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

Anisotropy in wavelet-based phase field models

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.
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

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