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Aerodynamic Shape Optimization Using “Turbulent” Adjoint And Robust Design in Fluid Mechanics

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Engineering and Applied Sciences Optimization

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 38))

Abstract

This article presents adjoint methods for the computation of the first- and higher-order derivatives of objective functions \(F\) used in optimization problems governed by the Navier–Stokes equations in aero/hydrodynamics. The first part of the chapter summarizes developments and findings related to the application of the continuous adjoint method to turbulence models, such as the Spalart-Allmaras and k-\(\varepsilon \) ones, in either their low- or high-Reynolds number (with wall functions) variants. Differentiating the turbulence model, over and above to the differentiation of the mean–flow equations, leads to the computation of the exact gradient of \(F\), by overcoming the frequently made assumption of neglecting turbulence variations. The second part deals with higher-order sensitivity analysis based on the combined use of the adjoint approach and the direct differentiation of the governing PDEs. In robust design problems, the so-called second-moment approach requires the computation of second-order derivatives of \(F\) with respect to (w.r.t.) the environmental or uncertain variables; in addition, any gradient-based optimization algorithm requires third-order mixed derivatives w.r.t. both the environmental and design variables; various ways to compute them are discussed and the most efficient is adopted. The equivalence of the continuous and discrete adjoint for this type of computations is demonstrated. In the last part, some other relevant recent achievements regarding the adjoint approach are discussed. Finally, using the aforementioned adjoint methods, industrial geometries are optimized. The application domain includes both incompressible or compressible fluid flow applications.

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References

  1. P. Spalart and S. Allmaras: A one-equation turbulence model for aerodynamic flows. AIAA Paper, 04(39), 1992

    Google Scholar 

  2. Launder BE, Sharma BI (1974) Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett Heat Mass Transf 1:131–137

    Article  Google Scholar 

  3. Lee BJ, Kim C (2007) Automated design methodology of turbulent internal flow using discrete adjoint formulation. Aerosp Sci Technol 11:163–173

    Article  MATH  Google Scholar 

  4. Mavriplis DJ (2007) Discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes. AIAA J 45:740–750

    Article  Google Scholar 

  5. Pironneau O (1974) On optimum design in fluid mechanics. J Fluid Mech 64:97–110

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Anderson WK, Venkatakrishnan V (1997) Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation. AIAA Paper 97–0643

    Google Scholar 

  8. Papadimitriou DI, Giannakoglou KC (2007) A continuous adjoint method with objective function derivatives based on boundary integrals for inviscid and viscous flows. J Comput Fluids 36:325–341

    Article  MATH  Google Scholar 

  9. Othmer C (2008) A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Int J Numer Meth Fluids 58:861–877

    Article  MATH  MathSciNet  Google Scholar 

  10. Zymaris AS, Papadimitriou DI, Giannakoglou KC, Othmer C (2009) Continuous adjoint approach to the Spalart-Allmaras turbulence model for incompressible flows. Comput Fluids 38:1528–1538

    Article  MATH  Google Scholar 

  11. Bueno-Orovio A, Castro C, Palacios F, Zuazua E (2012) Continuous adjoint approach for the SpalartAllmaras model in aerodynamic optimization. AIAA J 50:631–646

    Article  Google Scholar 

  12. Zymaris AS, Papadimitriou DI, Giannakoglou KC, Othmer C (2010) Adjoint wall functions: a new concept for use in aerodynamic shape optimization. J Comput Phys 229:5228–5245

    Article  MATH  MathSciNet  Google Scholar 

  13. Papoutsis-Kiachagias EM, Zymaris AS, Kavvadias IS, Papadimitriou DI, Giannakoglou KC (2014) The continuous adjoint approach to the k-\(\epsilon \) turbulence model for shape optimization and optimal active control of turbulent flows. Eng Optim. doi:10.1080/0305215X.2014.892595

  14. Asouti VG, Trompoukis XS, Kampolis IC, Giannakoglou C (2011) Unsteady CFD computations using vertex-centered finite volumes for unstructured grids on graphics processing units. Int J Numer Meth Fluids 67(2):232–246

    Article  MATH  MathSciNet  Google Scholar 

  15. Trompoukis XS, Tsiakas KT, Ghavami Nejad M, Asouti VG Giannakoglou KC (2014) The continuous adjoint method on graphics processing units for compressible flows. In: OPT-i, international conference on engineering and applied sciences optimization, Kos Island, Greece, 4–6 June 2014

    Google Scholar 

  16. Kavvadias IS, Papoutsis-Kiachagias EM, Dimitrakopoulos G, Giannakoglou KC (2014) The continuous adjoint approach to the k\(\omega \) SST turbulence model with applications in shape optimization. Eng Optim. doi:10.1080/0305215X.2014.979816

  17. Kavvadias IS, Karpouzas GK, Papoutsis-Kiachagias EM, Papadimitriou DI, Giannakoglou KC (2013) Optimal flow control and topology optimization using the continuous adjoint method in unsteady flows. In: EUROGEN conference 2013, Las Palmas de Gran Canaria, Spain

    Google Scholar 

  18. Griewank A, Walther A (2000) Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans Math Softw 26(1):19–45

    Article  MATH  Google Scholar 

  19. Vezyris CK, Kavvadias IS, Papoutsis-Kiachagias EM, Giannakoglou KC (2014) Unsteady continuous adjoint method using POD for jet-based flow control. In: ECCOMAS conference 2014, Barcelona, Spain

    Google Scholar 

  20. Papoutsis-Kiachagias EM, Kontoleontos EA, Zymaris AS, Papadimitriou DI, Giannakoglou KC (2011) Constrained topology optimization for laminar and turbulent flows, including heat transfer. In: EUROGEN Conference 2011, Capua, Italy

    Google Scholar 

  21. Papadimitriou DI, Giannakoglou KC (2008) Computation of the Hessian matrix in aerodynamic inverse design using continuous adjoint formulations. Comput Fluids 37:1029–1039

    Article  MATH  Google Scholar 

  22. Papadimitriou DI, Giannakoglou KC (2008) The continuous direct-adjoint approach for second-order sensitivities in viscous aerodynamic inverse design problems. Comput Fluids 38:1539–1548

    Article  Google Scholar 

  23. Papadimitriou DI, Giannakoglou KC (2008) Aerodynamic shape optimization using adjoint and direct approaches. Arch Comput Meth Eng 15:447–488

    Article  MATH  MathSciNet  Google Scholar 

  24. Papoutsis-Kiachagias EM, Papadimitriou DI, Giannakoglou KC (2012) Robust design in aerodynamics using third-order sensitivity analysis based on discrete adjoint. Application to quasi-1D flows. Int J Numer Meth Fluids 69:691–709

    Article  MathSciNet  Google Scholar 

  25. Papadimitriou DI, Giannakoglou KC (2013) Third-order sensitivity analysis for robust aerodynamic design using continuous adjoint. Int J Numer Meth Fluids 71:652–670

    Article  MathSciNet  Google Scholar 

  26. Papoutsis-Kiachagias EM, Papadimitriou DI, Giannakoglou KC (2011) On the optimal use of adjoint methods in aerodynamic robust design problems. In: CFD and OPTIMIZATION, (2011) ECCOMAS thematic conference. Antalya, Turkey 2011

    Google Scholar 

  27. Papadimitriou DI, Giannakoglou KC (2012) Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach. Int J NumerMeth Fluids 68:724–739

    Article  MATH  Google Scholar 

  28. Papadimitriou DI, Giannakoglou KC (2006) Compressor blade optimization using a continuous adjoint formulation. In: ASME TURBO EXPO, GT2006/90466, Barcelona, Spain, 8–11 May 2006

    Google Scholar 

  29. Papoutsis-Kiachagias EM, Kyriacou SA, Giannakoglou KC (2014) The continuous adjoint method for the design of hydraulic turbomachines. Comput Meth Appl Mech Eng 276:621–639

    Article  MathSciNet  Google Scholar 

  30. Othmer C, Papoutsis-Kiachagias E, Haliskos K (2011) CFD Optimization via sensitivity-based shape morphing. In: 4th ANSA & \(\mu \)ETA international conference, Thessaloniki, Greece, 2011

    Google Scholar 

  31. Menter FR (1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA 32(8):269–289

    Google Scholar 

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Acknowledgments

Parts of the research related to the exact differentiation of the turbulence models were funded by Volkswagen AG (Group Research, K-EFFG/V, Wolfsburg, Germany) and Icon Technology and Process Consulting Ltd. The support of Volkswagen AG for the development of the adjoint method for flow control applications is also acknowledged.

Research related to topology optimization was partially supported by a Basic Research Project funded by the National Technical University of Athens.

The authors would like to acknowledge contributions to topology optimization from Dr. E Kontoleontos and to “turbulent” adjoint by Dr. A. Zymaris. They are also indebted to Dr. Carsten Othmer, Volkswagen AG (Group Research, K-EFFG/V), for his support, some interesting discussions on the continuous adjoint method and his contributions in several parts of this work.

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Authors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Kyriakos C. Giannakoglou, Dimitrios I. Papadimitriou, Evangelos M. Papoutsis-Kiachagias & Ioannis S. Kavvadias

Authors
  1. Kyriakos C. Giannakoglou
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  2. Dimitrios I. Papadimitriou
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  3. Evangelos M. Papoutsis-Kiachagias
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  4. Ioannis S. Kavvadias
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Correspondence to Kyriakos C. Giannakoglou .

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Editors and Affiliations

  1. Institute of Structural Analysis & Seismic Research, National Technical University of Athens, Athens, Greece

    Nikos D. Lagaros

  2. Institute of Structural Analysis & Antiseismic Research, National Technical University of Athens, Athens, Greece

    Manolis Papadrakakis

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Giannakoglou, K.C., Papadimitriou, D.I., Papoutsis-Kiachagias, E.M., Kavvadias, I.S. (2015). Aerodynamic Shape Optimization Using “Turbulent” Adjoint And Robust Design in Fluid Mechanics. In: Lagaros, N., Papadrakakis, M. (eds) Engineering and Applied Sciences Optimization. Computational Methods in Applied Sciences, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-18320-6_16

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  • DOI: https://doi.org/10.1007/978-3-319-18320-6_16

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