Elastic Wave Propagation Problems Research Articles

A 2.5D hybrid SBM-MFS methodology for elastic wave propagation problems

This paper proposes a novel hybrid methodology that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) for the computational simulation of elastic wave propagation. Particularly, the methodology aims to address radiation or scattering problems of longitudinally invariant systems in the wavenumber–frequency domain involving boundaries with intricate geometries. The approach uses the SBM to deal with the complex parts of these geometries and the MFS for the smooth ones. The method is studied in the framework of three case studies involving longitudinally infinite cavities in a homogeneous full-space with circular, square-shaped and five-cusped hypocycloid cross-sections. These three examples are selected to assess the accuracy and robustness of the hybrid SBM-MFS approach in comparison with alternative modelling strategies. The comparisons show that the proposed method inherits the accuracy of the MFS while keeping the robustness of the SBM when dealing with complex geometries. The method is found to be computationally more efficient than the SBM or the boundary element method (BEM). Moreover, the hybrid approach naturally mitigates the effect of fictitious eigenfrequencies, a feature that neither conventional versions of the SBM nor the BEM have.

A review of temporal and spatial dispersions of linear and quadratic finite elements in linear elastic wave propagation problems

A review of temporal and spatial dispersions of linear and quadratic finite elements in linear elastic wave propagation problems

An implicit–explicit time discretization for elastic wave propagation problems in plates

AbstractWe propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest.

A Kronecker product linear-cost solver for the high-order generalized-α method for multi-dimensional hyperbolic systems

The main result of the work is the generalization of the concept of the variational splitting from [3] to the higher-order generalized-α schemes proposed in [5] in the context of hyperbolic and elastic wave propagation problems. It is possible to factorize the matrix of the linear system of equations (M+ηK) in the linear computational cost. Here M is the mass matrix, and K is the stiffness matrix. We discretize using the isogeometric finite element method. Following the idea proposed in [3], factorizing the matrix (M+ηK), we obtained an additional error of the order of η2. This paper introduces a technique to couple the possibility of fast factorization of (M+ηK) with higher-order generalized-α schemes proposed in [5] for hyperbolic and elastic wave propagation problems. Thus, we introduce a variationally separable splitting technique for the high-order accuracy generalized-α method. We use tensor-product meshes to develop the splitting method, which results in the linear cost for multi-dimensional problems. We consider standard C0 finite elements as well as smoother isogeometric analysis for spatial discretization. The direction splitting method requires the mass and the stiffness matrices to have the Kronecker product structure. For other problems, the corresponding Kronecker product solver can be employed as a preconditioner as long as a Kronecker product can sufficiently well approximate the problem matrix. We also study the spectrum of the amplification matrix to establish the unconditional stability of the method. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. In numerical tests, we compute the L2 and H1 norms to show the optimal convergence of the discrete method in space and high-order accuracy in time.

Dynamic stress analysis of elliptical inclusions in an inhomogeneous medium with modulus and density variations under the action of shear horizontal (SH) waves

In this paper, in connection with the problem of elastic wave propagation in a continuously inhomogeneous medium, the dynamic stress concentration phenomenon caused by the elliptical inclusions in a complex, continuously inhomogeneous medium under the action of shear horizontal (SH) waves is studied. First, the auxiliary function and the shear modulus function are introduced to construct the wave field function, and the modulus is reduced in order and simplified using a high-order fluctuating wave equation with variable coefficients that has power function variations; then, the auxiliary function and the modulus function are introduced once again to construct the density function; finally, the conformal mapping method is used to carry out coordinate transformation and to solve and obtain the standard form of the Helmholtz equation. Based on this combined problem model, the wave field and the stress field are derived, and the expression for the dynamic stress concentration factor (DSCF) is given. The influence of the inhomogeneous parameters, the ratio of the long and short axes of the elliptical inclusions, and the relative properties of the elliptical inclusions and the matrix on the DSCF around the elliptical inclusions is analyzed through calculation examples.

Optimisation strategies for multi-layered armour plates

ABSTRACT A set of non-linear optimisation algorithms are combined with a finite element simulation code to analyse the energy absorption and elastic stress wave propagation problem in multilayer/multimaterial armour systems under ballistic impacts. An Abaqus Python script is used to simulate the ballistic event and to generate the variables and post-processing outputs necessary for the integration with the optimisation algorithms. A number of modelling strategies are considered and three optimisation algorithms are used: Particle Swarm Optimisation (PSO), Genetic Algorithm (GA) and Simulated Annealing (SA). The performance and efficiency of each algorithm are assessed through four benchmark tests with different levels of complexity. A multi-objective optimisation procedure is proposed that uses the most efficient algorithm based on every single-objective formulation, variables and constraints from the benchmark tests, resulting in a highly non-linear problem. The proposed optimisation methods successfully achieve the study purposes both in the simulation of generic ballistic impacts and in the quality of the optimised solutions, demonstrating the potential for this type of optimisation method on terminal ballistic applications, serving as a standpoint for further studies into higher energy impacts and material non-linearities.

ADI-based, conditionally stable schemes for seismic P-wave and elastic wave propagation problems

ADI-based, conditionally stable schemes for seismic P-wave and elastic wave propagation problems

Partially pivoted ACA based acceleration of the energetic BEM for time-domain acoustic and elastic waves exterior problems

We consider acoustic and elastic wave propagation problems in 2D unbounded domains, reformulated in terms of space-time Boundary Integral Equations (BIEs). For their numerical solution, we employ a weak formulation related to the energy of the system and we discretize the weak problems by a Galerkin-type Boundary Element Method (BEM): this approach, called Energetic BEM, has revealed accurate and stable even on large time intervals of analysis. In particular, it results that, when standard Lagrangian basis functions are considered, the BEM matrices have Toeplitz lower triangular block structure, where blocks for growing time become fully populated; hence the overall memory cost of the energetic BEM is O(M2N), M and N being the number of the space degrees of freedom and the total number of performed time steps, respectively. This drawback prevents the application of such method to large scale problems. As a possible remedy, we propose a fast technique based on the Adaptive Cross Approximation (ACA). The core of this procedure is the approximation of sufficiently large time blocks of the energetic BEM matrix through the partially pivoted ACA algorithm, which allows to compute only few of the original entries. This leads to reduced assembly time, which for the energetic BEM is generally relevant, coupled with reduced memory storage requirements. Additionally, the consequent acceleration of the matrix/vector multiplication together with a marching on time procedure, leads to remarkable reduction of the computational solution time. The effectiveness of the proposed method is theoretically proved and several numerical results are presented and discussed.

Incident waves at the surface of micropolar porous materials

The problems of elastic wave propagation in the micropolar porous materials have been attempted under the interaction of micro-rotation tensor and porous distribution in this material. We obtain the existence of three coupled longitudinal and two coupled shear waves propagating with different phase speeds. Numerically, the phase speed and attenuation of five basic waves are computed. We consider the incident plane waves at the boundary of micropolar porous materials, then the dispersive nature of the incident waves is studied and the existence of critical angle for the incident shear wave is found. The amplitude and energy ratios of various reflected waves for the incident coupled longitudinal and shear waves are obtained analytically and numerically. Further, the influence of different material parameters is studied using dispersion relation and the relevant numerical values.

A modal-based Partition of Unity Finite Element Method for elastic wave propagation problems in layered media

The time-harmonic propagation of elastic waves in layered media is simulated numerically by means of a modal-based Partition of Unity Finite Element Method (PUFEM). Instead of using the standard plane waves or the Bessel solutions of the Helmholtz equation to design the discretization basis, the proposed modal-based PUFEM explicitly uses the tensor-product expressions of the eigenmodes (the so-called Love and interior modes) of a spectral elastic transverse problem, which fulfil the coupling conditions among layers. This modal-based PUFEM approach does not introduce quadrature errors since the coefficients of the discrete matrices are computed in closed-form. A preliminary analysis of the high condition number suffered by the proposed method is also analyzed in terms of the mesh size and the number of eigenmodes involved in the discretization. The numerical methodology is validated through a number of test scenarios, where the reliability of the proposed PUFEM method is discussed by considering different modal basis and source terms. Finally, some indicators are introduced to select a convenient discrete PUFEM basis taking into account the observability of cracks located on a coupling boundary between two adjacent layers.

Bending/shear wave dispersion analysis of granular chains – Discrete and enriched continuous Cosserat modelling

The current study focuses on the wave dispersion analysis of granular systems by investigating a paradigmatic one-dimensional discrete chain. Most of the analysis of the discrete problem are based on the discrete granular chain without additional confinement. This work is motivated by the detection of standing waves and negative velocity of acoustic and optical waves observed in discrete granular models. This study allows to understand better the incorporation between the theoretical models, granular materials and the included effective parameters. Concerning wave propagation, granular elements create a structured wave-guide network through which mechanical energy is transferred. In the presence of internal (microstructural) length scales, the elastic wave propagation problem involves an interplay between wave dispersion and structural features. The unidimensional granular chain is composed of uniform spherical grains elastically connected with both shear and rotational springs. The presented structural system can be considered as an elastic lattice model or a Cosserat chain model with shear interaction. We show how the elastic foundation affects the wave dispersion response of the system. The length scale (grain diameter) at which the system is probed, is an important issue bridging together multi-scale behavior and heterogeneity. In a first step, the wave dispersion of the discrete system is derived from the uncoupled mixed differential-difference equation of deflection. We further show that for infinite number of grains the dispersion equation of the discrete model converges toward the continuum model of Bresse-Timoshenko.The wave dispersive properties of this discrete model are investigated also in the Brillouin zone. Next, the discrete to continuous model is approximated by the methods of Taylor polynomial development and Padé rational expansion. Finally, a comprehensive dispersive analysis is done through the dispersion equations of the discrete model, the discrete to continuous approximation beam and the continuous one. Some experimental results for Carbon Nanotubes (CNTs) are replicated by the model. We show that the model is able to predict accurately the positive and negative group of wave velocity as well as the standing wave. Scale effects and wave dispersive characteristics of the granular chain are captured by the continuous gradientelasticity model.

An effective numeric method for different formulations of the elastic wave propagation problem in isotropic medium.

This paper shows how the Generalized Finite Difference Method allows the same schemes in differences to be used for different formulations of the wave propagation problem. These formulations present pros and cons, depending on the type of boundary and initial conditions at our disposal and also the variables we want to compute, while keeping additional calculations to a minimum. We obtain the explicit schemes of this meshless method for different possible formulations in finite differences of the problem. Criteria for stability and convergence of the schemes are given for each case. The study of the dispersion of the phase and group velocities presented in previuos paper of the authors is also completed here. We show the application of the propounded schemes to the wave propagation problem and the comparison of the efficiency, convenience and accuracy of the different formulations.

An analytical benchmark for a 2D problem of elastic wave propagation in a solid

A two-dimensional problem of elastic wave propagation in a rectangular solid is considered. It is first approached using the classical apparatus of mathematical physics, with a fully analytical model produced as the result. The same system is then analysed numerically using two commercial finite element software packages, COMSOL and ANSYS APDL, with the focus on the bulk and guided wave propagation. The results produced by the numerical models are compared with those produced by the analytical one, and the superiority of the latter in terms of the computational costs is established, while a good overall quantitative agreement is observed. The proposed analytical model is therefore shown to be a very attractive choice as a benchmark for validation of fully-numerical models. A procedure for the inverse use of the model is established and tested as well.

Finite element method based computational time reversal in elastodynamics: Refocusing, reconstruction and its numerical sensitivity

In this paper, we study the accuracy and robustness of the computational time reversal approach based on the explicit finite element method for application in nondestructive testing in solids. The main goal of this paper is to find a methodology for correct and accurate reconstruction of the original source time history. For numerical modeling of frontal (forward) and reverse (backward) problems of elastic wave propagation, we use the finite element method and explicit time integration with the lumped mass matrix. The suggested methodology is applicable in each finite element open source or commercial software. A special attention is paid to prescription of boundary conditions/loading for the reverse problem for accurate reconstruction of time history of the original source. For evaluation of the reconstruction quality, we suggest certain cost functions. Based on several numerical tests, we show effects of prescription of boundary conditions/loading in time reversal, effect of mesh size and time step size, an unknown obstacle, a number of sources, and environmental disturbance (noise) on the correctness of reconstruction of the original source.

Mixed variational one-dimensional dynamic thermo-viscoplasticity for wave propagation

In view of recent developments in advanced manufacturing and nanomaterial technologies, which have enabled realization of novel material architectures at small-scale, a mixed variational temperature-dependent rate-dependent thermo-viscoplastic continuum element is proposed and developed for elastic, plastic, and thermal wave propagation problems. The variational scheme is based on a Hamiltonian approach, while for the temporal and spatial discretization scheme a discrete calculus of variations approach is adopted. Within this framework, an exponential temperature dependence for material properties and second sound effects is included. The developed framework sets the foundations that can lead to new class of temperature dependent models for capturing thermoplastic behavior and wave propagation of different materials. The robustness of the proposed variational approach, and aspects of coupled dynamic temperature-dependent thermoplastic response are demonstrated through several case studies. Applications include elastic, plastic, and thermal wave propagation and reflection in semi-infinite and finite length space at small temporal scales.

Numerical approach of free and forced elastic vibrations using high-regularity Hermitian partition of unities

In this paper, shape functions with regularity up to $$C^{2}$$ were developed for the four-node quadrilateral finite element considering the partition of unity property. This high-regularity approximation space was applied to approach the natural frequencies of free vibration of some in-plane elastic problems as well as the elastic wave response of forced vibration caused by the application of impulsive loading. The obtained results show that it was possible to numerically approximate a greater number of accurate natural frequencies with the devised procedure when a comparison is established with the $$C^{0}$$ Lagrangian and serendipity elements of 4, 8 (serendipity), 16, and 25 nodes. Also, the numerical predictions for the elastic wave propagation problem using the derived approximation space presented small oscillations improving the representation of the displacement field.

Local Dirichlet-type absorbing boundary conditions for transient elastic wave propagation problems

In this paper a new collocation technique for constructing time-dependent absorbing boundary conditions (ABCs) applicable to elastic wave motion is devised. The approach makes use of plane waves which satisfy the governing equations of motion to construct the absorbing boundary conditions. The plane waves are adjusted so that they can cope with the satisfaction of radiation boundary conditions. The proposed technique offers some advantages and exhibits the following features: it is easy to implement; its approximation scheme is local in space and time and thus it does not deal with any routine schemes such as Fourier and Laplace transform, making the method computationally less demanding; as the employed basis functions used to construct the absorbing boundary condition are residual-free, it requires neither any differential operator (to approximate the wave dispersion relation), nor any auxiliary variables; it constructs Dirichlet-type ABCs and hence no derivatives of the field variables are required for the imposition of radiation conditions. In this study, we apply the proposed technique to the solution procedure of a collocation approach based on the finite point method which proceeds in time by an explicit velocity-Verlet algorithm. It contributes to developing a consistent meshless framework for the solution of unbounded elastodynamic problems in time domain. We also apply the proposed method to a standard finite element solver. The performance of the method in solution of some 2D examples is examined. We shall show that the method exhibits appropriate results, conserves the energy almost exactly, and it performs stably in time even in the case of long-term computations.

Анализ деформаций при прохождении волн через контактный слой упругих тел

The stress-strain state on the interface of the elastic solids is investigated. The studied interface presents a contact layer which is characterized by dimension and the set of physics mechanical parameters. The models of layered and block medium are used for the description this boundary. In the framework of these models the problem of elastic wave propagation through the interface is considered. Analytical expressions for the refraction and reflection coefficients allowing us to determine the strains on the interface and strains distribution in the contact layer are found. Corresponding strains amplitudes depending on the layer thickness are calculated at the different elastic parameters of contacting solids and boundary. The strain laws on the interface which is described by the layered and block medium models are analyzed. The regions of equivalent use these models are determined in the case of strain analysis on the boundary and the strain distribution in the contact layer.

Mathematical analysis of a penalisation strategy for incompressible elastodynamics

This work addresses the mathematical analysis – by means of asymptotic analysis – of a penalisation strategy for the full discretisation of elastic wave propagation problems in quasi-incompressible media that has been recently developed by the authors. We provide a convergence analysis of the solution to the continuous version of the penalised problem towards its formal limit when the penalisation parameter tends to infinity. Moreover, as a fundamental intermediate step we provide an asymptotic analysis of the convergence of solutions to quasi-incompressible problems towards solutions to purely incompressible problems when the incompressibility parameter tends to infinity. Finally, we further detail the regularity assumptions required to guarantee that the mentioned convergence holds.

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.