We develop a new mathematical model for describing a dynamical system at
limited resolution (or finite scale),and we give precise meaning to the notion
of a dynamical system having some property at finite resolution. Open covers
are used to approximate the topology of the phase space in a finite way,and the
dynamical system is represented by means of a combinatorial multivalued map.We
translate notions of transitivity and mixing known for general dynamical
systems into the finite setting in a consistent way. Moreover, we formulate
equivalent conditions for these properties in terms of graphs,and provide
effective algorithms for their verification. As an application we show that the
Henon attractor is topologically mixing at all resolutions coarser than 10^-5.
martes, 13 de octubre de 2009
Finite resolution dynamics. (arXiv:0910.2319v1 [math.DS])
Adaptive finite element computation of dielectric and mechanical intensity factors in piezoelectrics with impermeable cracks
The paper deals with the application of an adaptive, hierarchic-iterative finite element technique to solve two-dimensional electromechanical boundary value problems with impermeable cracks in piezoelectric plates. In order to compute the dielectric and mechanical intensity factors, the interaction integral technique is used. The iterative finite element solver takes advantage of a sequence of solutions on hierarchic discretizations. Based on an a posteriori error estimation, the finite element mesh is locally refined or coarsened in each step. Two crack configurations are investigated in an infinite piezoelectric plate: A finite straight crack and a finite kinked crack. Fast convergence of the numerical intensity factors to the corresponding analytical solution is exemplarily proved during successive adaptive steps for the first configuration. Similar tendency can be observed for the second configuration. Furthermore, the computed intensity factors for the kinks are found to coincide well with the corresponding analytical values. In order to simulate the kinks spreading from a straight crack, the finite element mesh is modified automatically with a specially developed algorithm. This forms the basis for a fully adaptive simulation of crack propagation. Copyright © 2009 John Wiley & Sons, Ltd.
An approximate solution to an initial boundary value problem to the one-dimensional Kuramoto-Sivashinsky equation
The aim of this paper is to present an approximate solution to the initial boundary-valued problem for the one-dimensional Kuramoto-Sivashinsky equation. The Fourier Method is combined with the Adomian's Decomposition Method in order to provide an approximate solution. One example of application is presented. Copyright © 2009 John Wiley & Sons, Ltd.
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