Isogeometric analysis of the Cahn–Hilliard phase-field model

• Published in: Computer methods in applied mechanics and engineering Vol. 197; no. 49; pp. 4333 - 4352
• Main Authors: Gómez, Héctor, Calo, Victor M., Bazilevs, Yuri, Hughes, Thomas J.R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15-09-2008; Elsevier
• Subjects: Cahn–Hilliard; Classical and quantum physics: mechanics and fields; Classical mechanics of continuous media: general mathematical aspects; Condensed matter: structure, mechanical and thermal properties; Equations of state, phase equilibria, and phase transitions; Exact sciences and technology; Isogeometric analysis; Isoperimetric problem; NURBS; Phase-field; Physics; Specific phase transitions; Steady state solutions; Isogeometric analysis; Phase-field; Steady state solutions; Isoperimetric problem; Cahn–Hilliard; NURBS; Experimental study; Phase field method; Finite element method; Non uniform rational B spline; Cahn Hilliard equation; Phase separation; Smooth function; Variational calculus; Modelling; Differential operator; Robustness; Non linear effect; Cahn-Hilliard
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2008.05.003

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
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-continuous. There are a very limited number of two-dimensional finite elements possessing
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-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
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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.