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In recent years, there has been increasing attention on the development of non-destructive evaluation (NDE) methods using guided waves for long-length materials such as thin plates and pipes. The guided waves are capable of long-distance propagation in thin plates and pipes, and they exhibit properties such as multimodality and dispersion. These characteristics of the guided waves make inspection using guided waves challenging. In this study, we apply a 2-D elastodynamic time-reversal method to detect surface breaking cracks of a thin plate where guided waves are present. The finite element method (FEM) is used to calculate the scattered waves from surface breaking cracks and their corresponding time-reversal waves. We also employ topological sensitivity as an assessment index for detecting surface breaking cracks using the time-reversal method. As numerical examples, we demonstrate guided wave propagation, scattering, and the time-reversal wave propagation obtained by using the FEM. Finally, we present the results of surface breaking crack detection in a thin plate and discuss the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present an immersed finite element (IFE) method for solving the elastodynamics interface problems on interface-unfitted meshes. For spatial discretization, we use vector-valued P 1 and Q 1 IFE spaces. We establish some important properties of these IFE spaces, such as inverse inequalities, which will be crucial in the error analysis. For temporal discretization, both the semi-discrete and the fully discrete schemes are derived. The proposed schemes are proved to be unconditionally stable and enjoy optimal rates of convergence in the energy, L 2 and semi- H 1 norms. Numerical examples are designed to verify our theoretical analysis and to demonstrate the stability and robustness of our schemes. [ABSTRACT FROM AUTHOR]

This paper deals with a numerical approximation of the elasticity vibration problem based on a potentials decomposition. Decomposing the displacements field into potentials is a well-known tool in elastodynamics that takes advantage of the decoupling of pressure waves and shear waves inside a homogeneous isotropic media. In the spectral problem on a bounded domain, this decomposition decouples the elasticity equations into two Laplacian-like equations that only interact at the boundary. We show that spurious eigenvalues appear when Lagrangian finite elements are used to discretize the problem. Then, we propose an alternative weak formulation which avoids this drawback. A finite element discretization of this weak formulation based again on Lagrangian finite elements is proposed and tested by means of some numerical experiments, which show convergence and absence of spurious modes. [ABSTRACT FROM AUTHOR]

We consider wave propagation problems in 2D unbounded isotropic homogeneous elastic media, with rigid boundary conditions. For their solution, we propose and compare two alternative numerical approaches, both obtained by coupling the differential equation with the associated space-time boundary integral equation. The latter is defined on an artificial boundary, chosen to surround the (bounded) exterior computational domain of interest. The integral equation defines a boundary condition which is non-reflecting for incoming and also for outgoing waves. In both approaches, the differential equations are discretized by applying a finite element method combined with the Crank Nicolson time marching scheme, while the discretization of the integral equation is obtained by coupling a time convolution quadrature with a space collocation boundary element method. The construction of the two approaches is described and discussed. Some numerical tests are also presented. [ABSTRACT FROM AUTHOR]

Standard finite element formulation and implementation in solid dynamics at large strains usually relies upon and indicial-tensor Voigt notation to factorized the weighting functions and take advantage of the symmetric structure of the algebraic objects involved. In the present work, a novel component-free approach, where no reference to a basis, axes or components is made, implied or required, is adopted for the finite element formulation. Under this approach, the factorisation of the weighting function and also of the increment of the displacement field, can be performed by means of component-free operations avoiding both the use of any index notation and the subsequent reorganisation in matrix Voigt form. This new approach leads to a straightforward implementation of the formulation where only vectors and second order tensors in R 3 are required. The proposed formulation is as accurate as the standard Voigt based finite element method however is more efficient, concise, transparent and easy to implement. [ABSTRACT FROM AUTHOR]

• BBFEM is extended to solve elastic wave problems. • Analytical rules are used to set up the local FE matrices for affine triangular elements. • The tensorial construction of Bernstein polynomials is exploited via sum factorisation for curved triangular elements. • The p BBFEM is more effective in coping with the pollution effect and resolving high order evanescent P and S wave modes. • Non uniform p refinement reduces the computational cost and enhances accuracy. This work presents a high-order Bernstein-Bézier finite element (FE) discretisation to accurately solve time harmonic elastic wave problems on unstructured triangular mesh grids. Although high-order FEs possess many advantages over standard FEs, the computational cost of matrix assembly is a major issue in high-order computations. A key ingredient to address this drawback is to resort to low complexity procedures in building the local high order FE matrices. This is achieved in this work by exploiting the tensorial property of Bernstein polynomials and applying the sum factorisation method for curved elements. An efficient implementation of the analytical rules for affine elements is also proposed. Furthermore, element-level static condensation of the interior degrees of freedom is performed to reduce the memory requirements. Additionally, the applicability of the method with a variable polynomial order, based on a simple a priori indicator, is investigated. The computational complexities of sum factorisation, analytical rules and standard quadrature are first evaluated, in terms of the CPU time against the polynomial order. The analysis shows that the achieved numerical complexities compare favourably to those expected theoretically. A significant runtime saving is also obtained by using analytical rules and sum factorisation. The performance of the Bernstein-Bézier FEs is then assessed on various benchmark tests, over a wide range of frequencies. Results from the elastic wave scattering problem demonstrate the effectiveness of this method in coping with the pollution error, and its accuracy in resolving high order evanescent wave modes. Additionally, a wave transmission problem with high wave speeds contrast and a curved interior interface is considered, where a simple a priori indicator is proposed to assign the variable polynomial order. The study provides evidence of the great benefit of a non uniform p -refinement in reducing the computational cost and enhancing accuracy. [ABSTRACT FROM AUTHOR]

The scaled boundary finite element method is known for its capability in reproducing highly-detailed solution fields over large subdomains, and also for its flexibility in terms of spatial discretization, enabling us to define subdomains with arbitrary numbers of boundary elements and hanging nodes. The former, however, is not readily attainable in the cases whose closed-form solution does not exist. Those cases invoke the use of a cascade of expensive computations for the recovery of inner-subdomain solution fields, leaving one of the main assets of the method almost unused. As a remedy, we propose a new solution scheme by which the interior fields of subdomains can be easily recovered so that the full potential of the scaled boundary finite element method can be harvested. No auxiliary variables are introduced to the global algebraic system and the dimensions of the matrices remain constant. Most of the computations are carried out prior to the global algebraic system assembly, and in a completely decoupled manner on the element level, rendering the introduced method to be embarrassingly parallelizable. The inner-subdomain information is not required for the solution process and it can be computed locally for the selected regions and subdomains, also using parallel computing. [ABSTRACT FROM AUTHOR]

Tangled (non-convex) elements, i.e. elements with negative Jacobian determinant, can lead to erroneous results in the standard finite element method (FEM). Constructing tangle-free, well-structured meshes for complex geometries is often impossible. Hence there is a need to explore analysis methods that can directly handle such tangled meshes. In this paper, we propose the isoparametric tangled finite element method (i-TFEM) for free and forced vibration problems over tangled meshes. By employing piece-wise invertible mapping, a variational formulation is derived, leading to a simple modification of the standard FEM stiffness and mass matrices with the incorporation of additional compatibility constraints. Moreover, i-TFEM reduces to standard FEM for non-tangled (regular) meshes. The proposed method is implemented for three types of elements: 4-node quadrilateral, 9-node quadrilateral, and 8-node hexahedral elements. The numerical results demonstrate that i-TFEM is able to consistently handle general tangled (non-convex) elements, enabling convenient meshing for complex geometries. • Isoparametric tangled finite element method (i-TFEM) is proposed for free and forced vibration problems over tangled meshes. • To handle concave elements using i-TFEM, simple modifications to the standard FEM stiffness and mass matrices are needed. • I-TFEM is implemented for three element types: 4-node quadrilateral, 9-node quadrilateral, and 8-node hexahedral elements. • The numerical results demonstrate that i-TFEM consistently handles general non-convex elements. [ABSTRACT FROM AUTHOR]

• The added degree of freedom method (ADM) is proposed to solve the absorbing boundary problem. • The principle of a mass damper is employed in the ADM to achieve the attenuation of stress waves. • The ADM can absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Absorbing boundaries are essential in engineering simulations, especially for elastodynamic problems, to ensure accuracy, reliability and numerical stability. In this paper, the added degree of freedom method (ADM) is developed as a novel approach to address absorbing boundary challenges in finite element analysis. In ADM, additional degrees of freedom (DOFs) are introduced within the absorbing domain to attenuate outgoing elastic waves. For the proposed ADM to implement the absorbing boundary, the stiffness and mass properties of both the added DOFs and conventional DOFs within the absorbing domain are adjusted. This adjustment aims to reduce the propagation speed of elastic waves within the medium, and to prolong the duration of interaction between elastic waves and the surrounding medium. Consequently, the vibrations of nodes can be effectively attenuated by applying damping forces to the added DOFs. The numerical results show that the ADM has the ability to absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Therefore, the proposed ADM offers an innovative solution for modeling absorbing boundaries in various scientific and engineering applications, addressing the challenges of simulating wave propagation within finite computational domains. [ABSTRACT FROM AUTHOR]

3D dynamic interaction of two adjacent elastic foundations embedded in a finite layered soil region rested in a homogeneous elastic isotropic half‐space with a transient dynamic source is studied. The hybrid computational model, the corresponding numerical scheme and the accompanied software are developed, verified and inserted in detail parametric study. The hybrid approach is based on (a) finite element method (FEM) describing the scattered wave field in a finite layered soil profile with two foundations; (b) boundary element method (BEM) considering waves radiating from a dynamic transient source in elastic semi‐infinite range. The aim is to propose an efficient hybrid methodology for evaluation of the dynamic response of a foundation‐soil‐foundation system, taking into account (a) the whole wave path from the dynamic source, through the layered soil region, till the underground structures; (b) the damage state of the geological material. The BEM model is based on the 3D elastodynamic fundamental solution in Fourier domain and it is applied in order to obtain frequency‐dependent stiffness matrix and load vector of the dynamically active semi‐infinite geological zone. Once the BEM model is formulated, it is inserted as a macro‐finite element in the FEM software package ABAQUS. The frequency‐dependent FEM model describing the wave field in finite layered soil profile with two elastic foundations is realized by ABAQUS. Solutions in time domain are obtained through application of the inverse fast Fourier transform. As a final result an efficient hybrid model comprising all in one: dynamic (seismic or other type) source‐homogeneous elastic half‐space‐finite heterogeneous by layers and foundations soil profile, is presented. Parametric study illustrating the sensitivity of the dynamic field to key factors such as the type and properties of the load, the soil layering, the material properties of the far‐ and near‐field soil regions, the damaged state of the geological material and the foundation‐soil‐foundation interaction is shown and discussed. [ABSTRACT FROM AUTHOR]

A finite element procedure is proposed for wave propagation in elastic media. The method is based on an alternative formulation for the equations of motion that can systematically be constructed for linear evolutionary partial differential equations. A weak formulation—corresponding to convolutional variational principles—is then defined, which paves the way for introducing a particular type of time-wise shape functions. Next, some mathematical characteristics of the method are investigated, and upon those properties, a new solution procedure for elastodynamics problems is proposed. Subsequently, several numerical examples are considered, including a single degree of freedom mass-spring-damper system as the prototype of structural dynamics along with 1d and 2d elastodynamics problems for the case of the wave motion in elastic solids. The present method can be considered as an alternative approach for time integration and time-marching algorithms, e.g., Newmark's algorithm, to solve time-domain problems in elastic media. [ABSTRACT FROM AUTHOR]

This paper presents an extended/generalized finite element method for bridging scales in linear elastodynamics in the absence of scale separation. More precisely, the GFEMgl framework is expanded to enable the numerical solution of multiscale problems through the automated construction of specially-tailored shape functions, thereby enabling high-fidelity finite element modeling on simple, fixed finite element meshes. This introduces time-dependencies in the shape functions in that they are subject to continuous adaptation with time. The temporal aspects of the formulation are investigated by considering the Newmark- β time integration scheme, and the efficacy of mass lumping strategies is explored in an explicit time-stepping scheme. This method is demonstrated on representative wave propagation examples as well as a dynamic fracture problem to assess its accuracy and flexibility. [ABSTRACT FROM AUTHOR]

The numerical solution of the elastodynamic problem with kinematic boundary conditions is considered using mixed finite elements for the space discretization and a staggered leap-frog scheme for the discretization in time. The stability of the numerical scheme is shown under the usual CFL condition. Using the general form of Robin-type boundary conditions some cases of debonding and the resulting acoustic emission are studied. The methodology presented finds applications to geophysics such as in seismic waves simulation with dynamic rupture and energy release. In this paper, we focus on problems of fracture occurring at the interface of composite materials. Our goal is to study both the mechanism of crack initiation and propagation, as well as the acoustic emission of the released structure-borne energy which may be used in structural health monitoring and prognosis applications. [ABSTRACT FROM AUTHOR]

An extension of the two-step staggered time discretization of linear elastodynamics in stress-velocity form to systems involving internal variables subjected to a possibly non-linear dissipative evolution is proposed. The original scheme is thus enhanced by another step for the internal variables which, in general, is implicit, although even this step might be explicit if no spatial gradients of the internal variables are involved. Using an abstract Banach-space formulation, a priori estimates and convergence are proved under a CFL condition. The developed three-step scheme finds applications in various problems of continuum mechanics at small strain. Here, we consider in particular plasticity, viscoelasticity (creep), diffusion in poroelastic media, and damage. [ABSTRACT FROM AUTHOR]

A virtual element framework for nonlinear elastodynamics is outlined within this work. The virtual element method (VEM) can be considered as an extension of the classical finite element method. While the finite element method (FEM) is restricted to the usage of regular shaped elements, VEM allows to use non‐convex shaped elements for the spatial discretization [1]. It has been applied to various engineering problems in elasticity and other areas, such as plasticity or fracture mechanics as outlined in [3, 4]. This work deals with the extension of VEM to dynamic problems. Low‐order ansatz functions in two and three dimensions, with elements being arbitrary shaped, are used in this contribution. The formulations considered in this framework are based on minimization of energy, where a pseudo potential is used for the dynamic behavior. While the stiffness‐matrix needs a suitable stabilization, the mass‐matrix can be calculated fully through the projection part. For the implicit time integration, Newmark‐Method is used. To show the performance of the method, various numerical examples in 2D and 3D are presented. [ABSTRACT FROM AUTHOR]

The truncation of an unbounded medium into a finite domain and the elimination of reflections from artificial medium boundaries is important in numerical simulations of elastodynamic equations. The perfectly matched layer (PML) technique as a novel absorbing boundary has demonstrated very high efficiency for elastic wave equation models. Based on the stretched coordinate concept, an efficient and unsplit-field perfectly matched layer equation for elastic waves is formulated in a cylindrical coordinate system. By introducing integrated complex variables for radial direction and auxiliary functions, the PML formulation is extended in cylindrical coordinates based on the second-order elastic wave equation with displacements as basic unknowns. Finite-element time-domain modeling of the second-order unsplit PML for the elastodynamic equation is presented, which is a standard displacement-based formulation for the computational domain including a PML region. The formulas for the special cases with 2-D axisymmetric coordinates and polar coordinates are also given. The efficiency of the proposed formulations is illustrated by a numerical example using the finite-element method. [ABSTRACT FROM AUTHOR]

This paper deals with seismic response evaluation of a finite multilayered geological structure with non-parallel boundaries rested on semi-infinite elastic half-plane subjected to transient seismic wave. The hybrid modelling approach is applied. It is based on the decomposition of the whole domain under consideration into two subregions: a finite-sized near-field zone and the open semi-infinite far-field region. The near-field range is presented by a finite geological profile, which is heterogeneous due to non-parallel layering. The far-field region is semi-infinite elastic isotropic homogeneous medium where the finite layered structure is located. An efficient 2D hybrid computational tool based on the boundary element method (BEM) for the far-field zone and the finite element method (FEM) for the finite layered domain is developed and verified in frequency domain. The BEM model of the seismically active far-field zone is inserted as a macro-finite element (MFE) in the FEM commercial program ANSYS. The hybrid model is of the FEM hosted type, where the boundary element domain (BED) is converted into macro-finite element, the nodal forces along the interface between two zones are expressed through the BED nodal tractions and the entire BEM formulation is converted to FEM-like approach. This hybrid model is applied for modelling of a real geological profile of the main wall of the "Liulyakovitsa" tailing pond, near Panagyurishte town, Bulgaria. Simulation results are given in the form of free surface synthetic seismograms stemming from the real seismic signals recorded at bedrock so as to establish the tailing pond's seismic response. [ABSTRACT FROM AUTHOR]

In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters. [ABSTRACT FROM AUTHOR]

The Note is concerned with a feasibility study of time reversal in a non-homogeneous elastic medium, from data recorded in an acoustic medium. Our aim here is to determine the presence and some physical properties of elastic "inclusions" (unknown, not observable solid objects, characterized by their elastic properties) from partial observations of acoustic waves scattered by these inclusions. A finite element numerical method, based on a variational acousto-elastodynamics formulation, is derived and used to solve the forward, and then, the time-reversed problem. A criterion, derived from the reverse time migration framework, is introduced, to help construct images of the inclusions to be determined. Numerical illustrations on configurations that mimic the breast cancer configuration are proposed, and show that one can differentiate between two inclusions, even with different properties. [ABSTRACT FROM AUTHOR]

It has been anticipated and recently observed that axial cracks can occur at keyway corners of graphite bricks in the UK Advanced Gas-cooled Reactors (AGR). The crack initiation is triggered by internal stress reversal during reactor operation and may potentially lead to brick more cracks through a mechanism known as Prompt Secondary Cracking (PSC). To assess the likelihood of this occurring, a method for modelling dynamic fracture, known as eXtended Finite Element Method with cohesive elements (XCZM), is used to analyse whole 3D component fragmentation. XCZM is used here to model primary crack interactions with a methane hole, the effect of graphite heterogeneity, and secondary crack profiles in a 3D brick slice under external hoop stress loading. The results show that methane holes in bricks can be beneficial for brick integrity, as these can lead to crack path deflection with associated increase of dissipated energy. Further, the results suggest that graphite heterogeneity does not affect significantly the crack profile and therefore the integrity assessment. Finally, it is demonstrated that PSC may readily occur and the secondary crack propagates towards regions of high kinetic energy. • Methodology for dynamic initiation and propagation of multiple crack in quasi-brittle media. • Application to analysis of cracking and fragmentation of nuclear graphite bricks. • Methane holes shown to be beneficial for brick integrity via increased energy dissipation. • Dynamic effects, stress amplification and kinetic energy, control secondary cracking. [ABSTRACT FROM AUTHOR]

Abstract Rock media may be exposed to dynamic forces and time-dependent stresses. Dynamic phenomena in rock such as blasting and earthquakes need to propagate waves and dynamic stresses through the body. Therefore, stresses, strains, and displacements in rocks should be defined as time-dependent due to the dynamic phenomena. In this study, a general Green's function solution of an elastic wave propagation due to rock blasting has been analytically derived. The Navier's equations of motion have been assumed as governing equations and a general two-dimensional elastodynamic Green's function has been obtained in terms of displacement. The strain and stress fields related to the displacement Green's function have also been obtained using the theory of elasticity. Then, a generic two-dimensional problem containing an explosion in a blast hole has been solved using the resulting analytical solution. Moreover, a similar numerical modeling has been conducted using Finite Difference Method (FDM). Eventually, results from the analytical scheme have been compared with the finite difference numerical modeling and the valuable results from a previous experimental and numerical study. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a coupled peridynamic-finite element method for modeling mechanical behavior and damage growth in materials is developed. Peridynamics is a nonlocal continuum model which, in contrast to other continuum models, uses integration instead of spatial derivations in its governing equations. Utilizing integration instead of derivatives is advantageous in modeling fracture since the governing equations remain valid even after the initiation or growth of discontinuities. Although peridynamics can capture material fracture effectively, however due to its nonlocal formulation peridynamics is computationally expensive. To reduce the computational costs, we propose to couple peridynamics with finite elements and use peridynamics only in small zones where higher accuracy is needed. The main challenge in developing such a coupling method is to eliminate the artifacts introduced by the interface of the two subdomains. One of the main issues is spurious wave reflections which occurs because high frequency waves traveling from peridynamic region cannot enter the finite element zone and spuriously reflect back into the peridynamic zone. This will lead to an increase in the energy of the peridynamic zone and will drastically reduce the computational accuracy. The main feature of the proposed method is eliminating the spurious wave reflections such that the coupled method is as accurate as the pure peridynamics. Highlights • A coupled peridynamic-finite element method is developed to model dynamic fracture. • Spurious wave reflections are effectively suppressed. • Arlequin method is used to impose the mechanical compatibility between finite element and peridynamic sub regions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss the time-dependent generalized membrane shell model, which has not been addressed numerically in literature. We show that the solution of this model exists and is unique. We first provide a numerical method for the time-dependent generalized membrane shell. Concretely, we semi-discretize the space variable and fully discretize the problem using time discretization by the Newmark scheme. The corresponding numerical analyses of existence, uniqueness, stability and convergence with a priori error estimates are given. Finally, we present numerical experiments with a portion of the conical shell and a portion of the hyperbolic shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme. Highlights • Existence and uniqueness of the time-dependent generalized membrane shell. • Newmark scheme for the time-dependent generalized membrane shell. • Analyses of existence, uniqueness, stability, convergence and a priori error estimate for elastodynamic problem. [ABSTRACT FROM AUTHOR]

Highlights • Features of a quasi-Rayleigh wave are analyzed based on the Partial Wave Method. • Two types of quasi-Rayleigh are defined, with discussion of Viktorov's condition. • Qualitative discussion of the quasi-Rayleigh wave's definition w.r.t. its partial wave composition. Abstract The Partial Wave Method is unique in that it establishes a foundation on which various elastodynamic guided waves can be compared. In this paper, the method is used to compare quasi-Rayleigh waves and Rayleigh waves, and investigate the eccentricities of the Partial Wave Method at phase velocities equal to the Rayleigh wave speed. The comparison results in the definition of two types of quasi-Rayleigh waves and an explanation for quasi-Rayleigh wave behavior reported in the literature at frequencies that do not satisfy Viktorov's quasi-Rayleigh wave condition. These conclusions are also verified by the superposition of A0 and S0 mode wave-structures calculated using the semi-analytical finite element method. [ABSTRACT FROM AUTHOR]

Abstract In this work, an investigation is carried out concerning the performance of proposed enriched quadrilateral C0 element, developed by Generalized Finite Element Method, in dynamic elastoplastic models. The enriched quadrilateral C0 element is formulated by considering conventional shape function as global PU basis and trigonometric functions as the local basis. The enriched mesh distortion sensitivity is investigated in an elastodynamic problem. Other numerical aspects are also investigated in several benchmark analyses, involving stress concentration effect. The condition numbers of the stiffness and the consistent mass matrices are calculated for each enrichment level in different situations, and the error in norm L 2 is calculated for displacement and stress. The results are compared with other numerical formulations encountered in the literature. Several numerical aspects of enriched quadrilateral C0 element formulated by GFEM are discussed and observed in the applications, and it performs competitiveness in comparison to other numerical formulations. Highlights • GFEM approximations are proposed for the plane elastodynamics analysis. • The application of GFEM C0 element in the elastoplastic dynamic analysis is original. • Different levels of enrichment are investigated systematically for different degrees of mesh distortion. • Satisfactory results presented by enriched element in analysis with stress concentration effect. [ABSTRACT FROM AUTHOR]

Abstract We propose a computational method which improves the identification of sources and scatterers in a two-dimensional elastic medium using the Time Reversal (TR) technique. The identification procedure makes use of partial numerical data available at a small number of sensors. While in realistic applications these data are obtained by lab or field measurements, in the present work they are synthesized by solving the corresponding forward problem. We introduce an "augmentation" procedure that is utilized in addition to the TR technique and investigate its benefits. The augmentation procedure involves the solution of a suitable elliptic problem at selected time steps. In practice, one needs to solve, as a pre-process, a small number of "fundamental" elliptic problems which is equal to the number of measured quantities. A version of the augmentation procedure, which is easier to implement and involves almost no computational cost, is also presented. The local elastic energy in the domain is employed to decide whether the TR solution has properly refocused at the source. Cost functionals, whose minima are sought, are constructed for the source and scatterer identification problems. We conclude that augmentation improves the identification performance, in particular in the presence of noisy measurements. In some cases augmented TR succeeds in the identification whereas non-augmented TR fails. Highlights • The Time Reversal (TR) method for scatterer identification is considered. • An improved TR method is proposed for cases of scarce or very noisy data. • The improvement is achieved by solving an elliptic problem at chosen times. • The associated added computational cost is marginal. • In some cases, the method succeeds in the identification whereas standard TR fails. [ABSTRACT FROM AUTHOR]

Abstract 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 h p -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. Highlights • A new polygonal discontinuous Galerkin method is introduced for elastodynamics. • Stability and hp-version error estimates in a suitable energy norm are proved. • A dispersion analysis is carried out. • Dispersion errors are compared with those of classical spectral element methods. • The application of the method to benchmarks and realistic case studies is presented. [ABSTRACT FROM AUTHOR]

In this paper, a time-discontinuous finite element method (TD-FEM) based on a new class of shape functions is suggested for solving two-dimensional elastodynamic problems. The main idea is to use spherical Hankel shape functions derived from corresponding radial basis functions instead of classical Lagrange shape functions. Among the advantages of the proposed interpolation, containing the first and second kind of Bessel functions in complex space as well as the polynomial ones can be pointed out, while the classic Lagrange interpolation is only able to contain polynomial functions. The validity and accuracy of the present method are fully demonstrated using several numerical examples. For this purpose, five numerical examples are considered, and the results obtained from the proposed method (Hankel-based TD-FEM) and the Lagrange-based TD-FEM are compared with the exact analytical solutions. The results show that the use of Hankel functions in TD-FEM leads to more accurate and stable solutions rather than those obtained from the classic TD-FEM. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we explain why beyond a certain critical frequency the extreme ends of a beam under flexural oscillations remain sometimes plane and in other cases become warped. We have seen this to be the case when the flexural oscillations of a beam are studied experimentally. Using the plane stress elastodynamic theory as well as a more precise calculation based on a 3D finite element method, we show for the first time that when the transverse displacement has a derivative equal to zero near the beam extremes, the ends can indeed become warped. In addition it is shown, from an energetic point of view, that warped modes result from a constructive spatial interference of the two components of flexural oscillations while flat modes result from a destructive interference of these two components. [ABSTRACT FROM AUTHOR]

An alternative approach to local transmitting boundaries for wave propagation problems in unbounded media is the application of the Perfectly Matched Layers (PML) around the finite region of interest. The PML model consists of absorbing layers that absorb almost perfectly propagating waves of all non-tangential angels-of-incident and all non-zero frequencies. This paper presents and verifies the concept of FE/PML methodology on the basis of two different formulations for the PML model in the context of time harmonic wave propagation analysis. A finite element numerical scheme is developed in which both PML formulations are successfully implemented as a macro-finite element in a commercial FEM program. Moreover, the parameters that influence the performance of the FE/PML method are extensively discussed and recommendations are given for the proper selection. Classical soil-structure interaction problems of surface or embedded rigid footing on homogeneous or heterogeneous half-plane containing layers and tunnels are investigated and the high accuracy and the easy implementation of FE/PML computational tool is demonstrated. [ABSTRACT FROM AUTHOR]

Highlights • A new integrated surface unit-cell model at the mesoscopic scale is proposed for 3D four-directional braided composites. • A novel macro-meso finite element modeling strategy for predicting the elastic properties is presented by the periodic boundary conditions. • The key finite element modeling procedures based on the solid unit-cell models are given in detail. Abstract Finite element method based on the mesoscopic scale periodic unit-cell models was often employed to analyze the mechanical properties of 3D braided composites with obvious shell-core structural feature. To obtain the accurate predicted results, the contribution of the interior and surface unit-cell models to the overall mechanical properties should be considered for the finite element modeling. However, how to apply reasonable boundary conditions to the surface unit-cell model is still an interesting issue worth studying, which has great effect on the prediction precision of surface unit-cell model. Meanwhile, it is valuable to develop a novel macro-meso finite element modeling approach combined with the shell-core structural feature of 3D braided composites, which can contribute to improving the prediction precision and saving the computational costs. In this paper, a new integrated surface unit-cell model with the true yarn microstructure at the mesoscopic scale is proposed for 3D four-directional braided composites firstly. Secondly, a novel macro‑meso scale finite element modeling strategy for predicting the elastic properties and the micromechanical responses is presented. Thirdly, the key modeling procedures based on the solid unit-cell models are given. Finally, comparison of the calculated results with the available experimental data validates the effectiveness of the macro‑meso finite element model. The effects of the braiding angle and the fiber volume fraction of specimens on the elastic properties are discussed in detail. Discussion results indicate that the present macro‑meso finite element modeling strategy can be utilized to efficiently predict the mechanical performances of 3D braided composites. [ABSTRACT FROM AUTHOR]

Abstract: A hybrid model for evaluation of the seismic response of a complex poroelastic soil region containing an underground structure is developed. The model is based on an efficient computational technique unifying the benefits of both boundary element method (BEM) and finite element method (FEM). The mechanical model takes the whole seismic wave path from the seismic source, through the heterogeneous geological saturated deposits, till the local site with underground structure into consideration. The seismic load comprises time‐harmonic or transient P‐ and SV‐wave, thus covering the plane strain case. A viscoelastic isomorphism to Biot's equations of dynamic poroelasticity presented by Bardet's model is used. The direct BEM is applied for the unbounded part of the geological region, while the FEM is used for the local finite geological profile containing an underground structure. The BEM model is inserted as a macro finite element in commercial program ABAQUS used for solution of the entire problem. The simulation results reveal that the seismic response is sensitive to the site conditions such as the existence of surface relief and layers, to soil inhomogeneity and poroelasticity, and most importantly to the soil‐underground structure interaction. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate, analytically and numerically, that degenerate states of a beam with free ends, with anti-symmetrical transverse mode shape and occurring in the first-degeneracy points above the cutoff frequency, tend asymptotically to the thickness-shear mode in the infinitely long beam limit, which has not been previously reported. We compare our numerical results, obtained using a Timoshenko–Ehrenfest beam theory of second order, with those obtained with the Finite Element method and plane stress elastodynamics theory, obtaining an excellent agreement. We show that anti-symmetric states within the vicinity of these first-degeneracy states are very similar and closely resemble the so-called thickness-shear mode but for a finite beam with free ends under flexural vibrations. • Mode shapes of degenerate states are studied in Timoshenko–Ehrenfest beams. • First-degeneracy asymmetric states turn out very similar to the thickness-shear mode. • Instead, it is shown that transverse oscillations of the cutoff states are non-negligible. • Results for a beam with free ends are corroborated with the Finite Element method. • It is demonstrated that analogs to the thickness-shear mode exist for finite beams. [ABSTRACT FROM AUTHOR]

A polygonal scaled boundary finite element method (SBFEM) is proposed for linear elastodynamics in two dimensions. The domain is divided into non-overlapping polygonal elements, and the scaled boundary finite element approach is employed over each polygon. The advantages are that an arbitrary interpolation order can be used on each polygon and we can choose the optimal element order on each edge individually. Moreover, the SBFEM simplifies the numerical integration over the polygons when compared to conventional approaches. The dynamic stiffness matrix is computed by employing the continued fraction expansion. The influence of the shape of the polygon and the element order on the boundary of each polygon are studied. The robustness and the accuracy of the approach are demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

Hyperelastic materials possess the appealing property that they may be employed as elastic wave manipulation devices and cloaks by imposing pre-deformation. They provide an alternative to microstructured metamaterials and can be used in a reconfigurable manner. Previous studies indicate that exact elastodynamic invariance to pre-deformation holds only for neo-Hookean solids in the antiplane wave scenario and the semi-linear material in the in-plane compressional/shear wave context. Furthermore, although ground cloaks have been considered in the acoustic context they have not yet been discussed for elastodynamics, either by employing microstructured cloaks or hyperelastic cloaks. This work therefore aims at exploring the possibility of employing a range of hyperelastic materials for use as antiplane ground cloaks (AGCs). The use of the popular incompressible Arruda-Boyce and Mooney-Rivlin nonlinear materials is explored. The scattering problem associated with the AGC is simulated via finite element analysis where the cloaked region is formed by an indentation of the surface. Results demonstrate that the neo-Hookean medium can be used to generate a perfect hyperelastic AGC as should be expected. Furthermore, although the AGC performance of the Mooney-Rivlin material is not particularly satisfactory, it is shown that the Arruda-Boyce medium is an excellent candidate material for this purpose. [ABSTRACT FROM AUTHOR]

Parallel kinematic machines exhibit strong pose-dependent static and dynamic behaviors, which makes accurate and rapid compliance analysis over the entire workspace an important issue in the design optimization. This paper presents a general approach for elastodynamic modeling of parallel kinematic machines using substructure synthesis technique. Firstly, the whole system is decomposed into two groups of substructures, i.e. component substructures and joint substructures. Then, the degrees of freedom of component substructures are reduced using modal reduction technique, and joint substructures are modeled by virtual springs with contact stiffness between adjacent component substructures. Finally, two threads are merged at junction surfaces to offer the equations of motion of the system as a whole, allowing the static and dynamic performances to be rapidly predicted with sufficient accuracy. The dynamic model of a 5-DOF hybrid robot is developed using the proposed method and the computational results show that rigidity and lower mode natural frequencies over the workspace match well with those obtained by the full finite element analysis. [ABSTRACT FROM AUTHOR]

The elastodynamic behavior of polycrystalline cubic materials is studied through the fundamental propagation properties, the attenuation and wave speed, of a longitudinal wave. Predictions made by different analytical models are compared to both numerical and experimental results. The numerical model is based on a three-dimensional Finite Element (FE) simulation which provides a full-physics solution to the scattering problem. The three main analytical models include the Far-Field Approximation (FFA), the Self-Consistent Approximation (SCA) to the reference medium, and the herein derived Second Order Approximation (SOA). The classic Stanke and Kino model is also included, which by comparison to the SOA, reveals the importance of the distribution of length-scales described in terms of the two-point correlation function in determining scattering behavior. Further comparison with the FE model demonstrates that the FFA provides a simple but satisfactory approximation, whereas the SOA shows all-around excellent agreement. The experimental wave velocity data evaluated against the SOA and SC reveal a better agreement when the Voigt reference is used in second order models. The use of full-physics numerical simulations has enabled the study of wave behavior in these random media which will be important to inform the ongoing development of analytical models and the understanding of observations. [ABSTRACT FROM AUTHOR]

Guided waves can effectively be used for inspection of large scale structures. Surface corrosion is often found as major defect type in large scale structures such as pipelines. Guided wave interaction with surface corrosion can provide useful information for sizing and classification. In this paper, the elastodynamic reciprocity theorem is used to formulate and solve complicated scattering problems in a simple manner. The approach has already been applied to scattering of Rayleigh and Lamb waves by defects to produce closed form solutions of amplitude of scattered waves. In this paper, the scattering of the lowest axially symmetric torsional mode, which is widely used in commercial applications, is analyzed by the reciprocity theorem. In the present paper, the theorem is used to determine the scattering of the lowest torsional mode by a tapered defect that was earlier considered experimentally and numerically by the finite element method. It is shown that by the presented method it is simple to obtain the ratio of amplitudes of scattered torsional modes for a tapered notch. The results show a good agreement with earlier numerical results. The wave field superposition technique in conjunction with the reciprocity theorem simplifies the solution of the scattering problem to yield a closed form solution which can play a significant role in quantitative signal interpretation. [ABSTRACT FROM AUTHOR]

Within the framework of linear two-dimensional elastodynamics, stress wave intensity attenuators are studied under material and boundary condition discontinuities collectively. The influence of various parameters on the efficiency of stress wave attenuators is investigated thoroughly and a comprehensive understanding of the response is developed under dynamic loadings for a wide range of frequencies. In particular, the effect of in-plane and out-of-plane dimensions, incident wave frequencies (wavelength), rigidity of the host structure, and impedance mismatch between different layers have been examined. The dependence of stress wave attenuator efficiency and robustness are found to be a complex function of all relevant parameters, and performance is observed to vary significantly for various combinations. To illustrate the significance of combined effects of various parameters on the potential efficiency of the stress wave intensity attenuators, an optimization problem is solved. An optimal material set-up of a 12-layered structure, subjected to transient loadings with varying durations and wide range of frequency contents, is presented. A coupled genetic algorithm-finite element methodology is developed specifically for the optimal design of layered structures. This methodology is highly suitable for investigating the solution space that is too large to be explored by an exhaustive parametric study. The results of the optimal designs evidently show that the efficiency of the stress wave attenuators depends significantly on the duration of transient loading, and high efficiency can be attained for short durations. [ABSTRACT FROM AUTHOR]

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually well-posed family of broken variational formulations of the original PDE. It can be exploited to solve challenging problems in a variety of physical scenarios where stability or a particular mode of convergence is desired in a part of the domain. The linear elasticity equations are solved in this work, but the approach can be applied to other equations as well. The broken variational formulations, which are essentially extensions of more standard formulations, are characterized by the presence of mesh-dependent broken test spaces and interface trial variables at the boundaries of the elements of the mesh. This allows necessary information to be naturally transmitted between adjacent subdomains, resulting in coupled variational formulations which are then proved to be globally well-posed. They are solved numerically using the DPG methodology, which is especially crafted to produce stable discretizations of broken formulations. Finally, expected convergence rates are verified in two different and illustrative examples. [ABSTRACT FROM AUTHOR]

A new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain. Our method is based on equal-order interpolation discrete spaces for both the velocity (or displacement) and stress (or strain) tensor variables. The mixed form is derived using either the velocity/stress or velocity/strain pair of unknowns, the latter being instrumental in extensions of the method to nonlinear mechanics. The proposed approach works equally well on hexahedral or tetrahedral grids and, for this reason, it is suitable for time-domain engineering applications in complex geometry. The peculiarity of the proposed approach is the use of the rate form of the stress update equation, which yields a set of governing equations with the structure of a non-dissipative space/time Friedrichs’ system. We complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity (or displacement). Weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes, because they do not require dedicated data structures for the imposition of the boundary degrees of freedom, but, rather, they utilize the structure of the variational formulation. We also show how the framework of weakly enforced boundary conditions can be used to develop variational forms for multi-domain simulations of heterogeneous media. A complete analysis including stability and convergence proofs is included, in the case of a space–time variational approach. A series of computational tests are used to demonstrate and verify the performance of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper considers elastic wave propagation in seven-wire parallel strands. Under a transverse distributed force, the stress state of seven-wire strands is derived from the Hertzian theory, which is regarded as the prestress state for the dynamic analysis. According to the wave finite element method, the elastodynamic equation with prestress is discretized corresponding to harmonic wave motion. Firstly, the results are verified by the semi-analysis finite element method. Then, dispersion properties in seven-wire parallel strands are computed and compared to those of double cylindrical rods. Besides, the results from parallel strands and helical strands indicate that wave propagation in the former is significantly different from that in the latter. For the deep understanding of wave characteristics, the displacement vectors are used to identify the propagating modes. It is found that there is only one torsional-like mode which rotates with a twist center in the whole region. Meanwhile, it is found that the variation of prestress has large influence on the torsional-like mode for low frequencies, but little on propagating modes for high frequencies. In addition, there appears the notch frequency and the effect of prestress on it is discussed in detail. The notch frequency decreases with the rising of the prestress in a certain range. [ABSTRACT FROM AUTHOR]

This paper developed a new multiphysics finite element method for the elastodynamic analysis of space debris deorbit by a bare flexible electrodynamic tether. Orbital motion limited theory and dynamics of flexible electrodynamic tethers are discretized by the finite element method, where the motional electric field is variant along the tether and coupled with tether deflection and motion. Accordingly, the electrical current and potential bias profiles of tether are solved together with the tether dynamics by the nodal position finite element method. The newly proposed multiphysics finite element method is applied to analyze the deorbit dynamics of space debris by electrodynamic tethers with a two-stage energy control strategy to ensure an efficient and stable deorbit process. Numerical simulations are conducted to study the coupled effect between the motional electric field and the tether dynamics. The results reveal that the coupling effect has a significant influence on the tether stability and the deorbit performance. It cannot be ignored when the libration and deflection of the tether are significant. [ABSTRACT FROM AUTHOR]