One-dimensional Wave Propagation Problems Research Articles

An improved algorithm for Finite Particle Method considering Lagrange-type remainder

The Finite Particle Method (FPM) is a significant improvement to the traditional Smoothed Particle Hydrodynamics method (SPH), which can greatly improve the computational accuracy of the entire computational domain. However, unstable calculation results and long computational time are still major obstacles to the development of FPM. Based on matrix decomposition, the fundamental equations of the traditional Finite Particle Method are rewritten and a Generalized Finite Particle Method (GFPM) is derived by introducing Lagrange-type remainder. By deriving and rewriting the fundamental equations, the GFPM method can be theoretically proven to be always stable. Numerical examples show that the GFPM method can utilize a smaller computational scale to achieve the same computational accuracy as the FPM method, with a corresponding reduction in computational time. Finally, the GFPM method is applied to a one-dimensional stress wave propagation problem and a one-dimensional heat conduction problem, and the computational results are compared with those of the SPH method and the FPM method, which verify that the GFPM has higher computational accuracy and stability.

Phase-preserving theory and its linearization approximation for forward scattering field of scalar acoustic wave equation

The conventional wave-equation linearization methods, such as the first-order Born or Rytov approximation, always implicitly imply a weak-scattering assumption, making it valid only for weak perturbation models. To extend the wave-equation linearization theory to strong perturbation models, we consider a scenario that the reference model is smooth within the scale of the incident wave length, and propose a phase-preserving method which can predict the phase perturbation of forward scattering wave field. First, we introduce the WKBJ approximation to the scattered- and incident wave fields so that the integral of the unknown solution (i.e. the scattered field) in the nonlinear Ricatti integral equation can be replaced by the integral of scattering-angle and model perturbation, yielding an explicit expression of the scattered field. Theoretical derivation shows that the proposed phase-preserving method can accurately predict the phase-perturbation of forward scattered wave field regardless of the strength of velocity perturbations for one-dimensional wave propagation problem. To apply the phase-preserving approximation to the inverse problem, we further consider a scenario of small-angle forward propagation. In this case, the phase-preserving approximation can be linearized by neglecting the influence of scattering angles, leading to a linear relation between the scattered field and the model perturbation, which we refer to as the generalized Rytov approximation. Numerical experiments demonstrate that the generalized Rytov approximation can predict the phase perturbation of the scattered field with higher accuracy for small-angle forward propagation, and is suitable for strong model perturbations. The generalized Rytov approximation extends the validity and the scope of application of the traditional Rytov approximation. In specific application fields such as the seismic traveltime tomography or medical ultrasonic transmission imaging, a new traveltime/phase sensitivity kernel can be derived by replacing the conventional Rytov approximation with the proposed method, which can increase the inversion accuracy and speed up the convergence.

A fractional nonlocal elastic model for lattice wave analysis

This paper investigates the link between lattice elasticity and nonlocal continuum mechanics from one-dimensional and multi-dimensional wave propagation problems. Eringen (1983) closed the bridge between one-dimensional linear lattice elasticity (called Lagrange lattice) and nonlocal elastic continuum. The Born-Kármán wave dispersive properties of the one-dimensional lattice are accurately fitted by Eringen's nonlocal differential elastic law using asymptotic or phenomenological arguments. A fractional version of Eringen's nonlocal model based on the introduction of a fractional Laplacian for the nonlocal differential operators can improve the accuracy in matching the wave dispersive curve of lattice dynamics. A multi-dimensional generalization of Lagrange lattice for cubic crystals is the lattice of Gazis et al. composed of central and angular interactions. We show in this paper that the differential Eringen's nonlocal elasticity differs from the lattice-based nonlocal elasticity continuum. In particular, Eringen's nonlocal elasticity preserves the isotropic property, whereas the cubic lattice is only isotropic in the low frequency regimes. The fractional generalization of Eringen's model accurately fitted the wave dispersive properties of the cubic crystal in the first Brillouin zone. The fractional generalization of Eringen's model highlights an isotropic response in the low frequency regimes and a cubic symmetry in the high frequency regime.

Micro-mechanical fracture dynamics and damage modelling in brittle materials.

This study explores the interplay between wave propagation and damage in brittle materials. The damage models, based on micro-mechanical fracture dynamics, capture any possible unstable growth of micro-cracks, introducing a macroscopic loss of stability. After stating the non-dimensional mathematical problem describing the wave propagation with damage, we introduce a non-dimensional number, called the microscopic evolution index, which links the micro and macro scales and discriminates the microscopic scale behaviour. For large values of microscopic evolution index, corresponding to a microscopic quasi-static process coupled with a macroscopic dynamic one, the macroscopic dynamic system could lose its hyperbolicity or become very stiff and generate shock waves. A semi-analytical solution to the one-dimensional wave propagation problem with damage, which could be very useful in the accuracy evaluation of the numerical schemes, was constructed. Concerning the asymptotic behaviour of the dynamic exact solution on the microscopic evolution index (or on the strain rate), an important strain rate sensitivity was found: the pulse loses its amplitude for decreasing strain rate and, starting with a critical value, the micro-scale model is rate independent. A possible regularization technique to smooth the shock waves at low and moderate strain rates is discussed. Finally, some numerical results analyse the role played by the the friction on the micro-cracks in the damage modelling of blast wave propagation. This article is part of the theme issue 'Fracture dynamics of solid materials: from particles to the globe'.

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.

Seismic harmonic response of inhomogeneous soil: scaling analysis

The study of the seismic response of inhomogeneous soil deposits, underlying rigid bedrock, has traditionally been framed as a one-dimensional wave propagation problem in a linear viscoelastic medium. When it comes to modelling soil inhomogeneity, the current trend to model the continuous variation of soil stiffness with depth employs a ‘generalised parabolic function’. Exact solutions have already been obtained, but these come in terms of Bessel functions, which muddle the interpretation of results and obstruct assessment of the parameters’ influence. This also hinders the definition of an ‘equivalent homogeneous’ soil deposit (a relevant concept utilised in many seismic codes). It is shown herein that the so-called inhomogeneity factor plays a secondary role in determining the response in many cases; thus straightforward guidelines are suggested for simplifying the problem, leading to elementary scaling relations. The scalings provide simple yet meaningful relations that reveal and explain the fundamental traits of the dynamic behaviour of these systems.

One-dimensional scalar wave propagation in multi-region domains by the boundary element method

Two boundary element method formulations are presented for the analysis of the one-dimensional scalar wave propagation problem in multi-region domains. One of the formulations employs the time-domain fundamental solution; the other, the fundamental solution related to the static problem. The problem domain is constituted of sub-domains with different material properties and, consequently, with different wave propagation velocities. Thus, the sub-region technique, akin to the boundary element method, is used to deal with the non-homogeneous problem. At the end of the article, some examples are presented, illustrating the characteristics of each formulation and their accuracy.

A stable high-order FC-based methodology for hemodynamic wave propagation

A high-order numerical algorithm is proposed for the solution of one-dimensional arterial pulse wave propagation problems based on use of an accelerated “Fourier continuation” (FC) methodology for accurate Fourier expansion of non-periodic functions. The solver provides high-order accuracy, mild Courant-Friedrichs-Lewy (CFL) constraints on the time discretization and, importantly, results that are essentially free of spatial dispersion errors—enabling fast and accurate resolution of clinically-relevant problems requiring simulation of many cardiac cycles or vascular segments. The left ventricle-arterial model that is employed presents a particularly challenging case of ordinary differential equation (ODE)-governed boundary conditions that include a hybrid ODE-Dirichlet model for the left ventricle and an ODE-based Windkessel model for truncated vasculature. Results from FC-based simulations are shown to capture the important physiological features of pressure and flow waveforms in the systemic circulation. The robustness of the proposed solver is demonstrated through a number of numerical examples that include performance studies and a physiologically-accurate case study of the coupled left ventricle-arterial system. The results of this paper imply that the FC-based methodology is straightforwardly applicable to other biological and physical phenomena that are governed by similar hyperbolic partial differential equations (PDEs) and ODE-based time-dependent boundaries.

A dispersive homogenization model for composites and its RVE existence

An asymptotic homogenization model considering wave dispersion in composites is investigated. In this approach, the effect of the microstructure through heterogeneity-induced wave dispersion is characterised by an acceleration gradient term scaled by a “dispersion tensor”. This dispersion tensor is computed within a statistically equivalent representative volume element (RVE). One-dimensional and two-dimensional elastic wave propagation problems are studied. It is found that the dispersive multiscale model shows a considerable improvement over the non-dispersive model in capturing the dynamic response of heterogeneous materials. To test the existence of an RVE for a realistic microstructure for unidirectional fiber-reinforced composites, a statistics study is performed to calculate the homogenized properties with increasing microstructure size. It is found that the convergence of the dispersion tensor is sensitive to the spatial distribution pattern. A calibration study on a composite microstructure with realistic spatial distribution shows that convergence is found although only with a relatively large micromodel.

Adaptive simulation of wave propagation problems including dislocation sources and random media

An adaptive Tikhonov regularization is integrated with an h-adaptive grid-based scheme for simulation of elastodynamic problems, involving seismic sources with discontinuous solutions and random media. The Tikhonov method is adapted by a newly-proposed detector based on the MINMOD limiters and the grids are adapted by the multiresolution analysis (MRA) via interpolation wavelets. Hence, both small and large magnitude physical waves are preserved by the adaptive estimations on non-uniform grids. Due to developing of non-dissipative spurious oscillations, numerical stability is guaranteed by the Tikhonov regularization acting as a post-processor on irregular grids. To preserve waves of small magnitudes, an adaptive regularization is utilized: using of smaller amount of smoothing for small magnitude waves. This adaptive smoothing guarantees also solution stability without over smoothing phenomenon in stochastic media. Proper distinguishing between noise and small physical waves are challenging due to existence of spurious oscillations in numerical simulations. This identification is performed in this study by the MINMOD limiter based algorithm. Finally, efficiency of the proposed concept is verified by: 1) three benchmarks of one-dimensional (1-D) wave propagation problems; 2) P-SV point sources and rupturing line-source including a bounded fault zone with stochastic material properties.

Interface-preserving level set method for simulating dam-break flows

An interface-preserving level set method that solves advection and re-initialization equations for simulating three-dimensional dam-break flows is developed. This method solves mass transfer problems on a uniform staggered Cartesian grid. The advection equation that is used to advect the level set function for capturing the interface is discretized by a proposed fourth-order spatial discretization scheme. This scheme is dispersion-relation-preserving and is compact-reconstruction weighted essentially non-oscillatory (DRP-CRWENO4). This scheme is compared with a previous fifth-order, weighted, essentially non-oscillatory (WENO5) scheme and can represent an interface more accurately, while exactly preserving mass conservation. This level set approach introduces a mass correction term into the re-initialization equation based on local interface-preserving conditions. An explicit Adams–Bashforth algorithm on a staggered Eulerian grid is used for the Navier–Stokes solver. The point successive over-relaxation method is then employed to solve the resulting linear system. Two one-dimensional wave propagation problems are simulated to verify the proposed DRP-CRWENO4 scheme, which is shown to be capable of effectively capturing large gradients with fourth-order accuracy. To demonstrate their resolution, the two advection schemes (WENO5 and DRP-CRWENO4) are applied in two two-dimensional benchmark cases, i.e., a vortex deforming problem and Zalesak's disk problem, where simulation results of both schemes are compared against each other. Demonstration study is then further extended to three-dimensional cases of the vortex deforming problem and Zalesak's sphere problem, and simulation results agree well with those using hybrid particle level set method. Finally, several dam-break problems with and without obstacles are investigated to validate the coupled two-phase incompressible flow and level set method solver. The results for the predicted flow structure and mass conservation properties are compared with the reported experimental data or numerical simulations.

Analytical D’Alembert Series Solution for Multi-Layered One-Dimensional Elastic Wave Propagation with the Use of General Dirichlet Series

A general initial-boundary value problem of one-dimensional transient wave propagation in a multi-layered elastic medium due to arbitrary boundary or interface excitations (either prescribed tractions or displacements) is considered. Laplace transformation technique is utilised and the Laplace transform inversion is facilitated via an unconventional method, where the expansion of complex-valued functions in the Laplace domain in the form of general Dirichlet series is used. The final solutions are presented in the form of finite series involving forward and backward travelling wave functions of the d’Alembert type for a finite time interval. This elegant method of Laplace transform inversion used for the special class of problems at hand eliminates the need for finding singularities of the complex-valued functions in the Laplace domain and it does not need utilising the tedious calculations of the more conventional methods which use complex integration on the Bromwich contour and the techniques of residue calculus. Justification for the solutions is then considered. Some illustrations of the exact solutions as time-histories of stress or displacement of different points in the medium due to excitations of arbitrary form or of impulsive nature are presented to further investigate and interpret the mathematical solutions. It is shown via illustrations that the one-dimensional wave motions in multi-layered elastic media are generally of complicated forms and are affected significantly by the changes in the geometrical and mechanical properties of the layers as well as the nature of the excitation functions. The method presented here can readily be extended for three-dimensional problems. It is also particularly useful in seismology and earthquake engineering since the exact time-histories of response in a multi-layered medium due to arbitrary excitations can be obtained as finite sums.

A quadratic b-spline based isogeometric analysis of transient wave propagation problems with implicit time integration method

• Wave dispersion error with both the spatial isogeometric elements and direct time integration method is first studied. • New generalized numerical algorithm is proposed to calculate the dispersion errors. • New isogeometric analysis for wave problems are proposed with desirable accuracy and computation efficiency. A uniform quadratic b-spline isogeometric element is exclusively considered for wave propagation problem with the use of desirable implicit time integration scheme. A generalized numerical algorithm is proposed for dispersion analysis of one-dimensional (1-D) and two-dimensional (2-D) wave propagation problems where the quantified influence of the defined CFL number on wave velocity error is analyzed and obtained. Meanwhile, the optimal CFL (Courant–Friedrichs–Lewy) number for the proposed 1-D and 2-D problems is suggested. Four representative numerical simulations confirm the effectiveness of the proposed method and the correctness of dispersion analysis when appropriate spatial element size and time increment are adopted. The desirable computation efficiency of the proposed isogeometric method was confirmed by conducting time cost and calculation accuracy analysis of a 2-D numerical example where the referred FEM was also tested for comparison.

Improved quadratic isogeometric element simulation of one-dimensional elastic wave propagation with central difference method

Two improved isogeometric quadratic elements and the central difference scheme are used to formulate the solution procedures of transient wave propagation problems. In the proposed procedures, the lumped matrices corresponding to the isogeometric elements are obtained. The stability conditions of the solution procedures are also acquired. The dispersion analysis is conducted to obtain the optimal Courant-Friedrichs-Lewy (CFL) number or time-step sizes corresponding to the spatial isogeometric elements. The dispersion analysis shows that the isogeometric quadratic element of the fourth-order dispersion error (called the isogeometric analysis (IGA)-f quadratic element) provides far more desirable numerical dissipation/dispersion than the element of the second-order dispersion error (called the IGA-s quadratic element) when appropriate time-step sizes are selected. The numerical simulations of one-dimensional (1D) transient wave propagation problems demonstrate the effectiveness of the proposed solution procedures.

Multi-modal transport prediction for linear one-dimensional wave propagation problems

Considering a linear one-dimensional wave propagation problem, a technique to predict wave transport across gaps between computational subdomains is proposed. The observed waves at the “source” domain are disassembled into a modal description, and their propagation into the “destination” domain is predicted using the system’s dispersion relation. This approach is extended into a multi-modal wave prediction functionality which enables efficient domain decomposition for time-critical computation problems. Simulation results on a railway catenary’s contact wire simulation highlights the attained high accuracy of the prediction. This technique can serve as a cornerstone for the development of real-time control tasks or any applications where domain decomposition is applied to reduce computational costs.

Propagation of one-dimensional non-stationary waves in viscoelastic half space

The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is considered. Relaxation kernel is considered exponential. Initial conditions are set to zero, displacement is determined on the half space boundary. The solution is represented in the form of perturbation and surface Green function resultant. For determination of Green function time Laplace transform is used. Its inverse transform is carried out both analytically expanding Green function in series and numerically. A good coincidence of analytical and numerical Green function calculation results is shown. The final solution is determined analytically. Examples of calculations are represented.

A tent pitching scheme motivated by Friedrichs theory

Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space–time finite element scheme of the tent pitching type, with spaces that conform to the continuity property, is designed. Numerical results for a model one-dimensional wave propagation problem are presented.

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

In this paper, a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves. The spatial domain is discretized by the finite element method. To obtain accurate results, the standard finite element method requires a very fine mesh, which is why the computational effort can be very time consuming. The use of high-order finite element methods–such as the spectral element method based on Lagrange polynomials through Gauss–Lobatto–Legendre points or the iso-geometric analysis using non-uniform rational B-splines–can reduce the enormous computational costs significantly, compared to the standard finite element method. However, explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave. The procedure proposed here is basically similar to the explicit method of Noh and Bathe, but the semi-discrete equation of motion is modified by introducing a damping parameter to suit the high-order FEM analysis of the discontinuous wave propagation. The performance due to this modification is tested for one-dimensional wave propagation problems using both lumped and consistent mass matrices. Then, the idea of a combination of the consistent and row sum lumped mass matrices is evaluated. The proposed method is studied also in two-dimensions by considering the Lamb problem of wave propagation. The results are promising enough to provide a better simulation of the discontinuous wave propagation problem.

Boundary element method formulations for the solution of the scalar wave equation in one-dimensional problems

This work is concerned with the development of two boundary element method (BEM) formulations for the solution of one-dimensional scalar wave propagation problems. The first formulation is called TD-BEM, TD meaning time-domain, as it employs a time-dependent fundamental solution. The second formulation is called D-BEM, D meaning domain, and employs the fundamental solution from the static problem. The Houbolt and the Newmark methods are employed for the time-marching in the D-BEM approach. Two examples, constituted of five analyses, are included.

Back-calculation of elastic modulus of soil and subgrade from portable falling weight deflectometer measurements

In the field of nondestructive testing of pavement, the portable deflectometer devices have gained, in the recent years, a wide use for in situ assessment of elastic properties of soils, subgrade and pavement foundations. However, if the use of the elasto-static model based on the Boussinesq’s theory does not constitute a shortcoming in the back-calculation procedure of homogenous elastic modulus, questions have been arose about the reliability and accuracy of the peak value method commonly used to extract the static stiffness of soils and subgrade from the dynamic transient data. This paper deals with the use of minimization technique, based on least square algorithm as an alternative method for data analysis and soil elastic stiffness identification. Details of the mathematical basis, the implementation and different steps for elastic modulus back calculation, are presented. In this method, an equivalent spring-mass-dashpot system, where the complex dynamic soil behavior is reduced to a viscoelastic one-dimensional wave propagation problem, is used for modeling the loading plate/soil system. The comparative study shows the suitability and the accuracy of the minimization method for soil elastic stiffness identification and elastic modulus back calculation.