Aerodynamic Shape Optimization in Three-Dimensional Turbulent Flows Using a Newton-Krylov Approach

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1 Aerodynamic Shape Optimization in Three-Dimensional Turbulent Flows Using a Newton-Krylov Approach Lana M. Osusky and David W. Zingg University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, ON, M3H 5T6, Canada lana@oddjob.utias.utoronto.ca ABSTRACT A parallel Newton-Krylov discrete-adjoint method is presented for gradient-based aerodynamic shape optimization in fully turbulent three-dimensional flow. The use of an integrated B-spline parameterization and mesh movement scheme allows for fast and accurate mesh updates that can accommodate large shape changes. Flow analyses are performed using a parallel Newton-Krylov algorithm with approximate-schur preconditioning. Turbulence is modelled using the Spalart-Allmaras one-equation model. We validate the algorithm with an inverse design optimization. Examples are presented to demonstrate the overall effectiveness of this approach to aerodynamic shape optimization. INTRODUCTION Over the last ten years, fuel costs have eclipsed labour costs to become the dominant operational expenditure of airlines around the world. Given the current economic climate, the airline industry could benefit immensely from a fleet of more efficient aircraft with reduced drag that consequently burn less fuel. Increased demand is being placed on numerical methods to solve aerodynamic design problems accurately and efficiently. Gradient-based optimization has become a popular approach to determining optimal aerodynamic designs, particularly since the introduction of adjointbased methods by Pironneau [] and Jameson [2], which makes the cost of a gradient evaluation virtually independent of the number of design variables. PhD Candidate, AIAA Student Member Professor and Director, Tier Canada Research Chair in Computational Fluid Dynamics, J. Armand Bombardier Foundation Chair in Aerospace Flight Jameson, Reuther, and colleagues applied the adjoint method to two-dimensional inviscid aerodynamic design problems [3] and then extended their work to three-dimensional constrained multi-point problems [4, 5, 6]. Jameson et al. [7] used the continuous adjoint approach to optimize wings and wing-body configurations based on the compressible Navier-Stokes equations. Anderson and Bonhaus [8] performed airfoil optimizations on unstructured grids using a discreteadjoint approach to the Navier-Stokes equations coupled with the one-equation Spalart-Allmaras turbulence model. Nemec and Zingg [9, ] demonstrated the efficiency of a Newton-Krylov scheme in performing two-dimensional aerodynamic shape optimization based on the Navier-Stokes equations and the Spalart- Allmaras turbulence model. Laminar-turbulent transition prediction was incorporated into a gradient-based Newton-Krylov optimization algorithm by Driver and Zingg [] and was used to design a series of highlift airfoils. Hicken and Zingg [2] performed a series of drag minimization studies of non-planar wing configurations in three-dimensional Euler flow; the work presented in this paper is an extension of the Eulerbased optimizer to include viscous and turbulent flow phenomena. 2 GEOMETRY PARAMETERIZATION AND MESH MOVEMENT Each block of a multi-block computational mesh is approximated with a B-spline tensor volume. The subset of B-spline control points associated with the aerodynamic surface geometry is used as the set of design variables for optimization, either individually for local changes, or coupled via a set of planform variables, such as sweep or twist. Mesh movement is carried out using a hybrid approach that combines a robust lin-

2 ear elasticity approach and an algebraic scheme. The nodes of the coarse B-spline control mesh are updated based on the principles of linear elasticity [3, 4]: M (i) (b (i ),b (i) )=K (i) (b (i ) )[b (i) b (i ) ] f (i) =, 3 FLOW ANALYSIS We solve the three-dimensional Navier-Stokes equations using a Newton-Krylov method with a flexible GMRES linear solver and approximate-schur parallel preconditioning [6, 7]. The governing equations are expressed in terms of the curvilinear coordinates by i=,...,m () wherem (i) is the mesh movement residual, b (i) is the set of B-spline control point coordinates for the entire computational domain, K is the global stiffness matrix at increment i, and m is the number of increments. The force vector f (i) at each increment is implicitly defined by the displacements of the surface and far-field control points. We typically perform the control mesh updates of () in five increments, which strikes a balance between speed and robustness in accommodating large changes in the geometry [4]. Because the control mesh is typically hundreds of times smaller than the fine computational mesh, the linear elasticity-based mesh movement requires minimal computational time and reduces the size of the mesh adjoint problem compared to applying the same method to the fine mesh. The fine mesh is updated algebraically based on the equations defining the B-spline volume mappings: x(ξ)= N i N j N k i= j= k= B i jk N (p) i (ξ)n (p) j (η)n (p) k (ζ), (2) where x(ξ) represents the set of Cartesian coordinates of the B-spline volume, N (p) i, j,k are the pth -order basis functions in each of the three coordinate directions, and B i jk are the B-spline control points for the given B-spline volume. We use p = 4, which produces cubic B-spline volumes. Using this hybrid approach allows us to perform mesh updates in a matter of seconds. It has been found that that substantial fitting error is incurred when obtaining B-spline volumes for computational domains with very fine grid spacings, which are required in order to capture turbulent flow features [5]. We avoid this issue by obtaining the B-spline for a relatively coarse mesh with an off-wall spacing of approximately 3 and subsequently refining the computational mesh (not the B-spline control mesh) by either redistributing the existing nodes such that the desired spacings are obtained, inserting additional nodes, or both. t ˆQ+ ξ Ê+ η ˆF+ ζ Ĝ= Re ( ξ Ê v + η ˆF v + ζ Ĝ v ), (3) where Re is the Reynolds number, ˆQ holds the set of conservative flow variables, (Ê, ˆF, Ĝ) are the inviscid fluxes, and (Ê v, ˆF v, Ĝ v ) are the viscous fluxes. Turbulence is represented by the Spalart-Allmaras one-equation turbulence model [8], which can be expressed as ν t = M( ν) ν+p( ν) ν D( ν) ν+t, (4) where M( ν) ν represents advection and diffusion, P( ν) ν is a production source term, D( ν) ν is a wall destruction source term, and T is a trip function representing the transition point at which the flow becomes turbulent. The work in this paper assumes fully turbulent flow; consequently, the trip term is not used. The governing equations are discretized using secondorder accurate summation-by-parts (SBP) operators [9] with the scalar artificial dissipation model first developed by Jameson et al. [2] and refined by Pulliam [2]. Simultaneous Approximation Terms (SATs) are used to enforce boundary conditions and the coupling of adjoining blocks in the computational mesh. The use of SATs allows us to solve the governing equations for a given block with less information required from neighbouring blocks compared to methods that make use of halo nodes. The subsequent reduction in inter-processor communication results in a faster solution method. Mesh generation is also made easier, as the SAT implementation eliminates the need for slope continuity at block interfaces. Further information on the development and implementation of the SBP-SATs can be found in references [6, 7, 22, 23]. The discretized governing equations produce the large, sparse system of linear equations given by R(v,b (m),q)=, (5) where R is the residual vector, v is the vector of design variables, b (m) represents the B-spline control

3 point coordinates for each volume at the completion of the mesh movement, and q is the vector of conservative flow variables. Applying an implicit-euler time-marching method to (5) results in the sparse linear system given by ( ) I t A(n) q (n) = R (n), (6) where n is the outer iteration, t is the time step,i is an identity matrix,r (n) =R(q (n) ), q (n) = q (n+) q (n), and A (n) is the flow Jacobian matrix: A (n) i j = R(n) i. (7) q j The system is solved using GMRES with approximate- Schur preconditioning [24, 6, 25]. The solution scheme makes use of an approximate-newton start-up phase, which involves a first-order approximation of the flow Jacobian as well as a spatially-varying time step, to form a suitable initial iterate. After the residual has been reduced by 4 orders of magnitude, the solver switches over to an inexact-newton method, which reduces the residual norm to a relative tolerance of. Deep convergence of the flow solver is required for the convergence of the optimizer. 4 GRADIENT-BASED OPTIMIZATION We use the gradient-based sequential quadratic programming optimization algorithm SNOPT [26], which finds the stationary point of an augmented Lagrangian merit function that enforces linear constraints exactly and incorporates non-linear constraints into the merit function via sets of adjoint variables. The use of a gradient-based optimizer necessitates the efficient and accurate computation of the objective and constraint gradients. In this work, the gradients are evaluated using the discrete-adjoint approach [27, 7]. Since the time required for one computation of the gradient is virtually independent of the number of design variables, this method is an attractive one for large-scale optimizations with many design variables. SNOPT finds the stationary point of a Lagrangian merit function of the form L(v,b (m),q, λ (i) m i=, ψ)=j(v,b(m),q) + m i= λ (i)t M (i) (v,b (i ),b (i) ) + ψ T R(v,b (m),q), (8) where the mesh residual and flow residual equations, M and R, respectively, are incorporated as non-linear constraints via the adjoint variables λ and ψ. Additionally, J is the objective function, v is the vector of design variables, b (i) represents the set of B-spline control points at the i th increment of the mesh movement, and q is the vector of flow variables. Any other nonlinear constraints, such as lift, volume, or area, would be included in a similar manner. The gradient of (8) can be expressed as ( ) G = J m v + λ (i)t M (i) + ψ T R i= v v, (9) but, in practice, is broken down into a series of intermediate systems and solved sequentially. After (5) is solved for the updated set of flow variables, q, the flow adjoint system given by ( ) R T ( ) J T ψ= () q q is solved to a tolerance of using a flexible and simplified variant of GCROT [28] for the flow adjoint variables, ψ. Next, we obtain the mesh adjoint variables, λ (m), corresponding to the final increment of the mesh movement by using the conjugate gradient method preconditioned with ILU() to solve ( M (m) b (m) ) T ( ) λ (m) J T ( ) R T = b (m) b (m) ψ. () The mesh adjoint variables, λ (i), corresponding to mesh movement increments through m are obtained by using the preconditioned conjugate gradient method to solve ( M (i) b (i) ) T λ (i) = ( M (i+) b (i) ) T λ (i+), i {m,m 2,...,}. (2) In each of the mesh adjoint equations, the residual is reduced by 2 orders of magnitude.

4 5 RESULTS 5. Wing in Turbulent Transonic Flow with Section Design Variables We minimize the drag of a rectangular planar wing with a semi-span of 2., a root chord of., and, initially, RAE2822 sections in turbulent transonic flow. The Mach number is.8, the Reynolds number is 6.5 million, and the initial angle of attack is.5 degrees; at these flow conditions, shocks are present on the upper and lower surfaces of the wing. The lift coefficient, calculated using the surface area as a reference area, is constrained to its initial value of.2722, and the wing volume is constrained to be greater than or equal to its initial value of.58. The 24-block computational mesh is made up of 2.94 million nodes and the grid node redistribution technique discussed in Section 2 is used to obtain an offwall spacing of.23 6 root chord units. Each of the 24 control volumes is made up of control points; the z-coordinates of the control points corresponding to the wing surface are used as design variables, with the exception of the control points near the leading and trailing edges. The angle of attack is also free, resulting in a total of 54 design variables. The optimizer reduces the drag coefficient by 5.7% and virtually eliminates the surface shocks while maintaining the desired lift coefficient and minimum volume targets. The initial and final drag coefficients are.74 and.46, respectively. There is minimal change in the angle of attack. These results are obtained after 33 function evaluations, where one function evaluation is defined as one flow solve and one gradient evaluation. Approximately 5 hours of CPU time were required using 24 processors. Comparisons of the coefficient of pressure and section geometries of the initial and optimized wings are shown in Figures and 2, respectively. The convergence history is shown in Figure 3, where we see a decrease in the optimality, which is a measure of the gradient, of approximately one order of magnitude. 5.2 Wing in Turbulent Transonic Flow with Planform Design Variables A second lift-constrained drag minimization is performed for an initially rectangular planar wing with NACA2 cross sections. The wing dimensions and computational mesh are the same as those of the case discussed in the preceding section. The Mach number is.8, the Reynolds number is 6.5 million, and the initial angle of attack is 2.8 degrees. The lift coefficient is constrained to a value of.292 and the surface area is constrained to be no less than its starting value of The B-spline control points corresponding to the wing surface are coupled to a set of planform design variables: leading edge sweep, trailing edge sweep, span, and the vertical position (defined in terms of dihedral angles) of the leading- and trailing-edge control points at 8 span-wise stations along the wing (which can effectively create dihedral and twist). The angle of attack is an additional design variable, for a total of 2 design variables. All of the design variables are scaled based on their respective initial objective gradient values. The sweep and dihedral angle variables all have an upper limit of 3 degrees imposed on them in order to prevent excessively large shape changes that could keep the mesh movement algorithm from converging. The span has an upper limit of root chord units. Over the course of 26 function evaluations, which required approximately 48 hours on 24 processors, the drag coefficient was reduced from.27 to.29, a 4.8% improvement, while satisfying the lift and surface area constraints. The drag reduction can be seen in the convergence plots in Figure 4, along with a reduction in optimality of three orders of magnitude. Note that the merit function represents drag values that are unscaled by the reference area. The initial and optimized geometries are compared in Figure 5, where the initial geometry is shown above the optimized geometry. The locations of the B-spline control points are shown as red spheres on the initial wing and blue spheres on the optimized wing. The leading edge sweep reached its upper limit of 3. degrees, while the trailing edge sweep increased to 29.4 degrees. The maximum span-wise dihedral angles along the leading- and trailing-edges are reached at the wing tip, where the leading-edge dihedral is 2.7 degrees and the trailing-edge dihedral is 3. degrees. The angle of attack reaches a final value of 7.32 degrees. There is no change in the span. 6 CONCLUSIONS We have presented an efficient numerical tool for aerodynamic shape optimization in three-dimensional turbulent flow. A unique integrated geometry parameterization and mesh movement scheme approximates the computational mesh with B-spline volumes and subsequently moves these volumes based on the equations of linear elasticity. Flow analysis is performed using a parallel Newton-Krylov flow solver with approximate-

5 Schur preconditioning. Objective and constraint gradients are calculated using the discrete-adjoint approach and used by the gradient-based optimizer SNOPT to find an optimum in the design space. We have demonstrated the performance of the optimization algorithm through two lift-constrained drag minimization studies in turbulent transonic flow involving the use of section and planform design variables. In each case, a substantial reduction in drag is achieved while maintaining a target lift coefficient. Future work will include multipoint optimization, studies of alternative wing tip treatments, and an investigation into multi-modality in the design space. Ultimately, this work will be used within an aero-structural optimization framework. 7 ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial assistance provided by the Natural Sciences and Engineering Research Council (NSERC), the Canada Research Chairs program, Bombardier Aerospace, MITACS, Zonta International, and the University of Toronto. REFERENCES [] O. Pironneau. On optimum design in fluid mechanics. Journal of Fluid Mechanics, 64():97, 974. [2] A. Jameson. Aerodynamic design via control theory. Journal of Scientific Computing, 3(3):233 26, 988. [3] A. Jameson and J. Reuther. Control theory based airfoil design using Euler equations. In AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. Panama City Beach, September 994. [4] J. Reuther, A. Jameson, J. Farmer, L. Martinelli, and D. Saunders. Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation. In The 34rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA Reno, Nevada, 996. [5] J. J. Reuther, A. Jameson, J. J. Alonso, M. J. Rimlinger, and D. Saunders. Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part. AIAA Journal, 36():5 6, January 999. [6] J. J. Reuther, A. Jameson, J. J. Alonso, M. J. Rimlinger, and D. Saunders. Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2. AIAA Journal, 36():6 74, January 999. [7] A. Jameson, L. Martinelli, and N. A. Pierce. Optimum aerodynamic design using the Navier- Stokes equations. Theoretical and Computational Fluid Dynamics, :23 237, 998. [8] W. K. Anderson and D. L. Bonhaus. Airfoil design on unstructured grids for turbulent flows. AIAA Journal, 37(2):85 9, February 999. [9] M. Nemec. Optimal Shape Design of Aerodynamic Configurations: A Newton-Krylov Approach. Ph.D. thesis, University of Toronto, Toronto, Ontario, Canada, 23. [] M. Nemec, D. W. Zingg, and T. H. Pulliam. Multipoint and multi-objective aerodynamic shape optimization. AIAA Journal, 42(6):57 65, 24. [] J. Driver and D. W. Zingg. Numerical aerodynamic optimization incorporating laminarturbulent transition prediction. AIAA Journal, 45(8):8 88, August 27. [2] J. Hicken and D. Zingg. Induced drag minimization of nonplanar geometries based on the Euler equations. AIAA Journal, 48(): , 2. [3] A. H. Truong, C. A. Oldfield, and D. W. Zingg. Mesh movement for a discrete-adjoint Newton- Krylov algorithm for aerodynamic optimization. AIAA Journal, 46(7):695 74, July 28. [4] J. E. Hicken and D. W. Zingg. Aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement. AIAA Journal, 48(2):4 43, February 2. [5] L. Osusky and D. W. Zingg. A Novel Aerodynamic Shape Optimization Approach for Three- Dimensional Turbulent Flows. In 5th AIAA Aerospace Sciences Meeting, AIAA Nashville, Tennessee, United States, January 22. [6] J. E. Hicken and D. W. Zingg. A parallel Newton- Krylov solver for the Euler equations discretized using simultaneous approximation terms. AIAA Journal, 46(): , November 28. [7] M. Osusky, J. E. Hicken, and D. W. Zingg. A parallel Newton-Krylov-Schur flow solver for the Navier-Stokes equations using the SBP-SAT approach. In 48th AIAA Aerospace Sciences Meeting and Aerospace Exposition, AIAA 2 6. Orlando, Florida, United States, January 2. [8] P. R. Spalart and S. R. Allmaras. A one-equation turbulence model for aerodynamic flows January 992. [9] M. Svärd, K. Mattsson, and J. Nordström. Steady-state computations using Summation-by-

6 Parts operators. Journal of Scientific Computing, 24():79 95, July 25. [2] A. Jameson, W. Schmidt, and E. Turkel. Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. In 4th Fluid and Plasma Dynamics Conference. Palo Alto, CA, 98. AIAA Paper [2] T. H. Pulliam. Efficient solution methods for the Navier-Stokes equations. Technical report, Lecture Notes for the von Kármán Inst. for Fluid Dynamics Lecture Series: Numerical Techniques for Viscous Flow Computation in Turbomachinery Bladings, Rhode-Saint-Genèse, Belgium, January 986. [22] M. Osusky and D. W. Zingg. A parallel Newton- Krylov-Schur flow solver for the Navier-Stokes equations using the SBP-SAT approach. In 5th AIAA Aerospace Sciences Meeting, AIAA Nashville, Tennessee, United States, January 22. [23] J. Nordström, J. Gong, E. van der Weide, and M. Svärd. A stable and conservative high order multi-block method for the compressible Navier- Stokes equations. Journal of Computational Physics, 228(24):92 935, 29. [24] Y. Saad and M. Sosonkina. Solution of distributed sparse linear systems using psparslib. In Applied Parallel Computing: Large Scale Scientific and Industial Problems, fourth International Workshop, PARA98, pages Springer Verlag, 998. [25] J. E. Hicken, M. Osusky, and D. W. Zingg. Comparison of parallel preconditioners for a Newton- Krylov flow solver. In 6th International Conference on Computational Fluid Dynamics. St. Petersburg, Russia, 2. [26] P. E. Gill, W. Murray, and M. A. Saunders. SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Journal on Optimization, 2(4):979 6, 22. [27] O. Pironneau. Optimal shape design for elliptic systems. Springer-Verlag, 983. [28] J. E. Hicken and D. W. Zingg. A simplified and flexible variant of GCROT for solving nonsymmetric linear systems. SIAM Journal on Scientific Computing, 32(3): , June 2.

7 - Initial Optimized (a) 2% span (b) 44% span (c) 65% span (d) 8% span (e) 9% span (f) 95% span Figure : Coefficient of pressure for wing optimization based on section shape.6 Initial Optimized (a) 2% span (b) 44% span (c) 65% span (d) 8% span (e) 9% span (f) 95% span Figure 2: Section shape comparison for wing optimization based on section shape

8 .36 Merit Function.35 Optimality Function evaluations 2 3 Function evaluations Figure 3: Convergence history for wing optimization based on section shape.45 Optimality Merit Function Function evaluations 2 Function evaluations Figure 4: Convergence history for wing optimization based on planform variables Z Y Z X Y X Z Y Z X X (a) Leading-edge view (b) 3D view Figure 5: Geometry comparison for wing optimization based on planform variables Y