Skip to main content

A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

Abstract

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.

This is a preview of subscription content, access via your institution.

References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford University Press, USA (2000)

    MATH  Google Scholar 

  2. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)

    MathSciNet  Article  Google Scholar 

  3. Bello, J. A., Fernándex-Cara, E., Lemoine, J., Simon, J.: The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier–Stokes flow. SIAM J. Control Optim. 35(2), 626–640 (1997)

    MathSciNet  Article  Google Scholar 

  4. Blank, L., Hecht, C., Garcke, H., Rupprecht, C.: Sharp interface limit for a phase field model in structural optimization. SIAM J. Control Optim. 54, 1558–1584 (2016)

    MathSciNet  Article  Google Scholar 

  5. Blank, L., Rupprecht, C.: An extension of the projected gradient method to a Banach space setting with application in structural topology optimization. SIAM J. Control Optim. 55, 1481–1499 (2017)

    MathSciNet  Article  Google Scholar 

  6. Boisgérault, S., Zolésio, J.: Shape derivative of sharp functionals governed by Navier–Stokes flow. In: Jäger, W., Nečas, J., John, O., Najzar, K., Stará, J. (eds.) Partial Differential Equations: Theory and Numerical Solution, pp 49–63. Chapman and Hall/CRC (1993)

  7. Borrvall, T., Petersson, J.: Topology optimization of fluids in Stokes flow. Internat. J. Numer. Methods Fluids 41(1), 77–107 (2003)

    MathSciNet  Article  Google Scholar 

  8. Bourdin, B., Chambolle, A.: Design-dependent loads in topology optimization. ESAIM Control Optim. Calc. Var. 9, 19–48 (2003)

    MathSciNet  Article  Google Scholar 

  9. Brandenburg, C., Lindemann, F., Ulbrich, M., Ulbrich, S.: A Continuous Adjoint Approach to Shape Optimization for Navier Stokes Flow. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds.) Optimal Control of Coupled Systems of Partial Differential Equations, International Series of Numerical Mathematics, vol. 158, pp 35–56. Birkhäuser (2009)

    Chapter  Google Scholar 

  10. Evans, L., Gariepy, R.: Measure theory and fine properties of functions. Studies in advanced mathematics. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  11. Garcke, H., Hecht, C.: Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow. ESAIM: Control Optim. Calc. Var. 22, 309–337 (2016)

    MathSciNet  Article  Google Scholar 

  12. Garcke, H., Hecht, C.: Shape and topology optimization in Stokes flow with a phase field approach. Appl. Math. Optim. 73, 23–70 (2016)

    MathSciNet  Article  Google Scholar 

  13. Garcke, H., Hecht, C., Hinze, M., Kahle, C.: Numerical approximation of phase field based shape and topology optimization for fluids. SIAM J. Sci. Comput. 37(4), A1846–A1871 (2015)

    MathSciNet  Article  Google Scholar 

  14. Garcke, H., Hecht, C., Hinze, M., Kahle, C., Lam, K.F.: Shape optimization for surface functionals in Navier–Stokes flow using a phase field approach. Interfaces Free Bound. 18(2), 219–261 (2016)

    MathSciNet  Article  Google Scholar 

  15. Giles, M., Larson, M., Levenstam, M., Süli, E.: Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow. Technical Report NA-79/06 Oxford University Computing Laboratory (1997)

  16. Giusti, E.: Minimal surfaces and functions of bounded variation, Monographs in mathematics, vol. 80. Birkhäuser, Basel (1984)

    Book  Google Scholar 

  17. Goldberg, H., Kampowsky, W., Tröltzsch, F.: On Nemytskij operators in L p-spaces of abstract functions. Math. Nachr. 155, 127–140 (1992)

    MathSciNet  Article  Google Scholar 

  18. Hecht, C.: Shape and topology optimization in fluids using a phase field approach and an application in structural optimization. Ph.D. thesis, University of Regensburg (2014)

  19. Hintermüller, M., Hinze, M., Kahle, C.: An adaptive finite element Moreau–Yosida-based solver for a coupled Cahn–Hilliard/Navier–Stokes system. J. Comput. Phys. 235, 810–827 (2013)

    MathSciNet  Article  Google Scholar 

  20. Hintermüller, M., Hinze, M., Kahle, C., Keil, T.: A goal-oriented dual-weighted adaptive finite elements approach for the optimal control of a Cahn–Hilliard–Navier–Stokes system. Preprint Hamburger Beiträge zur Angewandten Mathematik 2016-25 (2016)

  21. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constraints. Mathematical Modelling: Theory and Applications. Springer, Netherlands (2009)

    MATH  Google Scholar 

  22. Hoffman, J., Johnson, C.: Adaptive finite element methods for incompressible fluid flow. In: Barth, T., Deconinck, H. (eds.) Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, vol. 25, pp 95–157. Springer, Berlin Heidelberg (2003)

    Chapter  Google Scholar 

  23. Kawohl, B., Pironneay, O., Tartar, L., Zolesio, J. P.: Optimal Shape Design: Lectures Given at the Joint C.I.M./C.I.M.E. Summer School Held in Troia (Portugal), June 1-6, 1998. Lecture Notes in Mathematics / C.I.M.E. Foundation Subseries. Springer-Verlag, Berlin Heidelberg (2000)

    Book  Google Scholar 

  24. Kondoh, T., Matsumori, T., Kawamoto, A.: Drag minimization and lift maximization in laminar flows via topology optimization employing simple objective function expressions based on body force integration. Struct. Multidiscip. Optim. 45 (5), 693–701 (2012)

    MathSciNet  Article  Google Scholar 

  25. Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98(2), 123–142 (1987)

    MathSciNet  Article  Google Scholar 

  26. Murat, F.: Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients. Ann. Mat. Pura Appl., Serie 4 112(1), 49–68 (1977)

    MathSciNet  Article  Google Scholar 

  27. Penzler, P., Rumpf, M., Wirth, B.: A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: Control Optim. Calc. Var. 18, 229–258 (2012)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  29. Plotnikov, P., Sokolowski, J.: Shape derivative of drag functional. SIAM J. Control Optim. 48(7), 4680–4706 (2010)

    MathSciNet  Article  Google Scholar 

  30. Robinson, S.: Stability theorems for systems of inequalities, Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13(4), 497–513 (1976)

    MathSciNet  Article  Google Scholar 

  31. Schmidt, S., Schulz, V.: Shape derivatives for general objective functions and the incompressible Navier–Stokes equations. Control Cybernet. 39(3), 677–713 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Simon, J.: Domain variation for drag in Stokes flow. In: Control Theory of Distributed Parameter Systems and Applications, Lecture Notes in Control and Information Sciences, vol. 159. Springer, Berlin (1991)

  33. Sohr, H.: The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts. Springer-Verlag, Berlin (2001)

    MATH  Google Scholar 

  34. Sturm, K., Hintermüller, M., Hömberg, D.: Distortion compensation as a shape optimization problem for a sharp interface model. Comput. Optim. Appl. 64, 557–588 (2016)

    MathSciNet  Article  Google Scholar 

  35. Takezawa, A., Nishiwaki, S., Kitamura, M.: Shape and topology optimization based on the phase field method and sensitivity analysis. J. Comput. Phys. 229, 2697–2718 (2010)

    MathSciNet  Article  Google Scholar 

  36. Tartar, L.: Problemes de Controle des Coefficients Dans des Equations aux Derivees Partielles. In: Bensoussan, A., Lions, J. (eds.) Control Theory, Numerical Methods and Computer Systems Modelling, Lecture Notes in Economics and Mathematical Systems, vol. 107, pp 420–426. Springer, Berlin Heidelberg (1975)

    Google Scholar 

  37. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate studies in mathematics. AMS, Providence (2010)

    Book  Google Scholar 

  38. Wang, M., Zhou, S.: Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33, 89–111 (2007)

    MathSciNet  Article  Google Scholar 

  39. Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in banach spaces. Appl. Math. Optim. 5, 49–62 (1979)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) through the grants GA695/6-2 (first and fourth author) and HI689/7-1 (second and third author) within the priority program SPP1506 “Transport processes at fluidic interfaces”. The third author additionally gratefully acknowledges the support by the DFG through the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Harald Garcke.

Additional information

Communicated by: Jon Wilkening

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Garcke, H., Hinze, M., Kahle, C. et al. A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints. Adv Comput Math 44, 1345–1383 (2018). https://doi.org/10.1007/s10444-018-9586-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-018-9586-8

Keywords

  • Topology optimization
  • Shape optimization
  • Phase field approach
  • Navier–Stokes flow
  • Integral state constraints

Mathematics Subject Classification (2010)

  • 35Q35
  • 35Q56
  • 35R35
  • 49Q10
  • 49Q12
  • 65M22
  • 65M60
  • 76S05