Advertisement

NKS Method for the Implicit Solution of a Coupled Allen-Cahn/Cahn-Hilliard System

  • Chao Yang
  • Xiao-Chuan Cai
  • David E. Keyes
  • Michael Pernice
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 98)

Abstract

The coupled Allen-Cahn/Cahn-Hilliard system consists of high-order partial differential equations that make explicit methods hard to apply due to the severe restriction on the time step size. In order to relax the restriction and obtain steady-state solution(s) in an efficient way, we use a fully implicit method for the coupled system and employ a Newton-Krylov-Schwarz algorithm to solve the nonlinear algebraic equations arising at each time step. In the Schwarz preconditioner we impose low-order homogeneous boundary conditions for subdomain problems. We investigate several choices of subdomain solvers as well as different overlaps. Numerical experiments on a supercomputer with thousands of processor cores are provided to show the scalability of the fully implicit solver.

Keywords

Time Step Size Processor Core Newton Iteration Explicit Method Inexact Newton Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported in part by DE-FC02-06ER25784. The first author also received supports from NSFC under 61170075, 91130023 and 61120106005, and from 973 Program of China under 2011CB309701.

References

  1. 1.
    Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual. Technical Report ANL-95/11 - Revision 3.3, Argonne National Laboratory (2012)Google Scholar
  2. 2.
    Barrett, J.W., Blowey, J.F.: Finite element approximation of a degenerate Allen-Cahn/Cahn-Hilliard system. SIAM J. Numer. Anal. 39, 1598–1624 (2002)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cahn, J., Novick-Cohen, A.: Evolution equations for phase separation and ordering in binary alloys. J. Stat. Phys. 76, 877–909 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cai, X.-C., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 792–797 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gropp, W.D., Kaushik, D.K., Keyes, D.E., Smith, B.: Performance modeling and tuning of an unstructured mesh CFD application. In: Proceedings of Supercomputing 2000. IEEE Computer Society, Washington (2000)Google Scholar
  6. 6.
    Millett, P.C., Rokkam, S., El-Azab, A., Tonks, M., Wolf, D.: Void nucleation and growth in irradiated polycrystalline metals: a phase-field model. Model. Simul. Mater. Sci. Eng. 17, 064003 (2009)CrossRefGoogle Scholar
  7. 7.
    Mulder, W.A., Leer, B.V.: Experiments with implicit upwind methods for the Euler equations. J. Comput. Phys. 59, 232–246 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Rokkam, S., El-Azab, A., Millett, P., Wolf, D.: Phase field modeling of void nucleation and growth in irradiated metals. Model. Simul. Mater. Sci. Eng. 17, 064002 (2009)CrossRefGoogle Scholar
  9. 9.
    Tonks, M.R., Gaston, D., Millett, P.C., Andrs, D., Talbot, P.: An object-oriented finite element framework for multiphysics phase field simulations. Comput. Mater. Sci. 51, 20–29 (2012)CrossRefGoogle Scholar
  10. 10.
    Wang, L., Lee, J., Anitescu, M., Azab, A.E., Mcinnes, L.C., Munson, T., Smith, B.: A differential variational inequality approach for the simulation of heterogeneous materials. In: Proceedings of SciDAC 2011 (2011)Google Scholar
  11. 11.
    Xia, Y., Xu, Y., Shu, C.W.: Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. Commun. Comput. Phys. 5, 821–835 (2009)MathSciNetGoogle Scholar
  12. 12.
    Yang, C., Cai, X.-C., Keyes, D.E., Pernice, M.: Parallel domain decomposition methods for the 3D Cahn-Hilliard equation. In: Proceedings of SciDAC 2011 (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Chao Yang
    • 1
    • 2
  • Xiao-Chuan Cai
    • 3
  • David E. Keyes
    • 4
  • Michael Pernice
    • 5
  1. 1.Institute of SoftwareChinese Academy of SciencesBeijingChina
  2. 2.State Key Laboratory of Computer ScienceChinese Academy of SciencesBeijingChina
  3. 3.Department of Computer ScienceUniversity of Colorado BoulderBoulderUSA
  4. 4.CEMSE DivisionKing Abdullah University of Science and TechnologyThuwalSaudi Arabia
  5. 5.Idaho National LaboratoryIdaho FallsUSA

Personalised recommendations