American Institute of Mathematical Sciences

Advanced
June  2016, 21(4): 1167-1187. doi: 10.3934/dcdsb.2016.21.1167

Anisotropy in wavelet-based phase field models

Maciek Korzec 1, , Andreas Münch 2, , Endre Süli 3, and  Barbara Wagner 4,

1. 

Technische Universität Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany
2. 

Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
3. 

Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliff e Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
4. 

Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany

Received  May 2015 Revised  January 2016 Published  March 2016

When describing the anisotropic evolution of microstructures in solids using phase-field models, the anisotropy of the crystalline phases is usually introduced into the interfacial energy by directional dependencies of the gradient energy coefficients. We consider an alternative approach based on a wavelet analogue of the Laplace operator that is intrinsically anisotropic and linear. The paper focuses on the classical coupled temperature/Ginzburg--Landau type phase-field model for dendritic growth. For the model based on the wavelet analogue, existence, uniqueness and continuous dependence on initial data are proved for weak solutions. Numerical studies of the wavelet based phase-field model show dendritic growth similar to the results obtained for classical phase-field models.
Keywords: Phase-field model, free boundaries., wavelets, sharp interface model.
Mathematics Subject Classification: Primary: 34E13, 74N20; Secondary: 74E1.
Citation: Maciek Korzec, Andreas Münch, Endre Süli, Barbara Wagner. Anisotropy in wavelet-based phase field models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1167-1187. doi: 10.3934/dcdsb.2016.21.1167
References:
[1]

J. W. Barrett, H. Garcke and R. Nürnberg, Stable phase field approximations of anisotropic solidification,, IMA J. Numer. Anal., 34 (2014), 1289.  doi: 10.1093/imanum/drt044.  Google Scholar

[2]

A. Braides, Gamma-Convergence for Beginners,, Oxford University Press, (2002).  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[3]

E. Burman and J. Rappaz, Existence of solutions to an anisotropic phase-field model,, Math. Meth. Appl. Sci., 26 (2003), 1137.  doi: 10.1002/mma.405.  Google Scholar

[4]

W. K. Burton, N. Cabrera and F. C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces,, Phil. Trans. R. Soc. Lond. A, 243 (1951), 299.  doi: 10.1098/rsta.1951.0006.  Google Scholar

[5]

G. Caginalp, Penrose-Fife modification of solidification equations has no freezing or melting,, Appl. Math. Lett., 5 (1992), 93.  doi: 10.1016/0893-9659(92)90120-X.  Google Scholar

[6]

C. Cattani, Harmonic wavelets towards the solution of nonlinear PDE,, Comp. Math. Appl., 50 (2005), 1191.  doi: 10.1016/j.camwa.2005.07.001.  Google Scholar

[7]

W. Dahmen, Wavelet and multiscale methods for operator equations,, Acta Num., 6 (1997), 55.  doi: 10.1017/S0962492900002713.  Google Scholar

[8]

I. Daubechies, Ten Lectures on Wavelets,, SIAM, (1992).  doi: 10.1137/1.9781611970104.  Google Scholar

[9]

J. A. Dobrosotskaya and A. L. Bertozzi, A wavelet-laplace variational technique for image deconvolution and inpainting,, IEEE Trans. Imag. Proc., 17 (2008), 657.  doi: 10.1109/TIP.2008.919367.  Google Scholar

[10]

J. A. Dobrosotskaya and A. L. Bertozzi, Wavelet analogue of the Ginzburg-Landau energy and its Gamma-convergence,, Interf. Free Boundaries, 12 (2010), 497.  doi: 10.4171/IFB/243.  Google Scholar

[11]

J. A. Dobrosotskaya and A. L. Bertozzi, Analysis of the wavelet Ginzburg-Landau energy in image applications with edges,, SIAM J. Imaging Sci., 6 (2013), 698.  doi: 10.1137/100812859.  Google Scholar

[12]

M. E. Glicksman, Principles of Solidification,, Springer, (2011).  doi: 10.1007/978-1-4419-7344-3.  Google Scholar

[13]

C. Herring, Some theorems on the free energies of crystal surfaces,, Phys. Rev., 82 (1951), 87.  doi: 10.1103/PhysRev.82.87.  Google Scholar

[14]

M. Holmström, Solving hyperbolic PDEs using interpolating wavelets,, SIAM J. Sci. Comput., 21 (1999), 405.  doi: 10.1137/S1064827597316278.  Google Scholar

[15]

M. Holmström and J. Waldén, Adaptive wavelet methods for hyperbolic PDEs,, J Sci. Comp., 13 (1998), 19.  doi: 10.1023/A:1023252610346.  Google Scholar

[16]

L. Jameson, A wavelet-optimized, very high order adaptive grid and order numerical method,, SIAM J. Sci. Comput., 19 (1998), 1980.  doi: 10.1137/S1064827596301534.  Google Scholar

[17]

A. Karma and W.-J. Rappel, Numerical simulation of three-dimensional dendritic growth,, Phys. Rev. Lett., 77 (1996).  doi: 10.1103/PhysRevLett.77.4050.  Google Scholar

[18]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth,, Physica D, 63 (1993), 410.  doi: 10.1016/0167-2789(93)90120-P.  Google Scholar

[19]

B. Li, J. Lowengrub, A. Rätz and A. Voigt, Geometric evolution laws for thin crystalline films: Modeling and numerics,, Commun. Comput. Phys., 6 (2009), 433.   Google Scholar

[20]

S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way,, Academic Press, (2009).   Google Scholar

[21]

G. B. McFadden, Phase-field models of solidification,, in Recent Advances in Numerical Methods for Partial Differential Equations and Applications, 306 (2002), 107.  doi: 10.1090/conm/306/05251.  Google Scholar

[22]

G. B. McFadden, A. A. Wheeler, R. J. Braun, S. R. Coriell and R. F. Sekerka, Phase-field models for anisotropic interfaces,, Phys. Rev. E, 48 (1993), 2016.  doi: 10.1103/PhysRevE.48.2016.  Google Scholar

[23]

A. Miranville, Some mathematical models in phase transitions,, DCDS-S, 7 (2014), 271.  doi: 10.3934/dcdss.2014.7.271.  Google Scholar

[24]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 123.  doi: 10.1007/BF00251230.  Google Scholar

[25]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Physica D, 43 (1990), 44.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

[26]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a ''thermodynamically consistent'' phase field model,, Physica D, 69 (1993), 107.  doi: 10.1016/0167-2789(93)90183-2.  Google Scholar

[27]

K. Schneider and O. V. Vasilyev, Wavelet methods in computational fluid dynamics,, Ann. Rev. Fluid Mech., 42 (2010), 473.  doi: 10.1146/annurev-fluid-121108-145637.  Google Scholar

[28]

I. Steinbach, Phase-field models in materials science,, Mod. Sim. Mater. Sci. Eng., 17 (2009).  doi: 10.1088/0965-0393/17/7/073001.  Google Scholar

[29]

O. V. Vasilyev and S. Paolucci, A fast adaptive wavelet collocation algorithm for multidimensional PDEs,, J. Comp. Phys., 138 (1997), 16.  doi: 10.1006/jcph.1997.5814.  Google Scholar

[30]

O. V. Vasilyev, S. Paolucci and M. Sen, A multilevel wavelet collocation method for solving partial differential equations in a finite domain,, J. Comp. Phys., 120 (1995), 33.  doi: 10.1006/jcph.1995.1147.  Google Scholar

[31]

S.-L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun and G. B. McFadden, Thermodynamically-consistent phase-field models for solidification,, Physica D, 69 (1993), 189.  doi: 10.1016/0167-2789(93)90189-8.  Google Scholar

[32]

A. A. Wheeler, B. T. Murray and R. J. Schaefer, Computation of dendrites using a phase field model,, Physica D, 66 (1993), 243.  doi: 10.1016/0167-2789(93)90242-S.  Google Scholar

[33]

G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflösung der krystallflächen,, Zeitschrift f. Krystall. Mineral., 34 (1901), 449.   Google Scholar

[34]

S.-M. Zheng, Nonlinear Evolution Equations,, Pitman series Monographs and Survey in Pure and Applied Mathematics 133, (2004).  doi: 10.1201/9780203492222.  Google Scholar

show all references

References:
[1]

J. W. Barrett, H. Garcke and R. Nürnberg, Stable phase field approximations of anisotropic solidification,, IMA J. Numer. Anal., 34 (2014), 1289.  doi: 10.1093/imanum/drt044.  Google Scholar

[2]

A. Braides, Gamma-Convergence for Beginners,, Oxford University Press, (2002).  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[3]

E. Burman and J. Rappaz, Existence of solutions to an anisotropic phase-field model,, Math. Meth. Appl. Sci., 26 (2003), 1137.  doi: 10.1002/mma.405.  Google Scholar

[4]

W. K. Burton, N. Cabrera and F. C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces,, Phil. Trans. R. Soc. Lond. A, 243 (1951), 299.  doi: 10.1098/rsta.1951.0006.  Google Scholar

[5]

G. Caginalp, Penrose-Fife modification of solidification equations has no freezing or melting,, Appl. Math. Lett., 5 (1992), 93.  doi: 10.1016/0893-9659(92)90120-X.  Google Scholar

[6]

C. Cattani, Harmonic wavelets towards the solution of nonlinear PDE,, Comp. Math. Appl., 50 (2005), 1191.  doi: 10.1016/j.camwa.2005.07.001.  Google Scholar

[7]

W. Dahmen, Wavelet and multiscale methods for operator equations,, Acta Num., 6 (1997), 55.  doi: 10.1017/S0962492900002713.  Google Scholar

[8]

I. Daubechies, Ten Lectures on Wavelets,, SIAM, (1992).  doi: 10.1137/1.9781611970104.  Google Scholar

[9]

J. A. Dobrosotskaya and A. L. Bertozzi, A wavelet-laplace variational technique for image deconvolution and inpainting,, IEEE Trans. Imag. Proc., 17 (2008), 657.  doi: 10.1109/TIP.2008.919367.  Google Scholar

[10]

J. A. Dobrosotskaya and A. L. Bertozzi, Wavelet analogue of the Ginzburg-Landau energy and its Gamma-convergence,, Interf. Free Boundaries, 12 (2010), 497.  doi: 10.4171/IFB/243.  Google Scholar

[11]

J. A. Dobrosotskaya and A. L. Bertozzi, Analysis of the wavelet Ginzburg-Landau energy in image applications with edges,, SIAM J. Imaging Sci., 6 (2013), 698.  doi: 10.1137/100812859.  Google Scholar

[12]

M. E. Glicksman, Principles of Solidification,, Springer, (2011).  doi: 10.1007/978-1-4419-7344-3.  Google Scholar

[13]

C. Herring, Some theorems on the free energies of crystal surfaces,, Phys. Rev., 82 (1951), 87.  doi: 10.1103/PhysRev.82.87.  Google Scholar

[14]

M. Holmström, Solving hyperbolic PDEs using interpolating wavelets,, SIAM J. Sci. Comput., 21 (1999), 405.  doi: 10.1137/S1064827597316278.  Google Scholar

[15]

M. Holmström and J. Waldén, Adaptive wavelet methods for hyperbolic PDEs,, J Sci. Comp., 13 (1998), 19.  doi: 10.1023/A:1023252610346.  Google Scholar

[16]

L. Jameson, A wavelet-optimized, very high order adaptive grid and order numerical method,, SIAM J. Sci. Comput., 19 (1998), 1980.  doi: 10.1137/S1064827596301534.  Google Scholar

[17]

A. Karma and W.-J. Rappel, Numerical simulation of three-dimensional dendritic growth,, Phys. Rev. Lett., 77 (1996).  doi: 10.1103/PhysRevLett.77.4050.  Google Scholar

[18]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth,, Physica D, 63 (1993), 410.  doi: 10.1016/0167-2789(93)90120-P.  Google Scholar

[19]

B. Li, J. Lowengrub, A. Rätz and A. Voigt, Geometric evolution laws for thin crystalline films: Modeling and numerics,, Commun. Comput. Phys., 6 (2009), 433.   Google Scholar

[20]

S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way,, Academic Press, (2009).   Google Scholar

[21]

G. B. McFadden, Phase-field models of solidification,, in Recent Advances in Numerical Methods for Partial Differential Equations and Applications, 306 (2002), 107.  doi: 10.1090/conm/306/05251.  Google Scholar

[22]

G. B. McFadden, A. A. Wheeler, R. J. Braun, S. R. Coriell and R. F. Sekerka, Phase-field models for anisotropic interfaces,, Phys. Rev. E, 48 (1993), 2016.  doi: 10.1103/PhysRevE.48.2016.  Google Scholar

[23]

A. Miranville, Some mathematical models in phase transitions,, DCDS-S, 7 (2014), 271.  doi: 10.3934/dcdss.2014.7.271.  Google Scholar

[24]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rat. Mech. Anal., 98 (1987), 123.  doi: 10.1007/BF00251230.  Google Scholar

[25]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Physica D, 43 (1990), 44.  doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

[26]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a ''thermodynamically consistent'' phase field model,, Physica D, 69 (1993), 107.  doi: 10.1016/0167-2789(93)90183-2.  Google Scholar

[27]

K. Schneider and O. V. Vasilyev, Wavelet methods in computational fluid dynamics,, Ann. Rev. Fluid Mech., 42 (2010), 473.  doi: 10.1146/annurev-fluid-121108-145637.  Google Scholar

[28]

I. Steinbach, Phase-field models in materials science,, Mod. Sim. Mater. Sci. Eng., 17 (2009).  doi: 10.1088/0965-0393/17/7/073001.  Google Scholar

[29]

O. V. Vasilyev and S. Paolucci, A fast adaptive wavelet collocation algorithm for multidimensional PDEs,, J. Comp. Phys., 138 (1997), 16.  doi: 10.1006/jcph.1997.5814.  Google Scholar

[30]

O. V. Vasilyev, S. Paolucci and M. Sen, A multilevel wavelet collocation method for solving partial differential equations in a finite domain,, J. Comp. Phys., 120 (1995), 33.  doi: 10.1006/jcph.1995.1147.  Google Scholar

[31]

S.-L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun and G. B. McFadden, Thermodynamically-consistent phase-field models for solidification,, Physica D, 69 (1993), 189.  doi: 10.1016/0167-2789(93)90189-8.  Google Scholar

[32]

A. A. Wheeler, B. T. Murray and R. J. Schaefer, Computation of dendrites using a phase field model,, Physica D, 66 (1993), 243.  doi: 10.1016/0167-2789(93)90242-S.  Google Scholar

[33]

G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflösung der krystallflächen,, Zeitschrift f. Krystall. Mineral., 34 (1901), 449.   Google Scholar

[34]

S.-M. Zheng, Nonlinear Evolution Equations,, Pitman series Monographs and Survey in Pure and Applied Mathematics 133, (2004).  doi: 10.1201/9780203492222.  Google Scholar

[1]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[2]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[3]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[4]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035
[5]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[6]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[7]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[8]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161
[9]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101
[10]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
[11]

Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014
[12]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
[13]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[14]

Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209
[15]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[16]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[17]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623
[18]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[19]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[20]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences