Time-dependent adjoint-based aerodynamic shape optimization applied to helicopter rotors

Transcription

1 ime-dependent adjoint-based aerodynami shape optimization applied to heliopter rotors Asitav Mishra Karthik Mani Dimitri Mavriplis Jay Sitaraman Department of Mehanial Engineering,University of Wyoming, Laramie, WY ABSRAC 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. he fully oupled adjoint formulation is then used to perform rotor blade design optimization for a four bladed HAR2 rotor in hover onditions started impulsively from rest. he effet of time step size and mesh resolution on optimization results is investigated. Results indiate that good optimization results an be obtained using time steps as large as 2 degrees, and that optimizations obtained on the oarse mesh level an be used to initialize fine mesh optimization problems in order to redue overall omputational effort. Future work will fous on aeroelasti optimizations for forward flight ases. INRODUCION 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) (Refs. 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 twodimensional problems by Mani and Mavriplis (Ref. 7) and Postdotoral Researh Assoiate; amishra3@uwyo.edu Assoiate Researh Sientist; kmani@uwyo.edu Professor; mavripl@uwyo.edu Assistant Professor; jsitaram@uwyo.edu Presented at the AHS 7th Annual Forum, Montreal, Quebe,CA May 2 22, 214. Copyright 214 by the Amerian Heliopter Soiety International, In. All rights reserved. 1 also by Rumpfkeil and Zingg (Ref. 8). Preliminary demonstration of the method s feasibility in three-dimensional problems was done by Mavriplis (Ref. 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 (Refs. 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 rotary 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 (Refs. 12, 13). 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 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 (Ref. 14). his formulation was subsequently extended to time-dependent three-dimensional aeroelasti problems in referene (Ref. 15). his work built upon a previously demonstrated time-dependent aerodynami optimization apability that was applied to heliopter rotors in referene (Ref. 16) through the addition of a Hodges-Dowell

2 type beam finite-element model to simulate the rotor struture, and the development of the fully oupled disrete adjoint of the resulting aeroelasti system. his previous work emphasized the formulation, implementation, and verifiation of the adjoint sensitivity analysis approah for time dependent oupled aeroelasti problems. In the urrent work, we demonstrate the effetiveness of using this approah for performing aeroelasti optimization of a representative rotorraft onfiguration. Beause high fidelity time-dependent optimization represents a omputationally intensive approah, obtaining a suitable optimization result with manageable omputational requirements is an important onsideration. herefore, a partiular aspet of this work onsiders the suitability of optimization results obtained on relatively oarse meshes and using larger time steps, in order to redue overall omputational effort. AERODYNAMIC ANALYSIS AND SENSIIVIY FORMULAION 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 time-dependent flow problems. As suh, only a onise desription of these formulations will be given in this paper, with additional details available in previous referenes (Refs ). 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 Cartesian 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 (Ref. 19) 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) 2 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 (Ref. 2) 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 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 = (6) 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: [ ] 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 lineimpliit agglomeration multigrid sheme that an also be used as a preonditioner for a GMRES Krylov solver (Ref. 21). 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)

3 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. 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 (Refs. 22, 23). 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. AERODYNAMIC SENSIIVIY ANALYSIS FORMULAION he basi sensitivity analysis implementation follows the strategy developed in referenes (Refs. 9, 17). 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) 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: G(x,D) = (14) R(U,x) = (15) 3 whih when linearized with respet to the design inputs yields: G U 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 = G U U = (17) (18) 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 pre-multiplying 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 U Λ x = Λ U U (19) 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 Λ u

4 able 1. Comparison of Hart-II Natural Frequenies (Ref. 25) Modes Present Model UMARC DLR Flap Flap Flap orsion Fig degrees of freedom beam element with flap, lag, torsional and axial degrees of freedom. as: Λ U = U U G Λ x = Λ U + (21) (22) 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 (Ref. 14), where the proedure has been used to perform aerodynami shape optimization for a rigid rotor. BEAM MODEL: ANALYSIS AND ADJOIN OPIMIZAION 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 subiterations 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 strutural [I] [M] 1 [K] [M] 1 [C] 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 (Ref. 15) for the standard Hart-2 rotor ase (Ref. 26) 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 (Ref. 27) and DLR (Ref. 28) 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 (Ref. 29). 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 (Refs. 23, 24). 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: [M] q + [C] q + [K] = F (23) 4 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 dd = + as: 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

5 Fig. 2. Fan plot omparing Beam model with UMARC (Ref. 29) 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. 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 = + J F dd ] [ J J F ] (29) his requires solving for an adjoint vetor Λ Q defined as: [ ] 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 5 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) (Ref. 3) as a preursor to their use in fully oupled aero-strutural optimization problems. FULLY COUPLED FLUID-SRUCURE ANALYSIS FORMULAION 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 FSI an be written in residual form as: S(F b,q,f(x,u)) = F b [ (Q)]F(x,u) = (31) S (x s,q) = x s [ (Q)] Q = (32) 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 pointwise 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. 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)) = (33) R(u,x) = (34) S(F B,Q,F(x,u)) = (35) J(Q,F B ) = (36) S (x s,q) = (37)

6 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 ) (38) [ ] u = R(u 1,x ) (39) 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 ) (4) [ ] J Q = J(Q,F b ) (41) x s = [ ] Q (42) 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 aerostrutural problem at the given time step. SENSIIVIY ANALYSIS FOR COUPLED AEROELASIC 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 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 = ] (43) where the individual disiplinary sensitivities are given as the solution of the oupled system: G F F G s I S F S F b J F b S J S S s F F b s = 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 [ ] = G 1 G s = (44) (45) followed by the expliit evaluation of the surfae fore sensitivities as: F = F + F (46) 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 (47) F b G 6

7 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 (48) 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 pre-multiplying equation (44) by the inverse of the large oupling matrix and substituting this into equation (43), transposing the entire system, and defining adjoint variables as solutions to the following oupled system: G F F I G s S F S F b S J F b J Λ x Λ u Λ F Λ Fb Λ Q Λ xs = S S s (49) his system an be solved starting with the last equation and proeeding to the first equation as: Λ xs = G 1 Λ x s followed by the solution of the strutural adjoints S F b S J F b J 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 (52) and ending with the solution of the mesh and flow adjoints as: G Λ x = Λ u + F Λ F + F Λ F (53) 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. 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: f (x + ih) = f (x) + ih f (x) + (54) from whih the derivative f (x) an be easily determined as: f (x) = Im[ f (x + ih)] h (55) 7

8 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 aerostrutural 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. RESULS ime Dependent Analysis Problem he hosen test ase onsists of the four bladed HAR2 rotor with a 5 degree olletive in a hover ondition. he rotor is impulsively started from rest, in an initially quiesent flow field, and rotated with the mesh as a solid body for a fixed number of revolutions. his problem is solved both for a rigid blade model (using no strutural model), as well as for a flexible blade mode (using the beam strutural model). For the latter, the flow is solved in tight oupling mode with the beam solver. A series of progressively finer time step simulations are performed, and the simulation is arried out on a oarse and a fine mesh, in order to assess the level of temporal and spatial disretization errors. he baseline simulation (oarse mesh) makes use of a mixed element mesh made up of prisms, pyramids and tetrahedra onsisting of approximately 2.32 million grid points and is shown in Figures (3(a)), (3(b)) and (3()), where the rigid blade simulation is ompared with the oupled CFD/CSD simulation. he simulations are run for 1 rotor revolutions using a 2 degree time-step for 1, 8 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 (4(a)) and (4(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) at 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 8 (a) Computational Domain (b) Planform view () Zoomed in view Fig. 3. HAR2 rotor mesh onsisting of 2.32 million points used in the optimization example.

9 (a) Planform view Fig. 5. Flow and turbulene residual onvergene at a given time step for rigid analysis (b) Zoomed in view Fig. 4. 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. 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 at this level of resolution requires approximately one hour of wall lok time per rotor revolution. Figures 5 through 8 summarize the overall onvergene of the rigid and aero-elasti oupling analysis formulations. Figure 5 shows the typial flow and turbulene residual onvergene within a single time step for the rigid rotor ase (no strutural model), while Figure 6 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 Fig. 6. Flow and turbulene residual onvergene at a given time step for oupled aeroelasti analysis Fig. 7. Mesh deformation residual onvergene at a given time step for oupled aeroelasti analysis 9

10 Fig. 8. Residual onvergene of beam and overall FSI in one oupling iteration (a) Blade Deformation (b) Blade tip vs time Fig. 9. HAR2 blade deformation 1 values at the start of new oupling iterations 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 7 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 8 illustrates the onvergene of the oupled beam/fsi residual (i.e. equations (35) and (36)), 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. 8. he effet of the CFD/CSD aeroelasti oupling is learly demonstrated in Figure 9, whih ompares the deformed blade shape from the oupled simulation with that from the rigid blade simulation. As Figure 9(b) shows, the blade initially deforms onto larger tip flap values ( 16m) before settling into a lower value of 6m. Further Figure 1 ompares thrust, torque and Figure of Merit (FM) 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. 1(a)) as well as lower total torque magnitude (Fig. 1(b)). herefore, it is evident that the rigid body model might lead to over predition of required rotor torque. It is noteworthy that relatively large differenes are observed between the rigid and flexible rotor simulations in the first rotor revolution, even though these diminish as the rotor approahes the steady hover ondition. herefore, it is of partiular interest to test the oupled aeroelasti optimization apability in this region of the simulation. In order to study the effet of time-step size and grid resolution, analogous simulations were performed using smaller time steps and a finer grid. In total, three simulations using the 2.3 million point grid desribed above were performed using time step sizes of t = 2. and 1. and.5. he number of subiterations per time step was held onstant (6 oupling iterations with 1 flow subiterations per oupling iteration) for all three runs, thus providing slightly lower onverged residual levels for the smaller time step runs. In order to assess the effet of grid resolution, a simulation using a 2. time step on a finer mesh with 11.4 million grid nodes with mixed elements was also performed. For the fine mesh, the number of subiterations per oupling iteration was raised to 25, in order to obtain the same relative derease in residuals at eah time step ompared to the equivalent oarse grid ase. Figures 11 ompares omputed thrust and torque histories from all these ases for the first revolution of the rotor, where transient effets are important. he

11 airload time histories obtained using the three different time step sizes on the oarse grid are in relatively lose agreement, with the differene between the 1. and.5 time step ases being smaller than that between the two larger time step ases, whih is indiative of temporal error onvergene. More appreiable differenes are observed between the oarse and fine mesh runs, suggesting that the 2.3 million point mesh may be too oarse for aurate predition of airloads for this ase. However, one of the objetives of this work is to investigate the effetiveness of performing optimizations using less ostly oarser meshes and larger time steps. ime Dependent Optimization Problem (a) hrust oeffiient vs time (b) orque oeffiient vs time 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 (12(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 (12(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. Unsteady Objetive Funtion Formulation A timeintegrated 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 timeintegrated 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) () Figure of Merit vs time Fig. 1. HAR2 rotor performane time history: 2.32M, t = 2 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 11

12 (a) Blade design parameters (a) hrust oeffiient vs time (b) Baseline strutured blade mesh overlap with CFD mesh (b) orque oeffiient vs time Fig. 12. Illustration of (a) baseline blade with design parameters and (b) overlap in tip region between baseline blade strutured mesh and CFD surfae unstrutured mesh. oeffiients. he global or time-integrated objetive is then onstruted using equal unit weights at eah time-step as: () Figure of Merit vs time Fig. 11. HAR2 rotor performane time history all ases 12 L g = 1 n=n tl n (59) n=1 his so-alled penalty formulation for onstraining the thrust produed by the rotor is less rigorous than imposing a hard onstraint on thrust that must be maintained at eah design yle, sine small dereases in thrust an be traded off for dereases in torque using the penalty formulation. However, this approah is omputationally more effiient than the use of a hard onstraint formulation, sine this latter approah would require the omputation of two separate adjoint problems at eah design yle, as opposed to the single adjoint required in the urrent formulation. Fully Coupled CFD/CSD Adjoint: Unsteady Sensitivity Verifiation he fully oupled CFD/CSD adjoint formulation was verified by first omparing the tangent sensitivities

13 able 2. Unoupled forward linearization verifiation 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 a n=number of time step with those obtained from omplex step method. Subsequently, the adjoint formulation was verified by omparing its sensitivities with those obtained from the tangent as well as the omplex step method. 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. 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. 13 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 Heliopter Blade Optimization he optimization framework is applied to the flexible HAR2 rotor using the same time-dependent test ase as desribed in the analysis setion. he simulation is run for one full rotor revolution, starting impulsively from rest in quiesent flow. he objetive onsists of the time-integrated torque with thrust penalty, as desribed previously. However, the objetive to be minimized is only integrated over the last 6 degrees of revolution, in order to avoid the optimization proess from fousing on start-up transients. Limiting the optimization problem to a single rotor revolution redues overall simulation ost. However, it is lear that the rotor has not ahieved its steady state hover ondition within this simulation time frame, and thus the urrent approah may not be ideal for optimizing hover onditions. However, setting up the optimization problem in this manner allows the assessment of the ability of the optimization framework to handle problems with large transients in both aerodynamis and strutural defletions. he optimization problem is solved on the oarse grid (2.3 million points) using the three different time step sizes (2, 1, and.5 degrees) as well as on the fine grid (11 million points) using the 2 degree time step. One of the objetives of this study is to assess sensitivity of the optimization proess to time step sizes and mesh resolution, and to determine the effetiveness of optimizations performed using lower ost oarser resolution simulations. he optimization proedure used is the L-BFGS-B bounded redued Hessian algorithm (Ref. 3). 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 ±5% hord for eah defining airfoil setion was set on the Hiks-Henne bump funtions, and a bound of ±1. of twist was set on the root and tip twist definitions. he oarse grid optimizations were performed on the Mount Moran omputer luster at the Advaned Researh

14 able 4. Comparison of simulation parameters for 2.3M grid ase t (deg) t a f (hr:min) Iter b nf 2. 1: : : a t f =time per funtion/gradient all b Iter=maximum number of design iterations b nf=maximum number of funtion alls Computing Center (ARCC) at the University of Wyoming with the simulations (analysis/adjoint) running in parallel on 512 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. able 4 summarizes various parameters from the simulations performed for the three time step sizes onsidered for the oarse grid run. As observed from the table, reduing the time step size by a fator of 2 requires twie as long to omplete every funtion/gradient all. his results in fewer design iterations being ompleted within a given amount of wall lok time. Fig. 13. Residual onvergene in a typial adjoint time step Figures 13 shows the residual onvergene for a typial unsteady adjoint time step for t = 2. he figure shows the residual drops by 5 orders of magnitude over 6 oupling yles. Figures 14(a) and 14(b) show the funtional and gradient onvergene over design yles for all the time step sizes. he figures show that the funtional values have ahieved signifiant onvergene in all the three ases and the values are not signifiantly different from eah other. he gradient values for the largest time step ( t = 2 ), having run for largest number of design iterations (72), has onverged beyond two orders of magnitude. he gradient values for the other two smaller time steps have dropped by around one and half orders of magnitude. Overall, there is good onvergene for all the time steps onsidered. Figures 15 ompare the optimized Hart2 rotor airload time histories with those from the baseline rotor for t = 2. he optimized rotor results in a thrust penalty of 1.8% (Fig. 15(a)) and an improved torque redution by 4.9% (Fig. 15(b)). he overall performane gain in terms of Figure of Merit is 2.2% (Fig. 15(). he figures also show the range in the time domain over whih the objetive funtion is being omputed and optimized. his range orresponds to the last 6 of rotor revolution. All the optimization ases disussed in this work onsistently use this same range for omputing their respetive objetive funtions. Figures 16 and 17 further ompare the airloads from the t = 1. and.5 ases, respetively. he results are onsistent with what was observed from the run with 2 time step. Eah of these optimization results in redued final torque oeffiient values (5.3% and 5.16%, respetively) along with 14 (a) Funtional Convergene (b) Gradient Convergene Fig. 14. Convergene in design optimization; Grid size: 2.32M

15 (a) hrust oeffiient vs time (b) orque oeffiient vs time () Figure of Merit vs time Fig. 15. Performane time history on optimized and baseline HAR2 rotor; 2.32 million grid; t = 2. smaller thrust penalties (2.3% and 2.4%, respetively). he overall Figure of Merit improvement (2.2% and 1.6%, respetively) is also onsistent with the results from the optimization ase with 2 time step. Finally, Figures 18 ompare the optimized rotor blade setions with the baseline setions for the three different optimization ases using different time step sizes. he design proess seeks to redue the thikness of the outboard blade setions while produing smaller hanges on the inboard setions. he blade geometry modifiations are onsistent with an aerodynamially designed effiient rotor blade geometry, where the inboard stations are thiker to provide larger Cl max values near low dynami pressure root setions and the outboard stations are thinner so as to redue ompressibility effets at the tip. he differenes in the final design shapes obtained using the three different time steps are small, and all designs display similar trends. At most setions, the 1 degree and.5 degree time step designs are losest to eah other, although this is not observed at the inner most station. Some of the differenes in the design state are likely due to differenes in onvergene of the design proess, as well as to differenes in the omputed objetive and sensitivities due to the hange in time step value. However, the blade design optimization studies performed on this grid reveal that the overall design proess is relatively insensitive to the time step size (over the range onsidered). herefore, in order to redue overall omputational effort, design optimization using a relatively large time step of 2 appears to be feasible. Furthermore, the performane gains realized through the optimization proess based on a single rotor revolution appear to result in equivalent gains in the long-time integration of the rotor in hover. his is illustrated in Figure 19, where the performane improvements of the optimized geometry are maintained for the rotor after 1 revolutions in hover ondition. An improved Hart2 rotor blade design optimization is performed using the finer mesh of 11.4 million grid points with a time step of t = 2. Use of the larger time step on the fine grid is motivated by the results obtained on the oarser grid, as well as by the overall desire to redue omputational ost. wo types of design iterations are performed: i) one that starts with baseline blade design (referred to in figures as Dbase) and ii) another one that starts with the onverged blade design from the oarse mesh (2.32 million) obtained using t = 2. he optimizations were performed on the NCAR-Wyoming superomputer with the simulations (analysis/adjoint) running in parallel on 248 ores. Eah time step in the analysis problem employed 6 oupling yles. Eah oupling yle used 25 nonlinear iterations with eah nonlinear iteration requiring 3 linear multigrid yles per Krylov vetor. A typial design yle (funtion/gradient omputation) required approximately 5 hours total run time. Figures 2(a) and 2(b) ompare the funtional and gradient onvergene for the two fine mesh optimization ases desribed earlier. Both the funtional and gradient for the restarted ase start out at lower values than those of the base- 15