Numerical Approximation of Multi-Phase Penrose–Fife Systems

Abstract We consider a non-isothermal multi-phase field model. We subsequently discretize implicitly in time and with linear finite elements. The arising algebraic problem is formulated in two variables where one is the multi-phase field, and the other contains the inverse temperature field. We solve this saddle point problem numerically by a non-smooth Schur–Newton approach using truncated non-smooth Newton multigrid methods. An application in grain growth as occurring in liquid phase crystallization of silicon is considered.

[1]  A. Visintin.  Models of Phase Transitions , 1996 .

[2]  Ralf Kornhuber,et al.  A posteriori error estimates for elliptic problems in two and three space dimensions , 1996 .

[3]  Qingsong Zou,et al.  Efficient and reliable hierarchical error estimates for the discretization error of elliptic obstacle problems , 2010, Math. Comput..

[4]  Ralf Kornhuber,et al.  Numerical simulation of coarsening in binary solder alloys , 2014 .

[5]  M. Brokate,et al.  Hysteresis and Phase Transitions , 1996 .

[6]  Peter Deuflhard,et al.  Concepts of an adaptive hierarchical finite element code , 1989, IMPACT Comput. Sci. Eng..

[7]  Harald Garcke,et al.  Nonlocal Allen–Cahn systems: analysis and a primal–dual active set method , 2013 .

[8]  Andreas Veeser,et al.  Hierarchical error estimates for the energy functional in obstacle problems , 2011, Numerische Mathematik.

[9]  Ralf Kornhuber,et al.  Nonsmooth Schur–Newton methods for multicomponent Cahn–Hilliard systems , 2015 .

[10]  Ralf Kornhuber,et al.  A posteriori error estimates for elliptic variational inequalities , 1996 .

[11]  Carsten Gräser,et al.  Convex minimization and phase field models , 2011 .

[12]  C. Baiocchi,et al.  Variational and quasivariational inequalities: Applications to free boundary problems , 1983 .

[13]  F. Bornemann,et al.  Adaptive multivlevel methods in three space dimensions , 1993 .

[14]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[15]  Irena Pawlow,et al.  A mathematical model of dynamics of non-isothermal phase separation , 1992 .

[16]  Randolph E. Bank,et al.  A posteriori error estimates based on hierarchical bases , 1993 .

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  Jürgen Bey,et al.  Simplicial grid refinement: on Freudenthal's algorithm and the optimal number of congruence classes , 2000, Numerische Mathematik.

[19]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[20]  John W. Barrett,et al.  Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy , 1997 .

[21]  R. Hoppe,et al.  Adaptive multilevel methods for obstacle problems , 1994 .

[22]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE , 2008, Computing.

[23]  O. Sander,et al.  Truncated nonsmooth Newton multigrid methods for simplex-constrained minimization problems , 2014 .

[24]  Josef Stoer,et al.  Numerische Mathematik 2 , 1990 .

[25]  Oliver Sander,et al.  Truncated Nonsmooth Newton Multigrid Methods for Convex Minimization Problems , 2009 .

[26]  Björn Stinner.  Weak solutions to a multi-phase field system of parabolic equations related to alloy solidification , 2007 .

[27]  Bodo Erdmann Ralf Kornhuber.  Adaptive Multilevel Methods in Three Space Dimensions Folkmar Bornemann , 2011 .

[28]  Charles M. Elliott,et al.  The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis , 1991, European Journal of Applied Mathematics.

[29]  Ralf Kornhuber,et al.  Time discretizations of anisotropic Allen–Cahn equations , 2013 .

[30]  C. Mészáros.  Fast Cholesky factorization for interior point methods of linear programming , 1996 .

[31]  Harald Garcke,et al.  A Diffuse Interface Model for Alloys with Multiple Components and Phases , 2004, SIAM J. Appl. Math..

[32]  Paul C. Fife,et al.  Thermodynamically consistent models of phase-field type for the kinetics of phase transitions , 1990 .

[33]  Charles M. Elliott,et al.  Numerical analysis of a model for phase separation of a multi- component alloy , 1996 .

[34]  C. M. Elliott,et al.  Computation of geometric partial differential equations and mean curvature flow , 2005, Acta Numerica.

[35]  O. Zienkiewicz,et al.  The hierarchical concept in finite element analysis , 1983 .

[36]  Sören Bartels,et al.  Numerical Methods for Nonlinear Partial Differential Equations , 2015 .

[37]  Jeffrey S. Ovall,et al.  An efficient, reliable and robust error estimator for elliptic problems in R3 , 2011 .

[38]  J. Crank.  Free and moving boundary problems , 1984 .

[39]  Lavinia Sarbu,et al.  Primal‐dual active set methods for Allen–Cahn variational inequalities with nonlocal constraints , 2010 .

[40]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[41]  Harald Garcke,et al.  Phase-field model for multiphase systems with preserved volume fractions. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework , 2008, Computing.

[43]  Kunibert G. Siebert,et al.  A Unilaterally Constrained Quadratic Minimization with Adaptive Finite Elements , 2007, SIAM J. Optim..

[44]  Andreas Dedner,et al.  A Generic Grid Interface for Adaptive and Parallel Scientific Computing. Part II: Implementation and Tests in DUNE , 2007 .

[45]  Ralf Kornhuber,et al.  On Hierarchical Error Estimators for Time-Discretized Phase Field Models , 2010 .

[46]  Ralf Kornhuber,et al.  Nonsmooth Newton Methods for Set-Valued Saddle Point Problems , 2009, SIAM J. Numer. Anal..

[47]  D. Amkreutz,et al.  Electron‐beam crystallized large grained silicon solar cell on glass substrate , 2011 .

[48]  Martin A. Green,et al.  Large Grained, Low Defect Density Polycrystalline Silicon on Glass Substrates by Large-area Diode Laser Crystallisation , 2012 .

[49]  Bernd Rech,et al.  Towards monocrystalline silicon thin films grown on glass by liquid phase crystallization , 2015 .

[50]  Ralf Kornhuber,et al.  Robust Multigrid Methods for Vector-valued Allen–Cahn Equations with Logarithmic Free Energy , 2006 .

[51]  Oliver Sander,et al.  The dune-subgrid module and some applications , 2009, Computing.

[52]  Robert Nürnberg,et al.  Numerical simulations of immiscible fluid clusters , 2009 .