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Journal of Engineering Mathematics

, Volume 80, Issue 1, pp 173–188 | Cite as

Topology optimization in Bernoulli free boundary problems

  • Jukka I. Toivanen
  • Raino A. E. Mäkinen
  • Jaroslav Haslinger
Article

Abstract

In this work we consider the topology optimization of systems governed by the external Bernoulli free boundary problem arising, for example, from the mathematical modelling of electro-chemical machining. In this work we combine, for the first time, the so-called pseudo-solid approach to the solution of governing free boundary problems and the level set method, which is used to define the design domain. Previous studies of the problem showed a tendency towards topological changes in the design, which can now automatically take place thanks to level set parametrization. The scalar function used in the level set method is parametrized using radial basis functions, converting the problem into a parametric optimization problem, which is solved using a gradient-based method.

Keywords

Bernoulli problem Electro-chemical machining Level set method Shape optimization 

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References

  1. 1.
    Nilson RH, Tsuei YG (1974) Inverted Cauchy problem for the Laplace equation in engineering design. J Eng Math 8: 329–337MATHCrossRefGoogle Scholar
  2. 2.
    Patil SS, Yadava V (2007) Finite element prediction of tool shapes in electro-chemical machining. Int J Des Eng 1: 21–40Google Scholar
  3. 3.
    Toivanen JI, Haslinger J, Mäkinen RAE (2008) Shape optimization of systems governed by Bernoulli free boundary problems. Comput Methods Appl Mech Eng 197: 3803–3815ADSMATHCrossRefGoogle Scholar
  4. 4.
    Flucher M, Rumpf M (1997) Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J Für die reine Angew Math 486: 165–204MathSciNetMATHGoogle Scholar
  5. 5.
    Kärkkäinen K, Tiihonen T (1999) Free surfaces: shape sensitivity analysis and numerical methods. Int J Numer Methods Eng 44(8): 1079–1098MATHCrossRefGoogle Scholar
  6. 6.
    Cuvelier C, Schulkes RMSM (1990) Some numerical methods for the computation of capillary free boundaries governed by the Navier–Stokes equations. SIAM Rev 32(3): 355–423MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Haslinger J, Mäkinen RAE (2003) Introduction to shape optimization: theory, approximation, and computation. SIAM, PhiladelphiaMATHCrossRefGoogle Scholar
  8. 8.
    Cairncross RA, Schunk PR, Baer TA, Rao RR, Sackinger PA (2000) A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion. Int J Numer Methods Fluids 33: 375–403MATHCrossRefGoogle Scholar
  9. 9.
    Souli M, Zolesio JP (2001) Arbitrary Lagrangian–Eulerian and free surface methods in fluid mechanics. Comput Methods Appl Mech Eng 191: 451–466ADSMATHCrossRefGoogle Scholar
  10. 10.
    Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79(1): 12–49MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. Springer, New YorkMATHGoogle Scholar
  12. 12.
    Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and level-set methods. J Comput Phys 194: 363–393MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Wang S, Wang MY (2006) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65: 2060–2090MATHCrossRefGoogle Scholar
  14. 14.
    Chen J, Shapiro V, Suresh K, Tsukanov I (2007) Shape optimization with topological changes and parametric control. Int J Numer Methods Eng 71(3): 313–346MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Belytschko T, Xiao SP, Parimi C (2003) Topology optimization with implicit functions and regularization. Int J Numer Methods Eng 57: 1177–1196MATHCrossRefGoogle Scholar
  16. 16.
    Neittaanmäki P, Pennanen A, Tiba D (2009) Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions. Inverse Probl 25(5): 055003ADSCrossRefGoogle Scholar
  17. 17.
    Luo Z, Wang MY, Wang S, Wei P (2008) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76: 1–26MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Wendland H (2005) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4(1): 389–396MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bischof CH, Khademi PM, Bouaricha A, Carle A (1996) Efficient computation of gradients and Jacobians by dynamic exploitation of sparsity in automatic differentiation. Optim Methods Softw 7: 1–39CrossRefGoogle Scholar
  20. 20.
    Griewank A, Walther A (2008) Evaluating derivatives: principles and techniques of algorithmic differentiation. 2. SIAM, PhiladelphiaMATHCrossRefGoogle Scholar
  21. 21.
    Toivanen JI, Mäkinen RAE (2011) Implementation of sparse forward mode automatic differentiation with application to electromagnetic shape optimization. Optim Methods Softw 26: 601–616MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Geuzaine C, Remacle J-F (2009) Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Eng 79: 1309–1331MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Spellucci P (1998) An SQP method for general nonlinear programs using only equality constrained subproblems. Math Program 82:413–448. http://www.mathematik.tu-darmstadt.de/fbereiche/numerik/staff/spellucci/DONLP2/. Retrieved September 7, 2012Google Scholar
  24. 24.
    Demmel JW, Eisenstat SC, Gilbert JR, Li XS, Liu JWH (1999) A supernodal approach to sparse partial pivoting. SIAM J Matrix Anal Appl 30: 720–755MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Jukka I. Toivanen
    • 1
  • Raino A. E. Mäkinen
    • 1
  • Jaroslav Haslinger
    • 2
  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  2. 2.Department of Numerical Mathematics, Faculty of Mathematics and PhysicsCharles University PraguePrague 8Czech Republic

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