Elastodynamic Problems Research Articles

FE‐Meshfree QUAD4 Element with Modified Radial Point Interpolation Function for Structural Dynamic Analysis

The partition‐of‐unity method based on FE‐Meshfree QUAD4 element synthesizes the respective advantages of meshfree and finite element methods by exploiting composite shape functions to obtain high‐order global approximations. This method yields high accuracy and convergence rate without necessitating extra nodes or DOFs. In this study, the FE‐Meshfree method is extended to the free and forced vibration analysis of two‐dimensional solids. A modified radial point interpolation function without any supporting tuning parameters is applied to construct the composite shape functions. The governing equations of elastodynamic problem are transformed into a standard weak formulation and then discretized into time‐dependent equations which are solved via Bathe time integration scheme to conduct the forced vibration analysis. Several numerical test problems are solved and compared against previously published numerical solutions. Results show that the proposed FE‐Meshfree QUAD4 element owns greater tolerance for mesh distortion and provides more accurate solutions.

Element differential method for free and forced vibration analysis for solids

Abstracts In this work a new strong-form numerical method, element differential method (EDM), is proposed to perform free and forced vibration analysis of elastodynamic problems. The present method establishes the global algebraic system equations directly based on the strong form of the equilibrium equations without using any variational principles or energy principles. In this method, the isoparametric elements with a node inside them are utilized to discrete the geometries. And the direct differentiation of their shape functions is used to characterize the geometry and physical variables. A novel collocation technique is then proposed to generate the system of equations, in which the dynamic equilibrium equations are collocated only at the internal nodes of elements, and the traction equilibrium equations are collocated at the interface and outer surface nodes. Compared with the standard finite element method, no integrals are involved to form the coefficients of the system and a reduced and lamped mass matrix can be directly obtained from the density properties of the target problem. Since the mass term only exists at the internal nodes of elements, the dynamic coupling of final system equations of the structure can be reduced, which will greatly save the computational resources. Numerical examples about free and forced 2D and 3D dynamic problems are given to demonstrate the correctness and efficiency of the proposed method.

Transient elastodynamic analysis with a combination of convolution quadrature method and pseudo-initial condition method

Purpose The Convolution Quadrature Method (CQM) has been widely applied to solve transient elastodynamic problems because of its stability and generality. However, the CQM suffers from the problems of huge memory requirement in case of direct implementation in time domain or CPU time in case of its reformulation in Laplace domain. The purpose of this paper is to combine the CQM with the pseudo-initial condition method (PICM) to achieve a good balance between memory requirement and CPU time. Design/methodology/approach The combined methods first subdivide the whole analysis into a few sub-analyses, which is dealt with the PICM, namely, the results obtained by previous sub-analysis are used as the initial conditions for the next sub-analysis. In each sub-analysis, the time interval is further discretized into a number of sub-steps and dealt with the CQM. For non-zero initial conditions, the pseudo-force method is used to transform them into equivalent body forces. The boundary face method is employed in the numerical implementation. Three examples are analyzed. Results are compared with analytical solutions or FEM results and the results of reformulated CQM. Findings Results demonstrate that the computation time and the storage requirement can be reduced significantly as compared to the CQM, by using the combined approach. Originality/value The combined methods can be successfully applied to the problems of long-time dynamic response, which requires a large amount of computer memory when CQM is applied, while preserving the CQM stability. If the number of time steps is high, then the accuracy of the proposed approach can be deteriorated because of the pseudo-force method.

Domain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source terms

The direct application of the classical method of fundamental solutions (MFS) is restricted to homogeneous linear partial differential equations (PDEs). The use of fundamental solutions with different frequencies allowed the extension of the MFS to non-homogeneous PDEs, in particular, for Poisson or Helmholtz equations and for elastostatic or elastodynamic problems. This method has been called method of fundamental solutions for domains (MFS-D), but it faces an approximation problem when the non-homogeneous term presents discontinuities, because the fundamental solutions are analytic functions outside the source point set. In this paper we analyze two domain decomposition techniques for overcoming this approximation problem. The problem is set in the context of the modified Helmholtz equation, and we also establish the missing density results that justify both the MFS and the MFS-D approximations. Numerical results are presented comparing a direct and an iterative domain decomposition technique, with simulations in non-trivial domains.

Isogeometric analysis for explicit elastodynamics using a dual-basis diagonal mass formulation

We propose a method to obtain diagonal mass matrices for NURBS-based approximation spaces by a “dual lumping” method. The use of lumped mass matrices is of great importance in elastodynamics problems, as they can be employed in explicit time integration schemes which do not require the solution of a linear system. In finite elements, several well-established methods, such as row-sum, diagonal scaling, or nodal quadrature methods have been used to obtain lumped mass matrices for different applications. However, for higher-order and higher continuity approximation spaces such as those derived from NURBS, these approaches have only limited (second-order) accuracy.In this work, we derive a dual basis which has optimal approximation and dispersion properties, while maintaining local support. The dual space has discontinuities at the element boundaries (knots) and it is used to provide the test functions in the context of a Petrov–Galerkin method. This results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high-order accuracy. Numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity.

Explicit Verlet time-integration for a Nitsche-based approximation of elastodynamic contact problems

The aim of the present paper is to study theoretically and numerically the Verlet scheme for the explicit time-integration of elastodynamic problems with a contact condition approximated by Nitsche’s method. This is a continuation of papers (Chouly et al. ESAIM Math Model Numer Anal 49(2), 481–502, 2015; Chouly et al. ESAIM Math Model Numer Anal 49(2), 503–528, 2015) where some implicit schemes (theta-scheme, Newmark and a new hybrid scheme) were proposed and proved to be well-posed and stable under appropriate conditions. A theoretical study of stability is carried out and then illustrated with both numerical experiments and numerical comparison to other existing discretizations of contact problems.

Time-Discontinuous Finite Element Analysis of Two-Dimensional Elastodynamic Problems using Complex Fourier Shape Functions

This paper reformulates a time-discontinuous finite element method (TD-FEM) based on a new class of shape functions, called complex Fourier hereafter, for solving two-dimensional elastodynamic problems. These shape functions, which are derived from their corresponding radial basis functions, have some advantages such as the satisfaction of exponential and trigonometric function fields in complex space as well as the polynomial ones simultaneously, that make them a better choice than classic Lagrange shape functions, which only can satisfy polynomial function field. To investigate the validity and accuracy of the proposed method, three numerical examples are provided and the results obtained from the present method (complex Fourier-based TD-FEM) and the classic Lagrange-based TD-FEM are compared with the exact analytical solutions. According to them, using complex Fourier functions in TD-FEM leads to more accurate and stable solutions rather than those obtained from the classic TD-FEM.

The continuous-discontinuous cellular automaton method for elastodynamic crack problems

In this paper, the recently proposed continuous-discontinuous cellular automaton method (CDCA) is further developed for elastodynamic problems in 2D elastic solids. A continuity-to-discontinuity enriched function technique is developed for elastodynamics and combined with a cellular automaton formulation; the cellular automaton theory for elastodynamics, in which fast adaptive updating rules for the cell and its neighbors are developed, and the time domain CDCA are proposed. In this method, the cracks are independent of the integral meshes, and no assembled global matrix is needed for the whole process. Time-dependent equations for the cells are discretized by the Newmark method, and rapidly solved by the cellular automaton method. To analyze the fracture behaviors of elastic solids with dynamic loads, mixed-mode dynamic stress intensity factors are obtained by the interaction integral method. Some numerical examples for elastodynamic problems are studied to validate the accuracy and efficiency of the present method, and solutions from analytical method and some other methods are employed to show the high accuracy of the CDCA.

A time-dependent discontinuous Galerkin finite element approach in two-dimensional elastodynamic problems based on spherical Hankel element framework

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.

On the performance of GFEM with trigonometric enrichment in bidimensional dynamic elastoplastic modelling

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 L2 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.

Concurrent coupling of peridynamics and classical elasticity for elastodynamic problems

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.

Local uniqueness of the density from partial boundary data for isotropic elastodynamics

We consider an inverse problem in elastodynamics arising in seismic imaging. We prove locally uniqueness of the density of a non-homogeneous, isotropic elastic body from measurements taken on a part of the boundary. We measure the Dirichlet to Neumann map, only on a part of the boundary, corresponding to the isotropic elasticity equation of a 3D object. In earlier works it has been shown that one can determine the sheer and compressional speeds on a neighborhood of the part of the boundary (accessible part) where the measurements have been taken. In this article we show that one can determine the density of the medium as well, on a neighborhood of the accessible part of the boundary.

Parameter identification and model updating in the context of nonlinear mechanical behaviors using a unified formulation of the modified Constitutive Relation Error concept

The motivation of this work is to propose a general methodology to deal with complex nonlinear mechanical behaviors in the context of identification and model updating problems. We follow here the principle of the modified Constitutive Relation Error that is an energy-based functional suited to the solution of inverse problems, and originally used in the context of elasticity and elasto-dynamics problems with potentially highly corrupted experimental data. In the paper, we develop and analyze an extended formulation of the modified Constitutive Relation Error functional in order to deal with a wide class of nonlinear mechanical behaviors. The general idea in the construction of the functional is to ensure a strong mechanical content by referring to the thermodynamical framework. In addition, a dedicated numerical process sharing similarities with the LaTIn method is proposed in order to allow effective inversion with reasonable computational cost. The performance of the developed procedure is analyzed on 2D elastic-damage and 3D elasto-visco-plastic cases.

On the stability of direct time-domain boundary element methods for elastodynamics

Numerical stability has been a fundamental challenge in direct time-domain boundary element methods (TD-BEM) for elastodynamics. In this paper, an analytical framework for the evaluation of the critical aspect is presented. By casting a convolution integral-based TD-BEM algorithm in the form of a linear multi-step method with a hybrid amplification matrix and the incorporation of some fundamental characteristics of commonly-used transient Green's functions, a rigorous assessment of the problem is shown to be reducible to a standard spectral analysis in matrix theory by which the stability threshold can be clearly defined. Apt to be relevant to the evaluation of other TD-BEMs, the approach is applied to a regularized time-domain direct boundary element method with optional collocation weights and orders of solution variable projections as illustration. By virtue of the proposed formalism and a systematic parametric study, a proper resolution of the critical aspect for some past schemes as well as the versatility of the generalized TD-BEM algorithm as they pertain to the benchmark finite-domain square-bar and the infinite-domain cavity elastodynamic problems are given as examples.

Free vibrations of elastic beams by modified nonlocal strain gradient theory

Size-dependent axial and flexural free vibrations of Bernoulli–Euler nano-beams are investigated by the modified nonlocal strain gradient elasticity model presented in (Barretta & Marotti de Sciarra, 2018). The corresponding elastodynamic problem, with the natural constitutive boundary conditions, is solved by an effective analytical methodology. Axial and flexural fundamental frequencies are determined for cantilever and fully-clamped nano-beams. Effects of nonlocal and gradient scale parameters on fundamental frequencies are examined and compared with those obtained by Eringen's local/nonlocal mixture model. New benchmarks are found for vibrations of beams. The adopted nonlocal strain gradient model, with the appropriate constitutive boundary conditions, is capable of capturing both softening and stiffening dynamical responses. Accordingly, it provides an advantageous approach for design and optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS).

On the numerical solution of the exterior elastodynamic problem by a boundary integral equation method

A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth closed simply connected 2-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-depended problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting to a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

On the numerical convergence and performance of different spatial discretization techniques for transient elastodynamic wave propagation problems

We present a systematic investigation of several discretization approaches for transient elastodynamic wave propagation problems. This comparison includes a Finite Difference, a Finite Volume, a Finite Element, a Spectral Element and the Scaled Boundary Finite Element Method. Numerical examples are given for simple geometries with normalized parameters, for heterogeneous materials as well as for structures with arbitrarily shaped material interfaces. General conclusions regarding the accuracy of the methods are presented. Based on the essential numerical examples an expansion of the results to a wide range of problems and thus to numerous fields of application is possible.

A method based on 3D stiffness matrices in Cartesian coordinates for computation of 2.5D elastodynamic Green's functions of layered half-spaces

This article elaborates on an extension to the classical stiffness matrix method to obtain the Green's functions for two-and-a-half dimensional (2.5D) elastodynamic problems in homogeneous and horizontally layered half-spaces. Exact expressions for the three-dimensional (3D) stiffness matrix method for isotropic layered media in Cartesian coordinates are used to determine the stiffness matrices for a system of horizontal layers underlain by an elastic half–space. In the absence of interfaces, virtual interfaces are considered at the positions of external loads. The analytic continuation is used to find the displacements at any receiver point placed within a layer. The responses of a horizontally layered half-space subjected to a unit harmonic load obtained using the present method are compared with those calculated using a well-established methodology, achieving good agreement.

Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models

Size-dependent vibrational behavior of functionally graded (FG) Timoshenko nano-beams is investigated by strain gradient and stress-driven nonlocal integral theories of elasticity. Hellinger-Reissner's variational principle is preliminarily exploited to establish the equations governing the elastodynamic problem of FG strain gradient Timoshenko nano-beams. Differential and boundary conditions of dynamical equilibrium of FG Timoshenko nano-beams, with nonlocal behavior described by the stress-driven integral theory, are formulated. Free vibrational responses of simple structures of technical interest, associated with nonlocal stress-driven and strain gradient strategies, are analytically evaluated and compared in detail. The stress-driven nonlocal model for FG Timoshenko nano-beams provides an effective tool for dynamical analyses of stubby composite parts of Nano-Electro-Mechanical Systems.

4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems

In this article, a Space-Time Finite Element Method (STFEM) is proposed for the resolution of mechanical problems involving three dimensions in space and one in time. Special attention will be paid to the non-separation of the space and time variables because this kind of interpolation is well suited to mesh adaptation. For that purpose, we have developed a technique of 4D mesh generation adapted to space-time remeshing. A difficulty arose in the representation of 4D finite elements and meshes. This original technique does not require coarse-to-fine and fine-to-coarse mesh-to-mesh transfer operators and does not increase the size of the linear systems to be solved, compared to traditional finite element methods. Space-time meshes are composed of simplex finite elements. Computations are carried out in the context of the continuous Galerkin method. We have tested the method on a linearized elastodynamics problem. Our technique of mesh adaptation was validated on elementary examples and applied to a problem of mobile loading. The convergence and stability of the method are studied and compared with existing methods. This work is a first implementation of 4D space-time remeshing. A stability criterion for the method is established, as well as a convergence rate of about two. Using simplex elements, it is possible to develop a technique of mesh adaptation able to follow a mobile loading zone.