Isogeometric analysis of the Cahn-Hilliard phase-field model

Héctor Gómez*, Victor M. Calo, Yuri Bazilevs, Thomas J.R. Hughes

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

523 Scopus citations


The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions are not common because primal variational formulations of fourth-order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally C1-continuous. There are a very limited number of two-dimensional finite elements possessing C1-continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of C1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

Original languageEnglish (US)
Pages (from-to)4333-4352
Number of pages20
JournalComputer Methods in Applied Mechanics and Engineering
Issue number49-50
StatePublished - Sep 15 2008
Externally publishedYes


  • Cahn-Hilliard
  • Isogeometric analysis
  • Isoperimetric problem
  • Phase-field
  • Steady state solutions

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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