A stable implementation measure of multi-transmitting formula in the numerical simulation of wave motion

The Multi-Transmitting Formula (MTF) proposed by Liao et al. is a local artificial boundary condition widely used in numerical simulations of wave propagation in an infinite medium, while the drift instability is usually caused in its numerical implementation. In view of a physical interpretation of the Gustafsson, Kreiss and Sundström criterion on numerical solutions of initial-boundary value problems in the hyperbolic partial differential equations, the mechanism of the drift instability of MTF was discussed, and a simple measure for eliminating the drift instability was proposed by introducing a modified operator into the MTF. Based on the theory of spherical wave propagation and damping effect of medium, the physical implication on modified operator was interpreted. And the effect of the modified operator on the reflection coefficient of MTF was discussed. Finally, the validity of the proposed stable implementation measure was verified by numerical tests of wave source problem and scattering problem.

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Ignaczak equation of elastodynamics

Mathematics and Mechanics of Solids ◽

10.1177/1081286518757284 ◽

2018 ◽

Vol 24 (11) ◽

pp. 3674-3713 ◽

Cited By ~ 4

Author(s):

Martin Ostoja-Starzewski

Keyword(s):

Wave Propagation ◽

Nonlinear Problems ◽

Initial Boundary ◽

Initial Boundary Value Problems ◽

Elastic Bodies ◽

Displacement Equation ◽

History Of ◽

Displacement Based ◽

Initial Boundary Value ◽

Main Ideas

The development of linear elastodynamics in pure stress-based formulation began over half-a-century ago as an alternative to the classical displacement-based treatment that came into existence two centuries ago in the school of mathematical physics in France. While the latter approach – fundamentally based on the Navier displacement equation of motion – remains the conventional setting for analysis of wave propagation in elastic bodies, the stress-based formulation and the advantages it offers in elastodynamics and its various extensions remain much less known. Since the key mathematical results of that formulation, as well as a series of applications, originated with J. Ignaczak in 1959 and 1963, the key relation is named the Ignaczak equation of elastodynamics. This review article presents the main ideas and results in the stress-based formulation from a common perspective, including (i) a history of early attempts to find a pure stress language of elastodynamics, (ii) a proposal to use such a language in solving the natural traction initial-boundary value problems of the theory, and (iii) various applications of the stress language to elastic wave propagation problems. Finally, various extensions of the Ignaczak equation of elastodynamics focused on dynamics of solids with interacting fields of different nature (classical or micropolar thermoelastic, fluid-saturated porous, piezoelectro-elastic) as well as nonlinear problems are reviewed.

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Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(68)90038-3 ◽

1968 ◽

Vol 24 (2) ◽

pp. 372-387 ◽

Cited By ~ 97

Author(s):

Kenneth L Cooke ◽

David W Krumme

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Boundary Value Problems ◽

Difference Equations ◽

Boundary Value ◽

Initial Boundary ◽

Hyperbolic Partial Differential Equations ◽

Initial Boundary Value Problems ◽

Partial Differential ◽

Initial Boundary Value

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Rigorous 2D Model for Study of Pulsed and Monochromatic Waves Propagation Near the Earth’s Surface

International Journal of Antennas and Propagation ◽

10.1155/2014/158297 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Seil S. Sautbekov ◽

Yuriy K. Sirenko ◽

Nataliya P. Yashina ◽

Aleksey A. Vertiy

Keyword(s):

Wave Propagation ◽

Electromagnetic Wave ◽

Electromagnetic Wave Propagation ◽

Model Problem ◽

Initial Boundary ◽

Initial Boundary Value Problems ◽

Practical Applications ◽

Waves Propagation ◽

Initial Boundary Value ◽

2D Model

A model problem considered in the paper allows solving rather complex 2D problems of the electromagnetic wave propagation with a required accuracy using conventional personal computers. The problems are of great importance for the theory and practical applications. The association of FDTD schemes with exact absorbing conditions makes up the basis for constructing models of the kind. This approach reduces the original open initial boundary value problems to the equivalent closed problems which can be solved numerically using the standard grid methods.

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