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MARINE 2011, IV International Conference on Computational Methods in Marine Engineering pp 85–102Cite as

Shape-Newton Method for Isogeometric Discretizations of Free-Boundary Problems

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Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 29))

Abstract

We derive Newton-type solution algorithms for a Bernoulli-type free-boundary problem at the continuous level. The Newton schemes are obtained by applying Hadamard shape derivatives to a suitable weak formulation of the free-boundary problem. At each Newton iteration, an updated free boundary position is obtained by solving a boundary-value problem at the current approximate domain. Since the boundary-value problem has a curvature-dependent boundary condition, an ideal discretization is provided by isogeometric analysis. Several numerical examples demonstrate the apparent quadratic convergence of the Newton schemes on isogeometric-analysis discretizations with C 1-continuous discrete free boundaries.

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Notes

  1. 1.

    For domain-map linearization (linearization in the reference domain), domain-perturbations have a nonlocal effect, and this nonlocality depends on the particular domain map chosen [31, 32]).

  2. 2.

    We note that this derivation has been employed previously to obtain the linearized-adjoint operator; see [32].

References

  1. Alt HW, Caffarelli LA (1981) Existence and regularity for a minimum problem with free boundary. J Reine Angew Math 325:105–144

    MathSciNet  MATH  Google Scholar 

  2. Borden MJ, Scott MA, Evans JA, Hughes TJR (2011) Isogeometric finite element data structures based on Bézier extraction of NURBS. Int J Numer Methods Eng 87:15–47

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouchon F, Clain S, Touzani R (2005) Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput Methods Appl Mech Eng 194:3934–3948

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchon F, Clain S, Touzani R (2008) A perturbation method for the numerical solution of the Bernoulli problem. J Comput Math 26:23–36

    MathSciNet  Google Scholar 

  5. Cho S, Ha S-H (2009) Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidiscip Optim 38:53–70

    Article  MathSciNet  Google Scholar 

  6. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, Chichester

    Book  Google Scholar 

  7. Cox MG (1972) The numerical evaluation of B-splines. IMA J Appl Math 10:134–149

    Article  MATH  Google Scholar 

  8. Crank J (1987) Free and moving boundary problems. Oxford University Press, London

    MATH  Google Scholar 

  9. Cuvelier C, Schulkes RMSM (1990) Some numerical methods for the computation of capillary free boundaries governed by the Navier–Stokes equations. SIAM Rev 32:355–423

    Article  MathSciNet  MATH  Google Scholar 

  10. de Boor C (1972) On calculating with B-splines. J Approx Theory 6:50–62

    Article  MATH  Google Scholar 

  11. Delfour MC, Zolésio J-P (2001) Shapes and geometries: analysis, differential calculus, and optimization. SIAM series on advances in design and control, vol 4. Society for Industrial and Applied Mathematics, Philadelphia

    MATH  Google Scholar 

  12. Eppler K, Harbrecht H (2006) Efficient treatment of stationary free boundary problems. Appl Numer Math 56:1326–1339

    Article  MathSciNet  MATH  Google Scholar 

  13. Flucher M, Rumpf M (1997) Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J Reine Angew Math 486:165–204

    MathSciNet  MATH  Google Scholar 

  14. Harbrecht H (2008) A Newton method for Bernoulli’s free boundary problem in three dimensions. Computing 82:11–30

    Article  MathSciNet  MATH  Google Scholar 

  15. Haslinger J, Kozubek T, Kunisch K, Peichl G (2003) Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. Comput Optim Appl 26:231–251

    Article  MathSciNet  MATH  Google Scholar 

  16. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  MATH  Google Scholar 

  17. Kärkkäinen KT, Tiihonen T (1999) Free surfaces: shape sensitivity analysis and numerical methods. Int J Numer Methods Eng 44:1079–1098

    Article  MATH  Google Scholar 

  18. Kärkkäinen KT, Tiihonen T (2004) Shape calculus and free boundary problems. In: Neittaanmäki P, Rossi T, Korotov S, Oñate E, Périaux J, Knörzer D (eds) European Congress on computational methods in applied sciences and engineering, ECCOMAS 2004, Jyväskylä, 24–28 July 2004

    Google Scholar 

  19. Kiendl J, Bletzinger K-U, Linhard J, Wüchner R (2009) Isogeometric shell analysis with Kirchhoff–Love elements. Comput Methods Appl Mech Eng 198:3902–3914

    Article  MATH  Google Scholar 

  20. Kuster CM, Gremaud PA, Touzani R (2007) Fast numerical methods for Bernoulli free boundary problems. SIAM J Sci Comput 29:622–634

    Article  MathSciNet  MATH  Google Scholar 

  21. Mejak G (1994) Numerical solution of Bernoulli-type free boundary value problems by variable domain method. Int J Numer Methods Eng 37:4219–4245

    Article  MATH  Google Scholar 

  22. Mejak G (1997) Finite element solution of a model free surface problem by the optimal shape design approach. Comput Methods Appl Mech Eng 40:1525–1550

    MathSciNet  Google Scholar 

  23. Scott MA, Borden MJ, Verhoosel CV, Sederberg TW, Hughes TJR (2011) Isogeometric finite element data structures based on Bézier extraction of T-splines. Int J Numer Meth Eng 88

    Google Scholar 

  24. Sokołowski J, Zolésio J-P (1992) Introduction to shape optimization: shape sensitivity analysis. Springer series in computational mathematics, vol 16. Springer, Berlin

    Book  MATH  Google Scholar 

  25. Tiihonen T (1998) Fixed point methods for internal free boundary problems. Numer Funct Anal Optim 19:399–413

    Article  MathSciNet  MATH  Google Scholar 

  26. Toivanen JI, Haslinger J, Mäkinen RAE (2008) Shape optimization of systems governed by Bernoulli free boundary problems. Comput Methods Appl Mech Eng 197:3802–3815

    Article  Google Scholar 

  27. Tsai W-T, Yue DKP (1996) Computation of nonlinear free-surface flows. Annu Rev Fluid Mech 28:249–278

    Article  MathSciNet  Google Scholar 

  28. van Brummelen EH (2002) Numerical methods for steady viscous free-surface flows. PhD thesis, Universiteit van Amsterdam, Amsterdam, The Netherlands

    Google Scholar 

  29. van Brummelen EH, Raven HC, Koren B (2001) Efficient numerical solution of steady free-surface Navier–Stokes flow. J Comput Phys 174:120–137

    Article  MathSciNet  MATH  Google Scholar 

  30. van Brummelen EH, Segal A (2003) Numerical solution of steady free-surface flows by the adjoint optimal shape design method. Int J Numer Methods Fluids 41:3–27

    Article  MATH  Google Scholar 

  31. van der Zee KG, van Brummelen EH, de Borst R (2010) Goal-oriented error estimation and adaptivity for free-boundary problems: the domain-map linearization approach. SIAM J Sci Comput 32:1064–1092

    Article  MathSciNet  MATH  Google Scholar 

  32. van der Zee KG, van Brummelen EH, de Borst R (2010) Goal-oriented error estimation and adaptivity for free-boundary problems: the shape linearization approach. SIAM J Sci Comput 32:1093–1118

    Article  MathSciNet  MATH  Google Scholar 

  33. van der Zee KG, Verhoosel CV (2011) Isogeometric analysis-based goal-oriented error estimation for free-boundary problems. Finite Elem Anal Des 47:600–609

    Article  MathSciNet  Google Scholar 

  34. Wackers J, Koren B, Raven H, van der Ploeg A, Starke A, Deng G, Queutey P, Visonneau M, Hino T, Ohashi K (2011) Free-surface viscous flow solution methods for ship hydrodynamics. Arch Comput Methods Eng 18:1–42

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by NanoNextNL, a micro and nanotechnology consortium of the Government of the Netherlands and 130 partners. The research of K.G. van der Zee and C.V. Verhoosel is funded by the Netherlands Organisation for Scientific Research (NWO), VENI scheme.

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Authors and Affiliations

  1. Multiscale Engineering Fluid Dynamics, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    K. G. van der Zee, G. J. van Zwieten, C. V. Verhoosel & E. H. van Brummelen

Authors
  1. K. G. van der Zee
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  2. G. J. van Zwieten
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  3. C. V. Verhoosel
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  4. E. H. van Brummelen
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Corresponding author

Correspondence to K. G. van der Zee .

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Editors and Affiliations

  1. Mechanical Engineering, Instituto Superior Técnico, Avenida Rovisco Pais 1, Lisbon, 1049 001, Portugal

    Luís Eça

  2. CIMNE, Technical University of Catalonia (UPC), Gran Capitan, s/n, Barcelona, 08034, Spain

    Eugenio Oñate

  3. Universitat Politècnica de Catalunya, EDIFICI NT1 FNB-CIMNE PLAÇA PALAU 18, S/N, Barcelona, 08003, Spain

    Julio García-Espinosa

  4. Norwegian Univ. of Science & Technology, Alfred Getz vei 1, Trondheim, 7491, Norway

    Trond Kvamsdal

  5. Norwegian Univ. of Science & Technology, Alfred Getz vei 1, Trondheim, 7491, Norway

    Pål Bergan

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van der Zee, K.G., van Zwieten, G.J., Verhoosel, C.V., van Brummelen, E.H. (2013). Shape-Newton Method for Isogeometric Discretizations of Free-Boundary Problems. In: Eça, L., Oñate, E., García-Espinosa, J., Kvamsdal, T., Bergan, P. (eds) MARINE 2011, IV International Conference on Computational Methods in Marine Engineering. Computational Methods in Applied Sciences, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6143-8_5

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  • DOI: https://doi.org/10.1007/978-94-007-6143-8_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-6142-1

  • Online ISBN: 978-94-007-6143-8

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