Skip to main content
SpringerLink
Log in

CFD-based optimization of hovering rotors using radial basis functions for shape parameterization and mesh deformation

  • Published:
Optimization and Engineering Aims and scope Submit manuscript

Abstract

Aerodynamic shape optimization of a helicopter rotor in hover is presented, using compressible CFD as the aerodynamic model. An efficient domain element shape parameterization method is used as the surface control and deformation method, and is linked to a radial basis function global interpolation, to provide direct transfer of domain element movements into deformations of the design surface and the CFD volume mesh, and so both the geometry control and volume mesh deformation problems are solved simultaneously. This method is independent of mesh type (structured or unstructured) or size, and optimization independence from the flow solver is achieved by obtaining sensitivity information for an advanced parallel gradient-based algorithm by finite-difference, resulting in a flexible method of ‘wrap-around’ optimization. This paper presents results of the method applied to hovering rotors using local and global design parameters, allowing a large geometric design space. Results are presented for two transonic tip Mach numbers, with minimum torque as the objective, and strict constraints applied on thrust, internal volume and root moments. This is believed to be the first free form design optimization of a rotor blade using compressible CFD as the aerodynamic model, and large geometric deformations are demonstrated, resulting in significant torque reductions, with off-design performance also improved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. http://www.optimalsolutions.us/ (2008).

Abbreviations

c :

Aerofoil chord

C P :

Pressure coefficient

C T :

Thrust coefficient

C Z :

Sectional normal force coefficient

C mx ,C my ,C mz :

Hinge moment coefficients

C :

Constraint vector

D :

Number of domain element nodes

E :

Total energy

f :

Scalar function

F :

Flux vector

G :

Source term vector

i,j,k :

Matrix and vector component indices

i,j,k :

Unit vectors

J :

Objective function

m :

Number of constraints

M :

Number of polynomial terms

M Tip :

Tip Mach number

n :

Number of dimensions

n :

Unit normal vector

N :

Number of interpolation control points

p :

Polynomial term

P :

Static pressure

P :

Polynomial matrix

q :

Polynomial component

q :

Velocity vector

r :

Euclidean norm

r :

Cartesian coordinate vector

R :

Support radius

s :

Interpolated function

t :

Time

t/c :

Sectional thickness to chord ratio

u,v,w :

Velocity components

U :

Conserved quantity vector

V :

Cell volume

x,y,z :

Cartesian coordinates

x,y,z :

Vectors of Cartesian coordinates

X qc ,Z qc :

Sectional quarter chord coordinates

α :

Polynomial weighting coefficients

β :

Interpolation weighting coefficients

γ :

Ratio of specific heats

δ :

Design variables

λ :

Lagrange multipliers

ϕ :

Basis function

ρ :

Density

σ :

Rotor solidity

θ :

Section pitch angle

ω :

Angular velocity vector

ζ :

n-dimensional coordinate vector

Ω:

Angular velocity

References

  • Allen CB (2001) Multigrid acceleration of an upwind Euler code for hovering rotor flows. Aeronaut J 105(1051):517–524

    Google Scholar 

  • Allen CB (2004) An unsteady multiblock multigrid scheme for lifting forward flight rotor simulations. Int J Numer Methods Fluids 45(9):973–984

    Article  MATH  Google Scholar 

  • Allen CB (2007) Parallel universal approach to mesh motion and application to rotors in forward flight. Int J Numer Methods Eng 69(10):2126–2149

    Article  MATH  Google Scholar 

  • Allen CB (2008) Towards automatic structured multiblock mesh generation using improved transfinite interpolation. Int J Numer Methods Eng 74(5):697–733

    Article  MATH  Google Scholar 

  • Allen CB, Rendall TCS, Morris AM (2010) CFD-based twist optimisation of hovering rotors. J Aircr 47(6):2075–2085

    Article  Google Scholar 

  • Alonso JJ, Jameson A (1994) Fully implicit time-marching aeroelastic solutions. AIAA paper 1994-0056

  • Amoignon O (2010) AESOP—a numerical platform for aerodynamic shape optimization. Optim Eng 11:555–581

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson WK, Karman SL, Burdyshaw C (2009) Geometry parameterization method for multidisciplinary applications. AIAA J 17(6):1568–1578

    Article  Google Scholar 

  • Bloor MIG, Wilson MJ (1995) Efficient parameterization of generic aircraft geometry. J Aircr 32(6):1269–1275

    Article  Google Scholar 

  • Braibant V, Fleury C (1984) Shape optimal design using B-splines. Comput Methods Appl Mech Eng 44(3):247–267

    Article  MATH  Google Scholar 

  • Buhmann H (2005) Radial basis functions, 1st edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Caradonna FX, Tung C (1981) Experimental and analytical studies of a model helicopter rotor in hover. NASA TM-81232

  • Celi R (1999) Recent applications of design optimization to rotorcraft—a survey. J Aircr 36(1):176–189

    Article  Google Scholar 

  • Choi S, Pottsdam M, Lee KH, Iaccarino G, Alonso JJ (2008) Helicopter rotor design using a time-spectral and adjoint-based method. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference, Victoria, BC, Canada. AIAA paper 2008-5810

    Google Scholar 

  • Chung HS, Alonso J (2004) Multiobjective optimization using approximati on model-based genetic algorithms. In: 10th AIAA/ISSMO symposium on multidisciplinary analysis and optimization, Albany, NY. AIAA paper 2004-4325

    Google Scholar 

  • Dulikravich SG (1992) Aerodynamic shape design and optimization: status and trends. J Aircr 29(6):1020–1025

    Article  Google Scholar 

  • Dumont A, Pape AL, Peter J, Huberson S (2009) Aerodynamic shape optimization of hovering rotors using a discrete adjoint of the RANS equations. In: AHS 65th annual forum, Grapevine, TX

    Google Scholar 

  • Fasshauer GE (2007) Meshfree approximation methods with Matlab. Interdisciplinary Mathematical Sciences, vol 6. World Scientific, Singapore

    MATH  Google Scholar 

  • Gumbert CR, Hou G, Newman PA (2001a) Simultaneous aerodynamic analysis and design optimization (SAADO) for a 3-D flexible wing. In: Aerospace sciences meeting and exhibit, Reno, NV. AIAA paper 2001-1107

    Google Scholar 

  • Hicks RM, Henne PA (1978) Wing design by numerical optimization. J Aircr 15(1):407–412

    Article  Google Scholar 

  • Jakobsson S, Patriksson M, Rudholm J, Wojciechowski A (2009b) A method for simulation based optimization using radial basis functions. Optim Eng 11:1–32

    MathSciNet  Google Scholar 

  • Jameson A (1988) Aerodynamic design via control theory. J Sci Comput 3(3):233–260

    Article  MATH  Google Scholar 

  • Jameson A (2003) CFD for aerodynamic design and optimization: its evolution over the last three decades. In: 16th AIAA CFD conference, June 23–26, Orlando, FL. AIAA 2003-3438

    Google Scholar 

  • Jameson A, Leoviriyakit K, Shankaran S (2007) Multi-point aero-structural optimization of wings including variations. In: Aerospace sciences meeting and exhibit, Reno, NV

    Google Scholar 

  • Kulfan BM (2007) A universal parametric geometry representation method—CST. In: 45th AIAA aerospace sciences meeting and exibit, 8–11 Jan 2007, Reno, NV

    Google Scholar 

  • Li W, Huyse L, Padula S (2002) Robust airfoil optimization to achieve drag reduction over a range of Mach numbers. Struct Multidiscip Optim 24(1):38–50

    Article  Google Scholar 

  • Morris AM, Allen CB, Rendall TCS (2008) CFD-based optimization of aerofoils using radial basis functions for domain element parameterization and mesh deformation. Int J Numer Methods Fluids 58(8):827–860

    Article  MathSciNet  MATH  Google Scholar 

  • Morris AM, Allen CB, Rendall TCS (2009) Domain element method for aerodynamic shape optimization applied to modern transport wing. AIAA J 47(7):1647–1659

    Article  Google Scholar 

  • Nadarajah S, Castonguay P, Mousavi A (2007) Survey of shape parameterization techniques and its effect on three-dimensional aerodynamic shape optimization. In: 18th AIAA computational fluid dynamics conference, Miami, FL. AIAA paper 2007-3837

    Google Scholar 

  • Nadarajah S, Soucy O, Tatossian C (2008) Aerodynamic shape optimisation of hovering rotor blades using a NLFD approach. In: 46th AIAA aerospace sciences meeting and exibit, Reno, NV. AIAA paper 2008-0322

    Google Scholar 

  • Nielsen EJ, Lee-Rausch EM, Jones WT (2009) Adjoint-based design of rotors using the Navier-Stokes equations in a noninertial frame. In: AHS 65th annual forum, Grapevine, TX

    Google Scholar 

  • Pape AL, Beaumier P (2005) Numerical optimization of helicopter rotor aerodynamic performance in hover. Aerosp Sci Technol 9(3):111–201

    Google Scholar 

  • Parpia IH (1988) Van-Leer flux vector splitting in moving coordinates. AIAA J 26:113–115

    Article  MATH  Google Scholar 

  • Perry FJ (1987) Aerodynamics of the helicopter world speed record. In: 43rd annual national forum of the American helicopter society

    Google Scholar 

  • Pickett RM, Rubinstein MF, Nelson RB (1973) Automated structural synthesis using a reduced number of design coordinates. AIAA J 11(4):494–498

    Google Scholar 

  • Rendall TCS, Allen CB (2008a) Multi-dimensional aircraft surface pressure interpolation using radial basis functions. Proc Inst Mech Eng, G J Aerosp Eng 222:483–495

    Google Scholar 

  • Rendall TCS, Allen CB (2008b) Unified fluid-structure interpolation and mesh motion using radial basis functions. Int J Numer Methods Eng 74(10):1519–1559

    Article  MathSciNet  MATH  Google Scholar 

  • Rendall TCS, Allen CB (2009) Efficient mesh motion using radial basis functions with data reduction algorithms. J Comput Phys 228(17):6231–6249

    Article  MATH  Google Scholar 

  • Renzoni P, DAlascio A, Kroll N, Peshkin D, Hounjet MH, Boniface J, Vigevano L, Allen CB, Badcock KJ, Mottura L, Scholl E, Kokkalis A (2000) EROS—a common European Euler code for the analysis of the helicopter rotor flowfield. Prog Aerosp Sci 36(5):437–485

    Article  Google Scholar 

  • Reuther J, Jameson A, Farmer J, Martinelli L, Saunders D (1996) Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation. In: 34th aerospace sciences meeting and exhibit, Reno, NV. AIAA paper 96-0094

    Google Scholar 

  • Samareh JA (2001) Survey of shape parameterization techniques for high-fidelity multidisciplinary shape optimization. AIAA J 39(5):877–884

    Article  Google Scholar 

  • Smith RE, Bloor MIG, Wilson MJ, Thomas AT (1995) Rapid airplane parametric input design (rapid). In: Proceedings of 12th AIAA computational fluid dynamics conference. AIAA, Washington

    Google Scholar 

  • van Leer B (1982) Flux vector splitting for the Euler equations. In: Lecture notes in physics, vol 170, pp 507–512

    Google Scholar 

  • Watt A, Watt M (1992) Advanced animation and rendering techniques. Addison-Wesley, New York

    Google Scholar 

  • Wendland H (2005) Scattered data approximation, 1st edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Wong WS, Moigne AL, Qin N (2007) Parallel adjoint-based optimization of a blended wing body aircraft with shock control bumps. Aeronaut J 111(1117):165–174

    Google Scholar 

  • Zhou JL, Tits AL (1993) Nonmonotone line search for minimax problems. J Optim Theory Appl 76(3):455–476

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou JL, Tits AL, Lawrence CT (1997) Users guide for ffsqp version 3.7: a Fortran code for solving optimization programs, possibly minimax, with general inequality constraints and linear equality constraints, generating feasible iterates. Technical report SRC-TR-92-107r5, Institute for Systems Research, University of Maryland, College Park

Download references

Acknowledgements

The authors would like to thank Dr Asa Morris for his contribution to the development of the parallel optimizer.

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, University of Bristol, Bristol, BS8 1TR, UK

    Christian B. Allen & Thomas C. S. Rendall

Authors
  1. Christian B. Allen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas C. S. Rendall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian B. Allen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Allen, C.B., Rendall, T.C.S. CFD-based optimization of hovering rotors using radial basis functions for shape parameterization and mesh deformation. Optim Eng 14, 97–118 (2013). https://doi.org/10.1007/s11081-011-9179-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11081-011-9179-6

Keywords

Search

Navigation