Structural and Multidisciplinary Optimization

, Volume 48, Issue 5, pp 909–925 | Cite as

Isogeometric shape optimization in fluid mechanics

  • Peter Nørtoft
  • Jens Gravesen
Research Paper


The subject of this work is numerical shape optimization in fluid mechanics, based on isogeometric analysis. The generic goal is to design the shape of a 2-dimensional flow domain to minimize some prescribed objective while satisfying given geometric constraints. As part of the design problem, the steady-state, incompressible Navier-Stokes equations, governing a laminar flow in the domain, must be solved. Based on isogeometric analysis, we use B-splines as the basis for both the design optimization and the flow analysis, thereby unifying the models for geometry and analysis, and, at the same time, facilitating a compact representation of complex geometries and smooth approximations of the flow fields. To drive the shape optimization, we use a gradient-based approach, and to avoid inappropriate parametrizations during optimization, we regularize the optimization problem by adding to the objective function a measure of the quality of the boundary parametrization. A detailed description of the methodology is given, and three different numerical examples are considered, through which we investigate the effects of the regularization, of the number of geometric design variables, and of variations in the analysis resolution, initial design and Reynolds number, and thereby demonstrate the robustness of the methodology.


Shape optimization Isogeometric analysis Fluid mechanics Regularization Navier-Stokes equation Taylor-Couette flow Drag 



The authors would like to thank Allan Roulund Gersborg, Burmeister & Wain Energy A/S, Denmark, Thomas A. Grandine and Thomas A. Hogan, The Boeing Company, USA, and Mathias Stolpe, DTU Wind Energy, Denmark, for conceptual support and fruitful discussions during the course of this work. We also thank two anonymous reviewers for valuable comments on the manuscript.


  1. Akkerman I, Bazilevs Y, Calo VM, Hughes TJR, Hulshoff S (2010) The role of continuity in residual-based variational multiscale modeling of turbulence. Comput Mech 41:371–378MathSciNetCrossRefGoogle Scholar
  2. Azegami H, Fukumoto S, Aoyama T (2012) Shape optimization of contiua using NURBS as basis functions. Struct Multidisc Optim 47(2):247–258MathSciNetCrossRefGoogle Scholar
  3. Bazilevs Y, Hughes TJR (2008) NURBS-based isogeometric analysis for the computation of flows about rotating components. Comput Mech 43:143–150MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bletzinger KU, Firl M, Linhard J, Wüchner R (2010) Optimal shapes of mechanically motivated surfaces. Comput Methods Appl Mech Engrg 199:324–333CrossRefzbMATHGoogle Scholar
  5. Buffa A, de Falco C, Sangalli G (2011) IsoGeometric analysis: stable elements for the 2D Stokes equation. Int J Numer Meth Fluids 65:1407–1422CrossRefzbMATHGoogle Scholar
  6. Cho S, Ha SH (2009) Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidisc Optim 38:53–70MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cohen E, Martin T, Kirby RM, Lyche T, Riesenfeld RF (2010) Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. Comput Methods Appl Mech Engrg 199:334–356MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, ChichesterCrossRefGoogle Scholar
  9. Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, ChichesterCrossRefGoogle Scholar
  10. Gersborg–Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidisc Optim 30(3):181–192MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gill PE, Murray W, Saunders A (2008) User’s guide for SNOPT version 7: Software for large-scale nonlinear programming. Accessed 25 January, 2012
  12. Gravesen J, Evgrafov A, Gersborg AR, Nguyen DM, Nielsen PN (2010) Isogeometric analysis and shape optimisation. In: Eriksson A, Tibert G (eds) Proceedings of NSCM-23: the 23rd Nordic seminar on computational mechanics, pp 14–17Google Scholar
  13. Ha SH, Choi KK, Cho S (2010) Numerical method for shape optimization using T-spline based isogeometric method. Struct Multidisc Optim 42:417–428CrossRefzbMATHGoogle Scholar
  14. Hassani B, Khanzadi M, Tavakkoli SM (2012) An isogeometrical approach to structural topology optimization by optimality criteria. Struct Multidisc Optim 45:223–233MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Engrg 194:4135–4195MathSciNetCrossRefzbMATHGoogle Scholar
  16. Ivorra B, Hertzog DE, Mohammadi B, Santiago JS (2006) Semi-deterministic and genetic algorithms for global optimization of microfluidic protein-folding devices. Int J Numer Meth Eng 66:319–333MathSciNetCrossRefzbMATHGoogle Scholar
  17. Katamine E, Azegami H, Tsubata T, Itoh S (2005) Solution to shape optimization problems of viscous flow fields. Int J Comput Fluid D 19(1):45–51MathSciNetCrossRefGoogle Scholar
  18. Kim DW, Kim MU (1995) Minimum drag shape in two-dimensional viscous flow. Int J Numer Meth Fl 21:93–111CrossRefzbMATHGoogle Scholar
  19. Li K, Qian X (2011) Isogeometric analysis and shape optimization via boundary integral. Comput Aided Design 43:1427–1437CrossRefGoogle Scholar
  20. Mohammadi B, Pironneau O (2004) Shape optimization in fluid mechanics. Annu Rev Fluid Mech 36:255–279MathSciNetCrossRefGoogle Scholar
  21. Mohammadi B, Pironneau O (2010) Applied shape optimization for fluids, 2nd edn. Oxford University Press, New YorkGoogle Scholar
  22. Nagy AP, Abdalla MM, Gürdal Z (2010) Design of anisotropic composite shells using an isogeometric approach. In: Proceedings of the 13th AIAA/ISSMO multidisciplinary analysis optimization conference, Fort WorthGoogle Scholar
  23. Nagy AP, Abdalla MM, Gürdal Z (2010a) Isogeometric sizing and shape optimization of beam structures. Comput Methods Appl Mech Engrg 199:1216–1230MathSciNetCrossRefzbMATHGoogle Scholar
  24. Nagy AP, Abdalla MM, Gürdal Z (2010b) On the variational formulation of stress constraints in isogeometric design. Comput Methods Appl Mech Engrg 199:2687–2696MathSciNetCrossRefzbMATHGoogle Scholar
  25. Nagy AP, Abdalla MM, Gürdal Z (2011) Isogeometric design of elastic arches for maximum fundamental frequency. Struct Multidisc Optim 43:135–149CrossRefzbMATHGoogle Scholar
  26. Nguyen DM, Evgrafov A, Gersborg AR, Gravesen J (2011) Isogeometric shape optimization of vibrating membranes. Comput Methods Appl Mech Engrg 200:1343–1353MathSciNetCrossRefzbMATHGoogle Scholar
  27. Nielsen PN, Gersborg AR, Gravesen J, Pedersen NL (2011) Discretizations in isogeometric analysis of Navier-Stokes flow. Comput Methods Appl Mech Engrg 200:3242–3253MathSciNetCrossRefzbMATHGoogle Scholar
  28. Painchaud-Oullet S, Tribes C, Trepanier JY, Pelletier D (2006) Airfoil shape optimization using a nonuniform rational B-spline parametrization under thickness constraint. IAAA J 44(10):2170–2178Google Scholar
  29. Piegl L, Tiller W (1995) The NURBS Book. Springer-Verlag, Berlin Heidelberg New YorkCrossRefzbMATHGoogle Scholar
  30. Pironneau O (1973) On optimum profiles in stokes flow. J Fluid Mech 59:117–128MathSciNetCrossRefzbMATHGoogle Scholar
  31. Pironneau O (1974) On optimum design in fluid mechanics. J Fluid Mech 64:97–110MathSciNetCrossRefzbMATHGoogle Scholar
  32. Qian X (2010) Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput Methods Appl Mech Engrg 199:2059–2071MathSciNetCrossRefzbMATHGoogle Scholar
  33. Qian X, Sigmund O (2011) Isogeometric shape optimization of photonic crystals via Coons patches. Comput Methods Appl Mech Engrg 200:2237–2255MathSciNetCrossRefzbMATHGoogle Scholar
  34. Seo YD, Kim HJ, Youn SK (2010a) Isogeometric topology optimization using trimmed spline surfaces. Comput Methods Appl Mech Engrg 199:3270–3296MathSciNetCrossRefzbMATHGoogle Scholar
  35. Seo YD, Kim HJ, Youn SK (2010b) Shape optimization and its extension to topological design based on isogeometric analysis. Int J Solids Struct 47:1618–1640CrossRefzbMATHGoogle Scholar
  36. Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Engrg 197:2976–2988MathSciNetCrossRefzbMATHGoogle Scholar
  37. Xu G, Mourrain B, Duvigneau R, Galligo A (2011) Parameterization of computational domain in isogeometric analysis: methods and comparison. Comput Methods Appl Mech Engrg 200(2324):2021–2031MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.DTU ComputeTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations