Accurate, efficient, and (iso)geometrically flexible collocation methods for phase-field models

Journal of Computational Physics - Tập 262 - Trang 153-171 - 2014
Hector Gomez1, Alessandro Reali2, Giancarlo Sangalli3
1University of A Coruña, Department of Mathematical Methods, Campus de Elviña, s/n, 15192, A Coruña, Spain
2University of Pavia, Department of Civil Engineering and Architecture, Via Ferrata 3, 27100 Pavia, Italy
3University of Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata, 1, 27100, Pavia, Italy

Tài liệu tham khảo

Akkerman, 2007, The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech., 41, 371, 10.1007/s00466-007-0193-7 Anders, 2011, Numerical simulation of diffusion induced phase separation and coarsening in binary alloys, Comput. Mater. Sci., 50, 1359, 10.1016/j.commatsci.2010.03.030 Anders, 2011, Application of operator-scaling anisotropic random fields to binary mixtures, Philos. Mag., 91, 3766, 10.1080/14786435.2011.595378 Anders, 2012, Computational modeling of phase separation and coarsening in solder alloys, Int. J. Solids Struct., 49, 1557, 10.1016/j.ijsolstr.2012.03.018 Anderson, 1998, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 139, 10.1146/annurev.fluid.30.1.139 Asai, 2009, Dynamics and patter formation in thermally induced phase separation of polymer–solvent system, Comput. Mater. Sci., 47, 193, 10.1016/j.commatsci.2009.07.008 Auricchio, 2007, A fully locking-free isogeometric approach for plane linear elasticity problems: a stream function formulation, Comput. Methods Appl. Mech. Eng., 197, 160, 10.1016/j.cma.2007.07.005 Auricchio, 2010, Isogeometric collocation methods, Math. Models Methods Appl. Sci., 20, 2075, 10.1142/S0218202510004878 Auricchio, 2012, Isogeometric collocation for elastostatics and explicit dynamics, Comput. Methods Appl. Mech. Eng., 249–252, 1043 Auricchio, 2010, The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations, Comput. Methods Appl. Mech. Eng., 199, 314, 10.1016/j.cma.2008.06.004 Auricchio, 2013, Locking-free isogeometric collocation methods for spatial Timoshenko rods, Comput. Methods Appl. Mech. Eng., 262, 113, 10.1016/j.cma.2013.03.009 Badalasi, 2003, Computation of multiphase systems with phase field models, J. Comput. Phys., 190, 371, 10.1016/S0021-9991(03)00280-8 Bazilevs, 2010, Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Eng., 199, 229, 10.1016/j.cma.2009.02.036 Bazilevs, 2007, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Eng., 197, 173, 10.1016/j.cma.2007.07.016 Bazilevs, 2008, NURBS-based isogeometric analysis for the computation of flows about rotating components, Comput. Mech., 43, 143, 10.1007/s00466-008-0277-z Becker, 2003, Complex dewetting scenarios captured by thin-film models, Nat. Mater., 2, 59, 10.1038/nmat788 Beirao da Veiga, 2011, Some estimates for h-p-k-refinement in isogeometric analysis, Numer. Math., 118, 271, 10.1007/s00211-010-0338-z Beirao da Veiga, 2012, Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods, Comput. Methods Appl. Mech. Eng., 249–252, 2 Beirao da Veiga, 2012, Anisotropic NURBS approximation in isogeometric analysis, Comput. Methods Appl. Mech. Eng., 209–212, 1, 10.1016/j.cma.2011.10.016 Borden, 2012, A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Eng., 217, 77, 10.1016/j.cma.2012.01.008 Buffa, 2011, Isogeometric analysis: Stable elements for the 2D Stokes equation, Int. J. Numer. Methods Fluids, 65, 1407, 10.1002/fld.2337 Buffa, 2010, Isogeometric analysis in electromagnetics: B-splines approximation, Comput. Methods Appl. Mech. Eng., 199, 1143, 10.1016/j.cma.2009.12.002 Cahn, 1958, Free energy of a non-uniform system. I. Interfacial free energy, J. Chem. Phys., 28, 258, 10.1063/1.1744102 Cahn, 1959, Free energy of a non-uniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31, 688, 10.1063/1.1730447 Ceniceros, 2010, Three-dimensional, fully adaptive simulations of phase-field fluid models, J. Comput. Phys., 229, 6135, 10.1016/j.jcp.2010.04.045 Chen, 2002, Phase-field models for microstructural evolution, Annu. Rev. Mater. Res., 32, 113, 10.1146/annurev.matsci.32.112001.132041 Choksi, 2009, On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn–Hilliard functional, SIAM J. Appl. Math., 69, 1712, 10.1137/080728809 Cottrell, 2009 Cottrell, 2007, Studies of refinement and continuity in isogeometric structural analysis, Comput. Methods Appl. Mech. Eng., 196, 4160, 10.1016/j.cma.2007.04.007 Cottrell, 2006, Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Eng., 195, 5257, 10.1016/j.cma.2005.09.027 Cueto-Felgueroso, 2008, Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media, Phys. Rev. Lett., 101, 244504, 10.1103/PhysRevLett.101.244504 Cueto-Felgueroso, 2009, A phase-field model of unsaturated flow, Water Resour. Res., 45, W10409, 10.1029/2009WR007945 Cueto-Felgueroso, 2009, Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems, J. Comput. Phys., 228, 6536, 10.1016/j.jcp.2009.05.045 de Boor, 2001 de Falco, 2011, GeoPDEs: a research tool for isogeometric analysis of PDEs, Adv. Eng. Softw., 42, 1020, 10.1016/j.advengsoft.2011.06.010 Du, 1991, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28, 1310, 10.1137/0728069 Elder, 2004, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70, 051605, 10.1103/PhysRevE.70.051605 Elder, 2002, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88, 245701, 10.1103/PhysRevLett.88.245701 Elguedj, 2008, B¯ and F¯ projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements, Comput. Methods Appl. Mech. Eng., 197, 2732, 10.1016/j.cma.2008.01.012 Emmerich, 2003 Eyre Evans, 2009, n-widths, sup infs, and optimality ratios for the k-version of the isogeometric finite element method, Comput. Methods Appl. Mech. Eng., 198, 1726, 10.1016/j.cma.2009.01.021 Furihata, 2001, A stable and conservative finite difference scheme for the Cahn–Hilliard equation, Numer. Math., 87, 675, 10.1007/PL00005429 Gomez, 2008, Isogeometric analysis of the Cahn–Hilliard phase-field model, Comput. Methods Appl. Mech. Eng., 197, 4333, 10.1016/j.cma.2008.05.003 Gomez, 2013, Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium, J. Comput. Phys., 238, 217, 10.1016/j.jcp.2012.12.018 Gomez, 2010, Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations, Comput. Methods Appl. Mech. Eng., 199, 1828, 10.1016/j.cma.2010.02.010 Gomez, 2011, Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys., 230, 5310, 10.1016/j.jcp.2011.03.033 Gomez, 2012, An unconditionally energy-stable method for the phase-field crystal equation, Comput. Methods Appl. Mech. Eng., 249–252, 52, 10.1016/j.cma.2012.03.002 Gomez, 2012, A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional, Commun. Nonlinear Sci. Numer. Simul., 17, 4930, 10.1016/j.cnsns.2012.05.018 Gomez, 2011, Numerical simulation of asymptotic states of the damped Kuramoto–Sivashinsky equation, Phys. Rev. E, 83, 046703, 10.1103/PhysRevE.83.046702 Guillén-González, 2013, On linear schemes for a Cahn–Hilliard diffuse interface model, J. Comput. Phys., 234, 140, 10.1016/j.jcp.2012.09.020 Guo, 2012, An implicit parallel multigrid computing scheme to solve coupled thermal–solute phase-field equations for dendrite evolution, J. Comput. Phys., 231, 1781, 10.1016/j.jcp.2011.11.006 Gurtin, 1996, Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance, Physica D, 92, 178, 10.1016/0167-2789(95)00173-5 He, 2009, A class of stable spectral methods for the Cahn–Hilliard equation, J. Comput. Phys., 228, 5101, 10.1016/j.jcp.2009.04.011 Hu, 2009, Stable and efficient finite-difference nonlinear multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228, 5323, 10.1016/j.jcp.2009.04.020 Hughes, 2000 Hughes, 2005, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194, 4135, 10.1016/j.cma.2004.10.008 Hughes, 2013, Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems, Comput. Methods Appl. Mech. Eng. Hughes, 2008, Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS, Comput. Methods Appl. Mech. Eng., 197, 4104, 10.1016/j.cma.2008.04.006 Kim, 2007, Phase field computations for ternary fluid flows, Comput. Methods Appl. Mech. Eng., 196, 4779, 10.1016/j.cma.2007.06.016 Kiendl, 2009, Isogeometric shell analysis with Kirchhoff–Love elements, Comput. Methods Appl. Mech. Eng., 198, 3902, 10.1016/j.cma.2009.08.013 Lipton, 2010, Robustness of isogeometric structural discretizations under severe mesh distortion, Comput. Methods Appl. Mech. Eng., 199, 357, 10.1016/j.cma.2009.01.022 Lowengrub, 1998, Quasi-incompressible Cahn–Hilliard fluids and topological transitions, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 454, 2617, 10.1098/rspa.1998.0273 Mahapatra, 2006, Finite element analysis of phase transformation dynamics in shape memory alloys with a consistent Landau–Ginzburg free energy model, Mech. Adv. Mat. Struct., 13, 443, 10.1080/15376490600862863 Mahapatra, 2007, Finite element approach to modelling evolution of 3D shape memory materials, Math. Comput. Simul., 76, 141, 10.1016/j.matcom.2007.01.014 Pismen, 2005 Reali, 2006, An isogeometric analysis approach for the study of structural vibrations, J. Earthq. Eng., 10, 1, 10.1080/13632460609350626 Schillinger, 2013, Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations, Comput. Methods Appl. Mech. Eng., 267, 170, 10.1016/j.cma.2013.07.017 Shen, 2009, An efficient moving mesh spectral method for the phase-field model of two phase flows, J. Comput. Phys., 228, 2978, 10.1016/j.jcp.2009.01.009 Shin, 2011, A conservative numerical method for the Cahn–Hilliard equation in complex domains, J. Comput. Phys., 230, 7441, 10.1016/j.jcp.2011.06.009 Stogner, 2008, Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements, Int. J. Numer. Methods Eng., 76, 636, 10.1002/nme.2337 Tegze, 2009, Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients, J. Comput. Phys., 228, 1612, 10.1016/j.jcp.2008.11.011 Tremaine, 2003, On the origin of irregular structure in Saturnʼs rings, Astron. J., 125, 894, 10.1086/345963 Vuong, 2010, ISOGAT: A 2D tutorial MATLAB code for isogeometric analysis, Comput. Aided Geom. Des., 27, 644, 10.1016/j.cagd.2010.06.006 Xia, 2007, Local discontinuous Galerkin methods for the Cahn–Hilliard type equations, J. Comput. Phys., 227, 472, 10.1016/j.jcp.2007.08.001 Wodo, 2011, Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem, J. Comput. Phys., 230, 6037, 10.1016/j.jcp.2011.04.012