Numerical Simulations of Bifurcation Phenomena in Reaction Diffusion Systems

反応拡散系における分岐現象の数値シミュレーション

高石 武史

The bifurcation structures of two types of reaction diffusion systems are investigated.

A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic material is proposed. A phase field to represent the shape of the crack with a regularization parameter ε > 0 is introduced. The phase field model is derived as a gradient flow of this regularized energy that is approximated by the Francfort-Marigo type energy using the idea of Ambrosio and Tortorelli. Several numerical examples of the crack growth computed with an adaptive mesh finite element method are presented.

A simplified coupled reaction-diffusion system is derived from a diffusive membrane coupling of two reaction-diffusion systems of activator-inhibitor type. It is shown that the dynamics of the original decoupled systems persists for weak coupling, while new coupled stationary patterns of alternated type emerge at a critical strength of coupling and these become stable for strong coupling independently of the dynamics of the decoupled systems. The approach which is used here is singular perturbation techniques and complementarily numerical methods.

In this Thesis, the usefulness of the combination of the mathematical modeling and the numerical simulation for investigating the bifurcation phenomena on the nonlinear pattern formation in reaction diffusion system is found.

reaction-diffusion system

crack growth

phase field model

Turing pattern

diffusive coupling

numerical simulation

数学 [ 410 ]

英語

Copyright(c) by Author

(1) Phase Field Model for Mode III Crack Growth in Two Dimensional Elasticity, T. Takaishi and M. Kimura, Kybernetika 45(4) (2009), 605-614. ～を参照している

(2) Pattern Formation in Coupled Reaction-Diffusion System, T. Takaishi, M. Mimura and Y.Nishiura, Japan Journal of Industrial and Applied Mathematics 12(3) (1995) 385-424. ～を参照している

[URI] http://www.kybernetika.cz/content/2009/4/605 ～を参照している

[DOI] http://dx.doi.org/10.1007/BF03167236 ～を参照している

広島大学