Aerodynamic Shape Optimization of Benchmark Problems Using Jetstream

This work presents results from the application of an aerodynamic shape optimization code, Jetstream, to a suite of benchmark cases defined by the Aerodynamic Design Optimization Discussion Group. Geometry parameterization and mesh movement are integrated by fitting the multi-block structured grids with B-spline volumes and performing mesh movement based on a linear elastic model applied to the control points. Geometry control is achieved through two different approaches. Either the B-spline surface control points are taken as the design variables for optimization, or alternatively, the surface control points are embedded within free-form deformation (FFD) B-spline volumes, and the FFD control points are taken as the design variables. Spatial discretization of the Euler or Reynolds-averaged Navier-Stokes equations is performed using summation-by-parts operators with simultaneous approximation terms at boundaries and block interfaces. The governing equations are solved iteratively using a parallel Newton-Krylov-Schur algorithm. The discrete-adjoint method is used to calculate the gradients supplied to a sequential quadratic programming optimization algorithm. The first optimization problem studied is the drag minimization of a modified NACA 0012 airfoil at zero angle of attack in inviscid, transonic flow, with a minimum thickness constraint set to the initial thickness. The shock is weakened and moved downstream, reducing drag by 91%. The second problem is the liftconstrained drag minimization of the RAE 2822 airfoil in viscous, transonic flow. The shock is eliminated and drag is reduced by 48%. Both two-dimensional cases exhibit optimization convergence difficulties due to the presence of nonunique flow solutions. The third problem is the twist optimization for minimum induced drag at fixed lift of a rectangular wing in subsonic, inviscid flow. A span efficiency factor very close to unity and a near elliptical lift distribution are achieved. The final problem includes single-point and multi-point liftconstrained drag minimizations of the Common Research Model wing in transonic, viscous flow. Significant shape changes and performance improvements are achieved in all cases. Finally, two additional optimization problems are presented that demonstrate the capabilities of Jetstream and could be suitable additions to the Discussion Group problem suite. The first is a wing-fuselage-tail optimization with a prescribed spanwise load distribution on the wing. The second is an optimization of a box-wing geometry.

[1]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[2]  John C. Vassberg,et al.  Further Studies of Airfoils Supporting Non-Unique Solutions in Transonic Flow , 2011 .

[3]  John C. Vassberg,et al.  Airfoils Supporting Non-unique Transonic Solutions for Unsteady Viscous Flows , 2014 .

[4]  Ilan Kroo,et al.  DRAG DUE TO LIFT: Concepts for Prediction and Reduction , 2001 .

[5]  D. Zingg,et al.  Two-Level Free-Form and Axial Deformation for Exploratory Aerodynamic Shape Optimization , 2015 .

[6]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2005, SIAM Rev..

[7]  L. Prandtl.  Induced drag of multiplanes , 1924 .

[8]  Hugo Gagnon,et al.  High-fidelity Aerodynamic Shape Optimization of Unconventional Aircraft through Axial Deformation , 2014 .

[9]  D. Zingg,et al.  Mesh Movement for a Discrete-Adjoint Newton-Krylov Algorithm for Aerodynamic Optimization , 2007 .

[10]  David W. Zingg,et al.  Approach to Aerodynamic Design Through Numerical Optimization , 2013 .

[11]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[12]  Frank W. Spaid,et al.  Aerodynamic Design of High-Perf ormance Biplane Wings , 1975 .

[13]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[14]  Beckett Yx Zhou,et al.  Airfoil Optimization Using Practical Aerodynamic Design Requirements , 2010 .

[15]  David W. Zingg,et al.  Aerodynamic Shape Optimization of a Blended-Wing-Body Regional Transport for a Short Range Mission , 2013 .

[16]  Jason E. Hicken,et al.  Summation-by-parts operators and high-order quadrature , 2011, J. Comput. Appl. Math..

[17]  Jason E. Hicken.  Efficient Algorithms for Future Aircraft Design: Contributions to Aerodynamic Shape Optimization , 2009 .

[18]  David W. Zingg,et al.  A parallel Newton-Krylov-Schur flow solver for the Navier-Stokes equations discretized using summation-by-parts operators , 2013 .

[19]  David W. Zingg,et al.  Parallel Newton–Krylov–Schur Flow Solver for the Navier–Stokes Equations , 2013 .

[20]  D. P. Young,et al.  Study Based on the AIAA Aerodynamic Design Optimization Discussion Group Test Cases , 2015 .

[21]  Edward N. Tinoco,et al.  Summary of Data from the Fifth AIAA CFD Drag Prediction Workshop , 2013 .

[22]  Hugo Gagnon,et al.  Geometry Generation of Complex Unconventional Aircraft with Application to High-Fidelity Aerodynamic Shape Optimization , 2013 .

[23]  Thomas A. Reist,et al.  Drag Minimization Based on the Navier–Stokes Equations Using a Newton–Krylov Approach , 2015 .

[24]  David W. Zingg,et al.  Application of an efficient Newton-Krylov algorithm for aerodynamic shape optimization based on the Reynolds-Averaged Navier-Stokes equations , 2013 .

[25]  D. Zingg,et al.  Multimodality and Global Optimization in Aerodynamic Design , 2013 .

[26]  D. Zingg,et al.  Newton-Krylov Algorithm for Aerodynamic Design Using the Navier-Stokes Equations , 2002 .

[27]  R. H. Lange,et al.  Feasibility study of the transonic biplane concept for transport aircraft application , 1974 .

[28]  S. Nadarajah.  Adjoint-Based Aerodynamic Optimization of Benchmark Problems , 2015 .

[29]  Jason E. Hicken,et al.  A Simplified and Flexible Variant of GCROT for Solving Nonsymmetric Linear Systems , 2010, SIAM J. Sci. Comput..

[30]  Jason E. Hicken,et al.  Aerodynamic Optimization Algorithm with Integrated Geometry Parameterization and Mesh Movement , 2010 .

[31]  Jason E. Hicken,et al.  Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous-Approximation Terms , 2008 .

[32]  Hugo Gagnon,et al.  Aerodynamic Optimization Trade Study of a Box-Wing Aircraft Configuration , 2015 .

[33]  David W. Zingg,et al.  Application of Jetstream to a Suite of Aerodynamic Shape Optimization Problems , 2014 .

[34]  E. Sturler,et al.  Truncation Strategies for Optimal Krylov Subspace Methods , 1999 .

[35]  R. C. Swanson,et al.  On Central-Difference and Upwind Schemes , 1992 .

[36]  E. Sturler,et al.  Nested Krylov methods based on GCR , 1996 .