Solving Wave Propagation Problems Research Articles

Application of Slip Interface Conditions in the Elastic Parabolic Equation

The parabolic equation method is the most effective approach for solving wave propagation problems in environments with strong vertical variations and gradual horizontal variations. This approach efficiently provides accurate solutions for problems involving sloping fluid-fluid interfaces, sloping solid-solid interfaces, and sloping solid boundaries. There remains a need for improvement for the case of sloping fluid-solid interfaces. The most promising approach to date is the combination of a single-scattering approximation and the treatment of part of the fluid layer as a solid layer with low shear speed. This is a singular perturbation, which results in rapid oscillations near the interface between the low-speed layer and the true solid layer. It is demonstrated here that the rapid oscillations may be eliminated by using slip conditions at the interface between the low-speed layer and the true solid layer. Stability problems were encountered when attempting to directly implement the slip conditions, but this problem was resolved by using an equivalent condition involving the normal derivative of the tangential displacement. This progress does not fully resolve the case of the sloping fluid-solid interface. There remains a need to find a vertical interface condition that is compatible with the slip conditions and that provides accurate results for problems involving sloping fluid-solid interfaces.

A Fast Butterfly-Compressed Hadamard–Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources

We present a butterfly-compressed representation of the Hadamard–Babich (HB) ansatz for the Green’s function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all cell centers to each other. The overall CPU time and memory requirement scale as for any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency domain methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-of-the-art supercomputer at Lawrence Berkeley National Laboratory.

Modeling 3-D Elastic Wave Propagation in TI Media Using Discontinuous Galerkin Method on Tetrahedral Meshes

Transversely isotropic (TI) medium is a widely studied anisotropic solid medium in seismology. The numerical simulation of seismic wave propagation in TI media is an effective tool to analyze the mechanism of seismic waves in complex anisotropic media. In this study, we introduce a double-weighted Runge-Kutta discontinuous Galerkin (RKDG) method for numerically solving wave propagation problems in 3D TI media with surface topography. This method incorporates the discontinuous Galerkin spatial discretization with an explicit double-weighted two-step iteration time discretization. The local Lax-Friedrichs flux is used as the numerical flux in the formulations. This method can solve the first-order velocity-stress seismic wave equations including TI media as well as more general anisotropic media. Due to the large scale of 3D problems, the parallel technology is adopted. Regions with irregular boundaries are discretized into unstructured tetrahedral meshes. Numerical experiments for various anisotropic media including the vertical and tilted TI media are presented. The results demonstrate the effectiveness of the double-weighted RKDG method in wavefield simulations in 3D complicated anisotropic media.

Solving elastic wave equations in 2D transversely isotropic media by a weighted Runge-Kutta discontinuous Galerkin method

Accurate wave propagation simulation in anisotropic media is important for forward modeling, migration and inversion. In this study, the weighted Runge-Kutta discontinuous Galerkin (RKDG) method is extended to solve the elastic wave equations in 2D transversely isotropic media. The spatial discretization is based on the numerical flux discontinuous Galerkin scheme. An explicit weighted two-step iterative Runge-Kutta method is used as time-stepping algorithm. The weighted RKDG method has good flexibility and applicability of dealing with undulating geometries and boundary conditions. To verify the correctness and effectiveness of this method, several numerical examples are presented for elastic wave propagations in vertical transversely isotropic and tilted transversely isotropic media. The results show that the weighted RKDG method is promising for solving wave propagation problems in complex anisotropic medium.

A Generalized Method for Dispersion Analysis of Guided Waves in Multilayered Anisotropic Magneto-Electro-Elastic Structures

Based on the symplectic structure of the Hamiltonian matrix, the precise integration method (PIM), and the Wittrick–Williams (W-W) algorithm, a generalized method for computing the dispersion curves of guided waves in multilayered anisotropic magneto-electro-elastic (MEE) structures for different types of mechanical, electrical, and magnetical boundaries is developed. A strictly theoretical analysis shows that the W-W algorithm cannot be applied directly to the MEE structure. This is because a block of the Hamiltonian matrix is not positive definite for MEE structures so that the eigenvalue count of the sublayer is not zero when the divided sublayer is sufficiently thin. To overcome this difficulty, based on the symplectic structure of the Hamiltonian matrix, a symplectic transformation is introduced to ensure that the W-W algorithm can be applied conveniently to solve wave propagation problems in multilayered anisotropic MEE structures. The application of the PIM based on the mixed energy matrix to solve the wave equation can ensure the stability and efficiency of the method, and all eigenfrequencies are found without the possibility of any being missed using the W-W algorithm. This research provides the necessary insight to apply the W-W algorithm in wave propagation and vibration problems of MEE structures.

Introducing a moving load in a simulation in time over a truncated unbounded domain

When using Finite Element methods to solve wave propagation problems, the spatial domain must be truncated at a finite distance. This can be done using absorbing boundary conditions or layers, which effectively inhibit the reflection of outgoing waves back into the computational domain. They cannot however handle moving loads coming from outside of the (truncated) computational domain, for which some incoming waves must be allowed. After illustrating this fact, this paper proposes a technique to introduce such moving loads in time simulations over finite size domains. This technique involves the introduction of appropriate initial conditions within the computational domain. In general, these initial conditions can be computed numerically, and possibly re-used for various configurations. The interest of the method is illustrated on industrial cases of interest for the railway community.

On the numerical solution of the iterative parabolic equations by ETDRK pseudospectral methods in linear and nonlinear media

Iterative parabolic equations (IPEs) were recently introduced as a promising tool for solving wave propagation problems in complex linear and non-linear media. In this study a general approach to the numerical solution of iterative parabolic equations on the basis of powerful ETD pseudospectral method is developed. The solution of nth IPE requires the computation of the input term that is obtained by differentiation of the solution of n−1th IPE. The numerical noise resulting from such differentiation spoils the solutions of higher-order IPEs. It is shown that this noise can be suppressed via a filtering procedure involving Chebyshev polynomials of a discrete variable. In this study Padé type iterative parabolic approximations are introduced both for the case of linear and nonlinear media. It is shown that Padé-type approximations are more suitable for solving the problems of wave propagation.

Wave propagation in a three-dimensional half-space with semi-infinite irregularities

ABSTRACT Dynamic analysis of problems with complex geometries requires utilization of numerical methods. To completely capture the effects of seismic wave propagation in a system, one must consider the structure or irregularity within its encompassing half-space. Correct consideration of half-space in a numerical model is important specially when it comes to cases where the half-space contains semi-infinite irregularities. In this study, a generalized numerical methodology is presented for dynamic analysis of a half-space with semi-infinite irregularities. The methodology is first verified through comparison with analytical solution of known problems. Then the method is employed to solve the dynamic analysis of an arbitrary-shaped existing valley within a half-space under propagating plane harmonic SH wave with different angles of incidence. Results indicate complete capability of the methodology in solving wave propagation problems for complex systems without the usual restrictions of common existing methods.

Time series dual boundary element model for P and SV waves propagation in half-space

This paper details the implementation of a new time-domain boundary element analysis for solving wave propagation problems inside the half-space. Initially, by developing a new set of half-plane fundamental solutions and implementing them into the dual reciprocity boundary element analysis (DRBEM), the discretization of the ground surface boundaries were eliminated. To convert DRBEM inertia domain integrals into the boundary integrals, a set of Fourier series was selected as the displacement field and required fundamental solutions were obtained. The series displacement field, then was implemented into the 2-dimensional elastodynamic equilibrium Equations and the time-dependent series coefficients were calculated using an algorithm called TSBEM (Time Series Boundary Element Method). The accuracy and efficiency of the presented method were tested compared to the available analytical solutions by modelling several examples. The Results showed that, the method can be sufficiently used for analyzing wave propagation and also any other dynamic problems.

Development of composite sub-step explicit dissipative algorithms with truly self-starting property

This paper focuses mainly on the development of composite sub-step explicit algorithms for solving nonlinear dynamic problems. The proposed explicit algorithms are required to achieve the truly self-starting property, so avoiding computing the initial acceleration vector, and the controllable numerical dissipation at the bifurcation point, so eliminating spurious high-frequency components. With these two requirements, the single and two sub-step explicit algorithms with truly self-starting property and dissipation control are developed and analyzed. The present single sub-step algorithm shares the same spectral accuracy as the known Tchamwa–Wielgosz scheme, but the former possesses some advantages for solving wave propagation problems. The present two sub-step algorithm provides a larger stability limit, twice than those of single step schemes, due to explicit solutions of linear systems twice within each time increment. Numerical examples are also simulated to show numerical performance and superiority of two novel explicit methods over other explicit schemes.

Энергетический метод для решения волновых задач гибкой нити

The paper considers the behavior of a flexible deformable thread when longitudinal and transverse waves pass through it. The processes occurring in a thread during the passage of a wave perturbation through it are analyzed under the assumption that the wave front zone is limited comparing with the length of the thread but not assuming this quantity to be of infinite smallness. In contrast to other similar works, no assumptions were made in advance about the shape of the thread in the zone of the wave perturbation passage (the fracture of the thread) and the dependence of the tension force on its elongation. The only requirement is the implementation of the general theorems of dynamics. Formulas for the relation of the thread speed before and after the passage of the wave with a change in the angle of thread inclination are obtained making possible solving wave propagation problems in a new way. The method is illustrated by new solutions to known problems, which allows comparing the obtained results with known solutions and verifying the advantages of the proposed method.

Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays

A new methodology is developed in this work to solve a one-dimensional (1D) nonlinear wave propagation problem. In its response, the space variable is converted to time functions at different locations, and the resultant time functions are merged with the time variable. Since the merged variables are essentially time-delay variables, the main point of the methodology is that the governing equation of the nonlinear wave propagation problem can be converted to a corresponding nonlinear delay differential equation (DDE) with multiple delays. The new methodology is shown how to formulate the converted DDE, and a modified incremental harmonic balance method is used to solve the 1D nonlinear DDE by introducing a delay matrix operator, where a formula of the Jacobian matrix is derived and can be efficiently and automatically used in the Newton method. This new methodology is demonstrated by solving examples of 1D monatomic chains under the nonlinear Hertzian contact law and with cubic springs. Results well match those in previous works, but derivation effort of solutions and calculation time can be significantly reduced here. When the algorithm of the methodology is compiled as a computer program, there is no additional derivation required to solve wave propagation problems associated with different governing equations.

Dispersion Analysis of Multiscale Wavelet Finite Element for 2D Elastic Wave Propagation

AbstractRecently, the wavelet finite element has been introduced to solve wave propagation problems because of its outstanding compact support, multiscale, and multiresolution characteristics. In t...

An accurate method for guided wave propagation in multilayered anisotropic piezoelectric structures

The purpose of this paper is to extend and generalize the precise integration method (PIM) and the Wittrick–Williams (W–W) algorithm to analyze the dispersion of guided waves in multilayered anisotropic piezoelectric structures. The analysis shows that the W–W algorithm cannot be directly applied to piezoelectric materials. This is due to the fact that a submatrix of the Hamiltonian matrix is not positive definite for piezoelectric materials such that the eigenvalue count of sublayers is not zero when the divided sublayers are sufficiently small. The reason for this issue is explored by a theoretical analysis, and then, a symplectic transformation is introduced to ensure that the W–W algorithm can conveniently be applied to solve wave propagation problems in multilayered anisotropic piezoelectric structures. The present method not only guarantees that the computation is accurate and stable, but also finds all eigenfrequencies without being missed. Three numerical examples are provided to illustrate the performance of the method, and the results obtained by the method are compared with the published results and the results obtained by the semi-analytical finite element method. The effects of boundary conditions, wave propagation direction, thickness ratios and stacking sequences on the dispersion behavior of guided waves are discussed.

Transfer matrix method for the analysis of space-time-modulated media and systems

Space-time modulation adds another powerful degree of freedom to the manipulation of classical wave systems. It opens the door for complex control of wave behavior beyond the reach of stationary systems, such as nonreciprocal wave transport and realization of gain media. Here we generalize the transfer matrix method and use it to create a general framework to solve wave propagation problems in time-varying acoustic, electromagnetic, and electric circuit systems. The proposed method provides a versatile approach for the study of general space-time varying systems, which allows any number of time-modulated elements with arbitrary modulation profile, facilities the investigation of high order modes, and provides an interface between space-time modulated systems and other systems.

The hybrid element‐free Galerkin method for three‐dimensional wave propagation problems

SummaryThis paper presents a hybrid element‐free Galerkin (HEFG) method for solving wave propagation problems. By introducing the dimension split method, the three‐dimensional wave propagation problems are transformed into a series of two‐dimensional ones in other one‐dimensional directions. The two‐dimensional problems are solved using the improved element‐free Galerkin (IEFG) method, and the finite difference method is used in the one‐dimensional splitting direction and the time space. Then, the formulas of the HEFG method for three‐dimensional wave propagation problems are obtained. Numerical examples are selected to show the effectiveness and the advantage of the HEFG method. The convergence and error analysis of the HEFG method are discussed according to the numerical results under different splitting directions, weight functions, node distributions, scale parameters of the influence domain, penalty factors, and time steps. The numerical results are given to show the convergence and advantages of the HEFG method over the IEFG method. Comparing with the IEFG method, the HEFG method has greater computational precision and speed for three‐dimensional wave propagation problems.

A stochastic spectral finite element method for wave propagation analyses with medium uncertainties

• Wave propagation in a medium with stochastic properties is studied. • A stochastically enriched spectral finite element method (StSFEM) is applied. • StSFEM accelerates temporal integration schemes and decreases numerical dispersion. • It provides an efficient numerical solution for Fredholm equation of KL Expansion. • The StSFEM is developed to large-scale stochastic wave propagation phenomena. A stochastically enriched spectral finite element method (StSFEM) is developed to solve wave propagation problems in random media. This method simultaneously includes all features of spectral finite element and stochastic finite element methods, which leads to excellent accuracy and convergence by implementing Gauss–Lobatto–Legendre collocation points permitting to generate coarser meshes. In addition, the proposed StSFEM leads to diagonal mass matrices, which accelerates temporal integration schemes and provides desirable accuracy. Furthermore, it numerically solves the Fredholm integral equation arising from Karhunen–Loève Expansion with favorable accuracy and computing time. Here, the StSFEM is examined and developed to stochastic wave propagation phenomena through several numerical simulations. Results demonstrate successful performance of the StSFEM in the solved problems so that one can accomplish uncertainty quantification of time domain wave propagation within random continua by incorporating the StSFEM.

Dynamic Response of a Cracked Viscoelastic Anisotropic Plane Using Boundary Elements and Fractional Derivatives

Abstract The aim of this study is to develop an efficient numerical technique using the non-hypersingular, traction boundary integral equation method (BIEM) for solving wave propagation problems in an anisotropic, viscoelastic plane with cracks. The methodology can be extended from the macro-scale with certain modifications to the nano-scale. Furthermore, the proposed approach can be applied to any type of anisotropic material insofar as the BIEM formulation is based on the fundamental solution of the governing wave equation derived for the case of general anisotropy. The following examples are solved: (i) a straight crack in a viscoelastic orthotropic plane, and (ii) a blunt nano-crack inside a material of the same type. The mathematical modelling effort starts from linear fracture mechanics, and adds the fractional derivative concept for viscoelastic wave propagation, plus the surface elasticity model of M. E. Gurtin and A. I. Murdoch, which leads to nonclassical boundary conditions at the nano-scale. Conditions of plane strain are assumed to hold. Following verification of the numerical scheme through comparison studies, further numerical simulations serve to investigate the dependence of the stress intensity factor (SIF) and of the stress concentration factor (SCF) that develop in a cracked inhomogeneous plane on (i) the degree of anisotropy, (ii) the presence of viscoelasticity, (iii) the size effect with the associated surface elasticity phenomena, and (iv) finally the type of the dynamic disturbance propagating through the bulk material.

Generalized wave propagation problems and discrete exterior calculus

We introduce a general class of second-order boundary value problems unifying application areas such as acoustics, electromagnetism, elastodynamics, quantum mechanics, and so on, into a single framework. This also enables us to solve wave propagation problems very efficiently with a single software system. The solution method precisely follows the conservation laws in finite-dimensional systems, whereas the constitutive relations are imposed approximately. We employ discrete exterior calculus for the spatial discretization, use natural crystal structures for three-dimensional meshing, and derive a “discrete Hodge” adapted to harmonic wave. The numerical experiments indicate that the cumulative pollution error can be practically eliminated in the case of harmonic wave problems. The restrictions following from the CFL condition can be bypassed with a local time-stepping scheme. The computational savings are at least one order of magnitude.

Finite-Difference Solution of the Parabolic Equation Under Horizontal Polar Coordinates

The parabolic equation method has been widely used in the contexts of radar, remote sensing, and radio frequency planning, among other fields. It is an approximate means of solving wave propagation problems, and it has high computational efficiency compared to the Helmholtz full-wave equation. In this study, we derive the finite difference format with polar coordinates. By using this formula, the point source (i.e., line source) wave propagation problem can be solved. While considering various radio wave propagation environments, we make use of a scale model. In a series of experiments, we compare measurement results to calculations, then verify the feasibility and accuracy of the parabolic equation method in predicting the horizontal diffraction loss in the process of wave propagation.