Advertisement

Allen-Cahn and Cahn-Hilliard Variational Inequalities Solved with Optimization Techniques

  • Luise BlankEmail author
  • Martin Butz
  • Harald Garcke
  • Lavinia Sarbu
  • Vanessa Styles
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 160)

Abstract

Parabolic variational inequalities of Allen-Cahn and Cahn-Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We show several numerical computations also involving systems of parabolic variational inequalities.

Keywords

Optimization Phase-field method Allen-Cahn model Cahn-Hilliard model variational inequalities active set methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Banas and R. Nürnberg, A multigrid method for the Cahn-Hilliard equation with obstacle potential. Appl. Math. Comput. 213 (2009), 290–303.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    J.W. Barrett, H. Garcke, and R. Nürnberg, On sharp interface limits of Allen-Cahn/Cahn-Hilliard variational inequalities. Discrete Contin. Dyn. Syst. S 1 (1) (2008), 1–14.zbMATHGoogle Scholar
  3. 3.
    L. Blank, H. Garcke, L. Sarbu, T. Srisupattarawanit, V. Styles, and A. Voigt, Phase-field approaches to structural topology optimization, a contribution in this book.Google Scholar
  4. 4.
    L. Blank, M. Butz, and H. Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method, ESAIM: Control, Optimization and Calculus of Variations, DOI: 10.1051/COCV/2010032.Google Scholar
  5. 5.
    L. Blank, H. Garcke, L. Sarbu, and V. Styles, Primal-dual active set methods for Allen-Cahn variational inequalities with non-local constraints. DFG priority program 1253 “Optimization with PDEs”, Preprint No. 1253-09-01.Google Scholar
  6. 6.
    L. Blank, H. Garcke, L. Sarbu, and V. Styles, Non-local Allen-Cahn systems: Analysis and a primal dual active set method. Preprint No. 02/2011, Universität Regensburg.Google Scholar
  7. 7.
    L. Blank, L. Sarbu, and M. Stoll, Preconditioning for Allen-Cahn variational inequalities with non-local constraints. Preprint Nr.11/2010, Universität Regensburg, Mathematik.Google Scholar
  8. 8.
    J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy I. Mathematical analysis. European J. Appl. Math. 2, no. 3 (1991), 233–280.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy II. Mathematical analysis. European J. Appl. Math. 3, no. 2 (1991), 147–179.CrossRefMathSciNetGoogle Scholar
  10. 10.
    J.F. Blowey and C.M. Elliott, A phase-field model with a double obstacle potential. Motion by mean curvature and related topis (Trento, 1992), 1–22, de Gruyter, Berlin 1994.Google Scholar
  11. 11.
    L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation. Arch. Rat. Mech. Anal. 124 (1993), 355–379.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    L.Q. Chen, Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32 (2001), 113–140.CrossRefGoogle Scholar
  13. 13.
    H. Garcke, B. Nestler, B. Stinner, and F. Wendler, Allen-Cahn systems with volume constraints. Mathematical Models and Methods in Applied Sciences 18, no. 8 (2008), 1347–1381.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    H. Garcke, B. Nestler, and B. Stoth. Anisotropy in multi-phase systems: a phase field approach. Interfaces Free Bound. 1 (1999), 175–198.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    H. Garcke and V. Styles. Bi-directional diffusion induced grain boundary motion with triple junctions. Interfaces Free Bound. 6, no. 3 (2004), 271–294.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    C. Gräser and R. Kornhuber, On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints. Domain decomposition methods in science and engineering XVI, 91–102, Lect. Notes Comput. Sci. Engl. 55, Springer, Berlin 2007.Google Scholar
  17. 17.
    M. Hintermüller, K. Ito, and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), no. 3 (2003), 865–888 (electronic).CrossRefMathSciNetGoogle Scholar
  18. 18.
    M. Hintermüller, M. Hinze, and M.H. Tber, An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem. Matheon preprint Nr. 670, Berlin (2009), to appear in Optimization Methods and Software.Google Scholar
  19. 19.
    K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind. M2AN Math. Model. Numer. Anal. 37, no. 1 (2003), 41–62.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    R. Kornhuber and R. Krause. On multigrid methods for vector-valued Allen-Cahn equations with obstacle potential. Domain decomposition methods in science and engineering, 307–314 (electronic), Natl. Auton. Univ. Mex., México, 2003.Google Scholar
  21. 21.
    P.-L. Lions and B. Mercier, Splitting algorithms for the sum of two non-linear operators. SIAM J. Numer. Anal. 16 (1979), 964–979.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    C. Meyer, A. Rösch, and F. Tröltzsch, Optimal control problems of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, (2006), 209–228.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    I. Neitzel and F. Tröltzsch, On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. ESAIM: COCV 15 (2) (2009), 426–453.CrossRefzbMATHGoogle Scholar
  24. 24.
    A. Schmidt and K.G. Siebert, Design of adaptive finite element software. The finite element toolbar. ALBERTA, Lecture Notes in Computational Science and Engineering 42, Springer-Verlag, Berlin 2005.Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Luise Blank
    • 1
    Email author
  • Martin Butz
    • 1
  • Harald Garcke
    • 1
  • Lavinia Sarbu
    • 2
  • Vanessa Styles
    • 2
  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Department of MathematicsUniversity of SussexBrightonUK

Personalised recommendations