Numerical investigation of 1D continuum dynamical models of discrete chain

Abstract

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.

On the minimization of wave reflection at the interface of a discrete system and a dispersively similar continuum

“…This value of the coefficient A 21 has been earlier obtained in, for example, [12,23,24]. The dispersion curves for the chain, for the gradient continuum with A 21 given by Eq.…”

From Ziegler to Beck’s column: a nonlocal approach

“…The pseudo-differential Laplacian operator can be efficiently approximated by the Padé's approximant (see, for instance [2,32,33,39] or [3] for axial wave applications):…”

Nonstandard continualization of 1D lattice with next-nearest interactions. Low order ODEs and enhanced prediction of the dispersive behavior

“…Nonstandard continualization techniques have been also used [6][7][8][9]. These techniques consist in transforming the discrete equations into pseudodifferential equations, which are later expanded by Taylor series [6,7,9] or by Pad e approximants [8].…”

“…Nonstandard continualization techniques have been also used [6][7][8][9]. These techniques consist in transforming the discrete equations into pseudodifferential equations, which are later expanded by Taylor series [6,7,9] or by Pad e approximants [8]. Bacigalupo and Gambarotta [7] proposed an enhanced continualization scheme in which, besides using pseudodifferential operators, a central difference scheme is employed to discretize the first spatial derivative of the continuous displacement, achieving a continuous governing equation of higher order than that corresponding to the classical one.…”