A Dispersive Model for Wave Propagation in Periodic Heterogeneous Media Based on Homogenization With Multiple Spatial and Temporal Scales

2000 ◽  
Vol 68 (2) ◽  
pp. 153-161 ◽  
Author(s):  
W. Chen ◽  
J. Fish
Keyword(s):  
Wave Propagation ◽  
Heterogeneous Media ◽  
Initial Boundary ◽  
Multiple Scale ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Homogenization Theory ◽  
Temporal Scales ◽  
Spatial And Temporal Scales ◽  
Dispersive Model

A dispersive model is developed for wave propagation in periodic heterogeneous media. The model is based on the higher order mathematical homogenization theory with multiple spatial and temporal scales. A fast spatial scale and a slow temporal scale are introduced to account for the rapid spatial fluctuations as well as to capture the long-term behavior of the homogenized solution. By this approach the problem of secularity, which arises in the conventional multiple-scale higher order homogenization of wave equations with oscillatory coefficients, is successfully resolved. A model initial boundary value problem is analytically solved and the results have been found to be in good agreement with a numerical solution of the source problem in a heterogeneous medium.

scholarly journals Wave Propagation Across Acoustic/Biot’s Media: A Finite-Difference Method

2013 ◽  
Vol 13 (4) ◽  
pp. 985-1012 ◽  
Author(s):  
Guillaume Chiavassa ◽  
Bruno Lombard
Keyword(s):  
Wave Propagation ◽  
Numerical Methods ◽  
Heterogeneous Media ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Compressional Wave ◽  
Interface Conditions ◽  
Immersed Interface Method ◽  
Fluid Medium ◽  
Strang Splitting

AbstractNumerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media. Wave propagation is described by the usual acoustics equations (in the fluid medium) and by the low-frequency Biot’s equations (in the porous medium). Interface conditions are introduced to model various hydraulic contacts between the two media: open pores, sealed pores, and imperfect pores. Well-posedness of the initial-boundary value problem is proven. Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context: a fourth-order ADER scheme with Strang splitting for time- marching; a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory; and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution. Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions, demonstrating the accuracy of the approach.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

scholarly journals Damage Detection of A T-Shaped Panel by Wave Propagation Analysis in the Plane Stress / Wykrywanie Uszkodzen W Tarczy Typu T Z Uzyciem Analizy Propagacji Fal W Płaskim Stanie Naprezenia

2012 ◽  
Vol 58 (1) ◽  
pp. 3-24 ◽  
Author(s):  
M. Rucka ◽  
W. Witkowski ◽  
J. Chróscielewski ◽  
K. Wilde
Keyword(s):  
Wave Propagation ◽  
Plane Stress ◽  
Quadrature Rule ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Structural Element ◽  
Initial Excitation ◽  
Displacement Based ◽  
Initial Boundary Value ◽  
Uniformly Distributed Loads

Abstract A computational approach to analysis of wave propagation in plane stress problems is presented. The initial-boundary value problem is spatially approximated by the multi-node C0 displacement-based isoparametric quadrilateral finite elements. To integrate the element matrices the multi-node Gauss-Legendre-Lobatto quadrature rule is employed. The temporal discretization is carried out by the Newmark type algorithm reformulated to accommodate the structure of local element matrices. Numerical simulations are conducted for a T-shaped steel panel for different cases of initial excitation. For diagnostic purposes, the uniformly distributed loads subjected to an edge of the T-joint are found to be the most appropriate for design of ultrasonic devices for monitoring the structural element integrity

scholarly journals Higher Order Wave Equation with Logarithmic Source Term

2017 ◽  
Author(s):  
Sh. Hajrulla ◽  
L. Bezati ◽  
F. Hoxha
Keyword(s):  
Wave Equation ◽  
Boundary Value Problem ◽  
Source Term ◽  
Boundary Value ◽  
Initial Boundary ◽  
Global Weak Solution ◽  
Initial Boundary Value Problem ◽  
Logarithmic Sobolev Inequality ◽  
Higher Order ◽  
Initial Boundary Value

In this paper we study the initial boundary value problem for logarithmic Higher Order Wave equation. Introducing the Logarithmic Sobolev inequality and using the combination of Galerkin method, we consider the theorem of existence of a global weak solution to problem for the initial boundary value problem of the logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for logarithmic Higher Order Wave equation. The proof of the main theorem is given.

scholarly journals On Initial-Boundary Value Problem on Semiaxis for Generalized Kawahara Equation

2019 ◽  
Vol 65 (4) ◽  
pp. 683-699
Author(s):  
A. V. Faminskii ◽  
E. V. Martynov
Keyword(s):  
Boundary Value Problem ◽  
Global Solution ◽  
Existence And Uniqueness ◽  
Large Time ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Kawahara Equation ◽  
Initial Boundary Value

In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.

scholarly journals On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Said Mesloub
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Nonlocal Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Higher Order ◽  
Well Posedness ◽  
Initial Boundary Value

This paper is devoted to the study of the well-posedness of an initial boundary value problem for an odd higher order nonlinear pseudohyperbolic integrodifferential partial differential equation. We associate to the equationnnonlocal conditions andn+1classical conditions. Upon some a priori estimates and density arguments, we first establish the existence and uniqueness of the strongly generalized solution in a class of a certain type of Sobolev spaces for the associated linear mixed problem. On the basis of the obtained results for the linear problem, we apply an iterative process in order to establish the well-posedness of the nonlinear problem.

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