differene and an artifiial dissipation term as shown below, F e = 1 2{ F L (U L,V e,n e ) + F R (U R,V e,n e ) +κ (4) [ ] [λ] [ ] 1 { ( 2 U) L ( 2 U) R }} (4) where U L, U R are the left and right state vetors and ( 2 U) L, ( 2 U) R are the left and right undivided Laplaians omputed for any element i as ( 2 neighbors U) i = k=1 (U k U i ) (5) he time derivative term is disretized using a seond-order aurate bakward-differene formula (BDF2) sheme as: V U t = 3 2 V n U n 2V n 1 U n V n 2 U n 2 t (6) 2 of 23

3 he index n is used to indiate the urrent time-level as the onvention throughout the paper. he disretization of the BDF2 sheme shown in equation (6) is based on a uniform time-step size. Denoting the spatially disretized terms at time level n by the operator S n (U n ), the resulting system of non-linear equations to be solved for the analysis problem at eah time step an be written as: R n 3 2 = V n U n 2V n 1 U n V n 2 U n 2 + S n (U n ) = (7) t whih in simplified form exhibiting the funtional dependenies on U and x at different time levels is given as: R n (U n,u n 1,U n 2,x n,x n 1,x n 2 ) = (8) At eah time step n, the impliit residual is linearized with respet to the unknown solution vetor U n and solved for using Newton s method as: [ ] R k U k δu k = R k (9) U k+1 = U k + δu k δu k,u n = U k he Jaobian matrix is inverted iteratively using a line-impliit agglomeration multigrid sheme that an also be used as a preonditioner for a GMRES Krylov solver. 18 Although the above equation denotes the solution at a single time level n, for the remainder of this paper we will use the generalized notation: R(U,x) = (1) where the vetor U denotes the flow values at all time steps, and where eah (blok) row in this equation orresponds to the solution at a partiular time step as given in equation (8). Equation (1) denotes the simultaneous solution of all time steps and is solved in pratie by Newton s method using forward blok substitution (i.e. forward integration in time) sine eah new time step depends on the previous two time levels. II.B. Mesh deformation apability In order to deform the mesh for time-dependent problems a spring analogy and a linear elasti analogy mesh deformation approah have been implemented. he linear elastiity approah has proven to be muh more robust and is used exlusively in this work. In this approah, the mesh is modeled as a linear elasti solid with a variable modulus of elastiity that an be presribed either as inversely proportional to ell volume or to the distane of eah ell from the nearest wall. 19, 2 he resulting equations are disretized and solved on the mesh in its original undeformed onfiguration in response to surfae displaements using a line-impliit multigrid algorithm analogous to that used for the flow equations. he governing equations for mesh deformation an be written symbolially as: G(x,D) = (11) where x denotes the interior mesh oordinates and D denotes shape parameters that define the surfae geometry. III. Aerodynami Sensitivity Analysis Formulation he basi sensitivity analysis implementation follows the strategy developed in referenes. 9, 14 Consider an arbitrary objetive funtion L that is evaluated using the unsteady flow solution set U and unsteady mesh solution set x expressed as: L = L(U,x) (12) Assuming that the state variables (i.e.u, x) are dependent on some input design parameters D, the total sensitivity of the objetive funtion L to the set of design inputs an be expressed as the inner produt between the vetor of state sensitivities to design inputs and the vetor of objetive sensitivities to the state variables as: [ dl dd = U ] U (13) 3 of 23

4 he non-linear flow residual operator and the linear elastiity mesh residual operator as desribed earlier provide the onstraints whih an be expressed in general form over the whole spae and time domains as: whih when linearized with respet to the design inputs yields: G R R U G(x,D) = (14) R(U,x) = (15) U = G (16) hese onstitute the forward sensitivity or tangent sensitivity equations. In pratie these equations may be solved by forward substitution following: G = G R U U = R G represents the sensitivity of the surfae mesh points to a hange in the shape parameter D whih an be onsidered as a known input vetor. hus, the first equation may be solved to obtain the grid sensitivities whih an then be used in the solution of the seond equation to obtain the flow sensitivities U. In both ases, the required inversion of eah disiplinary Jaobian matrix an be aomplished following the same solution proedure used for the orresponding analysis problem, i.e. following equation (9) for the flow equations. Sine these equations represent the solution over the entire time domain, in pratie this proedure is performed at eah time step proeeding from the initial to final time step. he mesh and flow sensitivity vetors an then be substituted into equation (13) to obtain the omplete sensitivity of the objetive with respet to the design variable D. he forward sensitivity approah requires a new solution of equation (16) for eah design parameter D. On the other hand, the adjoint approah an obtain the sensitivities for any number of design inputs D at a ost whih is independent of the number of design variables. he adjoint problem an be obtained by premultiplying equation (16) by the inverse of the large oupled matrix and substituting the resulting expression for the sensitivities into equation (13) and defining adjoint variables as the solution of the system: G R R U Λ x = Λ U where Λ U and Λ x are the flow and mesh adjoint variables respetively. he final objetive sensitivities an be obtained as: [ ] dl = G dd Λ x (2) he adjoint system an be solved using bak-substitution as: R Λ U = U U G Λ x = R Λ u U Λ U + where one again the inversion of the transposed Jaobian matries an be aomplished using the same iterative tehniques as applied to the analysis and tangent sensitivity problems. Realling that equation (19) applies over the entire time domain, the bak-substitution proedure leads to a reverse integration in time, beginning with the last physial time step and proeeding to the initial time step. A more detailed desription of the omplete formulation is presented in, 12 where the proedure has been used to perform aerodynami shape optimization for a rigid rotor. (17) (18) (19) (21) (22) 4 of 23

5 IV. Beam model: Analysis and Adjoint Optimization A non-linear bend-twist beam model is a suitable and widely utilized strutural model for slender fixed and rotary wing airraft strutures within the ontext of an aeroelasti problem. A bend-twist beam model, desribed below, has previously been developed and oupled to the NSU3D unstrutured mesh Reynolds-averaged Navier-Stokes solver. 2, 21 IV.A. Beam Analysis Formulation he non-linear governing equations of a slender beam are disretized using the Finite-element method (FEM) in spae. Figure 1 shows a typial beam with 15 degrees of freedom for eah element to aommodate bend wise, lag wise, axial and torsional displaements. he seond order equation of motion for the beam an be expressed as: Figure degrees of freedom beam element with flap, lag, torsional and axial degrees of freedom. [M] q + [C] q + [K] = F (23) where [M], [C] and [K] are mass, damping and stiffness matries of the system of equations representing the beam. Vetor F = F(t) is the foring vetor. Vetor q represents the displaements along all degrees of freedom. his set of equations an be redued to a first order system and solved using a seond order bakward differene formula (BDF2) time integration with standard Newton-type linearization and sub-iterations to effiiently invert the impliit system: [I] Q + [A]Q = F (24) [ ] where [I] is the identity matrix, Q = [q, q], F = [, [M] 1 F] and [A] =. he residual of the [I] [M] 1 [K] [M] 1 [C] strutural equations an be defined as: J = [I] Q + [A] Q F =, and an be expressed in a simplified form as: J(Q,F) = (25) he beam model has been validated for the standard Hart-2 rotor ase 23 by omparing its natural frequeny preditions with the preditions from other reliable CSD models. As shown in able 1, the rotating natural frequenies ompare well with those predited by the UMARC 24 and DLR 25 strutural odes, for the first 3 flap and first torsional frequeny modes. Further, Fig. 2 shows that the frequeny predition over a range of operating rotor frequenies ompare well with those predited by the UMARC omprehensive ode of 23

6 able 1. Comparison of Hart-II Natural Frequenies 22 Modes Present Model UMARC DLR Flap Flap Flap orsion Figure 2. Fan plot omparing Beam model with UMARC 26 6 of 23

7 IV.B. Forward Sensitivity Formulation of Beam Model he beam tangent (forward sensitivity) linearization is similar to the analysis problem. For a given funtion, L, its sensitivity with respet to a blade design parameter, D an be written as: dd dl = +. his requires solving for sensitivity of the beam state (Q), whih an be obtained by differentiating Eqn. (25) with respet to the design variable D and rearranging as: [ ] J = J F F J (26) he last term on the right hand side is non zero for strutural design parameters suh as beam element stiffnesses, in whih ase the applied fore does not hange with the design parameter, making the first term on the right hand size zero. In the oupled aeroelasti ase, using aerodynami shape parameters that primarily affet the airloads on the struture, the first term on the right-hand size is non-zero while the seond term vanishes. Solving for in Eqn. (26), the forward sensitivity of the objetive funtion dd dl an be obtained. IV.C. Adjoint Formulation of Beam Model he adjoint formulation of the beam model an be derived by approahing the tangent formulation in the reverse (transpose) diretion. aking the transpose of the objetive funtional sensitivity yields: dl dd = + (27) his requires solving for the transpose sensitivity of the beam state (Q). he solution of an be derived from the transposed Eqn. (26): [ = J J ] F [ ] J (28) F Substituting the above into Eqn. (27): dl dd [ = + J his requires solving for an adjoint vetor Λ Q defined as: F ] [ J J F ] (29) [ ] J Λ Q = (3) he above forms the adjoint formulation of the beam model. It is observed here again that the left hand side Jaobian term of the adjoint step is just the transpose of the Jaobian in the forward linearization. In this work, forward and adjoint formulations of the beam solver have been implemented and verified for both strutural design parameters, and fore-based design parameters (as required for the oupled aeroelastiity problem). he implementation inludes sensitivities for both stati and dynami beam motion sensitivities. he adjoint implementation has been verified for a tip loaded (harmonially varying tip fore) unsteady beam. he adjoint derived sensitivities of the beam shape due to hanges in design variables, suh as element stiffnesses EI, GJ, and element masses, have been shown to ompare well with sensitivities omputed using omplex variable methods and forward sensitivity methods to within mahine preision (1 15 ). Having established the orretness and auray of the strutural adjoint formulation, the potential for using this approah to drive time-dependent strutural optimization problems has been explored using the large-sale bound onstraint optimization tool (L-BFGS-B) 27 as a preursor to their use in fully oupled aero-strutural optimization problems. IV.D. Optimization to Ahieve Presribed Unsteady Beam Defletion As a proof of onept we provide appliation of the developed methodology for a five-element beam model for an unsteady flap defletion problem. As an initial ondition, the beam is given a small tip perturbation (using a tip load F =.1N) and then allowed to vibrate naturally. he objetive is to optimize on the elemental design variables (suh 7 of 23

8 as, EI(i) i = [1 : nelem]) so as to attain a presribed flap defletion. o define the target tip flap defletion over a time period (q tipre f (t)), a linearly varying stiffness distribution is assigned (EI [75 : 125]Nm 2 ). hen the stiffness of the beam is initialized to EI(i) = 1, i = [1 : nelem] and the beam is optimized to obtain the target tip defletion (q re f ) variation over a time period. Design variables are defined as x = EI(i),i = [1 : nelem], and the objetive funtion for the unsteady problem as: nelem L ob j = q tip q tipre f 2 + β i EI(i) (31) where the first term f 1 = q tip q tipre f 2, refers to the seond order norm of tip flap defletion over the whole time period onsidered (i.e. over 1.6 revs or nstep = 1 time steps). he seond term ( f 2 = β nelem i EI(i),β = 1 9 ) refers to the weight penalty of the beam. he goal is : min(l ob j ), subjet to l < x < u; where l and u are the lower and upper bounds of design variables x (speifially EI values), taken as l = 75Nm 2,u = 125Nm 2. (a) Design variables x (EI) (b) Objetive () Sensitivity Figure 3. Convergene of EI distribution, L ob j and sensitivity for unsteady beam. (Red) L ob j (= f 1 + f 2 ); (Green) f1; (Blue) f2 (a) iteration 1 (b) iteration 8 Figure 4. Flap defletion (w) onvergene for unsteady beam (nelem=5) Figure 3(a) show the evolution of stiffness distribution and Fig. 3(b) show the onvergene of the objetive funtion for the five element unsteady beam. he overall objetive dereases only slightly over 8 design yles due to the onstraining effet of the penalty term. However the shape mathing omponent f 1 of the objetive dereases by several orders of magnitude, and the time history of the tip defletion losely mathes the target at the final design yle as shown in Figure 4. Additionally, Figure 3() shows that the sensitivity gradients are dereased by 2.5 orders of magnitude at the final design indiating that the proess is onverging towards an optimum. V.A. V. Fully Coupled Fluid-Struture Analysis Formulation Fluid-struture interfae (FSI) In addition to the solution of the aerodynami problem and the strutural dynamis problem, the solution of the fully oupled time-dependent aeroelasti problem requires the exhange of aerodynami loads from the CFD solver to the beam struture, whih in turn returns surfae displaements to the fluid flow solver. he governing equations for the 8 of 23

9 FSI an be written in residual form as: S(F b,q,f(x,u)) = F b [ (Q)]F(x,u) = (32) S (x s,q) = x s [ (Q)] Q = (33) respetively for the fores transfered to the strutural solver and displaements returned to the flow solver. In these equations, [ ] represents the transfer matrix whih projets point-wise CFD surfae fores F(x,u) onto the individual beam elements resulting in the beam fores F b. he transpose of this matrix is used to obtain the CFD surfae displaements x s from the beam degrees of freedom Q. Also note that [ ] is a funtion of Q sine the transfer patterns hange with the beam defletion. V.B. General solution proedure he aeroelasti problem onsists of multiple oupled sets of equations namely, the mesh deformation equations, the flow equations (CFD), the beam model-based strutural equations, and the fluid-struture interfae transfer equations. he system of equations to be solved at eah time step an be written as: G(x,x s (Q)) = (34) R(u,x) = (35) S(F B,Q,F(x,u)) = (36) J(Q,F B ) = (37) S (x s,q) = (38) where S and S represent the residuals of the FSI equations, and J represents the residual of the strutural analysis problem. Note that the mesh motion residual now depends also on any surfae defletions x s introdued by the strutural model. Within eah physial time step, solution of the fully oupled fluid struture problem onsists of performing multiple oupling iterations on eah disipline using the latest available values from the other disiplines. hus the oupled iteration strategy proeeds as: [ ] G x = G(x 1,x 1 s ) (39) [ R ] u = R(u 1,x ) (4) for the flow equations, where the supersript denotes the oupling iteration index, and the variables are updated as x = x 1 + x and u = u 1 + u. his is followed by the solution of the FSI and strutural model as: F b = [(Q )]F(x,u ) (41) [ ] J Q = J(Q,F b ) (42) x s = [ ] Q (43) In this implementation, subiterations are performed to onverge the first two equations simultaneously for the final values of Q, while the third equation orresponds to an expliit evaluation for the x s given the Q values. At the first oupling iteration, x s = and solution of the mesh deformation equation is trivial, although non zero values of x s are produed at subsequent oupling iterations as the beam deflets in response to the aero loads. From a disiplinary point of view, the aerodynami solver produes updated values of u and x, whih are used to ompute F(x, u) pointwise surfae fores. hese surfae fores are input to the FSI/strutural model whih returns surfae displaements x s. hese new surfae displaements are then fed bak into the mesh deformation equations and the entire proedure is repeated until onvergene is obtained for the full oupled aero-strutural problem at the given time step. VI. Sensitivity Analysis for Coupled Aeroelasti Problem In the formulation of the sensitivity analysis for the oupled aeroelasti problem, it is desirable to mimi as losely as possible the solution strategies and data strutures employed for the analysis problem. hus, analogous disiplinary 9 of 23

10 solvers an be reused for eah disiplinary sensitivity problem, and the analysis oupling strategy an be extended to the sensitivity analysis formulation. Furthermore, the data transfered between disiplinary solvers should onsist of vetors of the same dimension for the analysis, tangent and adjoint formulations. Starting with the forward sensitivity problem, the sensitivity of an objetive L an be written as: [ dl dd = where the individual disiplinary sensitivities are given as the solution of the oupled system: ] (44) G R F R F G s I S F S F b J F b S J S S s F F b s = G he first and seond equations orrespond to equations for the mesh and flow variable sensitivities, as previously desribed for the aerodynami solver, and the third equation orresponds to the onstrution of the surfae fore sensitivities given these two previous sensitivities. he fourth equation denotes the sensitivity of the FSI transfer from the fluid to the strutural solver, while the fifth equation orresponds to the sensitivity of the strutural solver. Finally, the last equation orresponds to the sensitivity of the FSI transfer from the strutural solver bak to the flow solver. his oupled system of sensitivities an be solved analogously to the oupled analysis problem as: [ ] G [ ] R = G 1 G s = R (45) (46) followed by the expliit evaluation of the surfae fore sensitivities as: F = F hese sensitivity evaluations are all implemented within the flow solver. hey are followed by the solution of the remaining omponents of the system as: S S F b F b J J = S F F (48) F b + F (47) 1 of 23

11 hese two equations are solved simultaneously to obtain the strutural sensitivities whih are then used to evaluate the surfae mesh sensitivities expliitly as: s = S (49) where the fat that S s = [I] (identity matrix) has been used. hese new surfae mesh sensitivities are then fed bak into the first equation in the system initiating the next oupling iteration. As an be seen, eah disiplinary solution proedure requires the inversion of the same Jaobian matrix as the orresponding analysis problem, whih is done using the same iterative solver. Furthermore, the fluid-struture oupling requires the transfer of the fore sensitivities F from the flow to the strutural solver, and the surfae mesh sensitivities s from the strutural solver bak to the fluid solver, whih are of the same dimension as the fore and surfae displaements transfered in the analysis problem, respetively. he orresponding adjoint problem an be obtained by premultiplying equation (45) by the inverse of the large oupling matrix and substituting this into equation (44), transposing the entire system, and defining adjoint variables as solutions to the following oupled system: G R R F F I G s S F S F b S J F b J S S s Λ x Λ u Λ F Λ Fb Λ Q Λ xs = his system an be solved starting with the last equation and proeeding to the first equation as: followed by the solution of the strutural adjoints S J F b F b S J Λ xs = G 1 Λ x s Λ Fb Λ Q = S Λ xs (5) (51) followed by the expliit onstrution of the pointwise CFD surfae fore adjoint: Λ F = S Λ Fb F and ending with the solution of the mesh and flow adjoints as: G R R Λ x = Λ u + F Λ F + F Λ F (52) (53) 11 of 23

12 One again, the solution of the various disiplinary adjoints requires the inversion of the orresponding disiplinary Jaobians (transposed in this ase) whih an be aomplished using the same iterative solvers as for the analysis and forward sensitivity problems. Additionally, the input to the strutural adjoint problem onsists of the variable Λ xs, whih is the same dimension as the surfae displaements output from the strutural analysis solver, while the output of the strutural adjoint solver onsists of the variable Λ F whih is of the same dimension as the fore inputs to the strutural solver in the analysis problem. VII. Verifiation of Coupled Aeroelasti Sensitivity he forward and adjoint sensitivities for the oupled aeroelasti problem are verified using the omplex step method. Any funtion f (x) operating on a real variable x an be utilized to ompute the derivative f (x) by redefining the input variable x and all intermediate variables used in the disrete evaluation of f (x) as omplex variables. For a omplex input, the funtion when redefined as desribed produes a omplex output. he derivative of the real funtion f (x) an be omputed by expanding the omplex operator f (x + ih) as: from whih the derivative f (x) an be easily determined as: f (x + ih) = f (x) + ih f (x) + (54) f (x) = Im[ f (x + ih)] h As in the ase of finite-differening, the omplex step-based differentiation also requires a step size. However, unlike finite-differening the omplex step method is insensitive to small step sizes sine no differening is required. In theory it is possible to verify forward and adjoint-based gradients using the omplex step method to mahine preision. With this in mind, a omplex version of the omplete oupled aero-strutural analysis ode has been onstruted through sripting of the original soure ode to redefine variables from real to omplex types and to overload a small number of funtions for use with omplex variables. VII.A. Unsteady est Problem he Hart2 rotor in hover is solved both for a rigid blade (no strutural model) as well as for a flexible blade (oupled with struture). For the latter, the flow is solved in tight oupling with the beam solver. he mixed element mesh made up of prizms, pyramids and tetrahedra onsists of approximately 2.32 million grid points and is shown in Figures (5(a)), (5(b)) and (5()), where the rigid blade simulation is ompared with the oupled CFD/CSD simulation. he simulations are run for three rotor revolutions using a 2 degree time-step for 54 time-steps starting from freestream initialization. For the rigid blade simulation, the time-dependent mesh motion is determined by rotating the entire mesh as a solid body at eah time step. he unsteady Reynolds-averaged Navier-Stokes equations are solved at eah time step in ALE form, using the Spalart-Allmaras turbulene model. Figures (6(a)) and (6(b)) shows a snapshot of the pressure oeffiient ontours on the rotor at the end of a single revolution for both the rigid and flexible blades. he oupled CFD/CSD simulation is run in a similar manner. However, the flow solution (CFD) is oupled with the beam solver (CSD) every time step by appropriately exhanging, a) airloads information from the flow domain to the beam and b) blade deformation information from the beam to the flow domain, at the fluid-struture interfae (i.e. blade surfae). In this oupled simulation, the mesh is first moved aording the deformations ditated by the new flexed blade oordinates determined from the strutural beam ode before the solid body rotation of the entire mesh is performed. hus, the flow now sees not only the rigidly rotated mesh (observed in rigid blade simulation), but also the deformed mesh around the blades. his oupled fluid-struture interation problem needs to be iterated until satisfatory onvergene is ahieved on both flow and mesh deformation within eah time step. his kind of CFD/CSD oupling done within every time step is known as tight oupling. he simulations were performed on the Yellowstone superomputer at the NCAR-Wyoming Superomputing Center (NWSC), with the analysis problem running in parallel on 512 ores. Eah time step used 6 oupling iterations, and eah oupling iteration used 1 non-linear flow iterations with eah non-linear iteration onsisting of a three-level line-impliit multigrid yle. he typial simulation for 3 rotor revolutions required around 3 hours of total run time. Figures 7 through 1 summarize the overall onvergene of the rigid and aero-elasti oupling analysis formulations. Figure 7 shows the typial flow and turbulene residual onvergene within a single time step for the rigid rotor ase (no strutural model), while Figure 8 depits onvergene of the flow and turbulene residuals at the same time step for the oupled aeroelasti ase. In this ase, the jumps in residual values at the start of new oupling iterations (55) 12 of 23

13 (a) Computational Domain (b) Planform view () Zoomed in view Figure 5. HAR2 rotor mesh onsisting of 2.32 million points used in the optimization example. 13 of 23

14 (a) Planform view (b) Zoomed in view Figure 6. C P ontours for the baseline HAR2 rotor, with flexible and rigid blades, in hover after one revolution. he mesh onsists of 2.32 million verties. Figure 7. Flow and turbulene residual onvergene at a given time step for rigid analysis 14 of 23

15 Figure 8. Flow and turbulene residual onvergene at a given time step for oupled aeroelasti analysis Figure 9. Mesh deformation residual onvergene at a given time step for oupled aeroelasti analysis Figure 1. Residual onvergene of beam and overall FSI in one oupling iteration 15 of 23

16 (a) Blade Deformation (b) Blade tip vs time Figure 11. HAR2 blade deformation are learly visible, although these jumps beome smaller as the oupling proedure onverges, and the overall residual histories losely follow those of the rigid rotor ase after the first few oupling iterations. (he drop in density residual at eah new oupling iteration is due to the imposition of a small start-up CFL value at eah new oupling iteration.) Figure 9 depits the onvergene of the mesh deformation residual for the same time step, also showing jumps in the residual at the start of eah new oupling iteration. Solution of the mesh deformation equations terminates when the residuals reah a presribed tolerane of 1.e-8, thus the variable number of iterations per oupling yle. Most notable is the fat that the initial mesh deformation residual dereases at eah new oupling iteration, providing a measure of the onvergene of the entire oupling proedure. Figure 1 illustrates the onvergene of the oupled beam/fsi residual (i.e. equations (36) and (37)), showing rapid onvergene to mahine zero in a small number of iterations within a single CFD/CSD oupling iteration. he orresponding beam residual drop is observed to be of 15 orders of magnitude, as shown in Fig. 1. he effet of the CFD/CSD aeroelasti oupling is learly demonstrated in Figure 11, whih ompares the deformed blade shape from the oupled simulation with that from the rigid blade simulation. As Figure 11(b) shows, the blade initially deforms onto larger tip flap values ( 16m) before settling into a lower value of 6m. Further Figure 12 ompares thrust and torque values from the oupled solution with the rigid blade solution. he flexed blade results in predition of more oning of the blade and slightly lower total thrust (Fig. 12(a)) as well as lower total torque magnitude (Fig. 12(b)). herefore, it is evident that the rigid body model might lead to over predition of required rotor torque. (a) hrust vs time (b) orque vs time Figure 12. ime history of airloads on HAR2 rotor 16 of 23

17 VII.B. Geometry Parameterization In order to obtain sensitivities with respet to a set of shape parameters that are well suited for design optimization purposes, a baseline blade is onstruted by staking 11 airfoil setion along the span. Eah airfoil ontains 1 Hiks- Henne bump funtions 5 on the upper surfae and 5 on the lower surfae that an be used to modify the airfoil shape. Additionally, the twist values of the blade at the root and tip airfoil setions are also used as design variables resulting in a total of 112 design variables. Figure (13(a)) provides an illustration of the baseline blade design setup. A high density strutured mesh is generated about this blade geometry, whih is then rotated and translated to math eah individual blade in the CFD mesh, as shown in Figure (13(b)). Interpolation patterns between eah unstrutured mesh surfae point and the baseline strutured mesh are determined in a preproessing phase. hese interpolation patterns are then used to interpolate shape hanges from the baseline blade to all four blades in the CFD mesh (as determined by hanges in the design variables) and to transfer sensitivities from the surfae CFD mesh points to the design variables using the hain rule of differentiation. (a) Blade design parameters (b) Baseline strutured blade mesh overlap with CFD mesh Figure 13. Illustration of (a) baseline blade with design parameters and (b) overlap in tip region between baseline blade strutured mesh and CFD surfae unstrutured mesh. VII.C. Unsteady Objetive Funtion Formulation A time-integrated objetive funtion based on the time variation of the thrust (C ) and torque (C Q ) oeffiients is used for this test ase. he goal of the optimization is to redue the time-integrated torque oeffiient while onstraining the time-integrated thrust oeffiient to the baseline rotor performane. he objetive funtion is based on the summation of the differenes between a target and a omputed objetive value at eah time level n. Mathematially the loal objetive funtion at eah time-step in the integration range is defined as: L n = (δc n ) 2 + 1(δC n Q )2 (56) δc n = (C n C n target ) (57) δc n Q = (Cn Q Cn Qtarget ) (58) where the target thrust oeffiient values at eah time-step in the integration range are set from the baseline HAR2 rotor values and the target torque values are set to zero. he weight of 1 on the torque oeffiient is neessary to equalize the differene in orders-of-magnitude between the thrust and torque oeffiients. he global or timeintegrated objetive is then onstruted using equal unit weights at eah time-step as: L g n=n = n=1 L n (59) 17 of 23

18 VIII. Fully Coupled CFD/CSD Adjoint: Unsteady Sensitivity Verifiation he fully oupled CFD/CSD adjoint formulation was verified by first omparing the tangent sensitivities with those obtained from omplex step method. Further the adjoint formulation was verified by omparing its sensitivities with those obtained from the tangent as well as the omplex step method. VIII.A. Forward Sensitivity Verifiation he tangent formulation was verified for perturbations on one geometri design parameter, namely, blade twist at the tip, for both unoupled (rigid blade) as well as oupled simulations. As desribed earlier, a omplex perturbation of size is introdued on twist at the beginning of the analysis run. he derivatives of the funtional (L n ) are evaluated and ompared with those obtained using the forward linearization proedure at every time instane for up to 5 time steps. In both ases, for the omplex step method, as well as for the forward linearization approah, the fully oupled aeroelasti problem is onverged to mahine zero at eah time step in order to avoid ontaminating the sensitivity values with errors due to inomplete onvergene. ables (2) and (3), respetively, ompare rigid aerodynami only derivatives and oupled aeroelasti derivatives ( n ) obtained from the omplex analysis run with those from the forward linearization run for the first 5 time steps. he unoupled tangent verifiation serves as a sanity hek of the new forward sensitivity formulation when the strutural ode is swithed off. As an be seen from the tables, the forward tangent sensitivities and the omplex step sensitivities agree to 12 signifiant digits for both the rigid and flexible aeroelasti rotor ases. able 2. Unoupled forward linearization verifiation a n=number of time step n a Method Unoupled (Rigid) 1 Complex E-6 angent E-6 2 Complex E-6 angent E-6 3 Complex E-6 angent E-6 4 Complex E-6 angent E-6 5 Complex E-6 angent E-6 VIII.B. Adjoint Sensitivity Verifiation Having validated the forward sensitivity with that from the omplex step method, the adjoint formulation was validated with these two formulations. able 3 ompares the three formulations for 5 unsteady time steps as before. he sensitivity values from the adjoint formulation mathes to twelve signifiant digits with the other two formulations. his provides the onfidene in the present methodology to be used as an effetive aeroelasti optimization tool. IX. Heliopter Blade Optimization he same Hart2 rotor blade onsidered for analysis is used for the optimization test problem. he solution methodology is similar to what was desribed for the analysis problem. However, for the optimization problem the simulation was performed for only one and a half rotor revolutions, using a 2 time step size. he optimization proedure used 18 of 23

19 able 3. Coupled adjoint linearization verifiation n Method(n) Coupled (Aeroelasti) 1 Complex E-6 angent E-6 Adjoint E-6 2 Complex E-6 angent E-6 Adjoint E-6 3 Complex E-6 angent E-6 Adjoint E-6 4 Complex E-6 angent E-6 Adjoint E-6 5 Complex E-6 angent E-6 Adjoint E-6 is the L-BFGS-B bounded redued Hessian algorithm. 27 Eah request by the optimization driver for a funtion and gradient value results in a single forward time-integration of the analysis solver and a single bakward integration in time of the adjoint solver. A bound of ±2% hord for eah defining airfoil setion was set on the Hiks-Henne bump funtions, and a bound of ±.5 of twist was set on the root and tip twist definitions. he optimization was performed on the Yellowstone superomputer at the NCAR-Wyoming Superomputing Center (NWSC) with the simulations (analysis/adjoint) running in parallel on 124 ores. Eah time step in the analysis problem employed 6 oupling yles. Eah oupling yle used 1 nonlinear iterations with eah nonlinear iteration requiring 3 linear multigrid yles per Krylov vetor. he time required for a single funtion/gradient all was approximately 1.4 hours of wall lok time. he optimization problem was run for 4 design yles whih had 5 funtion alls in total. Figures 14 shows the residual onvergene for a typial unsteady adjoint time step. he figure shows the residual drops by 5 orders of magnitude over 6 oupling yles. Figures 15(a) and 15(b) show the funtional and gradient onvergene over 4 design yles. he figures show that, the funtional is already dropping while the gradient value is being redued. Figures 16 ompare the optimized Hart2 rotor load time history with that from the baseline rotor. he optimized rotor results in almost same thrust values as the baseline rotor with only 1% redution in thrust (Fig. 16(a)) and ahieves a 2% redution in torque (Fig. 16(b)). Figure 17 further ompares the optimized rotor blade tip time history with the baseline rotor and shows that the optimized rotor results in a rotor blade with redued bending. Finally, Figures 18 ompare the optimized rotor blade setions with the baseline setions. As the figure shows, the root stations are modified more than the tip stations. X. Conlusions and Future Works In this work, a disrete adjoint formulation for time-dependent tightly oupled aeroelasti three-dimensional problems has been developed and demonstrated. he formulation is designed to reuse as muh as possible the original oupled aeroelasti data-strutures and solution strategies used for the analysis problem, thus simplifying implementation and verifiation. A omprehensive verifiation approah was used to determine the auray of the adjoint sensitivities. In a first step, the individual disiplinary omponents of the adjoint sensitivity analysis formulation were verified independently using the forward tangent and omplex step method. In a seond step, these individual omponents were linked together and verifiation of the fully oupled aeroelasti sensitivities was demonstrated to mahine preision. he adjoint sensitivities were used to perform a shape optimization problem for a flexible time-dependent rotor problem. he optimization problem used a relatively oarse mesh and was run for only several design iterations due to resoure limitations. Future work will onentrate on demonstrating the potential of this approah for aeroelasti design optimization problems involving a larger number of design iterations, using finer meshes and smaller time 19 of 23

20 Figure 14. Residue onvergene in a typial adjoint time step (a) Funtional Convergene (b) Gradient Convergene Figure 15. Convergene in design optimization 2 of 23

21 (a) hrust vs time (b) orque vs time Figure 16. ime history of airloads on optimized and baseline HAR2 rotor Figure 17. Optimized and baseline Hart2 rotor blade tip time history 21 of 23

22 Figure 18. Optimized (red dashes) and baseline (blue solid) Hart2 rotor blade setions (y/r = [.2 : 1.]) steps. More realisti design senarios, inluding multiple onstraints and multi-point optimization problems will also be onsidered. XI. Aknowledgements his work was partly funded by the Alfred Gessow Rotorraft Center of Exellene through a subontrat with the University of Maryland. Partial support was also provided by AFOSR SR Phase 1 ontrat FA955-12C-48. Computer resoures were provided by the University of Wyoming Advaned Researh Computing Center and by the NCAR-Wyoming Superomputer Center. Referenes 1 Jameson, A., Aerodynami Shape Optimization using the Adjoint Method, VKI Leture Series on Aerodynami Drag Predition and Redution, von Karman Institute of Fluid Dynamis, Rhode St Genese, Belgium, Jameson, A. and Vassberg, J., Computational Fluid Dynamis for Aerodynami Design: Its Current and Future Impat, Proeedings of the 39th Aerospae Sienes Meeting and Exhibit, Reno NV, AIAA Paper , Nadarajah, S. and Jameson, A., A Comparison of the Continuous and Disrete Adjoint Approah to Automati Aerodynami Optimization, Proeedings of the 38th Aerospae Sienes Meeting and Exhibit, Reno NV, AIAA Paper 2 667, 2. 4 Nielsen, E. and Anderson, W., Reent Improvements in Aerodynami Design Optimization on Unstrutured Meshes, AIAA Journal, Vol. 4-6, June 22, pp Jameson, A., Alonso, J., Reuther, J., Martinelli, L., and Vassberg, J., Aerodynami shape optimization tehniques based on ontrol theory, AIAA Paper , Giles, M., Duta, M., and Muller, J., Adjoint Code Developments Using Exat Disrete Approah, AIAA Paper , Mani, K. and Mavriplis, D. J., Unsteady Disrete Adjoint Formulation for wo-dimensional Flow Problems with Deforming Meshes, AIAA Journal, Vol. 46-6, June 28, pp Rumpfkeil, M. and Zingg, D., A General Framework for the Optimal Control of Unsteady Flows with Appliations, 45th AIAA Aerospae Sienes Meeting and Exhibit, Reno, NV, January, AIAA Paper , Mavriplis, D. J., Solution of the Unsteady Disrete Adjoint for hree-dimensional Problems on Dynamially Deforming Unstrutured Meshes, Proeedings of the 46th Aerospae Sienes Meeting and Exhibit, Reno NV, AIAA Paper , of 23

23 1 Nielsen, E. J., Diskin, B., and Yamaleev, N., Disrete Adjoint-Based Design Optimization of Unsteady urbulent Flows on Dynami Unstrutured Grids, AIAA Journal, Vol. 48-6, June 21, pp Nielsen, E. J., Lee-Raush, E. M., and Jones, W.., Adjoint-Based Design of Rotors in a Noninertial Referene Frame, Journal of Airraft, Vol. 47-2, Marh-April 21, pp Mani, K. and Mavriplis, D. J., Adjoint based sensitivity formulation for fully oupled unsteady aeroelastiity problems, AIAA Journal, Vol. 47, (8), August 29, pp Mani, K. and Mavriplis, D. J., Geometry Optimization in hree-dimensional Unsteady Flow Problems using the Disrete Adjoint, 51st AIAA Aerospae Sienes Meeting, Grapevine, X, AIAA Paper , January Mavriplis, D. J., Disrete Adjoint-Based Approah for Optimization Problems on hree-dimensional Unstrutured Meshes, AIAA Journal, Vol. 45-4, April 27, pp Mavriplis, D. J., Solution of the Unsteady Disrete Adjoint for hree-dimensional Problems on Dynamially Deforming Unstrutured Meshes, Proeedings of the 46th AIAA Aerospae Sienes Meeting, Reno, NV, AIAA Paper , Spalart, P. R. and Allmaras, S. R., A One-equation urbulene Model for Aerodynami Flows, La Reherhe Aérospatiale, Vol. 1, 1994, pp Mavriplis, D. J., Unstrutured-Mesh Disretizations and Solvers for Computational Aerodynamis, AIAA Journal, Vol. 46-6, June 28, pp Mavriplis, D. J., Multigrid Strategies for Visous Flow Solvers on Anisotropi Unstrutured Meshes, Journal of Computational Physis, Vol. 145, (1), September 1998, pp Yang, Z. and Mavriplis, D. J., A Mesh Deformation Strategy Optimized by the Adjoint Method on Unstrutured Meshes, AIAA Journal, Vol. 45, (12), 27, pp Mavriplis, D. J., Yang, Z., and Long, M., Results using NSU3D for the first Aeroelasti Predition Workshop, Proeedings of the 51st Aerospae Sienes Meeting and Exhibit, Grapevine X, AIAA Paper , Shuster, D. M., Chwalowski, P., Heeg, J., and Wieseman, C. D., Summary of Data and Findings from the First Aeroelasti Predition Workshop, Seventh International Conferene on Computational Fluid Dynamis (ICCFD7), July Kumar, A. A., Vishwamurthy, S. R., and Ganguli, R., Correlation of heliopter rotor aeroelastiresponse with HAR-II wind tunnel test data, Airraft Engineering and Aerospae ehnology: An International Journal, Vol. 82, (4), 21, pp Yu, Y. H., ung, C., van der Wall, B., Pausder, H.-J., Burley, C., Brooks,., Beaumier, P., Delrieux, Y., Merker, E., and Pengel, K., he HAR-II est: Rotor Wakes and Aeroaoustis with Higher-Harmoni Pith Control (HHC) Inputs -he Joint German/Frenh/Duth/US Projet-, 58th Amerian Heliopter Soiety Annual Forum, Montreal, Canada, June Chopra, I. and Bir, G., University of Maryland Advaned Rotor Code: UMARC, Amerian Heliopter Soiety Aeromehanis Speialists Conferene, January Hekmann, A. Otter, M., Dietz, S., and Lopez, J. D., he DLR FlexibleBodies Library to Model Large Motions of Beams and of flexible bodies exported from finite element programs, Modelia, September Smith, M. J., Lim, J. W., van der Wall, B. G., Baeder, J. D., Bierdron, R.., Boyd Jr., D. D., Jayaraman, B., Junk, S. N., and Min, B.-Y., Adjoint-based Unsteady Airfoil Design Optimization with appliation to Dynami Stall, 68th Amerian Heliopter Soiety Annual Forum, Fort Worth, X, May Zhu, C., Byrd, R. H., Lu, P., and Noedal, J., L-BFGS-B - FORRAN Subroutines for Large-sale Bound Constrained Optimization, ehnial report, Department of Eletrial Engineering and Computer Siene, Deember of 23