Waves In Unbounded Domains Research Articles

Peridynamic elastic waves in two-dimensional unbounded domains: Construction of nonlocal Dirichlet-type absorbing boundary conditions

The focus of this paper is on application of peridynamics (PD) to propagation of elastic waves in unbounded domains. We construct absorbing boundary conditions (ABCs) derived from a semi-analytical solution of the PD governing equation at the exterior region. This solution is made up of a finite series of plane waves, as fundamental solutions (modes), which satisfy the PD dispersion relations. The modes are adjusted to transmit the energy from the interior region (near field) to the exterior region (far field). The corresponding unknown coefficients of the series are found in terms of the displacement field at a layer of points adjacent to the absorbing boundary. This is accomplished through a collocation procedure at subregions (clouds) around each absorbing point. The proposed ABCs offer appealing advantages, which facilitate their application to PD. They are of Dirichlet-type, hence their implementation is relatively simple as no derivatives of the field variables are required. They are constructed in the time and space domains and thus application of Fourier and Laplace transforms, cumbersome for nonlocal models, is not required. At the discrete level, the modes satisfy the same numerical dispersion relations of the near field, which makes the far-field solution compatible with that of the near field. We scrutinize the performance of the proposed ABCs through several examples. Our investigation shows that the proposed ABCs perform stably in time with an appropriate level of accuracy even in problems characterized by highly-dispersive propagating waves, including crack propagation in semi-unbounded brittle solids.

Implementation of the novel perfectly matched layer element for elastodynamic problems in time-domain finite element method

For the numerical simulation of infinite domain problems in finite element method (FEM), unwanted reflection from the artificial boundaries of the model is a limiting factor. Therefore, in this paper, based on the theory of perfectly matched layer (PML), a novel element is derived to deal with absorbing boundary conditions in the framework of time-domain finite element method (TDFEM). Then, by combining with the viscoelastic model and modified Newmark algorithm, a novel method is proposed to absorb outgoing waves in unbounded domains. Numerical results show that the proposed method can efficiently absorb the elastic wave, and it can be employed to solve infinite domain problems.

A Time-Domain Formulation of Elastic Waves in Heterogeneous Unbounded Domains

A time-domain formulation and its numerical implementation of three-dimensional elastic waves in unbounded domains are discussed. To simulate elastic waves in unbounded domains, it is necessary to truncate the originally infinite extent and remove artificial reflections at truncated boundaries. In this work, perfectly matched layer (PML) is used to surround the truncated regular domain and to absorb outgoing waves beyond the truncated boundaries. To derive governing equations for elastic waves in the PML-truncated domain, complex coordinate stretching is applied to equilibrium, constitutive, and compatibility equations in the frequency domain. Then the stretched equations were converted to the time domain by applying the inverse Fourier transform to yield a system of mixed displacement-stress equations. Introducing a mixed finite element method leads to mixed semi-discrete equations of motion which can be integrated in time for displacements and stresses. Numerical experiments show that reflections from truncated boundaries are significantly reduced by using the developed PML method. For the reduction of computational cost in higher dimensions, an MPI-based parallel computation has been performed on assembly and time integration routines of the finite element wave simulation, which shows good scalability.

Numerical modelling of wave propagation in anisotropic soil using a displacement unit-impulse-response-based formulation of the scaled boundary finite element method

An efficient method for modelling the propagation of elastic waves in unbounded domains is developed. It is applicable to soil–structure interaction problems involving scalar and vector waves, unbounded domains of arbitrary geometry and anisotropic soil. The scaled boundary finite element method is employed to derive a novel equation for the displacement unit-impulse response matrix on the soil–structure interface. The proposed method is based on a piecewise linear approximation of the first derivative of the displacement unit-impulse response matrix and on the introduction of an extrapolation parameter in order to improve the numerical stability. In combination, these two ideas allow for the choice of significantly larger time steps compared to conventional methods, and thus lead to increased efficiency. As the displacement unit-impulse response approaches zero, the convolution integral representing the force–displacement relationship can be truncated. After the truncation the computational effort only increases linearly with time. Thus, a considerable reduction of computational effort is achieved in a time domain analysis. Numerical examples demonstrate the accuracy and high efficiency of the new method for two-dimensional soil–structure interaction problems.

Efficient and stable perfectly matched layer for CEM

An efficient unsplit perfectly matched layer for numerical simulation of electromagnetic waves in unbounded domains is derived via a complex change of variables. In order to surround a Cartesian grid with the PML, the time-dependent PML requires only one (scalar) auxiliary variable in two space dimensions and six (scalar) auxiliary variables in three space dimensions. It is therefore cheap and straightforward to implement. We use Fourier and energy methods to prove the stability of the PML. We extend the stability result to a semi-discrete PML approximated by central finite differences of arbitrary order of accuracy and to a fully discrete problem for the ‘Leap-Frog’ schemes. This makes precise the usefulness of the derived PML model for longtime simulations. Numerical experiments are presented, illustrating the accuracy and stability of the PML.

STRESS–VELOCITY COMPLETE RADIATION BOUNDARY CONDITIONS

A new high-order local Absorbing Boundary Condition (ABC) has been recently proposed for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. It is based on the stress–velocity formulation of the elastodynamics problem, and on the general Complete Radiation Boundary Condition (CRBC) approach, originally devised by Hagstrom and Warburton in 2009. The work presented here is a sequel to previous work that concentrated on the stability of the scheme; this is the first known high-order ABC for elastodynamics which is long-time stable. Stability was established both theoretically and numerically. The present paper focuses on the accuracy of the scheme. In particular, two accuracy-related issues are investigated. First, the reflection coefficients associated with the new CRBC for different types of incident and reflected elastic waves are analyzed. Second, various choices of computational parameters for the CRBC, and their effect on the accuracy, are discussed. These choices include the optimal coefficients proposed by Hagstrom and Warburton for the acoustic case, and a simplified formula for these coefficients. A finite difference discretization is employed in space and time. Numerical examples are used to experiment with the scheme and demonstrate the above-mentioned accuracy issues.

Long-time stable high-order absorbing boundary conditions for elastodynamics

Two new high-order local absorbing boundary conditions (ABCs) are devised for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. The elastic medium in the exterior domain is assumed to be homogeneous and isotropic. The first ABC is written directly with respect to the displacement vector field, using an operator involving high derivatives. The second ABC is applied to the problem written in a first-order conservation form, using stresses and velocities as variables, and is formulated as a sequence of recursive relations using auxiliary variables. The two ABCs are not equivalent but are closely related. In each case, the order of the ABC determines its accuracy and can be chosen to be arbitrarily high. Both ABCs involve a product of first-order differential operators; all of them are of the Higdon type, except one which is of the Lysmer–Kuhlemeyer type. The stability of both ABCs is analyzed theoretically. The second (stress–velocity) ABC is implemented, employing a finite difference discretization scheme in space and time. Numerical experiments demonstrate the performance of the scheme. To the best of the authors’ knowledge, these are the first existing local high-order ABCs for elastodynamics which are long-time stable.

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

SUMMARYA high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.

A finite element scheme with a high order absorbing boundary condition for elastodynamics

A high-order absorbing boundary condition (ABC) is devised on an artificial boundary for time-dependent elastic waves in unbounded domains. The configuration considered is that of a two-dimensional elastic waveguide. In the exterior domain, the unbounded elastic medium is assumed to be isotropic and homogeneous. The proposed ABC is an extension of the Hagstrom–Warburton ABC which was originally designed for acoustic waves, and is applied directly to the displacement field. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The initial boundary value problem including this ABC is written in second-order form, which is convenient for geophysical finite element (FE) analysis. A special variational formulation is constructed which incorporates the ABC. A standard FE discretization is used in space, and a Newmark-type scheme is used for time-stepping. A long-time instability is observed, but simple means are shown to dramatically postpone its onset so as to make it harmless during the simulation time of interest. Numerical experiments demonstrate the performance of the scheme.

Energy decay rates of elastic waves in unbounded domain with potential type of damping

In this paper we first show that the total energy of solutions for a semilinear system of elastic waves in R n with a potential type of damping decays in an algebraic rate to zero. We study the critical potential case and we assume that the initial data have a compact support. An application for the Euler–Poisson–Darboux type dissipation V ( t , x ) is obtained and in this case the compactness of the support on the initial data is not necessary. Finally, we shall discuss the energy concentration region for the linear system of elastic waves in an exterior domain.

The influence of magnetic field on short-wavelength instability of Riemann ellipsoids

We address the question of stability of the so-called S-type Riemann ellipsoids, i.e. a family of Euler flows in gravitational equilibrium with the vorticity and background rotation aligned along the principal axis perpendicular to the flow. The Riemann ellipsoids are the simplest models of self-gravitating, tidally deformed stars in binary systems, with the ellipticity of the flow modelling the tidal deformation. By the use of the WKB theory we show that mathematically the problem of stability of Riemann ellipsoids with respect to short-wavelength perturbations can be reduced to the problem of magneto-elliptic instability in rotating systems, studied previously by Mizerski and Bajer [K.A. Mizerski, K. Bajer, The magneto-elliptic instability of rotating systems, J. Fluid Mech. 632 (2009) 401–430]. In other words the equations describing the evolution of short-wavelength perturbations of the Riemann ellipsoids considered in Lagrangian variables are the same as those for the evolution of the magneto-elliptic-rotational ( MER) waves in unbounded domain. This allowed us to use the most unstable MER eigenmodes found in Mizerski et al. [K.A. Mizerski, K. Bajer, H.K. Moffatt, The α -effect associated with elliptical instability, J. Fluid Mech., 2010 (in preparation)] to provide an estimate of the characteristic tidal synchronization time in binary star systems. We use the idea of Tassoul [J.-L. Tassoul, On synchronization in early-type binaries, Astrophys. J. 322 (1987) 856–861] and that the interactions between perturbations significantly increase the effective viscosity and hence the energy dissipation in an Ekman-type boundary layer at the surface of the star. The results obtained suggest that if the magnetic field generated by (say) the secondary component of a binary system is strong enough to affect the flow dynamics in the primary, non-magnetized component, the characteristic tidal synchronization time can be significantly reduced.

The Large Core Limit of Spiral Waves in Excitable Media: A Numerical Approach

We modify the freezing method introduced by Beyn and Thümmler in 2004 for analyzing rigidly rotating spiral waves in excitable media. The proposed method is designed to stably determine the rotation frequency and the core radius of rotating spirals, as well as the approximate shape of spiral waves in unbounded domains. In particular, we introduce spiral wave boundary conditions based on geometric approximations of spiral wave solutions by Archimedean spirals and by involutes of circles. We further propose a simple implementation of boundary conditions for the case when the inhibitor is nondiffusive, a case which had previously caused spurious oscillations. We then utilize our method to numerically analyze the large core limit. The proposed method allows us to investigate the case close to criticality where spiral waves acquire infinite core radius $r_c$ and zero rotation frequency $\omega$, before they begin to develop into retracting fingers. We confirm the linear scaling regime of a drift bifurcation for the rotation frequency and the core radius of spiral wave solutions close to criticality. This regime is unattainable with conventional numerical methods.

A perfectly matched layer approach to the nonlinear Schrödinger wave equations

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrödinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrödinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrödinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrödinger equations and some Schrödinger-coupled systems in each spatial dimension.

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

AbstractA high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.

A representation of acoustic waves in unbounded domains

AbstractCompact, time‐harmonic, acoustic sources produce waves that decay too slowly to be square‐integrable on a line away from the sources. We introduce an inner product, arising directly from Green's second theorem, to form a Hilbert space of these waves and present examples of its computation.1 © 2005 Wiley Periodicals, Inc.

Generalized Galerkin approximations of elastic waves with absorbing boundary conditions

For the propagation of elastic waves in unbounded domains, absorbing boundary conditions (ABCs) at the fictitious numerical boundaries have been proposed. In this paper we focus on both first- and second-order ABCs in the framework of variational (weak) approximations, like those stemming from Galerkin method (or its variants) for finite element or spectral approximations [1]. In particular, we recover first order conditions as natural (or Neumann) conditions, whereas we propose a penalty residual method for the treatment of second order ABCs. The time discretization is based on implicit backward finite differences, whereas we use spectral Legendre collocation methods set in a variational form for the spatial discretization (treatment of finite element or spectral element approximations is completely similar). Numerical experiments exhibit that the present formulation of second-order ABCs improves the one based on first-order ABCs with regard to both the reduction of the total energy in the computational domain, and the Fourier spectrum of the displacement field at selected points of the elastic medium. A stability analysis is developed for the variational problem in the continuous case both for first- and second-order ABCs. A suitable treatment of ABCs at corners is also proposed.

Analyses of Waves in Unbounded Domains by the Use of a Direct Combination of Analytical Solutions and Finite Elements. (Two-Dimensional Acoustic Problems Containing Several Scattering Objects).

There is no single use of the FEM for the problems including unbounded domains because they require countless elements. Thus some combination methods of FEM and analytical solutions have been investigated. In most of them, a single artificial boundary such as a circle is assumed, and analytical solutions are used for the area outside of the artificial boundary. Nevertheless, in problems such as of a wave scattered by several objects, many elements are required to fill the space within the artificial boundary. If the artificial boundaries enclosing the individual objects can be assumed, the number of finite elements can be reduced. This paper deals with a method based on this idea for treating acoustic waves in two dimensions scattered by several objects.

Analyses of waves in unbounded domains by use of direct combination of analytical solutions and finite elements.

An analysis of wave-motion problems in unbounded or infinite domains is presented. Analytical solutions that express outgoing waves from an artificial boundary such as a circle or a sphere can be obtained easily. These solutions may be transformed into discretized relations between values of velocity potential and volumetric flux on the artificial boundary. These relations are in the same forms as the FEM equations. They can be joined directly to finite elements within the artificial boundary in order to construct an equation system for the whole domain. Some problems of radiation and scattering are calculated numerically by this method. The so-called fictitious interior eigenvalue failure does not arise in this method.

Partial Differential Equations with Periodic Coefficients and Bloch Waves in Crystals

Bloch waves are special solutions of Schrödinger's equation with a periodic real potential. They are plane waves multiplied by periodic functions. In this paper we prove the existence and completeness of Bloch waves and of the related Kohn-Luttinger waves in unbounded domains for a class of partial differential equations which includes the Schrödinger equation. In addition, we discuss the dependence of these waves and the corresponding eigenvalues on the wave vector of the associated plane wave. The results may be interpreted as the analogs for certain partial differential equations of Floquet's theory for ordinary differential equations or as the determination of the spectral representation of certain periodic Hamiltonian operators.