Wave Propagation In Unbounded Domains Research Articles

Time‐domain scaled boundary perfectly matched layer for elastic wave propagation

AbstractA novel artificial boundary method called the scaled boundary perfectly matched layer (SBPML) is proposed for the transient analysis of elastic wave propagation in unbounded domains. This method constructs a generalized perfectly matched layer (PML) around the finite interior subdomain based on a coordinate transformation inspired by the scaled boundary finite element method (SBFEM). It can model the infinite exterior subdomains of heterogeneous materials, featuring both parallel and radial straight‐line physical surfaces and interfaces. In addition, generally‐shaped (even nonconvex‐shape) artificial boundaries can be used to provide high flexibility in choosing the geometry of the finite interior subdomain. Two coordinate mapping technologies, namely the modified scaled boundary coordinate transformation from the SBFEM and the complex coordinate stretching from the PML, are successively utilized to construct the proposed method. By introducing the integral of stress as an auxiliary variable, the resulting SBPML subdomain is represented in the time domain using a mixed displacement‐stress unsplit‐field formulation. This has the added benefit of allowing for easy incorporation into standard dynamic finite element methods. The performance of the SBPML is demonstrated through several numerical examples, including P‐SV and SH wave propagation problems occurring in unbounded domains with complex geometries and heterogeneous material properties. The numerical results are evaluated in comparison to both reference solution and existing artificial boundary methods, ultimately demonstrating the flexibility, robustness, accuracy, and stability of the proposed approach.

The numerical manifold method for harmonic wave propagation in unbounded domains

The numerical manifold method (NMM) has been applied successfully to a wide variety of problems, including time-independent exterior problems, where problem domains are unbounded. However, of the utmost interest and significance is the analysis of wave propagation in unbounded domains, which is still an open issue because it is far more difficult than time-independent exterior problems. This study aims to fill partly this gap from the prospective of NMM. A new type of infinite patches and the local approximations are designed to construct the Galerkin approximation that satisfies the asymptotic behavior of waves at infinite. Different from the infinite element technique in the finite element method (FEM), where the shape functions are required to satisfy not only the continuity between finite and infinite elements but also the attenuation behavior of solution while approaching infinite, the NMM is demonstrated more straightforward and elegant in the construction of the Galerkin approximation. Some examples in propagation of elastic wave and surface water wave are investigated to verify the accuracy and applicability of the proposed method.

Wave propagation in inhomogeneous media via FE/PML method

AbstractThe Perfectly Matched Layer (PML) method is an efficient approach to imposing radiation conditions at the bounded region of interest in case of wave propagation in unbounded domains. This paper presents and validates 3D FE/PML numerical schemes based on two different PML formulations for homogeneous and inhomogeneous geological media exhibiting discrete or continuous inhomogeneity. In the equation of motion for the PML domain the applied stretching behavior is expressed either as complex material properties or as complex coordinates. Both PML formulations are implemented in the FEM and verified against analytical solutions. Three different types of material inhomogeneity are considered: layered half‐space, continuously inhomogeneous half‐space with linear velocity profile and continuously inhomogeneous half‐space with nonlinear velocity profile. Sensitivity analyses are conducted, and the performance of the developed numerical schemes is investigated taking into account a broad variation of the PML parameters. Recommendations are given for the optimal values of the PML parameters for the case of homogeneous and inhomogeneous geological media.

A new development of the scaled boundary finite element method for wave motion in layered half‐space

AbstractThe scaled boundary finite element method (SBFEM) developed by Wolf and Song has shown certain parallels to the finite element method (FEM) and boundary element method (BEM). Because of its semi‐analytical nature, SBFEM is particularly suitable for the analysis of wave propagation in unbounded domains. This paper makes a certain modification of the standard SBFEM. A new idea of scaling surface instead of a scaling center is introduced to formulate the governing SBFE equations for the analysis of wave propagation in multilayered half‐space, which leads to simplifying the modeling and saving considerably the computational effort. In addition, by employing the proposed approach, some problems encountered in engineering practice, which are difficult to deal with by the conventional SBFEM, for example, 3D foundation impedance on half‐space with irregular geographical features, can be effectively solved. The proposed approach also helps to simplify the solution of shell structures. Numerical examples are provided to validate the accuracy and efficiency of the proposed approach.

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.

Perfectly matched discrete layers for modeling unbounded domains

The method of perfectly matched layers (PML) is based on the elegant idea of introducing attenuation while keeping impedance unaltered, thus resulting in effective absorption of waves without spurious reflections. Due to its simplicity and effectiveness, PML has been widely used over the past 25 years to model wave propagation in unbounded domains. However, an acknowledged challenge is that PML no longer preserves impedance once the domain and the PML region are discretized, leading to spurious reflection at the interface. This issue has been fixed with the help of so-called impedance preserving discretization, leading to the method of perfectly matched discrete layers (PMDL), developed more than a decade ago. The idea is based on the observation that midpoint integration completely eliminates the discretization error in impedance that arises from linear finite element discretization. Beyond eliminating reflections at the interface, impedance preserving discretization is shown to be crucial to achieving stability of absorbing boundary conditions in situations with differing signs of phase and group velocities, e.g., anisotropic media. This talk will be an exposition of the development of the PMDL by the author and his collaborators, staring with fundamental ideas impedance preserving discretization followed by more advanced ABCs for anisotropic and periodic media.

Hybrid asynchronous absorbing layers based on Kosloff damping for seismic wave propagation in unbounded domains

This paper presents a novel approach for modeling infinite media, called Hybrid (different time integrators) Asynchronous (different time steps) Kosloff Absorbing Layers with Increasing Damping (HA-Kosloff ALID). By using strong forms of wave propagation in Kosloff media, its design equation is derived as well as optimal conditions between physical and absorbing domains. Explicit/Implicit co-simulation is adopted to reduce computation time. Examples of semi-infinite bar and Lamb’s test are implemented to illustrate the efficiency of our approach in terms of accuracy and CPU time, in comparison to Rayleigh ALID and PML. It turns out to be efficient and convenient for modeling unbounded domain.

A high-order doubly asymptotic continued-fraction solution for frequency-domain analysis of vector wave propagation in unbounded domains

The scaled boundary finite element (SBFE) equation in dynamic stiffness of unbounded domains is efficiently solved by the continued fraction expansion, which has a large convergence range and a high convergence rate. In this paper, a doubly asymptotic continued fraction solution algorithm for frequency-domain analysis of vector wave propagation in unbounded domains of arbitrary geometry is developed based on the high-frequency (singly) asymptotic approach. The exact solution over the whole frequency range can be calculated rapidly with increasing order of expansion. The coefficient matrices of the solution are determined recursively by satisfying the dynamic stiffness at both high (ω→∞) and low (ω→0) frequency limits. The solutions of out-plane motion of a semi-infinite wedge, in-plane motion of a semi-infinite wedge and a 3-D homogeneous half space in the frequency domain are presented. The results of three numerical examples demonstrate that the doubly asymptotic algorithm is more stable and superior than the high-frequency asymptotic approach.

Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: necessary and sufficient conditions of stability

In this work we consider the problem of modelling of 2D anisotropic dispersive wave propagation in unbounded domains with the help of perfectly matched layers (PMLs). We study the Maxwell equations in passive media with a frequency-dependent diagonal tensor of dielectric permittivity and magnetic permeability. An application of the traditional PMLs to this kind of problems often results in instabilities. We provide a recipe for the construction of new, stable PMLs. For a particular case of non-dissipative materials, we show that a known necessary stability condition of the perfectly matched layers is also sufficient. We illustrate our statements with theoretical and numerical arguments.

Hybrid Asynchronous Perfectly Matched Layer for seismic wave propagation in unbounded domains

Perfectly Matched Layer (PML) is recognized as a very effective tool for modeling unbounded domains. Nonetheless, the computation time required by the PML may be large, especially when an explicit time integration scheme is adopted for dealing with the wave propagation problem both in the domain of interest and in the PML medium. In this paper, it is proposed to investigate subdomain strategies enabling the appropriate time integration scheme in the PML with its own time step to be chosen, independently of the choice of the time scheme in the domain of interest. We focus on explicit time integrator in the physical subdomain (Central Difference scheme) associated with a fine time step satisfying the CFL stability criterion. The PML formulation proposed by Basu and Chopra (2004) [1] for 2D transient dynamics, has been coupled with the interior physical subdomain using the dual Schur approach proposed by Gravouil and Combescure (2001) [2]. Hybrid (implicit time integrator for the PML) asynchronous (multi time steps) PMLs have been derived. Their very good accuracy has been shown by considering the following numerical examples: Lamb׳s test, loaded rigid strip footing on an half space and a layered half space.

Design of an efficient multi-directional explicit/implicit Rayleigh absorbing layer for seismic wave propagation in unbounded domain using a strong form formulation

Design of an efficient multi-directional explicit/implicit Rayleigh absorbing layer for seismic wave propagation in unbounded domain using a strong form formulation

Use of Finite Point Method for Wave Propagation in Nonhomogeneous Unbounded Domains

Wave propagation in an unbounded domain surrounding the stimulation resource is one of the important issues for engineers. Past literature is mainly concentrated on the modelling and estimation of the wave propagation in partially layered, homogeneous, and unbounded domains with harmonic properties. In this study, a new approach based on the Finite Point Method (FPM) has been introduced to analyze and solve the problems of wave propagation in any nonhomogeneous unbounded domain. The proposed method has the ability to use the domain properties by coordinate as an input. Therefore, there is no restriction in the form of the domain properties, such as being periodical as in the case of existing similar numerical methods. The proposed method can model the boundary points between phases with trace of errors and the results of this method satisfy both conditions of decay and radiation.

Water wave propagation in unbounded domains. Part II: Numerical methods for fractional PDEs

This paper concerns numerical solutions for a fractional partial differential equation arising as a nonreflecting boundary condition in water wave propagation. The fractional derivative operator is written as divergence of a singular integrable convolution, which allows the equation to be viewed as a conservation law with a linear nonlocal flux. A semi-discrete finite volume scheme is presented, using conservative piecewise polynomial reconstruction of the solution. The convolution with the singular kernel is then integrated exactly. Time integration uses Runge–Kutta schemes of matching order. Stability is discussed, convergence is established and numerical examples are presented.

An extended Krylov subspace model-order reduction technique to simulate wave propagation in unbounded domains

In this paper we present a novel extended Krylov subspace reduced-order modeling technique to efficiently simulate time- and frequency-domain wavefields in open complex structures. To simulate the extension to infinity, we use an optimal complex-scaling method which is equivalent to an optimized perfectly matched layer in which the frequency is fixed. Wavefields propagating in strongly inhomogeneous open domains can now be modeled as a non-entire function of the complex-scaled wave operator. Since this function contains a square root singularity, we apply an extended Krylov subspace technique to construct fast converging reduced-order models. Specifically, we use a modified version of the extended Krylov subspace algorithm as proposed by Jagels and Reichel [14], since this algorithm allows us to balance the computational costs associated with computing powers of the wave operator and its inverse. Numerical experiments from electromagnetics and acoustics illustrate the performance of the method.

Water wave propagation in unbounded domains. Part I: Nonreflecting boundaries

This paper concerns non-reflecting boundary conditions for linearized 2D incompressible, irrotational, inviscid free surface flows in deep water. The linearization at flat water of the water wave equation is factored as a product of one-way equations. The one-way factors involve a nonlocal half derivative in space. They serve to construct essentially non-reflective and absorbing layers. The one way nature of the equation prevents errors at the boundary from propagating back and polluting the solution in the interior of the domain. Two versions of the one-way equations are discussed, with and without additional wave damping. Numerical results are presented.

Wave Propagation in Unbounded Domains under a Dirac Delta Function with FPM

Wave propagation in unbounded domains is one of the important engineering problems. There have been many attempts by researchers to solve this problem. This paper intends to shed a light on the finite point method, which is considered as one of the best methods to be used for solving problems of wave propagation in unbounded domains. To ensure the reliability of finite point method, wave propagation in unbounded domain is compared with the sinusoidal unit point stimulation. Results indicate that, in the case of applying stimulation along one direction of a Cartesian coordinate, the results of finite point method parallel to the stimulation have less error in comparison with the results of finite element method along the same direction with the same stimulation.

A Krylov Stability-Corrected Coordinate-Stretching Method to Simulate Wave Propagation in Unbounded Domains

The Krylov subspace projection approach is a well-established tool for the reduced-order modeling of dynamical systems in the time domain. In this paper, we address the main issues obstructing the application of this powerful approach to the time-domain solution of exterior wave problems. We use frequency-independent perfectly matched layers to simulate the extension to infinity. Pure imaginary stretching functions based on Zolotarev's optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest. A new Krylov-based solution method via stability-corrected operator exponents is presented which allows us to construct reduced-order models (ROMs) that respect the delicate spectral properties of the original scattering problem. The ROMs are unconditionally stable and are based on a renormalized bi-Lanczos algorithm. We give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two-dimensional electromagnetic wave propagation in unbounded domains, including a photonic waveguide example. The new algorithm outperforms the conventional finite-difference time-domain method for problems on large time intervals.

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.

Time‐domain analysis of wave propagation in unbounded domains based on the scaled boundary finite element method

AbstractTransient wave propagation in unbounded domains is considered in this paper. The scaled boundary finite element method is used to model the far field. This technique is originally formulated in the frequency‐domain. Two extensions of the method which lead to a direct description of the unbounded medium in the time‐domain are described, that is the rational approximation and the continued‐fraction expansion of the dynamic stiffness. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Transient soil-structure interaction with consistent description of radiation damping

Radiation damping due to wave propagation in unbounded domains may cause a significant reduction of structural vibrations when excited near resonance. Here a novel matrix-valued algebraic Pad<TEX>$\acute{e}$</TEX>-like stiffness formulation in the frequency-domain and a corresponding state equation in the time domain are elaborated for a soil-structure interaction problem with a layered soil excited in a transient manner by a flexible rotor during startup and shutdown. The contribution of radiation damping caused by a soil-layer upon a rigid bedrock is characterized by the corresponding amount of critical damping as it is used in structural dynamics.