February  2017, 10(1): 87-100. doi: 10.3934/dcdss.2017005

Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation

1. 

Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA), CNRS UMR 7352, Université de Picardie Jules Verne, 33, rue Saint Leu, 80039 Amiens, France

2. 

Laboratoire de Réactivité et de Chimie des Solides (LRCS), CNRS UMR 7314, Université de Picardie Jules Verne, 33, rue Saint Leu, 80039 Amiens, France

3. 

Réseau sur le Stockage Electrochimique de l'Energie (RS2E), FR CNRS 3459, France, ALISTORE European Research Institute, FR CNRS 3104,80039 Amiens, France

Received  March 2015 Revised  May 2016 Published  December 2016

Fund Project: Prof. Franco deeply acknowledges the Conseil Regional de Picardie and the European Regional Development Fund for the support through the project MASTERS, as well as the National Research Agency ANR for the support through the Project ALIBABA (reference ANR-11-PRGE-0002).

The identification of optimal structures in reaction-diffusion models is of great importance in numerous physicochemical systems. We propose here a simple method to monitor the number of interphases formed after long simulated times by using a boundary flux condition as a control parameter. We consider as an illustration a 1-D Allen-Cahn equation with Neumann boundary conditions. Numerical examples are provided and perspectives for the application of this approach to electrochemical systems are discussed.

Citation: Jean-Paul Chehab, Alejandro A. Franco, Youcef Mammeri. Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (1) : 87-100. doi: 10.3934/dcdss.2017005
References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. Mater., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[2]

S. Bartels, Numerical Methods for Nonlinear PDEs Springer, 2015. doi: 10.1007/978-3-319-13797-1.

[3]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.  doi: 10.1007/BF00254827.

[4]

J. W. Cahn and S. M. Allen, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1084-1095. 

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.

[6]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.

[7]

L. Q. Chen, Phase-field models for microstructure evolution, Annual Review of Material Research, 32 (2002), 113-140.  doi: 10.1146/annurev.matsci.32.112001.132041.

[8]

L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase-field equations, Comput. Phys. Comm., 108 (1998), 147-158.  doi: 10.1016/S0010-4655(97)00115-X.

[9]

L. Cherfils and M. Pierre, Non-global existence for an Allen-Cahn-Gurtin equation with logarithmic free energy, J. Evol. Equ., 8 (2008), 727-748.  doi: 10.1007/s00028-008-0412-5.

[10]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.

[11]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.

[12]

A. R. Conn, K. Scheinberg and L. N. Vicente, Introduction to Derivative-Free Optimization MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718768.

[13]

A. A. Franco and K. H. Xue, Carbon-based electrodes for lithium air batteries: Scientific and technological challenges from a modeling perspective, ECS Journal of Solid State Science and Technology, 2 (2013), 3084-3100.  doi: 10.1149/2.012310jss.

[14]

A. A. Franco, Multiscale modeling and numerical simulation of rechargeable lithium ion batteries: concepts, methods and challenges, RSC Advances, 3 (2013), 13027-13058. 

[15]

A. A. Franco, MS LIBER–T computational software, Available from: http://www.modeling-electrochemistry.com/ms-liber-t/

[16]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981.

[18]

L. IgnatA. Pozo and E. Zuazua, Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws, Math. Comp., 84 (2015), 1633-1662.  doi: 10.1090/S0025-5718-2014-02915-3.

[19]

H. Israel, Well-posedness and long time behavior of an Allen-Cahn type equation, Commun. Pure Appl. Anal., 12 (2013), 2811-2827.  doi: 10.3934/cpaa.2013.12.2811.

[20]

G. Karali and Y. Nagase, On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 127-137.  doi: 10.3934/dcdss.2014.7.127.

[21]

D. Kondepudi and I. Prigogine, Modern thermodynamics: From heat engines to dissipative structures, John & Wiley Sons Ltd., 8 (2014), p56.  doi: 10.3934/dcdss.2014.7.127.

[22]

H. G. Lee and J.-Y. Lee, A semi-analytical Fourier spectral method for the Allen-Cahn equation, Comput. Math. Appl., 68 (2014), 174-184.  doi: 10.1016/j.camwa.2014.05.015.

[23]

A. Makki and A. Miranville, Well-posedness for one-dimensional anisotropic Cahn-Hilliard and Allen-Cahn systems, Electron. J. Diff. Eq., 4 (2015), 15 pp.  doi: 10.1002/9781118698723.

[24]

K. MalekM. EikerlingQ. WangT. Navessin and Z. Liu, Self organization in catalyst layers of PEM Fuel Cells, J. Phys. Chem. C, 111 (2007), 13627-13634. 

[25]

K. Malek and A. A. Franco, Microstructural resolved modeling of aging mechanisms in PEMFC, J. Phys. Chem. B, 115 (2011), 8088-8101. 

[26]

Matlab's optimization toolbox, Available from: http://www.mathworks.fr/fr/products/optimization/index.html

[27]

A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277. 

[28]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs Series, Princeton University Press, 2005.

[29]

M. Pierre, Etude Numérique et Mathématique de Quelques Modéles de Transition de Phase, de Séparation de Phases et de Cristaux Liquides Habilitation à diriger les recherches, (in French), Université de Poitiers (Oct. 2011).

[30]

N. ProvatasJ. A. DantzigB. AthreyaP. ChanP. StefanovicN. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution, J. of the Minerals, Metals and Materials Society, 59 (2007), 83-90. 

[31]

C. SachsM. HildebrandS. VölkeningJ. Wintterlin and G. Ertl, Spatiotemporal self-organization in a surface reaction: From the atomic to the mesoscopic scale, Science, 293 (2001), 1635-1638.  doi: 10.1126/science.1062883.

[32]

A. SirimungkalaH.-D. Försterling and V. Dlask, Bromination Reactions Important in the Mechanism of the Belousov-Zhabotinsky System, J. Phys. Chem. A, 103 (1999), 1038-1043.  doi: 10.1021/jp9825213.

[33]

J. Shen and X. Yang, Numerical Approximations of Allen-Cahn and Cahn-Hilliard Equations, DCDS, Series A, 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.

show all references

References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. Mater., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[2]

S. Bartels, Numerical Methods for Nonlinear PDEs Springer, 2015. doi: 10.1007/978-3-319-13797-1.

[3]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.  doi: 10.1007/BF00254827.

[4]

J. W. Cahn and S. M. Allen, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1084-1095. 

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.

[6]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.

[7]

L. Q. Chen, Phase-field models for microstructure evolution, Annual Review of Material Research, 32 (2002), 113-140.  doi: 10.1146/annurev.matsci.32.112001.132041.

[8]

L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase-field equations, Comput. Phys. Comm., 108 (1998), 147-158.  doi: 10.1016/S0010-4655(97)00115-X.

[9]

L. Cherfils and M. Pierre, Non-global existence for an Allen-Cahn-Gurtin equation with logarithmic free energy, J. Evol. Equ., 8 (2008), 727-748.  doi: 10.1007/s00028-008-0412-5.

[10]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.

[11]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.

[12]

A. R. Conn, K. Scheinberg and L. N. Vicente, Introduction to Derivative-Free Optimization MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718768.

[13]

A. A. Franco and K. H. Xue, Carbon-based electrodes for lithium air batteries: Scientific and technological challenges from a modeling perspective, ECS Journal of Solid State Science and Technology, 2 (2013), 3084-3100.  doi: 10.1149/2.012310jss.

[14]

A. A. Franco, Multiscale modeling and numerical simulation of rechargeable lithium ion batteries: concepts, methods and challenges, RSC Advances, 3 (2013), 13027-13058. 

[15]

A. A. Franco, MS LIBER–T computational software, Available from: http://www.modeling-electrochemistry.com/ms-liber-t/

[16]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981.

[18]

L. IgnatA. Pozo and E. Zuazua, Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws, Math. Comp., 84 (2015), 1633-1662.  doi: 10.1090/S0025-5718-2014-02915-3.

[19]

H. Israel, Well-posedness and long time behavior of an Allen-Cahn type equation, Commun. Pure Appl. Anal., 12 (2013), 2811-2827.  doi: 10.3934/cpaa.2013.12.2811.

[20]

G. Karali and Y. Nagase, On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 127-137.  doi: 10.3934/dcdss.2014.7.127.

[21]

D. Kondepudi and I. Prigogine, Modern thermodynamics: From heat engines to dissipative structures, John & Wiley Sons Ltd., 8 (2014), p56.  doi: 10.3934/dcdss.2014.7.127.

[22]

H. G. Lee and J.-Y. Lee, A semi-analytical Fourier spectral method for the Allen-Cahn equation, Comput. Math. Appl., 68 (2014), 174-184.  doi: 10.1016/j.camwa.2014.05.015.

[23]

A. Makki and A. Miranville, Well-posedness for one-dimensional anisotropic Cahn-Hilliard and Allen-Cahn systems, Electron. J. Diff. Eq., 4 (2015), 15 pp.  doi: 10.1002/9781118698723.

[24]

K. MalekM. EikerlingQ. WangT. Navessin and Z. Liu, Self organization in catalyst layers of PEM Fuel Cells, J. Phys. Chem. C, 111 (2007), 13627-13634. 

[25]

K. Malek and A. A. Franco, Microstructural resolved modeling of aging mechanisms in PEMFC, J. Phys. Chem. B, 115 (2011), 8088-8101. 

[26]

Matlab's optimization toolbox, Available from: http://www.mathworks.fr/fr/products/optimization/index.html

[27]

A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277. 

[28]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs Series, Princeton University Press, 2005.

[29]

M. Pierre, Etude Numérique et Mathématique de Quelques Modéles de Transition de Phase, de Séparation de Phases et de Cristaux Liquides Habilitation à diriger les recherches, (in French), Université de Poitiers (Oct. 2011).

[30]

N. ProvatasJ. A. DantzigB. AthreyaP. ChanP. StefanovicN. Goldenfeld and K. R. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution, J. of the Minerals, Metals and Materials Society, 59 (2007), 83-90. 

[31]

C. SachsM. HildebrandS. VölkeningJ. Wintterlin and G. Ertl, Spatiotemporal self-organization in a surface reaction: From the atomic to the mesoscopic scale, Science, 293 (2001), 1635-1638.  doi: 10.1126/science.1062883.

[32]

A. SirimungkalaH.-D. Försterling and V. Dlask, Bromination Reactions Important in the Mechanism of the Belousov-Zhabotinsky System, J. Phys. Chem. A, 103 (1999), 1038-1043.  doi: 10.1021/jp9825213.

[33]

J. Shen and X. Yang, Numerical Approximations of Allen-Cahn and Cahn-Hilliard Equations, DCDS, Series A, 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.

Figure 1.  Solution of the Allen-Cahn equation (1) with $f(u)=u(u^2-1)$ for $\epsilon=0.004$ (A) and for $\epsilon=0.001$ (B)
Figure 2.  Multiphase decomposition of a signal
Figure 3.  Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, \ N=127$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)
Figure 4.  Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.005, \ T=0.5, \ N=127$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)
Figure 5.  Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, \ N=255$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)
Figure 6.  Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, \ N=511$ starting from the smooth initial datum $u_0=\cos(20\pi x)$. Solution obtained without control (C) and with the optimal control (D)
Figure 7.  Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.00001, \ T=0.001, \ N=127$ starting from a random initial datum. Solution obtained without control (C) and with the optimal control (D)
Figure 8.  Optimal control $\alpha(t)$ (A) and merit function (B) computed with $\epsilon=0.01, \ \Delta t = 0.00001, \ T=0.001, \ N=255$ starting from a random initial datum. Solution obtained without control (C) and with the optimal control (D)
Figure 9.  Optimal control $\alpha(t)$ (A) and merit function $J(u)=10 \parallel u-1\parallel +\parallel u+1\parallel$ (B) computed with $\epsilon=0.005, \ \Delta t = 0.00001, \ T=0.001, \ N=255$ starting from a random initial datum. Solution obtained without control (C) and with the optimal control (D)
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