Wave Propagation In Unbounded Media Research Articles

Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains

This review article revisits and outlines the perfectly matched layer (PML) method and its various formulations developed over the past 25 years for the numerical modeling and simulation of wave propagation in unbounded media. Based on the concept of complex coordinate stretching, an efficient mixed displacement-strain unsplit-field PML formulation for second-order (displacementbased) linear elastodynamic equations is then proposed for simulating the propagation and absorption of elastic waves in unbounded (infinite or semi-infinite) domains. Both time-harmonic (frequency-domain) and time-dependent (timedomain) PML formulations are derived for two-and three-dimensional linear elastodynamic problems. Through the introduction of only a few additional variables governed by low-order auxiliary differential equations, the resulting mixed timedomain PML formulation is second-order in time, thereby allowing the use of standard time integration schemes commonly employed in computational structural dynamics and thus facilitating the incorporation into existing displacementbased finite element codes. For computational efficiency, the proposed timedomain PML formulation is implemented using a hybrid approach that couples a mixed (displacement-strain) formulation for the PML region with a classical (displacement-based) formulation for the physical domain of interest, using a standard Galerkin finite element method (FEM) for spatial discretization and a Newmark time scheme coupled with a finite difference (Crank-Nicolson) time scheme for time sampling. Numerical experiments show the performances of the PML method in terms of accuracy, efficiency and stability for two-dimensional linear elastodynamic problems in single-and multi-layer isotropic homogeneous elastic media.

FE/PML numerical schemes for dynamic soil-structure interaction and seismic wave propagation analysis

The numerical solution of wave propagation in unbounded media is of great importance in many engineering applications. An alternative approach to the application of artificial boundaries in the FEM is the Perfectly Matched Layer (PML) technique. The main idea of this technique is that an absorbing layer is applied around the finite region of interest absorbing almost perfectly waves of all incident angles and frequencies propagating from the bounded region. A perfectly matched layer can be obtained by performing a complex-valued coordinate transformation inside the layer which can be expressed either as a change in the material properties or as a stretching of the coordinates. Both approaches are applied to the PML formulation and implemented in a commercial FEM program. All simulation results demonstrate the accuracy of the FE/PML models and the ability of the developed models to deal with complex dynamic soil-structure interaction systems.

Application of a perfectly matched layer in seismic wavefield simulation with an irregular free surface

Recently, an effective and powerful approach for simulating seismic wave propagation in elastic media with an irregular free surface was proposed. However, in previous studies, researchers used the periodic condition and/or sponge boundary condition to attenuate artificial reflections at boundaries of a computational domain. As demonstrated in many literatures, either the periodic condition or sponge boundary condition is simple but much less effective than the well-known perfectly matched layer boundary condition. In view of this, we intend to introduce a perfectly matched layer to simulate seismic wavefields in unbounded models with an irregular free surface. We first incorporate a perfectly matched layer into wave equations formulated in a frequency domain in Cartesian coordinates. We then transform them back into a time domain through inverse Fourier transformation. Afterwards, we use a boundary-conforming grid and map a rectangular grid onto a curved one, which allows us to transform the equations and free surface boundary conditions from Cartesian coordinates to curvilinear coordinates. As numerical examples show, if free surface boundary conditions are imposed at the top border of a model, then it should also be incorporated into the perfectly matched layer imposed at the top-left and top- right corners of a 2D model where the free surface boundary conditions and perfectly matched layer encounter; otherwise, reflections will occur at the intersections of the free surface and the perfectly matched layer, which is confirmed in this paper. So, by replacing normal second derivatives in wave equations in curvilinear coordinates with free surface boundary conditions, we successfully implement the free surface boundary conditions into the perfectly matched layer at the top-left and top-right corners of a 2D model at the surface. A number of numerical examples show that the perfectly matched layer constructed in this study is effective in simulating wave propagation in unbounded media and the algorithm for implementation of the perfectly matched layer and free surface boundary conditions is stable for long-time wavefield simulation on models with an irregular free surface.

A simple multi‐directional absorbing layer method to simulate elastic wave propagation in unbounded domains

AbstractThe numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries. This point is critical for the analysis of wave propagation in heterogeneous or layered solids. Various techniques such as Absorbing Boundary Conditions, infinite elements or Absorbing Boundary Layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this paper, a simple absorbing layer method is proposed: it is based on a Rayleigh/Caughey damping formulation which is often already available in existing Finite Element softwares. The principle of the Caughey Absorbing Layer Method is first presented (including a rheological interpretation). The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences. A comparison with the PML method is first performed for pure P‐waves and the method is shown to be reliable in a more complex 2D case involving various wave types and incidences. It may thus be used for various types of problems involving elastic waves (e.g. machine vibrations, seismic waves, etc.). Copyright © 2010 John Wiley & Sons, Ltd.

A simple numerical absorbing layer method in elastodynamics

The numerical analysis of elastic wave propagation in unbounded media may be difficult to handle due to spurious waves reflected at the model artificial boundaries. Several sophisticated techniques such as nonreflecting boundary conditions, infinite elements or absorbing layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this Note, a simple and efficient absorbing layer method is proposed in the framework of the Finite Element Method. This method considers Rayleigh/Caughey damping in the absorbing layer and its principle is presented first. The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types (surface waves, body waves) and incidences (normal to grazing). The method is shown to be efficient for different types of elastic waves and may thus be used for various elastodynamic problems in unbounded domains.

Finite element method for time dependent scattering: nonreflecting boundary condition, adaptivity, and energy decay

An adaptive finite element method is developed for acoustic wave propagation in unbounded media. The efficiency and high accuracy of the method are achieved by combining an exact nonreflecting boundary condition [SIAM J. Appl. Math. 55 (1995) 280; J. Comput. Phys. 127 (1996) 52] with space–time adaptivity [East–West J. Numer. Math. 7(4) (1999) 263]. Hence the computational effort is concentrated where needed, while the artificial boundary can be brought as close as desired to the scatterer. Both features combined yield high accuracy and keep the number of unknowns to a minimum. An energy inequality is derived for the initial-boundary value problem at the continuous level. Together with an implicit second order time discretization it guarantees unconditional stability of the semi-discrete system. The resulting fully discrete linear system that needs to be solved every time step is unsymmetric but can be transformed into an equivalent sequence of small nonsymmetric and large symmetric positive definite systems, which are efficiently solved by conjugate gradient methods. Numerical examples illustrate the high accuracy of the method, in particular in the presence of complex geometry.

A discontinuous Galerkin method for seismic soil–structure interaction analysis in the time domain

AbstractA technique for modeling transient wave propagation in unbounded media is extended and applied to seismic soil–structure interaction analysis in the time domain. The technique, based on the discontinuous Galerkin method, requires lower computational cost and less storage than the boundary element method, and the time‐stepping scheme resulting from Newmark's method in conjunction with the technique is unconditionally stable, allowing for efficient and robust time‐domain computations. To extend the technique to cases characterized by seismic excitation, the free‐field motion is used to compute effective forces, which are introduced on the boundary of the computational domain containing the structure and the soil in the vicinity of the structure. A numerical example on a dam–foundation system subjected to seismic excitation demonstrates the performance of the method. Copyright © 2003 John Wiley & Sons, Ltd.

Characteristics methods for transient analysis of wave propagation in unbounded media

A method based on characteristics was developed recently for the analysis of wave propagation in unbounded layered media [7]. In this paper, the method is extended to problems involving spherical and cylindrical computational boundaries. The procedure resulting from straightforward extension of the method meets with several difficulties in stability and accuracy. Modifications based on radial scaling and upwinding are applied to circumvent these difficulties. The resulting boundary conditions have all the properties desired in an artificial boundary condition. Numerical experiments are presented to illustrate the effectiveness of the procedure.

An optimal absorbing boundary condition for elastic wave modeling

Absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries (Lindman, 1975; Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan et al., 1985; Randall, 1988; Higdon, 1991). We are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage. This is practical for 3-D simulation of elastic wave propagation by a finite‐difference method. Peng and Toksöz (1994) developed a method to design a class of optimal absorbing boundary conditions for a given operator length. In this short note, we give a brief introduction to this technique, and we compare the optimal absorbing boundary conditions against those by Reynolds (1978) and Higdon (1991) using examples of 3-D elastic finite‐difference modeling on an nCUBE-2 parallel computer. In the Appendix, we also give explicit formulas for computing coefficients of the optimal absorbing boundary conditions.

TIME DOMAIN SOLUTION OF ELECTROMAGNETIC PROBLEMS

ABSTRACT After a brief review of the contemporary time-domain methods for solving electromagnetic problems, the FDTD (Finite Difference Time Domain) technique is discussed in detail. The visualization of electromagnetic field distribution, a unique feature of FDTD, is emphasized by applying the technique to simulate electromagnetic wave propagation in unbounded medium. Results obtained with various absorbing boundary conditions are compared. The field distributions in the vicinity of microwave integrated circuit components such as branch-line coupler, bandpass and lowpass filters as well as patch antennas are displayed. In addition, the mapping of the RF (Radio Frequency) magnetic field in a human head is also illustrated to show the applicability of FDTD to aid in the analyses of RF coils developed for magnetic resonance imaging. The extraction of frequency-domain information by Fourier transforming the transient time response generated by pulse excitation is another salient feature of the time-domain methods. The scattering parameters over a wide frequency band are obtained in this manner for microwave integrated circuit components. An efficient extrapolation technique based on Prony's method to generate accurate scattering parameters is also discussed.

Analysis of wave propagation in unbounded media

A new procedure is presented to account for the radiation condition of unbounded media analysed using the finite-element method in the time domain. The medium is divided into a finite region that is of interest in the analysis, and the remaining unbounded medium. The stiffness contribution of the latter is properly represented by its boundary influence matrix, which is directly calculated in the time domain. This procedure is suited for situations in which many analyses of the finite region must be performed.