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 and 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), 1289-1327. doi: 10.1093/imanum/drt044.

[2]

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

[3]

E. Burman and J. Rappaz, Existence of solutions to an anisotropic phase-field model, Math. Meth. Appl. Sci., 26 (2003), 1137-1160. doi: 10.1002/mma.405.

[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), 299-358. doi: 10.1098/rsta.1951.0006.

[5]

G. Caginalp, Penrose-Fife modification of solidification equations has no freezing or melting, Appl. Math. Lett., 5 (1992), 93-96. doi: 10.1016/0893-9659(92)90120-X.

[6]

C. Cattani, Harmonic wavelets towards the solution of nonlinear PDE, Comp. Math. Appl., 50 (2005), 1191-1210. doi: 10.1016/j.camwa.2005.07.001.

[7]

W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Num., 6 (1997), 55-228. doi: 10.1017/S0962492900002713.

[8]

I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, USA, 1992. doi: 10.1137/1.9781611970104.

[9]

J. A. Dobrosotskaya and A. L. Bertozzi, A wavelet-laplace variational technique for image deconvolution and inpainting, IEEE Trans. Imag. Proc., 17 (2008), 657-663. doi: 10.1109/TIP.2008.919367.

[10]

J. A. Dobrosotskaya and A. L. Bertozzi, Wavelet analogue of the Ginzburg-Landau energy and its Gamma-convergence, Interf. Free Boundaries, 12 (2010), 497-525. doi: 10.4171/IFB/243.

[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), 698-729. doi: 10.1137/100812859.

[12]

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

[13]

C. Herring, Some theorems on the free energies of crystal surfaces, Phys. Rev., 82 (1951), 87-93. doi: 10.1103/PhysRev.82.87.

[14]

M. Holmström, Solving hyperbolic PDEs using interpolating wavelets, SIAM J. Sci. Comput., 21 (1999), 405-420. doi: 10.1137/S1064827597316278.

[15]

M. Holmström and J. Waldén, Adaptive wavelet methods for hyperbolic PDEs, J Sci. Comp., 13 (1998), 19-49. doi: 10.1023/A:1023252610346.

[16]

L. Jameson, A wavelet-optimized, very high order adaptive grid and order numerical method, SIAM J. Sci. Comput., 19 (1998), 1980-2013. doi: 10.1137/S1064827596301534.

[17]

A. Karma and W.-J. Rappel, Numerical simulation of three-dimensional dendritic growth, Phys. Rev. Lett., 77 (1996), p4050. doi: 10.1103/PhysRevLett.77.4050.

[18]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D, 63 (1993), 410-423. doi: 10.1016/0167-2789(93)90120-P.

[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), 433-482.

[20]

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

[21]

G. B. McFadden, Phase-field models of solidification, in Recent Advances in Numerical Methods for Partial Differential Equations and Applications, Contemporary Mathematics, American Mathematical Society, 306 (2002), 107-145. doi: 10.1090/conm/306/05251.

[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), 2016-2024. doi: 10.1103/PhysRevE.48.2016.

[23]

A. Miranville, Some mathematical models in phase transitions, DCDS-S, 7 (2014), 271-306. doi: 10.3934/dcdss.2014.7.271.

[24]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.

[25]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[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), 107-113. doi: 10.1016/0167-2789(93)90183-2.

[27]

K. Schneider and O. V. Vasilyev, Wavelet methods in computational fluid dynamics, Ann. Rev. Fluid Mech., 42 (2010), 473-503. doi: 10.1146/annurev-fluid-121108-145637.

[28]

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

[29]

O. V. Vasilyev and S. Paolucci, A fast adaptive wavelet collocation algorithm for multidimensional PDEs, J. Comp. Phys., 138 (1997), 16-56. doi: 10.1006/jcph.1997.5814.

[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), 33-47. doi: 10.1006/jcph.1995.1147.

[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), 189-200. doi: 10.1016/0167-2789(93)90189-8.

[32]

A. A. Wheeler, B. T. Murray and R. J. Schaefer, Computation of dendrites using a phase field model, Physica D, 66 (1993), 243-262. doi: 10.1016/0167-2789(93)90242-S.

[33]

G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflösung der krystallflächen, Zeitschrift f. Krystall. Mineral., 34 (1901), 449-530.

[34]

S.-M. Zheng, Nonlinear Evolution Equations, Pitman series Monographs and Survey in Pure and Applied Mathematics 133, Chapman Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222.

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), 1289-1327. doi: 10.1093/imanum/drt044.

[2]

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

[3]

E. Burman and J. Rappaz, Existence of solutions to an anisotropic phase-field model, Math. Meth. Appl. Sci., 26 (2003), 1137-1160. doi: 10.1002/mma.405.

[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), 299-358. doi: 10.1098/rsta.1951.0006.

[5]

G. Caginalp, Penrose-Fife modification of solidification equations has no freezing or melting, Appl. Math. Lett., 5 (1992), 93-96. doi: 10.1016/0893-9659(92)90120-X.

[6]

C. Cattani, Harmonic wavelets towards the solution of nonlinear PDE, Comp. Math. Appl., 50 (2005), 1191-1210. doi: 10.1016/j.camwa.2005.07.001.

[7]

W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Num., 6 (1997), 55-228. doi: 10.1017/S0962492900002713.

[8]

I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, USA, 1992. doi: 10.1137/1.9781611970104.

[9]

J. A. Dobrosotskaya and A. L. Bertozzi, A wavelet-laplace variational technique for image deconvolution and inpainting, IEEE Trans. Imag. Proc., 17 (2008), 657-663. doi: 10.1109/TIP.2008.919367.

[10]

J. A. Dobrosotskaya and A. L. Bertozzi, Wavelet analogue of the Ginzburg-Landau energy and its Gamma-convergence, Interf. Free Boundaries, 12 (2010), 497-525. doi: 10.4171/IFB/243.

[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), 698-729. doi: 10.1137/100812859.

[12]

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

[13]

C. Herring, Some theorems on the free energies of crystal surfaces, Phys. Rev., 82 (1951), 87-93. doi: 10.1103/PhysRev.82.87.

[14]

M. Holmström, Solving hyperbolic PDEs using interpolating wavelets, SIAM J. Sci. Comput., 21 (1999), 405-420. doi: 10.1137/S1064827597316278.

[15]

M. Holmström and J. Waldén, Adaptive wavelet methods for hyperbolic PDEs, J Sci. Comp., 13 (1998), 19-49. doi: 10.1023/A:1023252610346.

[16]

L. Jameson, A wavelet-optimized, very high order adaptive grid and order numerical method, SIAM J. Sci. Comput., 19 (1998), 1980-2013. doi: 10.1137/S1064827596301534.

[17]

A. Karma and W.-J. Rappel, Numerical simulation of three-dimensional dendritic growth, Phys. Rev. Lett., 77 (1996), p4050. doi: 10.1103/PhysRevLett.77.4050.

[18]

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D, 63 (1993), 410-423. doi: 10.1016/0167-2789(93)90120-P.

[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), 433-482.

[20]

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

[21]

G. B. McFadden, Phase-field models of solidification, in Recent Advances in Numerical Methods for Partial Differential Equations and Applications, Contemporary Mathematics, American Mathematical Society, 306 (2002), 107-145. doi: 10.1090/conm/306/05251.

[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), 2016-2024. doi: 10.1103/PhysRevE.48.2016.

[23]

A. Miranville, Some mathematical models in phase transitions, DCDS-S, 7 (2014), 271-306. doi: 10.3934/dcdss.2014.7.271.

[24]

L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.

[25]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.

[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), 107-113. doi: 10.1016/0167-2789(93)90183-2.

[27]

K. Schneider and O. V. Vasilyev, Wavelet methods in computational fluid dynamics, Ann. Rev. Fluid Mech., 42 (2010), 473-503. doi: 10.1146/annurev-fluid-121108-145637.

[28]

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

[29]

O. V. Vasilyev and S. Paolucci, A fast adaptive wavelet collocation algorithm for multidimensional PDEs, J. Comp. Phys., 138 (1997), 16-56. doi: 10.1006/jcph.1997.5814.

[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), 33-47. doi: 10.1006/jcph.1995.1147.

[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), 189-200. doi: 10.1016/0167-2789(93)90189-8.

[32]

A. A. Wheeler, B. T. Murray and R. J. Schaefer, Computation of dendrites using a phase field model, Physica D, 66 (1993), 243-262. doi: 10.1016/0167-2789(93)90242-S.

[33]

G. Wulff, Zur frage der geschwindigkeit des wachstums und der auflösung der krystallflächen, Zeitschrift f. Krystall. Mineral., 34 (1901), 449-530.

[34]

S.-M. Zheng, Nonlinear Evolution Equations, Pitman series Monographs and Survey in Pure and Applied Mathematics 133, Chapman Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222.

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