Triangulation-based isogeometric analysis of the Cahn–Hilliard phase-field model

Abstract This paper presents triangulation-based Isogeometric Analysis of the Cahn–Hilliard phase-field model. The Cahn–Hilliard phase-field model is governed by a time-dependent fourth-order partial differential equation. The corresponding primal variational form involves second-order operators, making it difficult to be directly analyzed with traditional C 0 finite element analysis. In this paper, we construct C 1 Bernstein–Bezier simplicial elements through macro-element techniques, including various triangle-split based macro-elements in both 2D and 3D space. We extend triangulation-based isogeometric analysis to solving the primal variational form of the Cahn–Hilliard equation. We validate our method by convergence analysis, showing the nodal and degree-of-freedom advantages over C 0 Finite Element Analysis. We then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two-dimensional squares and three-dimensional cubes. We incorporate an adaptive time-stepping scheme in these numerical experiments. Our numerical study demonstrates that triangulation-based isogeometric analysis offers optimal convergence and time step stability, is applicable to complex geometry and allows local refinement.

[1]  John A. Evans,et al.  Isogeometric finite element data structures based on Bézier extraction of NURBS , 2011 .

[2]  Harald Garcke,et al.  Finite Element Approximation of the Cahn-Hilliard Equation with Degenerate Mobility , 1999, SIAM J. Numer. Anal..

[3]  Seunggyu Lee,et al.  Basic Principles and Practical Applications of the Cahn–Hilliard Equation , 2016 .

[4]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[5]  Thomas J. R. Hughes,et al.  Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology , 2012, Comput. Aided Des..

[6]  Xiaoping Qian,et al.  Kirchhoff–Love shell formulation based on triangular isogeometric analysis , 2019, Computer Methods in Applied Mechanics and Engineering.

[7]  Songtao Xia,et al.  Isogeometric shape optimization on triangulations , 2018 .

[8]  P. Roache.  Code Verification by the Method of Manufactured Solutions , 2002 .

[9]  Markus Kästner,et al.  Isogeometric analysis of the Cahn-Hilliard equation - a convergence study , 2016, J. Comput. Phys..

[10]  David Andrs,et al.  A quantitative comparison between C0 and C1 elements for solving the Cahn-Hilliard equation , 2013, J. Comput. Phys..

[11]  Ju Liu,et al.  Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow , 2013, J. Comput. Phys..

[12]  Peter Alfeld,et al.  Bivariate spline spaces and minimal determining sets , 2000 .

[13]  G. Hulbert,et al.  A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method , 2000 .

[14]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[15]  Peter Sternberg,et al.  Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problems , 2006 .

[16]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[17]  Yinnian He,et al.  On large time-stepping methods for the Cahn--Hilliard equation , 2007 .

[18]  Yan Xu,et al.  Local discontinuous Galerkin methods for the Cahn-Hilliard type equations , 2007, J. Comput. Phys..

[19]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

[20]  Larry L. Schumaker,et al.  Spline functions on triangulations , 2007, Encyclopedia of mathematics and its applications.

[21]  Paul Steinmann,et al.  Comparative computational analysis of the Cahn-Hilliard equation with emphasis on C1-continuous methods , 2016, J. Comput. Phys..

[22]  Anders Logg,et al.  The FEniCS Project Version 1.5 , 2015 .

[23]  Ming-Jun Lai,et al.  A Cr trivariate macro-element based on the Alfeld split of tetrahedra , 2013, J. Approx. Theory.

[24]  John E. Hilliard,et al.  Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 1959 .

[25]  Krishna Garikipati,et al.  A discontinuous Galerkin method for the Cahn-Hilliard equation , 2006, J. Comput. Phys..

[26]  Hendrik Speleers,et al.  A normalized basis for reduced Clough-Tocher splines , 2010, Comput. Aided Geom. Des..

[27]  Olga Wodo,et al.  Computationally efficient solution to the Cahn-Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem , 2011, J. Comput. Phys..

[28]  Michael Andreas Matt.  Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness , 2012 .

[29]  Xiaoping Qian,et al.  Isogeometric analysis on triangulations , 2014, Comput. Aided Des..

[30]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[31]  Hong Dong,et al.  Spaces of bivariate spline functions over triangulation , 1991 .

[32]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[33]  Xiaoping Qian,et al.  Isogeometric analysis with Bézier tetrahedra , 2017 .

[34]  Viktor Blåsjö,et al.  The Isoperimetric Problem , 2005, Am. Math. Mon..

[35]  T. Hughes,et al.  Solid T-spline construction from boundary representations for genus-zero geometry , 2012 .

[36]  Xiaoping Qian,et al.  Continuity and convergence in rational triangular Bézier spline based isogeometric analysis , 2015 .