Computational Mechanics

, Volume 56, Issue 4, pp 691–708

An enhanced finite element technique for diffuse phase transition

Original Paper

Abstract

We propose a finite element technique to enhance phase-field simulations. As adaptive p-method it and can be generally applied to finite element formulations. However, diffuse interfaces have non-linear gradients within regions typically smaller compared to the size of the overall model. Thus, enhanced field interpolation with higher polynomial functions on demand allows for coarser meshing or lower regularization length for the phase transition. Our method preserves \(C^0\) continuity of finite elements and is particularly advantageous in the context of parallelized computing. An analytical solution for the evolution of a phase-field variable governed by the Allen–Cahn equation is used to define an error measure and to investigate the proposed method. Several examples demonstrate the capability of this finite element technique.

Keywords

Phase-field modeling Phase transition Adaptive p-method Enhanced finite element method 

References

  1. 1.
    Boettinger WJ, Warren JA, Beckermann C, Karma A (2002) Phase-field simulation of solidification. Annu Mater Res 32:163–194CrossRefGoogle Scholar
  2. 2.
    Münch I, Krauß M, Wagner W, Kamlah M (2012) Ferroelectric nanogenerators coupled to an electric circuit for energy harvesting. Smart Mater Struct 21:115026-1–115026-8CrossRefGoogle Scholar
  3. 3.
    Schmitt R, Müller R, Kuhn C, Urbassek HM (2013) A phase field approach for multivariant martensitic transformations of stable and metastable phases. Arch Appl Mech 83:849–859MATHCrossRefGoogle Scholar
  4. 4.
    Rodney D, Le Bouar Y, Finel A (2003) Phase field methods and dislocations. Acta Mater 51:17–30CrossRefGoogle Scholar
  5. 5.
    Wang YU, Jin YM, Cuitino AM, Khachaturyan AG (2001) Phase field microelasticity theory and modeling of multiple dislocation dynamics. Appl Phys Lett 78:2324–2326CrossRefGoogle Scholar
  6. 6.
    Henry H, Levine H (2004) Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett 93:105504-1–105504-4CrossRefGoogle Scholar
  7. 7.
    Miehe C, Welschinger F, Hofacker M (2004) Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. Int J Numer Methods Eng 83:1273–1311MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kuhn C, Müller R (2004) A new finite element technique for a phase field model of brittle fractur. Int J Numer Methods Eng 83:1273–1311Google Scholar
  9. 9.
    Hesch C, Weinberg K (2014) Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture. Int J Numer Methods Eng 99:906–924MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kazaryan A, Wang Y, Patton BR (1999) Generalized phase field approach for computer simulation of sintering: incorporation of rigid-body motion. Scr Mater 41(5):487–492CrossRefGoogle Scholar
  11. 11.
    Wang YU (2006) Computer modeling and simulation of solid-state sintering: a phase field approach. Acta Mater 54:953–961CrossRefGoogle Scholar
  12. 12.
    Du Q, Liu C, Wang X (2006) Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J Comput Phys 212:757–777MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Biben T, Kassner K, Misbah C (2005) Phase-field approach to three-dimensional vesicle dynamics. Phys Rev E 72:041921-1–041921-15CrossRefGoogle Scholar
  14. 14.
    Chen L-Q (2002) Phase-field models for microstructure evolution. Annu Rev Mater Res 32:113–140CrossRefGoogle Scholar
  15. 15.
    Moelans N, Blanpain B, Wollants P (2008) An introduction to phase-field modeling of microstructure evolution. Comput Coupling Phase Diag Thermochem 32:268–294CrossRefGoogle Scholar
  16. 16.
    Provatas N, Goldenfeld N, Dantzig J (1998) Efficient computation of dendritic microstructures using adaptive mesh refinement. Phys. Rev. Lett. 80(15):3308–3311CrossRefGoogle Scholar
  17. 17.
    Bourdin B, Chambolle A (2000) Implementation of an adaptive finite-element approximation of the Mumford-Sah functional. Numer Math 85(4):609–646MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Welschinger F, Hofacker M, Miehe C (2010) Configurational-force-based adaptive fe solver for a phase field model of fracture. Proc Appl Math Mech 10:689–692CrossRefGoogle Scholar
  19. 19.
    Li R (2005) On multi-mesh H-adaptive methods. J Sci Comput 24(3):321–341MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Hu X, Li R, Tang T (2009) A multi-mesh adaptive finite element approximation to phase field models. Commun Comput Phys 5(5):1012–1029MathSciNetGoogle Scholar
  21. 21.
    Taylor RL, Zienkiewicz OC, Onate E (1998) A hierarchical finite element method based on the partition of unity. Comput Methods Appl Mech Eng 152:73–84MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Falk F (1983) Ginzburg-Landau theory of static domain walls in shape-memory alloys. Z Phys B 51:177–185CrossRefGoogle Scholar
  23. 23.
    Gurtin ME (1996) Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92:178–192MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Cahn JW, Allen SM (1977) A microscopic theory of domain wall motion and its experimental verification in Fe–Al alloy domain growth kinetics. J Phys Colloques 38(C7):51–54CrossRefGoogle Scholar
  25. 25.
    Wang J, Kamlah M, Zhang T-Y (2009) Phase field simulations of ferroelectric nanoparticles with different long-range-electrostatic and -elastic interactions. J Appl Phys 105:014104-1–014104-8Google Scholar
  26. 26.
    Padilla J, Zhong W, Vanderbilt D (1996) First-principle investigation of \(180^{\circ }\) domain walls in BaTiO\(_3\). Phys Rev B 5310:5969–5973CrossRefGoogle Scholar
  27. 27.
    Zienkiewicz OC, De JP, Gago SR, Kelly DW (1983) The hierarchical concept in finite element analysis. Comput Struct 16(1–4):53–65MATHCrossRefGoogle Scholar
  28. 28.
    Peano A, Rodin EY (1976) Hierarchies of conforming finite elements for plane elasticity and plate bending. Comput Math Appl 2:211–224MATHCrossRefGoogle Scholar
  29. 29.
    Babuska I, Szabo BA, Katz IN (1981) The p-version of the finite element method. SIAM J Numer Anal 18(3):515–545MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Methods Eng 10:1211–1219MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute for Structural AnalysisKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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