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A Heuristic Method Using Hessian Matrix for Fast Flow Topology Optimization

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Abstract

We propose a heuristic optimization method for a density-based fluid topology optimization using a Hessian matrix. In flow topology optimization, many researches use a gradient-based method. Convergence rate of a gradient method is linear, which means slow convergence near the optimal solution. For faster convergence, we utilize a Hessian matrix toward the end of the optimization procedure. In the present paper, we formulate a fluid optimization problem using the lattice Boltzmann method and heuristically solve the optimization problem with using an approximated sensitivity. In the formulation of a Hessian matrix, we use a heuristic trick in order to formulate it as a diagonal matrix. By the heuristics, the computation cost is decreased drastically. The validity of the method is studied via numerical examples.

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References

  1. Bendsøe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods and Applications. Springer, Berlin (2003)

    MATH  Google Scholar 

  2. Borrval, T., Peterson, J.: Topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Fluids 41(1), 77–107 (2003)

    Article  MathSciNet  Google Scholar 

  3. Aage, N., Poulsen, T.H., Hansen, A.G., Sigmund, O.: Topology optimization of large scale Stokes flow problems. Struct. Multidiscip. Optim. 35(2), 175–180 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Challis, V.J., Guest, J.K.: Level set topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Eng. 79(10), 1284–1308 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guest, J.K., Précost, J.H.: Topology optimization of creeping fluid flows using a Darcy–Stokes finite element. Int. J. Numer. Methods Eng. 66(3), 461–484 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bajaj, N., Subbarayan, G., Garimella, S.V.: Topological design of channels for squeeze flow optimization of thermal interface materials. Int. J. Heat Mass Transf. 55(13–14), 3375–3560 (2012)

    Google Scholar 

  7. Hansen, A.G., Sigmund, O., Haber, R.B.: Topology optimization of channel flow problem. Struct. Multidiscip. Optim. 30(3), 181–192 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Pingen, G., Maute, K.: Optimal design for non-Newtonian flows using a topology optimization approach. Comput. Math. Appl. 59(7), 2340–2350 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kreissl, S., Pingen, G., Evgrafov, A., Maute, K.: Topology optimization of flexible micro-fluidic devices. Struct. Multidiscip. Optim. 42(4), 495–516 (2010)

    Article  Google Scholar 

  10. Yaji, K., Yamada, T., Yoshino, M., Matsumoto, T., Izui, K., Nishiwaki, S.: Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions. J. Comput. Phys. 274, 158–181 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yonekura, K., Kanno, Y.: Topology optimization method for interior flow based on transient information of the lattice Boltzmann method with a level-set function. Jpn. J. Ind. Appl. Math. 34, 611–632 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Vicente, W.M., Picelli, R., Pavanello, R., Xie, Y.M.: Topology optimization of frequency responses of fluid-structure interaction systems. Finite Elem. Anal. Des. 98, 1–13 (2015)

    Article  Google Scholar 

  13. Yonekura, K., Kanno, Y.: A flow topology optimization method for steady state flow using transient information of flow field solved by lattice Boltzmann method. Struct. Multidiscip. Optim. 51(1), 159–172 (2015)

    Article  MathSciNet  Google Scholar 

  14. Yonekura, K., Kanno, Y.: Erratum to: A flow topology optimization method for steady state flow using transient information of flow field solved by lattice Boltzmann method. Struct. Multidiscip. Optim. 54(1), 193–195 (2016)

    Article  MathSciNet  Google Scholar 

  15. Özkaya, E., Gauger, N.: Single-step one-shot aerodynamic shape optimization. In: Kunisch, K., Sprekels, J., Leugering, G., Troltzsch, F. (eds.) Optimal Control of Coupled Systems of Partial Differential Equations, vol. 158, pp. 191–204. Birkhäuser Basel, Basel (2009)

    Chapter  Google Scholar 

  16. Barthold, F.J., Stander, N., Stein, E.: Performance comparison of SAM and SQP methods for structural shape optimization. Struct. Optim. 11(2), 102–112 (1996)

    Article  Google Scholar 

  17. Succi, S.: The Lattice Boltzmann Equation: for Fluid Dynamics and Beyond. Oxford University Press, Oxford (1992)

    MATH  Google Scholar 

  18. Bhatnager, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases, I: small amplitude processes in charged and neutral one-component system. Phys. Rev. 94(3), 511–525 (1954)

    Article  MATH  Google Scholar 

  19. Guo, Z., Zhao, T.S.: Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev. E 66, 036304 (2002)

    Article  Google Scholar 

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Acknowledgements

The study of the second author is partially supported by JSPS KAKENHI 17K06633.

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Authors and Affiliations

  1. IHI Corporation, Yokohama, Japan

    Kazuo Yonekura

  2. University of Tokyo, Bunkyo-ku, Japan

    Yoshihiro Kanno

Authors
  1. Kazuo Yonekura
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  2. Yoshihiro Kanno
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Correspondence to Kazuo Yonekura.

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Ilio Galligani.

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Yonekura, K., Kanno, Y. A Heuristic Method Using Hessian Matrix for Fast Flow Topology Optimization. J Optim Theory Appl 180, 671–681 (2019). https://doi.org/10.1007/s10957-018-1404-4

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  • DOI: https://doi.org/10.1007/s10957-018-1404-4

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