# zbMATH — the first resource for mathematics

Isogeometric analysis of the Cahn-Hilliard phase-field model. (English) Zbl 1194.74524
Summary: 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 $$\mathcal C^1$$-continuous. There are a very limited number of two-dimensional finite elements possessing $$\mathcal C^1$$-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 $$\mathcal C^1$$ 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.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74N20 Dynamics of phase boundaries in solids
Full Text:
##### References:
 [1] Abels, H.; Wilke, M., Convergence to equilibrium for the cahn – hilliard equation with a logarithmic free energy, Nonlinear anal.: theory methods appl., 67, 3176-3193, (2007) · Zbl 1121.35018 [2] Anderson, D.M.; McFadden, G.B.; Wheeler, A.A., Diffuse-interface methods in fluid mechanics, Annu. rev. fluid mech., 30, 139-165, (1998) · Zbl 1398.76051 [3] Akkerman, I.; Bazilevs, Y.; Calo, V.M.; Hughes, T.J.R.; Hulshoff, S., The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. mech., 41, 371-378, (2007) · Zbl 1162.76355 [4] Barret, J.W.; Blowey, J.F.; Garcke, H., Finite element approximation of the cahn – hilliard equation with degenerate mobility, SIAM J. numer. anal., 37, 286-318, (1999) · Zbl 0947.65109 [5] Barrett, J.W.; Garcke, H.; Nurnberg, R., Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid, Math. comput., 75, 7-41, (2006) · Zbl 1078.74050 [6] Batchelor, G.K., An introduction to fluid dynamics, (1967), Cambridge University Press · Zbl 0152.44402 [7] Bazilevs, Y.; Calo, V.M.; Cottrell, J.A.; Hughes, T.J.R.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. methods appl. mech. engrg., 197, 173-201, (2007) · Zbl 1169.76352 [8] Benderskaya, G.; Clemens, M.; De Gersem, H.; Weiland, T., Embedded runge – kutta methods for field-circuit coupled problems with switching elements, IEEE trans. magn., 41, 1612-1615, (2005) [9] Blowey, J.F.; Elliott, C.M., The cahn – hilliard gradient theory for phase separation with non-smooth free energy. part II: numerical analysis, Eur. J. appl. math., 3, 147-149, (1992) · Zbl 0810.35158 [10] Caginalp, G., Stefan and hele – shaw type models as asymptotic limits of the phase field equations, Phys. rev. A, 39, 5887-5896, (1989) · Zbl 1027.80505 [11] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961) [12] Cahn, J.W.; Hilliard, J.E., Free energy of a non-uniform system. I. interfacial free energy, J. chem. phys., 28, 258-267, (1958) [13] Cahn, J.W.; Hilliard, J.E., Free energy of a non-uniform system. III. nucleation in a two-component incompressible fluid, J. chem. phys., 31, 688-699, (1959) [14] Ceniceros, H.D.; Roma, A.M., A nonstiff adaptive mesh refinement-based method for the cahn – hilliard equation, J. comput. phys., 225, 1849-1862, (2007) · Zbl 1343.65109 [15] Chella, R.; Viñals, J., Mixing of a two-phase fluid by cavity flow, Phys. rev. E, 53, 3832-3840, (1996) [16] Chen, L.Q., Phase-field models for microstructural evolution, Annu. rev. mater. res., 32, 113-140, (2002) [17] Choksi, R.; Sternberg, P., Periodic phase separation: the periodic cahn – hilliard and the isoperimetric problems, Interfaces free bound., 8, 371-392, (2006) · Zbl 1109.35092 [18] Chung, J.; Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-$$\alpha$$ method, J. appl. mech., 60, 371-375, (1993) · Zbl 0775.73337 [19] Copetti, M.I.M.; Elliott, C.M., Numerical analysis of the cahn – hilliard equation with a logarithmic free energy, Numer. math., 63, 39-65, (1992) · Zbl 0762.65074 [20] Cottrell, J.A.; Hughes, T.J.R.; Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Comput. methods appl. mech. engrg., 196, 4160-4183, (2007) · Zbl 1173.74407 [21] Crank, J., Free and moving boundary problems, (1997), Oxford University Press [22] Debussche, A.; Dettori, L., On the cahn – hilliard equation with a logarithmic free energy, Nonlinear anal., 24, 1491-1514, (1995) · Zbl 0831.35088 [23] Dolcetta, I.C.; Vita, S.F.; March, R., Area preserving curve-shortening flows: from phase separation to image processing, Interfaces free bound., 4, 325-343, (2002) · Zbl 1021.35129 [24] Du, Q.; Nicolaides, R.A., Numerical analysis of a continuum model of phase transition, SIAM J. numer. anal., 28, 1310-1322, (1991) · Zbl 0744.65089 [25] Elliott, C.M.; French, D.A., Numerical studies of the cahn – hilliard equation for phase separation, IMA J. appl. math., 38, 97-128, (1987) · Zbl 0632.65113 [26] Elliott, C.M.; French, D.A.; Milner, F.A., A 2nd-order splitting method for the cahn – hilliard equation, Numer. math., 54, 575-590, (1989) · Zbl 0668.65097 [27] Elliott, C.M.; Garcke, H., On the cahn – hilliard equation with degenerate mobility, SIAM J. math. anal., 27, 404-423, (1996) · Zbl 0856.35071 [28] Elliott, C.M.; Zheng, S., On the cahn – hilliard equation, Arch. ration. mech. anal., 96, 339-357, (1986) · Zbl 0624.35048 [29] Engel, G.; Garikipati, K.; Hughes, T.J.R.; Larson, M.G.; Mazzei, L.; Taylor, R.L., Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. methods appl. mech. engrg., 191, 3669-3750, (2002) · Zbl 1086.74038 [30] Fan, D.N.; Chen, L.Q., Diffuse interface description of grain boundary motion, Philos. mag. lett., 75, 187-196, (1997) [31] Feng, X.; Prohl, A., Analysis of a fully discrete finite element method for phase field model and approximation of its sharp interface limits, Math. comput., 73, 541-547, (2003) [32] Fonseca, I.; Morini, M., Surfactants in foam stability: a phase field model, Arch. ration. mech., 183, 411-456, (2007) · Zbl 1107.76076 [33] H.B. Frieboes, J.P. Sinek, S. Sanga, F. Gentile, A. Granaldi, P. Decuzzi, C. Cosentino, F. Amato, M. Ferrari, V. Cristini, Towards multiscale modeling of nanovectored delivery of therapeutics to cancerous lesions, Biomed. Dev. in press. [34] Fried, E.; Gurtin, M.E., Dynamic solid – solid transitions with phase characterized by an order parameter, Physica D, 72, 287-308, (1994) · Zbl 0812.35164 [35] Furihata, D., A stable and conservative finite difference scheme for the cahn – hilliard equation, Numer. math., 87, 675-699, (2001) · Zbl 0974.65086 [36] Garcke, H.; Nestler, B., A mathematical model for grain growth in thin metallic films, Math. models methods appl. sci., 10, 895-921, (2000) · Zbl 1205.74138 [37] Garcke, H.; Nestler, B.; Stinner, B., A diffuse interface model for alloys with multiple components and phases, SIAM J. appl. math., 64, 775-799, (2004) · Zbl 1126.82025 [38] Garcke, H.; Nestler, B.; Stoth, B., On anisotropic order parameter models for multiphase systems and their sharp interface limits, Physica D, 115, 87-108, (1998) · Zbl 0936.82010 [39] Garcke, H.; Niethammer, B.; Rumpf, M.; Weikard, U., Transient coarsening behaviour in the cahn – hilliard model, Acta mater., 51, 2823-2830, (2003) [40] Garcke, H.; Novick-Cohen, A., A singular limit for a system of degenerate cahn – hilliard equations, Adv. differ. eq., 5, 401-434, (2000) · Zbl 0988.35019 [41] Gustafsson, K., Control-theoretic techniques for stepsize selection in implicit runge – kutta methods, ACM trans. math. software, 20, 496-517, (1994) · Zbl 0888.65096 [42] Hauswirth, L.; Pérez, J.; Romon, P.; Ros, A., The periodic isoperimetric problem, Trans. AMS, 356, 2025-2047, (2004) · Zbl 1046.52002 [43] He, Y.; Liu, Y.; Tang, T., On large time-stepping methods for the cahn – hilliard equation, Appl. numer. math., 57, 616-628, (2007) · Zbl 1118.65109 [44] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origin of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044 [45] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications Mineola, NY [46] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. methods appl. mech. engrg., 194, 4135-4195, (2005) · Zbl 1151.74419 [47] Hughes, T.J.R.; Feijóo, G.R.; Mazzei, L.; Quincy, J.-B., The variational multiscale method — a paradigm for computational mechanics, Comput. methods appl. mech. engrg., 166, 3-24, (1998) · Zbl 1017.65525 [48] Jansen, K.E.; Whiting, C.H.; Hulbert, G.M., A generalized-$$\alpha$$ method for integrating the filtered navier – stokes equations with a stabilized finite element method, Comput. methods appl. mech. engrg., 190, 305-319, (1999) · Zbl 0973.76048 [49] Jeong, J.-H.; Goldenfeld, N.; Dantzig, J.A., Phase field model for three-dimensional dendritic growth with fluid flow, Phys. rev. E, 64, 1-14, (2001) [50] Jingxue, Y.; Changchun, L., Regularity of solutions of the cahn – hilliard equation with concentration dependent mobility, Nonlinear anal., 45, 543-554, (2001) · Zbl 0984.35072 [51] Kay, D.; Welford, R., A multigrid finite element solver for the cahn – hilliard equation, J. comput. phys., 212, 288-304, (2006) · Zbl 1081.65091 [52] Kim, J.; Kang, K.; Lowengrub, J., Conservative multigrid methods for cahn – hilliard fluids, J. comput. phys., 193, 511-543, (2004) · Zbl 1109.76348 [53] Kim, Y.-T.; Provatas, N.; Goldenfeld, N.; Dantzig, J.A., Universal dynamics of phase field models for dendritic growth, Phys. rev. E, 59, 2546-2549, (1999) [54] Knuth, D.E., The art of computing programming, vol. 2, (1997), Addison-Wesley · Zbl 0191.17903 [55] D.J. Korteweg, Sur la forme que prenent les équations du mouvements des fluides si l’on tient compte des forces capilaires causées par des variations de densité considérables mains continues et sur la théorie de la capillarité dans l’hypothése d’une varation continue de la densité, Arch. Néerl Sci. Exactes Nat. Ser. II 6 (1901) 124. · JFM 32.0756.02 [56] Landau, L.D.; Ginzburg, V.I., On the theory of superconductivity, (), 626-633 [57] Lang, J., Two-dimensional fully adaptive solutions of reaction-diffusion equations, Appl. numer. math., 18, 223-240, (1995) · Zbl 0846.65044 [58] Lee, H.-G.; Lowengrub, J.S.; Goodman, J., Modeling pinchoff and reconnection in a hele – shaw cell. II. analysis and simulation in the nonlinear regime, Phys. fluids, 14, 514-545, (2002) · Zbl 1184.76317 [59] Lighthill, J., Waves in fluids, (1978), Cambridge University Press · Zbl 0375.76001 [60] Liu, C.; Shen, J., A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179, 211-228, (2003) · Zbl 1092.76069 [61] Lowengrub, J.; Truskinovsky, L., Quasi-incompressible cahn – hilliard fluids and topological transitions, Proc. roy. soc. London ser. A, 454, 2617-2654, (1998) · Zbl 0927.76007 [62] Modica, L., The gradient theory of phase transitions and the minimal interface criterion, Arch. ration. mech. anal., 98, 123-142, (1987) · Zbl 0616.76004 [63] Ostwald, W., Lehrbruck der allgemeinen chemie, 2, (1896) [64] Penrose, O.; Fife, P.C., Thermodynamically consistent models of phase field type for the kinetics of phase transition, Physica D, 43, 44-62, (1990) · Zbl 0709.76001 [65] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 856-869, (1986) · Zbl 0599.65018 [66] Sederberg, T.N.; Zheng, J.M.; Bakenov, A.; Nasri, A., T-splines and T-nurccss, ACM trans. graphics, 22, 477-484, (2003) [67] Sternberg, P., The effect of singular perturbation on nonconvex variational problems, Arch. ration. mech. anal., 101, 209-260, (1988) · Zbl 0647.49021 [68] Stogner, R.H.; Carey, G.F., $$C^1$$ macroelements in adaptive finite element methods, Int. J. numer. methods engrg., 70, 1076-1095, (2007) · Zbl 1194.76141 [69] Sun, Z.Z., A second order accurate linearized difference scheme for the two-dimensional cahn – hilliard equation, Math. comput., 64, 1463-1471, (1995) · Zbl 0847.65056 [70] Toral, R.; Chakrabarti, A.; Gunton, J.D., Large scale simulations of the two-dimensional cahn – hilliard model, Physica A, 213, 41-49, (1995) [71] Tremaine, S., On the origin of irregular structure in saturn’s rings, Astron. J., 125, 894-901, (2003) [72] Udaykumar, H.S.; Mittal, R.; Shyy, W., Computation of solid – liquid phase fronts in the sharp interface limit on fixed grids, J. comput. phys., 153, 535-574, (1999) · Zbl 0953.76071 [73] van der Houwen, P.J.; Sommeijer, B.P.; Couzy, W., Embedded diagonally implicit runge – kutta algorithms on parallel computers, Math. comput., 58, 135-159, (1992) · Zbl 0744.65050 [74] van der Waals, J.D., The thermodynamics theory of capillarity under the hypothesis of a continuous variation of density, J. stat. phys., 20, 197-244, (1979) [75] Vollmayr-Lee, B.P.; Rutemberg, A.D., Fast and accurate coarsening simulation with an unconditionally stable time step, Phys. rev. E, 68, 066703, (2003) [76] Wells, G.N.; Kuhl, E.; Garikipati, K., A discontinuous Galerkin method for the cahn – hilliard equation, J. comput. phys., 218, 860-877, (2006) · Zbl 1106.65086 [77] Xia, Y.; Xu, Y.; Shu, C.-W., Local discontinuous Galerkin methods for the cahn – hilliard type equations, J. comput. phys., 227, 472-491, (2007) · Zbl 1131.65088 [78] Ye, X.; Cheng, X., The Fourier spectral method for the cahn – hilliard equation, Appl. math. comput., 171, 345-357, (2005) · Zbl 1093.65099 [79] J. Zhu, L.-Q. Chen, J. Shen, V. Tikare, Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier-spectral method, Phys. Rev. E 60, 3564-3572.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.