相场法的准确性与边界曲率的关系研究
摘要
螺旋波的传播常常涉及零流边界的情况,有很多模拟真实系统的模型,它们的边界都是不规则的。如何准确地构造和求解这类不规则的零流边界,对各种真实系统的准确模拟和研究都有很重要的意义。
我们可以使用有限差分法,通过定义边界外附加的格点来解决这个问题。然而,这种处理方式一个不太理想的特征就是,同一个格点可能被多次赋值,这个格点上的值取决于最后更新了哪个邻近格点,而随着时间的变化,不同形状边界下的格点可能要通过不同的算法来确定。另一种常用的方法称为有限元法,这种方法可以很自然地处理零流边界条件。但是在相同格点间距的情况下,使用有限元法来进行数值模拟要比使用有限差分法更费时,也更难以应用。
我们在第二章介绍了一种新的算法,可以在任何几何形状的边界上实现零流条件。这种称为相场的方法已经成功运用于各种问题中,包括树状凝固(dendritic solidification)、指进现象(viscous fingering)、裂纹扩展(crack propagation)、囊泡塌缩(the tumbling of vesicles)以及细胞内部动力学(intracellular dynamics)。相场法的主要优点首先在于它能自动处理复杂几何结构中的边界问题,通过附加一个辅助的场,相场法就可以使边界满足零流条件。另一个优点是相场法可以应用于移动的边界。
在第三章和第四章,我们通过对Barkley模型和FHN模型的数值模拟,研究了相场法的准确性和边界曲率的关系。我们发现与有限差分法相比较,在曲率为0,即边界为一条直线的时候,两者的差别很小。而在边界曲率不为0的情形下,随着曲率的变化,相场法的准确性对曲率存在一定的依赖关系。我们可以说,相场法的准确性是和边界曲率密切相关的。
The propagation of spiral waves often refers to the no-flux boundary condition. There are many models simulating realistic systems, whose boundaries are irregular. How to construct and solve these irregular no-flux boundaries exactly is important to the simulation and the research of various realistic systems.
We can use finite difference method to overcome this difficulty by defining additional external grid points. However, the undesirable feature of this approach is that it is possible for the same grid cell to have different values depending on which neighboring cell is being updated.
Another commonly used technique, finite element method, is able to handle no-flux boundary condition naturally. However, these methods are generally slower than finite differences for equivalent grid spacing and are more cumbersome to implement.
We introduce a new method in the chapter 2 which can implement no-flux boundary condition in arbitrary geometries. The algorithm called phase field method has been applied successfully to a wide range of problems including dendritic solidification, viscous fingering, crack propagation, the tumbling of vesicles and intracellular dynamics. This method has the chief advantage that it avoids the need to track the interface explicitly to establishing no-flux boundary condition by introducing an auxiliary field that makes the interface spatially diffuse. What's more, phase field method can be extended to modeling moving boundaries.
In chapter 3 and chapter 4, we researched the relationship between the accuracy of phase field method and the curvature of boundary by simulating the Barkley model and the FHN model. When the curvature is 0, we found that the difference between the phase field method and the finite difference method is very small. When the curvature is not 0, the difference is concerned to the change of curvature. Thus, we can say there is a close correlation between the accuracy of phase field method and the boundary curvature.
我们可以使用有限差分法,通过定义边界外附加的格点来解决这个问题。然而,这种处理方式一个不太理想的特征就是,同一个格点可能被多次赋值,这个格点上的值取决于最后更新了哪个邻近格点,而随着时间的变化,不同形状边界下的格点可能要通过不同的算法来确定。另一种常用的方法称为有限元法,这种方法可以很自然地处理零流边界条件。但是在相同格点间距的情况下,使用有限元法来进行数值模拟要比使用有限差分法更费时,也更难以应用。
我们在第二章介绍了一种新的算法,可以在任何几何形状的边界上实现零流条件。这种称为相场的方法已经成功运用于各种问题中,包括树状凝固(dendritic solidification)、指进现象(viscous fingering)、裂纹扩展(crack propagation)、囊泡塌缩(the tumbling of vesicles)以及细胞内部动力学(intracellular dynamics)。相场法的主要优点首先在于它能自动处理复杂几何结构中的边界问题,通过附加一个辅助的场,相场法就可以使边界满足零流条件。另一个优点是相场法可以应用于移动的边界。
在第三章和第四章,我们通过对Barkley模型和FHN模型的数值模拟,研究了相场法的准确性和边界曲率的关系。我们发现与有限差分法相比较,在曲率为0,即边界为一条直线的时候,两者的差别很小。而在边界曲率不为0的情形下,随着曲率的变化,相场法的准确性对曲率存在一定的依赖关系。我们可以说,相场法的准确性是和边界曲率密切相关的。
The propagation of spiral waves often refers to the no-flux boundary condition. There are many models simulating realistic systems, whose boundaries are irregular. How to construct and solve these irregular no-flux boundaries exactly is important to the simulation and the research of various realistic systems.
We can use finite difference method to overcome this difficulty by defining additional external grid points. However, the undesirable feature of this approach is that it is possible for the same grid cell to have different values depending on which neighboring cell is being updated.
Another commonly used technique, finite element method, is able to handle no-flux boundary condition naturally. However, these methods are generally slower than finite differences for equivalent grid spacing and are more cumbersome to implement.
We introduce a new method in the chapter 2 which can implement no-flux boundary condition in arbitrary geometries. The algorithm called phase field method has been applied successfully to a wide range of problems including dendritic solidification, viscous fingering, crack propagation, the tumbling of vesicles and intracellular dynamics. This method has the chief advantage that it avoids the need to track the interface explicitly to establishing no-flux boundary condition by introducing an auxiliary field that makes the interface spatially diffuse. What's more, phase field method can be extended to modeling moving boundaries.
In chapter 3 and chapter 4, we researched the relationship between the accuracy of phase field method and the curvature of boundary by simulating the Barkley model and the FHN model. When the curvature is 0, we found that the difference between the phase field method and the finite difference method is very small. When the curvature is not 0, the difference is concerned to the change of curvature. Thus, we can say there is a close correlation between the accuracy of phase field method and the boundary curvature.
引文
[1]G. Nicolis and I. Prigogine, Self-organization in nonequilibrium systems, John Wileyand Sons,1977.
[2]A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. B, 1952,237:37-72.
[3]Q. Ouyang and H. L. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns, Nature (London),1991,352:610-612.
[4]欧阳颀,反应扩散系统中的斑图动力学.上海：上海科技教育出版社,2000.
[5]A. N. Zaikin and A. M. Zhabotinsky, Concentration wave propagation in two-dimensional liquid-phase self-oscillating system, Nature,1970,225:535-537.
[6]S. Jakubith, H. H. Rotermund, W. Engel, A. von Oertzen, and G. Ertl, Spatiotemporal concentration patterns in a surface reaction:Propagating and standing waves,rotating spirals, and turbulence, Phys. Rev. Lett.,1990,65:3013-3016.
[7]J. J. Tyson and J. D. Murray, Cyclic AMP waves during aggregation of Dictyosteliumamoebae, Development,1989,106:421-426.
[8]J. M. Davidenko, A. V. Pertsov, R. Salomonsz,W. Baxter, and J. Jalife, Stationary anddrifting spiral waves of excitation in isolated cardiac muscle, Nature,1992, 355:349-351.
[9]J.J. Tyson and J.P. Keener, Singular perturbation theory of traveling waves in excitable media, Physica D,1988,32:327-361.
[10]B. Lindner, J. Garc'ia-Ojalvo, A. Neiman, and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep.,2004,392:321-424.
[11]E.M. Izhikevich, Dynamical systems in neuroscience:The geometry of excitability andbursting, MTI Press,2007.
[12]P. Jung, A. Cornell-Bell, F. Moss, S. Kadar, J. Wang, and K. Showalter, Noise sustained waves in subexcitable media:From chemical waves to brain waves, Chaos, 1998,8:567-575
[13]A. T. Winfree, When time breaks down, NJ:Princeton,1987.
[14]V. Krinsky and H. Swinney (eds), Wave and patterns in biological and chemical excitable media. Amsterdam:North-Holland,1991.
[15]A.T.Winfree,Chaos 1,303(1991).
[16]V.S.Zykov,Modelling of Wave Precesses in Excitable Media(Manchester Univ,Press,Manchester,1988).
[17]J.J.Tyson and J.P.Keener,Physica 32D,327(1988).
[18]A.S.Mikhailov and V.S.Zykov,Physica 52D,379(1991).
[19]E.Meron,Phys,Reperts,218,1 (1992).
[20]R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,Biophysical J.,1961,1:445-466.
[21]J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon,1962, Proc IRE.50:2061-2070.
[22]A. V. Panfilov and J. P. Keener, "Re-entry in an anatomical model of the heart," Chaos, Solitons Fractals 5,681-689 s1995d.
[23]F. Xie, Z. Qu, J. Yang, A. Baher, J. N. Weiss, and A. Garfinkel, "A simulation study of the effects of cardiac anatomy in ventricular fibrillation," J. Clin. Invest.113, 686-693 s2004d
[24]J. M. Rogers, "Wave front fragmentation due to ventricular geometry in a model of the rabbit heart," Chaos 12,779-787 s2002d.
[25]J. M. Meunier, J. C. Eason, and N. A. Trayanova, "Termination of reentry by a long-lasting AC shock in a slice of canine heart:a computational study," J. Cardiovasc. Electrophysiol.13,1253-1261 s2002d.
[26]F. J. Vetter and A. D. McCulloch, "Three-dimensional stress and strain in passive rabbit left ventricle:a model study," Ann. Biomed. Eng.28,781-792 s2000d.
[27]A. Karma and W.-J. Rappel, "Quantitative phase-field modeling of dendritic growth in two and three dimensions," Phys. Rev. E 57,4323-4349s1998d.
[28]R. Folch, J. Casademunt, A. Hernandez-Machado, and L. Ramirez-Piscina, "Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast," Phys. Rev. E 60,1724-1733s1999d.
391. S. Aranson, V. A. Kalatsky, and V. M. Vinokur, "Continuum field description of crack propagation," Phys. Rev. Lett.85,118-121 s2000d.
[30]A. Karma, D. A. Kessler, and H. Levine, "Phase-field model of mode Ⅲ dynamic fracture," Phys. Rev. Lett.87,045501 s2001d.
[31]T. Biben and C. Misbah, "Tumbling of vesicles under shear flow within an advected-field approach," Phys. Rev. E 67,031908 s2003d.
[32]J. Kockelkoren, H. Levine, and W. J. Rappel, "Computational approach for modeling intra and extracellular dynamics," Phys. Rev. E 68,037702 s2003d.
[33]D.Y. Shao, W.J.Rappel, H.Levine, "Computational Model for Cell Morphodynamics," 10.1103/PhysRevLett.105.108104
[34]F.H.Fenton,E.M.Cherry,A.Karma,W.J.Rappel, "Modeling wave propagation in realistic heart geometries using the phase-field method/"CHAOS 15,013502 (2005)
[35]A.Bueno-Orovio,Victor.M.Perez-Garcia, "Spectral Smoothed Boundary Methods: The Role of External Boundary Conditions,"Wiley InterScience.DOI 10.1002/num.20103.
[36]. 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), 1084 1095.
[37]. A. N. Zaikin and A. M. Zhabotinsky, Concentration wave propagation in two-dimensional liquidphase self-oscillating system, Nature 225 (1970),535 537.
[38]A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J Physiol 117 (1952), 500 544.
[39]. G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, J Physiol 268(1977),177 210.
[40]. V. Iyer, R. Mazhari, and R. L. Winslow, A computational model of the human left ventricular epicardial myocyte, Biophys J 87 (2004),1507 1525.
[41]. F. Fenton and A. Karma, Vortex dynamics in three-dimensional continuous myocardium with fiber rotation:filament instability and fibrillation, Chaos 8 (1998), 20 47.
[2]A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. B, 1952,237:37-72.
[3]Q. Ouyang and H. L. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns, Nature (London),1991,352:610-612.
[4]欧阳颀,反应扩散系统中的斑图动力学.上海：上海科技教育出版社,2000.
[5]A. N. Zaikin and A. M. Zhabotinsky, Concentration wave propagation in two-dimensional liquid-phase self-oscillating system, Nature,1970,225:535-537.
[6]S. Jakubith, H. H. Rotermund, W. Engel, A. von Oertzen, and G. Ertl, Spatiotemporal concentration patterns in a surface reaction:Propagating and standing waves,rotating spirals, and turbulence, Phys. Rev. Lett.,1990,65:3013-3016.
[7]J. J. Tyson and J. D. Murray, Cyclic AMP waves during aggregation of Dictyosteliumamoebae, Development,1989,106:421-426.
[8]J. M. Davidenko, A. V. Pertsov, R. Salomonsz,W. Baxter, and J. Jalife, Stationary anddrifting spiral waves of excitation in isolated cardiac muscle, Nature,1992, 355:349-351.
[9]J.J. Tyson and J.P. Keener, Singular perturbation theory of traveling waves in excitable media, Physica D,1988,32:327-361.
[10]B. Lindner, J. Garc'ia-Ojalvo, A. Neiman, and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep.,2004,392:321-424.
[11]E.M. Izhikevich, Dynamical systems in neuroscience:The geometry of excitability andbursting, MTI Press,2007.
[12]P. Jung, A. Cornell-Bell, F. Moss, S. Kadar, J. Wang, and K. Showalter, Noise sustained waves in subexcitable media:From chemical waves to brain waves, Chaos, 1998,8:567-575
[13]A. T. Winfree, When time breaks down, NJ:Princeton,1987.
[14]V. Krinsky and H. Swinney (eds), Wave and patterns in biological and chemical excitable media. Amsterdam:North-Holland,1991.
[15]A.T.Winfree,Chaos 1,303(1991).
[16]V.S.Zykov,Modelling of Wave Precesses in Excitable Media(Manchester Univ,Press,Manchester,1988).
[17]J.J.Tyson and J.P.Keener,Physica 32D,327(1988).
[18]A.S.Mikhailov and V.S.Zykov,Physica 52D,379(1991).
[19]E.Meron,Phys,Reperts,218,1 (1992).
[20]R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,Biophysical J.,1961,1:445-466.
[21]J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon,1962, Proc IRE.50:2061-2070.
[22]A. V. Panfilov and J. P. Keener, "Re-entry in an anatomical model of the heart," Chaos, Solitons Fractals 5,681-689 s1995d.
[23]F. Xie, Z. Qu, J. Yang, A. Baher, J. N. Weiss, and A. Garfinkel, "A simulation study of the effects of cardiac anatomy in ventricular fibrillation," J. Clin. Invest.113, 686-693 s2004d
[24]J. M. Rogers, "Wave front fragmentation due to ventricular geometry in a model of the rabbit heart," Chaos 12,779-787 s2002d.
[25]J. M. Meunier, J. C. Eason, and N. A. Trayanova, "Termination of reentry by a long-lasting AC shock in a slice of canine heart:a computational study," J. Cardiovasc. Electrophysiol.13,1253-1261 s2002d.
[26]F. J. Vetter and A. D. McCulloch, "Three-dimensional stress and strain in passive rabbit left ventricle:a model study," Ann. Biomed. Eng.28,781-792 s2000d.
[27]A. Karma and W.-J. Rappel, "Quantitative phase-field modeling of dendritic growth in two and three dimensions," Phys. Rev. E 57,4323-4349s1998d.
[28]R. Folch, J. Casademunt, A. Hernandez-Machado, and L. Ramirez-Piscina, "Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast," Phys. Rev. E 60,1724-1733s1999d.
391. S. Aranson, V. A. Kalatsky, and V. M. Vinokur, "Continuum field description of crack propagation," Phys. Rev. Lett.85,118-121 s2000d.
[30]A. Karma, D. A. Kessler, and H. Levine, "Phase-field model of mode Ⅲ dynamic fracture," Phys. Rev. Lett.87,045501 s2001d.
[31]T. Biben and C. Misbah, "Tumbling of vesicles under shear flow within an advected-field approach," Phys. Rev. E 67,031908 s2003d.
[32]J. Kockelkoren, H. Levine, and W. J. Rappel, "Computational approach for modeling intra and extracellular dynamics," Phys. Rev. E 68,037702 s2003d.
[33]D.Y. Shao, W.J.Rappel, H.Levine, "Computational Model for Cell Morphodynamics," 10.1103/PhysRevLett.105.108104
[34]F.H.Fenton,E.M.Cherry,A.Karma,W.J.Rappel, "Modeling wave propagation in realistic heart geometries using the phase-field method/"CHAOS 15,013502 (2005)
[35]A.Bueno-Orovio,Victor.M.Perez-Garcia, "Spectral Smoothed Boundary Methods: The Role of External Boundary Conditions,"Wiley InterScience.DOI 10.1002/num.20103.
[36]. 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), 1084 1095.
[37]. A. N. Zaikin and A. M. Zhabotinsky, Concentration wave propagation in two-dimensional liquidphase self-oscillating system, Nature 225 (1970),535 537.
[38]A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J Physiol 117 (1952), 500 544.
[39]. G. W. Beeler and H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres, J Physiol 268(1977),177 210.
[40]. V. Iyer, R. Mazhari, and R. L. Winslow, A computational model of the human left ventricular epicardial myocyte, Biophys J 87 (2004),1507 1525.
[41]. F. Fenton and A. Karma, Vortex dynamics in three-dimensional continuous myocardium with fiber rotation:filament instability and fibrillation, Chaos 8 (1998), 20 47.