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.

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 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

On the Computations of Gas-Solid Mixture Two-Phase Flow

2014 ◽  
Vol 6 (01) ◽  
pp. 49-74 ◽  
Author(s):  
D. Zeidan ◽  
R. Touma
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Test Cases ◽  
Two Phase ◽  
Solid Mixture ◽  
Flow Problems ◽  
Model Equations ◽  
Initial Boundary Value

AbstractThis paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model. The HLL Riemann solver is applied to solve the Riemann problem for the model equations. This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture. Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results. To demonstrate the robustness, effectiveness and capability of these methods, the model results are compared with reference solutions. In addition to that, these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature. The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.

scholarly journals Modeling the Interaction of Solit-Like Pulse Signals with Electromagnetic Shields in the Form of Heterogeneous Media

2020 ◽  
pp. 1-5
Author(s):  
MA Aliseyko ◽  
OV Boiprav ◽  
NN Grinchik ◽  
AV Tarasevich
Keyword(s):  
Reflection Coefficient ◽  
Electromagnetic Radiation ◽  
Electromagnetic Waves ◽  
Heterogeneous Media ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Copper Plate ◽  
Spectral Lines ◽  
Matching Conditions ◽  
Initial Boundary Value

Modeling of electromagnetic radiation interaction with electromagnetic shields is an important problem that is solved during their development. By solving this problem, as a rule, it is possible to reduce the time and financial costs necessary to obtain electromagnetic shields, characterized by the required electromagnetic radiation attenuation and reflection coefficient. Currently, electromagnetic shields are usually developed in the form of heterogeneous media, which is due to lower the values of electromagnetic radiation reflection coefficient of such shields in comparison with the shields in the form of homogeneous media made in the form of continuous materials sheets. The authors of the article have proposed a new approach to modeling of electromagnetic radiation interaction with electromagnetic shields in the form of heterogeneous media. This approach is based on the use of difference schemes of end-to-end counting without explicitly distinguishing the interface between adjacent media, fulfilling the conditions of equality of full currents and charge flows on this boundary, and also on describing electromagnetic waves in the form of soliton-like signals, characterized by a greater penetration depth compared to other waves used in currently in the process of modeling (rectangular, sawtooth, etc). When using soliton-like signals that take into account broadening of spectral lines, the matching conditions for the first initial-boundary-value problem are satisfied. The existing software packages for electrodynamic tasks solving don’t take into account the matching conditions. On the base of the proposed approach, using the COMSOL Multiphysics software package, the authors first simulated the electromagnetic radiation interaction with a silver-based shield, the surface of which is rough and characterized by roughness sizes significantly smaller than the length of electromagnetic waves interacting with them, and with a shield in the form of a copper plate, the surface of which has slots, diameters and the depth of which is much less than the length of the electromagnetic waves interacting with them. The selection of these objects of study is due to the wide use of copper and silver for the electromagnetic shields manufacture, as well as the prospects for the development of shields formation technology, which consists in the heterogenization of the solid sheet metal materials surface.

Complex fractional Zener model of wave propagation in ℝ

2018 ◽  
Vol 21 (5) ◽  
pp. 1313-1334
Author(s):  
Teodor M. Atanacković ◽  
Marko Janev ◽  
Sanja Konjik ◽  
Stevan Pilipović
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Zener Model ◽  
Classical Wave ◽  
Classical Wave Equation ◽  
Fractional Zener Model ◽  
Initial Boundary Value ◽  
Derivatives Of

Abstract The classical wave equation is generalized within fractional framework, by using fractional derivatives of real and complex order in the constitutive equation, so that it describes wave propagation in one dimensional infinite viscoelastic rod. We analyze existence, uniqueness and properties of solutions to the corresponding initial-boundary value problem for generalized wave equation. Also, we provide a comparative analysis with the case of the same equation but considered on a bounded or half-bounded spatial domain. We conclude our investigation with a numerical example that illustrates obtained results.

Blow-up of error estimates in time-fractional initial-boundary value problems

2020 ◽  
Author(s):  
Hu Chen ◽  
Martin Stynes
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Error Bounds ◽  
Time Domain ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

Abstract Time-fractional initial-boundary value problems of the form $D_t^\alpha u-p \varDelta u +cu=f$ are considered, where $D_t^\alpha u$ is a Caputo fractional derivative of order $\alpha \in (0,1)$ and the spatial domain lies in $\mathbb{R}^d$ for some $d\in \{1,2,3\}$. As $\alpha \to 1^-$ we prove that the solution $u$ converges, uniformly on the space-time domain, to the solution of the classical parabolic initial-boundary value problem where $D_t^\alpha u$ is replaced by $\partial u/\partial t$. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as $\alpha \to 1^-$, as we demonstrate. We show that in some cases these analyses can be modified to obtain robust error bounds that do not blow up as $\alpha \to 1^-$.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

Sign in / Sign up

Export Citation Format

Share Document