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The 20-60-20 rule
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, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom |
4. | Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany |
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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