Solution Of Wave Propagation Problems Research Articles

A dispersion analysis of uniformly high order, interior and boundaries, mimetic finite difference solutions of wave propagation problems

A preliminary stability and dispersion study for wave propagation problems is developed for mimetic finite difference discretizations. The discretization framework corresponds to the fourth-order staggered-grid Castillo-Grone operators that offer a sextuple of free parameters. The parameter-dependent mimetic stencils allow problem discretization at domain boundaries and at the neighbor grid cells. For arbitrary parameter sets, these boundary and near-boundary mimetic stencils are lateral, and we here draw first steps on the parametric dependency of the stability and dispersion properties of such discretizations. As a reference, our analyses also present results based on Castillo-Grone parameters leading to mimetic operators of minimum bandwidth that have been previously applied in similar physical problems. The most interior parameter-dependent mimetic stencils exhibit a specific Toeplitz-like structure, which reduces to the standard central finite difference formula for staggered differentiation at grid interior. Thus, our results apply to the whole discretization grid. The study done for the 1-D problem could be applied to the discretization of a free surface boundary condition along an orthogonal gridline to this boundary.

IGA-Energetic BEM: An Effective Tool for the Numerical Solution of Wave Propagation Problems in Space-Time Domain

The Energetic Boundary Element Method (BEM) is a recent discretization technique for the numerical solution of wave propagation problems, inside bounded domains or outside bounded obstacles. The differential model problem is converted into a Boundary Integral Equation (BIE) in the time domain, which is then written into an energy-dependent weak form successively discretized by a Galerkin-type approach. Taking into account the space-time model problem of 2D soft-scattering of acoustic waves by obstacles described by open arcs by B-spline (or NURBS) parametrizations, the aim of this paper is to introduce the powerful Isogeometric Analysis (IGA) approach into Energetic BEM for what concerns discretization in space variables. The same computational benefits already observed for IGA-BEM in the case of elliptic (i.e., static) problems, is emphasized here because it is gained at every step of the time-marching procedure. Numerical issues for an efficient integration of weakly singular kernels, related to the fundamental solution of the wave operator and dependent on the propagation wavefront, will be described. Effective numerical results will be given and discussed, showing, from a numerical point of view, convergence and accuracy of the proposed method, as well as the superiority of IGA-Energetic BEM compared to the standard version of the method, which employs classical Lagrangian basis functions.

Accurate solution of wave propagation problems in elasticity

The accurate solution of wave propagations in general two- and three-dimensional solids is still difficult and frequently impossible to achieve with the current computational schemes and computers available. We present in this paper a solution scheme that has much promise for the accurate solution of wave propagations in general solids. The procedure is based on the use of “overlapping finite elements” and direct time integration. The overlapping finite elements are effective because the spatial dispersion error is relatively small and can be monotonically reduced using a finer mesh. Similarly, the time integration dispersion errors can also be reduced monotonically as the time step becomes smaller. Hence the key property of the solution scheme is that the total dispersion error in the simulation of multiple waves traveling through solids is monotonically reduced as the spatial discretization and time stepping become finer. We summarize the ingredients of the solution scheme and illustrate the characteristics in the solution of some wave propagation problems that are difficult to solve accurately. These solutions may be benchmark solutions to use in the evaluations of other computational schemes.

On dynamic behavior of composite plates using a higher-order Zig-Zag theory and exponential basis functions

In this paper, a boundary node method is presented to study wave propagation in laminated composite plates. The Zig-Zag theory of Cho and Parmerter is used for deriving the governing equations of laminated plates. To the best of the authors’ knowledge, this study is the first one employing the theory in non-stationary dynamic plate problems addressing wave propagation issues. For this theory, there is no information about the Green’s functions and thus the presented method can be considered as an alternative to the boundary integral method. With the use of exponential basis functions (EBFs), the response of the structure is first found in the frequency domain and finally the time-domain response is obtained using inverse Fourier transformation. The EBFs are found so that they satisfy the governing equations in the frequency domain. For the first time, in this paper, we shall present explicit relations for the EBFs in the frequency domain for Cho and Parmerter’s Zig-Zag theory. The coefficients of the EBFs are found through the collocation of the boundary conditions. The dynamic analysis of some composite laminated plates is presented, and the results are compared to those obtained from the dynamic analysis of two-/three-dimensional finite element method (FEM). We shall discuss the capabilities and limitations of the theory and the solution method. The capability of the method, in the analysis of problems excited by high-frequency loads, is shown in the solution of wave propagation problems for which the FEM needs excessive number of elements and thus it is practically impossible to be applied.

High-order Discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes

We propose and analyze a high-order Discontinuous Galerkin Finite Element Method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements. We analyze the well posedness of the resulting formulation, prove hp-version error a-priori estimates, and present a dispersion analysis, showing that polygonal meshes behave as classical simplicial/quadrilateral grids in terms of dispersion properties. The theoretical estimates are confirmed through various two-dimensional numerical verifications.

A comparative study of three composite implicit schemes on structural dynamic and wave propagation analysis

A comparative study of three composite implicit schemes including the Bathe scheme (Bathe and Baig, 2005), the TTBDF scheme (Chandra et al., 2015) and the Wen scheme (Wen et al., 2017) is conducted in this paper. The stability and accuracy characteristics of these schemes are studied and compared by analytical and numerical simulation analysis. In the solution of wave propagation problems, a demonstrative dispersion analysis is given and problems are solved to illustrate the capabilities of the schemes for the solution of wave propagation problems as well as the numerical dissipation characteristics of the schemes. The nonlinear dynamic behavior of the Wen scheme is studied by considering commonly used nonlinear dynamic problems where the nonlinear performance of the scheme is exclusively compared with the Bathe and TTBDF schemes. The priority ranks of three presented composite schemes for different types of dynamic problems are obtained by theoretical and numerical analysis.

Efficient iterative methods for multi-frequency wave propagation problems: A comparison study

In this paper we present a comparison study for three different iterative Krylov methods that we have recently developed for the simultaneous numerical solution of wave propagation problems at multiple frequencies. The three approaches have in common that they require the application of a single shift-and-invert preconditioner at a suitable seed frequency. The focus of the present work, however, lies on the performance of the respective iterative method. We conclude with numerical examples that provide guidance concerning the suitability of the three methods.

On the use of NURBS-based discretizations in the scaled boundary finite element method for wave propagation problems

We discuss the application of non-uniform rational B-splines (NURBS) in the scaled boundary finite element method (SBFEM) for the solution of wave propagation problems at rather high frequencies. We focus on the propagation of guided waves along prismatic structures of constant cross-section. Comparisons are made between NURBS-based discretizations and high-order spectral elements in terms of the achievable convergence rates. We find that for the same order of shape functions, NURBS can lead to significantly smaller errors compared with Lagrange polynomials. The difference becomes particularly important at very high frequencies, where spectral elements are prone to instabilities. Furthermore, we analyze the behavior of NURBS for the discretization of curved boundaries, where the benefit of exact geometry representation becomes crucial even in the low-frequency range.

Error Analysis and Comparative Study of Numerical Methods for the Parabolic Equation Applied to Tunnel Propagation Modeling

Parabolic equation (PE) methods have been widely applied to the modeling of wireless propagation in tunnel environments. However, the relevant literature does not include concrete guidelines for the choice of the parameters of these methods and the tradeoffs involved. This paper provides a comprehensive analysis of the two sources of error that arise when PE methods are employed for the modeling of radio-wave propagation scenarios: the well-known numerical dispersion error stemming from the finite-difference solvers for PE and the approximation error stemming from the use of PE for the solution of wave propagation problems that are subject to Maxwell's equations. The analysis is performed for four methods, three of which have been already used in PE-based propagation studies, namely, the Crank-Nicolson (CN) scheme, the alternative-direction-implicit (ADI) method, and its locally one-dimensional (LOD-ADI) version. The fourth method is the Mitchell-Fairweather (MF)-ADI scheme that has been recently shown to be a promising alternative technique for tunnel propagation modeling. The proposed method leads to robust criteria for the choice of spatial discretization in realistic propagation scenarios, as shown via numerical examples.

Accurate solutions of wave propagation problems under impact loading by the standard, spectral and isogeometric high-order finite elements. Comparative study of accuracy of different space-discretization techniques

For the first time, accurate numerical solutions to impact problems have been obtained with the standard, spectral, and isogeometric high-order finite elements. Spurious high-frequency oscillations appearing in numerical results are quantified and filtered out by the two-stage time-integration approach. We also use the 1-D impact problem with a simple analytical solution for the comparison of accuracy of the different space-discretization techniques used for transient acoustics and elastodynamics problems. The numerical results show the computational efficiency of the linear finite elements with reduced dispersion compared with other space-discretization techniques used for elastodynamics with implicit and explicit time-integration methods. We also show that for all space-discretization methods considered (except the linear finite elements with the lumped mass matrix), very small time increments which are much smaller than stability limit should be used in basic computations at large observation times. We should note that the size of time increments used at the filtering stage and calculated according to the special formulas defines the range of actual frequencies and can be used as a quantitative measure for the comparison and prediction of the accuracy of different space-discretization techniques. We also show that the new findings are valid in the multidimensional case.

An explicit time integration scheme for the analysis of wave propagations

A new explicit time integration scheme is presented for the solution of wave propagation problems. The method is designed to have small solution errors in the frequency range that can spatially be represented and to cut out high spurious frequencies. The proposed explicit scheme is second-order accurate for systems with and without damping, even when used with a non-diagonal damping matrix. The stability, accuracy and numerical dispersion are analyzed, and solutions to problems are given that illustrate the performance of the scheme.

Dynamic analysis of a poroelastic layered half-space using continued-fraction absorbing boundary conditions

A methodology for dynamic analysis of a poroelastic layered half-space is proposed. A poroelastic layered half-space is discretized with regular thin layers to a certain depth. Below the layers, continued-fraction absorbing boundary conditions (CFABCs) which are accurate and effective in modeling wave propagation in various unbounded domains are applied in order to represent the infinite extent of the half-space. With the representation, a spectral decomposition which is an effective approach in the solution of wave-propagation problems is utilized. Green functions of a poroelastic layered half-space are obtained accurately with the proposed numerical model. The Green functions can be applied to various dynamic problems in the half-space including foundation dynamics and soil–structure interaction problems.

Performance of an implicit time integration scheme in the analysis of wave propagations

In earlier work an effective implicit time integration scheme was proposed for the finite element solution of nonlinear dynamic problems [1,2]. The method, referred to as the Bathe method, was shown to possess unusual stability and accuracy characteristics for the solution of problems in linear and nonlinear structural dynamics [1–3]. In this paper we study the dispersion properties of the method, in comparison to those of the widely used Newmark trapezoidal rule, and show that the desired characteristics of the Bathe method for structural dynamics are also seen, and are very important, in the solution of wave propagation problems. A dispersion analysis is given and problems are solved to illustrate the capabilities of the scheme for the solution of wave propagation problems.

Accurate 3-D finite element simulation of elastic wave propagation with the combination of explicit and implicit time-integration methods

One of the big issues in finite element solutions of wave propagation problems is the presence of spurious high-frequency oscillations that may lead to divergent results at mesh refinement. The paper deals with the extension of the new two-stage time-integration technique developed in our previous papers to the solution of wave propagation problems with explicit time-integration methods.The explicit central difference method is used for accurate time-integration of the semi-discrete system of elastodynamics at the stage of basic computations and allows spurious high-frequency oscillations. To filter these oscillations, pre- or/and post-processing (the filtering stage) is applied using a few time increments of the implicit time-continuous Galerkin method with large numerical dissipation.A special calibration procedure is used for the selection of the minimum necessary amount of numerical dissipation (in terms of a time increment) at the filtering stage. In contrast to existing approaches that use a time-integration method with the same dissipation (or artificial viscosity) for all time increments, the new technique yields accurate and non-oscillatory results for wave propagation problems without interaction between user and computer code. The solutions of 3-D wave propagation and impact problems show the effectiveness of the new approach.

A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L 2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.

High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations

A new PDE solver was introduced recently, in Part I of this two-paper sequence, on the basis of two main concepts: the well-known Alternating Direction Implicit (ADI) approach, on one hand, and a certain ''Fourier Continuation'' (FC) method for the resolution of the Gibbs phenomenon, on the other. Unlike previous alternating direction methods of order higher than one, which only deliver unconditional stability for rectangular domains, the new high-order FC-AD (Fourier-Continuation Alternating-Direction) algorithm yields unconditional stability for general domains-at an O(Nlog(N)) cost per time-step for an N point spatial discretization grid. In the present contribution we provide an overall theoretical discussion of the FC-AD approach and we extend the FC-AD methodology to linear hyperbolic PDEs. In particular, we study the convergence properties of the newly introduced FC(Gram) Fourier Continuation method for both approximation of general functions and solution of the alternating-direction ODEs. We also present (for parabolic PDEs on general domains, and, thus, for our associated elliptic solvers) a stability criterion which, when satisfied, ensures unconditional stability of the FC-AD algorithm. Use of this criterion in conjunction with numerical evaluation of a series of singular values (of the alternating-direction discrete one-dimensional operators) suggests clearly that the fifth-order accurate class of parabolic and elliptic FC-AD solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations. To illustrate the FC-AD methodology in the hyperbolic PDE context, finally, we present an example concerning the Wave Equation-demonstrating sixth-order spatial and fourth-order temporal accuracy, as well as a complete absence of the debilitating ''dispersion error'', also known as ''pollution error'', that arises as finite-difference and finite-element solvers are applied to solution of wave propagation problems.

Lacunae based stabilization of PMLs

Perfectly matched layers (PMLs) are used for the numerical solution of wave propagation problems on unbounded regions. They surround the finite computational domain (obtained by truncation) and are designed to attenuate and completely absorb all the outgoing waves while producing no reflections from the interface between the domain and the layer. PMLs have demonstrated excellent performance for many applications. However, they have also been found prone to instabilities that manifest themselves when the simulation time is long. Hereafter, we propose a modification that stabilizes any PML applied to a hyperbolic partial differential equation/system that satisfies the Huygens’ principle (such as the 3D d’Alembert equation or Maxwell’s equations in vacuum). The modification makes use of the presence of lacunae in the corresponding solutions and allows us to establish a temporally uniform error bound for arbitrarily long-time intervals. At the same time, it does not change the original PML equations. Hence, the matching properties of the layer, as well as any other properties deemed important, are fully preserved. We also emphasize that besides the aforementioned PML instabilities per se, the methodology can be used to cure any other undesirable long-term computational phenomenon, such as the accuracy loss of low order absorbing boundary conditions.

Use of Exact Solutions of Wave Propagation Problems to Guide Implementation of Nonlinear Seismic Ground Response Analysis Procedures

One-dimensional nonlinear ground response analyses provide a more accurate characterization of the true nonlinear soil behavior than equivalent-linear procedures, but the application of nonlinear codes in practice has been limited, which results in part from poorly documented and unclear parameter selection and code usage protocols. In this article, exact (linear frequency-domain) solutions for body wave propagation through an elastic medium are used to establish guidelines for two issues that have long been a source of confusion for users of nonlinear codes. The first issue concerns the specification of input motion as “outcropping” (i.e., equivalent free-surface motions) versus “within” (i.e., motions occurring at depth within a site profile). When the input motion is recorded at the ground surface (e.g., at a rock site), the full outcropping (rock) motion should be used along with an elastic base having a stiffness appropriate for the underlying rock. The second issue concerns the specification of viscous damping (used in most nonlinear codes) or small-strain hysteretic damping (used by one code considered herein), either of which is needed for a stable solution at small strains. For a viscous damping formulation, critical issues include the target value of the viscous damping ratio and the frequencies for which the viscous damping produced by the model matches the target. For codes that allow the use of “full” Rayleigh damping (which has two target frequencies), the target damping ratio should be the small-strain material damping, and the target frequencies should be established through a process by which linear time domain and frequency domain solutions are matched. As a first approximation, the first-mode site frequency and five times that frequency can be used. For codes with different damping models, alternative recommendations are developed.

Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems

The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh, calls for the construction of local-time stepping algorithms. These schemes have already been developed for N-body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.

DGTD methods using modal basis functions and symplectic local time-stepping

The discontinuous Galerkin time domain (DGTD) methods are now widely used for the solution of wave propagation problems. Able to deal with unstructured meshes past complex geometries, they remain fully explicit with easy parallelization and extension to high orders of accuracy. Still, modal or nodal local basis functions have to be chosen carefully to obtain actual numerical accuracy. Concerning time discretization, explicit non-dissipative energypreserving time-schemes exist, but their stability limit remains linked to the smallest element size in the mesh. Symplectic algorithms, based on local-time stepping or local implicit scheme formulations, can lead to dramatic reductions of computational time, which is shown here on two-dimensional acoustics problems.