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

  • K. G. van der Zee
  • G. J. van Zwieten
  • C. V. Verhoosel
  • E. H. van Brummelen
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 29)


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.


Newton-type methods Bernoulli free-boundary problem Shape derivative Isogeometric analysis Smooth discrete boundaries Shape-linearized free-boundary problem 



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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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • K. G. van der Zee
    • 1
  • G. J. van Zwieten
    • 1
  • C. V. Verhoosel
    • 1
  • E. H. van Brummelen
    • 1
  1. 1.Multiscale Engineering Fluid DynamicsEindhoven University of TechnologyEindhovenThe Netherlands

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