Far field expansion for anisotropic wave equations

A necessary ingredient for the numerical simulation of many time dependent phenomena in acoustics and aerodynamics is the imposition of accurate radiation conditions at artificial boundaries. The asymptotic analysis of propagating waves provides a rational approach to the development of such conditions. A far field asymptotic expansion of solutions of anisotropic wave equations is derived. This generalizes the well known Friedlander expansion for the standard wave operator. The expansion is used to derive a hierarchy of radiation conditions of increasing accuracy. Two numerical experiments are given to illustrate the utility of this approach. The first application is the study of unsteady vortical disturbances impinging on a flat plate; the second is the simulation of inviscid flow past an impulsively started cylinder

An absorbing boundary formulation for the stratified, linearized, ideal MHD equations based on an unsplit, convolutional perfectly matched layer

Perfectly matched layers are a very efficient and accurate way to absorb waves in media. We present a stable convolutional unsplit perfectly matched formulation designed for the linearized stratified Euler equations. However, the technique as applied to the Magneto-hydrodynamic (MHD) equations requires the use of a sponge, which, despite placing the perfectly matched status in question, is still highly efficient at absorbing outgoing waves. We study solutions of the equations in the backdrop of models of linearized wave propagation in the Sun. We test the numerical stability of the schemes by integrating the equations over a large number of wave periods.Comment: 8 pages, 7 figures, accepted, A &

A new construction of perfectly matched layers for the linearized Euler equations

Based on a PML for the advective wave equation, we propose two PML models for the linearized Euler equations. The derivation of the first model can be applied to other physical models. The second model was implemented. Numerical results are shown.Comment: submitted for publication on February 1st 2005 What's new: interface conditions for the first PML model, a 3D section, more numerical result

High-order full discretization for anisotropic wave equations

Two-dimensional linear wave equation in anisotropic media, on a rectangular domain with initial conditions and periodic boundary conditions, is considered. The energy of the problem is contemplated. The space discretization is reached by means of finite differences on a uniform grid, paying attention to the mixed derivative of the equation. The discrete energy of the semi-discrete problem is introduced. For the time integration of the system of ordinary differential equations obtained, a fourth order exponential splitting method, which is a geometric integrator, is proposed. This time integrator is efficient and easy to implement. The stability condition for time step and space step ratio is deduced. Numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided.MTM2015-66837-P del Ministerio de Economía y Competitivida

Absorbing boundary conditions for 3D anisotropic media

Seismic methods of subsurface exploration are based on mechanical wave propagation and the numerical modeling of these phenomena is a worthy tool that can be applied as a complement. Since small regions of Earth’s crust are studied, it is necessary to consider absorbing boundary conditions for solving the wave equations efficiently. Therefore, this work presents a derivation of low-order absorbing boundary conditions at the artificial boundaries of the computational domain with the purpose of minimizing spurious reflections. Laboring on a surface S, which separates disturbed and undisturbed regions of the domain, the equations for the absorbing boundary conditons are derived from kinematic conditions, considering continuity of the displacements across S and dynamic conditions, using momentum equations of the wave fronts arriving normally to S and expressions for the strain energy density along S. The arguments to obtain non-reflecting artificial boundaries are carried out for the more general case, through the generalized Hooke’s law. In this way, an isotropic medium is included in this derivation. The performance of these absorbing boundary conditions is illustrated for different models of effective anisotropy -vertically and tilted transversely isotropic media- and, obviously, for isotropic media. The numerical simulations use these absorbing boundary conditions to propagate waves in anisotropic media using an iterative domain decomposition finite element procedure that is implemented in machines with parallel architecture.Publicado en: Mecánica Computacional vol. XXXV, no. 2Facultad de Ingenierí

Absorbing boundary conditions for 3D anisotropic media

Seismic methods of subsurface exploration are based on mechanical wave propagation and the numerical modeling of these phenomena is a worthy tool that can be applied as a complement. Since small regions of Earth’s crust are studied, it is necessary to consider absorbing boundary conditions for solving the wave equations efficiently. Therefore, this work presents a derivation of low-order absorbing boundary conditions at the artificial boundaries of the computational domain with the purpose of minimizing spurious reflections. Laboring on a surface S, which separates disturbed and undisturbed regions of the domain, the equations for the absorbing boundary conditons are derived from kinematic conditions, considering continuity of the displacements across S and dynamic conditions, using momentum equations of the wave fronts arriving normally to S and expressions for the strain energy density along S. The arguments to obtain non-reflecting artificial boundaries are carried out for the more general case, through the generalized Hooke’s law. In this way, an isotropic medium is included in this derivation. The performance of these absorbing boundary conditions is illustrated for different models of effective anisotropy -vertically and tilted transversely isotropic media- and, obviously, for isotropic media. The numerical simulations use these absorbing boundary conditions to propagate waves in anisotropic media using an iterative domain decomposition finite element procedure that is implemented in machines with parallel architecture.Publicado en: Mecánica Computacional vol. XXXV, no. 2Facultad de Ingenierí

Absorbing boundary conditions for 3D anisotropic media

Seismic methods of subsurface exploration are based on mechanical wave propagation and the numerical modeling of these phenomena is a worthy tool that can be applied as a complement. Since small regions of Earth’s crust are studied, it is necessary to consider absorbing boundary conditions for solving the wave equations efficiently. Therefore, this work presents a derivation of low-order absorbing boundary conditions at the artificial boundaries of the computational domain with the purpose of minimizing spurious reflections. Laboring on a surface S, which separates disturbed and undisturbed regions of the domain, the equations for the absorbing boundary conditons are derived from kinematic conditions, considering continuity of the displacements across S and dynamic conditions, using momentum equations of the wave fronts arriving normally to S and expressions for the strain energy density along S. The arguments to obtain non-reflecting artificial boundaries are carried out for the more general case, through the generalized Hooke’s law. In this way, an isotropic medium is included in this derivation. The performance of these absorbing boundary conditions is illustrated for different models of effective anisotropy -vertically and tilted transversely isotropic media- and, obviously, for isotropic media. The numerical simulations use these absorbing boundary conditions to propagate waves in anisotropic media using an iterative domain decomposition finite element procedure that is implemented in machines with parallel architecture.Publicado en: Mecánica Computacional vol. XXXV, no. 2Facultad de Ingenierí