Premium
Numerical investigation of 1D continuum dynamical models of discrete chain
Author(s) -
Andrianov I.V.,
Starushenko G.A.,
Weichert D.
Publication year - 2012
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201200057
Subject(s) - physics , boundary value problem , partial differential equation , continuum hypothesis , perturbation (astronomy) , mathematical analysis , mathematics , classical mechanics , statistical physics , quantum mechanics
This paper is devoted to the numerical analysis of the various continuum models of a 1D discrete media. Namely, we consider intermediate, quasi‐continuum, and improved quasi‐continuum models. The analysis of various receptions of continualization in a linear case is carried out. For this purpose, we consider the solution in the form of a traveling wave caused by the initial perturbation. The solution is obtained by Fourier transformation inverted numerically. Then we compare solutions of discrete and continuum models for wave velocity. Numerical calculations show that all the above described nonlocal theories qualitatively equally well describe the propagation of waves in a discrete media. The disadvantage of intermediate continuum model is the high order of differential equation. For finite chain this leads to the need for the formulation of boundary conditions that do not follow naturally from the source of the problem. The quasi‐continuum approximation yields results practically identical with those obtained on the basis of improved quasi‐continuum approximation. The most accurate approximation of a discrete medium gives improved quasi‐continuum approximation. The advantage of improved quasi‐continuum approximation is manifested in those cases where you need to describe the high modes of oscillation of the discrete finite chain.