Transient Elastodynamic Problems Research Articles

Meshless generalized finite difference method for two- and three-dimensional transient elastodynamic analysis

In this paper, a meshless collocation method is introduced for two-dimensional (2D) and three-dimensional (3D) transient elastodynamic problems by applying the generalized finite difference method (GFDM) in conjunction with the Houbolt scheme. Coupled equilibrium equations with a time-dependent loading are transformed into the static equations at time nodes by adopting the Houbolt method. After then, the solution of the static equations is achieved with the GFDM with second-order and fourth-order expansions. Several numerical examples involving complicated geometries and different initial and boundary conditions are simulated to validate the performance of the present approach.

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.

Approximation of the state variables of Navier’s differential equation in transient dynamic problems using finite element method based on complex Fourier shape functions

In this research, the analysis of 2D transient elastodynamic problems is performed using a reformulation of finite element method based on new complex Fourier shape functions, which are able to satisfy exponential and trigonometric function fields in addition to polynomial ones unlike classic Lagrange shape functions. Another superiority of these shape functions over classic Lagrange ones is their ability to obtain much better results using fewer degrees of freedom. Moreover, Runge phenomenon does not occur in the proposed shape functions when high degrees of freedom using equispaced elements are used, unlike classic Lagrange ones. In addition, the approximation of all kinds of surfaces, such as smooth and folded ones, is done perfectly by complex Fourier shape functions. In the end, six numerical examples are provided to show the efficiency and robustness of the suggested method. To do so, its results are compared with the classic Lagrange ones and analytical solutions (if available).

A coupled meshless finite point/Peridynamic method for 2D dynamic fracture analysis

In this paper, a new coupled meshless method for 2D transient elastodynamic problems involving dynamic crack propagation is developed. The method is based on an efficient coupling between the finite point method (FPM) and a discretized form of Peridynamics. The solution domain is partitioned in three parts: one discretized by Peridynamics, one by FPM, and a transition part discretized by both methods where the switching between the two approaches is performed. The coupling adopts a local/nonlocal framework that benefits from the full advantages of both methods. The parts of domain where cracks either exist or are likely to propagate are described by Peridynamics; the remaining part of the domain is described by FPM that requires less computational effort. The capabilities of the proposed approach are demonstrated by means of the solution of dynamic problems including dynamic fracture as well as a ghost force test.

Evaluation of the T-stress for cracks in functionally graded materials by the FEM

Using the quarter-point crack tip element, a simple formula is derived which provides a quick and expedient means to evaluate the elastic T-stress in the finite element method (FEM) fracture mechanics analysis of functionally graded materials (FGMs). The formula is derived by comparing the variation of displacements in the quarter-point element with the corresponding asymptotic expression for the displacement field in the vicinity of a crack-tip. The asymptotic fields in the crack-tip vicinity in a FGM with continuously varying material properties are the same as in a homogeneous one. The simplicity of the use of this formula for T-stress evaluation in FEM fracture mechanics analysis is demonstrated by its application to some transient elastodynamic and thermoelastic problems.

Elastodynamic Analysis of a Hollow Cylinder with Decagonal Quasicrystal Properties: Meshless Implementation of Local Integral Equations

A meshless approximation and local integral equation (LIE) formulation are proposed for elastodynamic analysis of a hollow cylinder made of quasicrystal materials with decagonal quasicrystal properties. The cylinder is assumed to be under shock loading. Therefore, the general transient elastodynamic problem is considered for coupled phonon and phason displacements and stresses. The equations of motion in the theory of compatible elastodynamics of wave type for phonons and wave-telegraph type for phasons are employed and can be easily modified to the elasto-hydro dynamic equations for quasicrystals (QCs). The angular dependence of the tensor of phonon–phason coupling coefficients handicaps utilization of polar coordinates, when the governing equations would be given by partial differential equations with variable coefficients. Despite the symmetry of the geometrical shape, the local weak formulation and meshless approximation are developed in the Cartesian coordinate system. The response of the cylinder in terms of both phonon and phason stress fields is obtained and studied in detail.

Effective transmission conditions for thin-layer transmission problems in elastodynamics. The case of a planar layer model

This article is concerned with the design, analysis, numerical approximation and implementation of effective transmission conditions (ETCs) for the propagation of elastic waves through a thin planar elastic layer with small uniform thickness η which is embedded in a reference elastic medium, under transient conditions, with both materials assumed to have isotropic properties. A family of ETCs of order k (i.e. whose approximation error is of expected order O(η k+1)) is formulated by deriving and exploiting a formal asymptotic expansion in powers of η of the transmission solution inside the layer. The second-order ETCs are then retained as the main focus for the remainder of the article, and given a full justification in terms of both the stability of the resulting transient elastodynamic problem and the error analysis. The latter is performed by establishing and justifying asymptotic expansions for the solutions of both the exact transmission problem and its approximation based on the second-order ETCs. As a result, the error (in energy norm) between those two solutions is shown to be, as expected, of order O(η 3). Finally, the numerical approximation of the proposed second-order ETC within the framework of spectral element methods is studied, with special attention devoted to the selection of a robust time-stepping scheme that is mostly explicit (and conditionally stable). Among these, a scheme that is implicit only for the interfacial degrees of freedom, termed semi-implicit, is shown to be stable under the same stability condition as for the layer-less configuration. The main theoretical results of this work are illustrated and validated by 2D and 3D numerical experiments under transient elastodynamic conditions.

Time-domain BEM for 3-D transient elastodynamic problems with interacting rigid movable disc-shaped inclusions

Formulation of time-domain boundary element method for elastodynamic analysis of interaction between rigid massive disc-shaped inclusions subjected to impinging elastic waves is presented. Boundary integral equations (BIEs) with time-retarded kernels are obtained by using the integral representations of displacements in a matrix in terms of interfacial stress jumps across the inhomogeneities and satisfaction of linearity conditions at the inclusion domains. The equations of motion for each inclusion complete the problem formulation. The time-stepping/collocation scheme is implemented for the discretization of the BIEs by taking into account the traveling nature of the generated wave field and local structure of the solution at the inclusion edges. Numerical results concern normal incidence of longitudinal wave onto two coplanar circular inclusions. The inertial effects are revealed by the time dependencies of inclusions' kinematic parameters and dynamic stress intensity factors in the inclusion vicinities for different mass ratios and distances between the interacting obstacles.

Space–time proper generalized decompositions for the resolution of transient elastodynamic models

In this paper, we investigate ability of proper generalized decomposition (PGD) to solve transient elastodynamic models in space–time domain. Classical methods use time integration schemes and an incremental resolution process. We propose here to use standard time integration methods in a non-incremental strategy. As a result, PGD gives a separated representation of the space–time solution as a sum of tensorial products of space and time vectors, that we interpret as space–time modes. Recent time integration schemes are based on multi-field formulations. In this case, separated representation can be constructed using state vectors in space and same vectors in time. However, we have experienced bad convergence order using this decomposition. Furthermore, temporal approximation must be the same for all fields. Thus, we propose an extension of classical separated representation for multi-field problems. This multi-field PGD (MF-PGD) uses space and time vectors that are different for each field. Calculation of decomposition is done using a monolithic approach in space and time, potentially allowing the use of different approximations in space and time. Finally, several simulations are performed with the transient elastodynamic problem with one dimension in space. Different approximations in time are investigated: Newmark scheme, single field time discontinuous Galerkin method and two fields time continuous and discontinuous Galerkin methods.

AN ADAPTIVE TIME-STEPPING PROCEDURE BASED ON THE SCALED BOUNDARY FINITE ELEMENT METHOD FOR ELASTODYNAMICS

This study develops an adaptive time-stepping procedure of Newmark integration scheme for transient elastodynamic problems, based on the semi-analytical scaled boundary finite element method (SBFEM). In each time step, a posteriori local error estimator based on the linear distributed acceleration is employed to estimate the error caused by the time discretization. The total energy of the domain, consisting of the kinetic energy and the strain energy, is calculated semi-analytically. The time increment is automatically adjusted according to a simple criterion. Three examples with stress wave propagation were modeled. The numerical results show that the developed method is capable of limiting the local error estimator within specified targets by using an optimal time increment in each time step.

Modeling transient elastodynamic problems using a novel semi-analytical method yielding decoupled partial differential equations

In this paper, a novel semi-analytical method is proposed for modeling of two-dimensional (2D) elastodynamic problems. In this regard, the domain boundary of the problem is discretized by specific non-isoparametric elements that are proposed for the first time in this paper. In these elements, special shape functions as well as higher-order Chebyshev mapping functions are implemented. For the shape functions, Kronecker Delta property is satisfied for displacement function and its derivatives, simultaneously. Moreover, the first derivatives of shape functions are assigned to zero at any given node. Finally, using a weak form of weighted residual method and implementing Clenshaw–Curtis numerical integration, coefficient matrices of the system of equations are converted into diagonal ones, which leads to a set of decoupled partial differential equations for solving the whole system. This means that the governing partial differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. Also, to assess the accuracy and efficiency of the proposed method, four benchmark problems of 2D elastodynamics are solved using a few numbers of DOFs. The numerical results present excellent agreement with the analytical solutions and the results from other numerical methods.

An adaptive scaled boundary finite element method by subdividing subdomains for elastodynamic problems

The scaled boundary finite element method (SBFEM) is a semi-analytical numerical method, which models an analysis domain by a small number of large-sized subdomains and discretises subdomain boundaries only. In a subdomain, all fields of state variables including displacement, stress, velocity and acceleration are semi-analytical, and the kinetic energy, strain energy and energy error are all integrated semi-analytically. These advantages are taken in this study to develop a posteriori h-hierarchical adaptive SBFEM for transient elastodynamic problems using a mesh refinement procedure which subdivides subdomains. Because only a small number of subdomains are subdivided, mesh refinement is very simple and efficient, and mesh mapping to transfer state variables from an old mesh to a new one is also very simple but accurate. Two 2D examples with stress wave propagation were modelled. The results show that the developed method is capable of capturing propagation of steep stress regions and calculating accurate dynamic responses, using only a fraction of degrees of freedom required by adaptive finite element method.

An h-hierarchical adaptive scaled boundary finite element method for elastodynamics

A posteriori h-hierarchical adaptive scaled boundary finite element method (ASBFEM) for transient elastodynamic problems is developed. In a time step, the fields of displacement, stress, velocity and acceleration are all semi-analytical and the kinetic energy, strain energy and energy error are all semi-analytically integrated in subdomains. This makes mesh mapping very simple but accurate. Adaptive mesh refinement is also very simple because only subdomain boundaries are discretised. Two 2D examples with stress wave propagation were modelled. It is found that the degrees of freedom needed by the ASBFEM are only 5%–15% as needed by adaptive FEM for the examples.

First kind Bessel function (J-Bessel) as radial basis function for plane dynamic analysis using dual reciprocity boundary element method

This paper presents a new radial basis function (RBF) for the boundary element method in the analysis of plane transient elastodynamic problems. The dual reciprocity method (DRM) is reconsidered by using the first kind Bessel (J-Bessel) function as a new generation of RBFs to approximate the inertia term. Employing the initial value theorem of Laplace transform, the particular solution kernels of the proposed RBFs corresponding to displacement and traction, with no singular terms, has been explicitly derived. Furthermore, the limiting values of the particular solution kernels have been evaluated. To illustrate the validity and accuracy of the present RBFs, three numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. In comparison with other RBFs, J-Bessel RBFs represent more accurate results, using a smaller degree of freedom, and hence they are more efficient.

A dual reciprocity BEM approach using new Fourier radial basis functions applied to 2D elastodynamic transient analysis

In this paper, a new boundary element analysis for two-dimensional (2D) transient elastodynamic problems is proposed. The dual reciprocity method (DRM) is reconsidered by employing new radial basis functions (RBFs) to approximate the domain inertia terms. These new RBFs, which are in the form of ζ+ κ sin ( ωr+ α), are called Fourier RBFs hereafter. Using the method of variation of parameters, the particular solution kernels of Fourier RBFs corresponding to displacement and traction, whose a few terms are singular, has been explicitly derived. Therefore, a new simple smoothing trick has been employed to resolve the singularity problem. Moreover, the limiting values of the particular solution kernels have been evaluated. In order to find the unknown parameters of Fourier RBFs, an optimization problem seeking for the optimum value of the Houbolt scheme parameter β that minimizes the mean squared error (MSE) function of the problem is established. Since the MSE function of the proposed RBFs is a function of five unknown parameters (i.e., ζ, κ, ω, α, and β), the genetic algorithm (GA) has been used to solve the necessary optimization problem. In order to illustrate the validity, accuracy, and superiority of the present study, several numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. Compared to other RBFs, Fourier RBFs show more accurate and stable results. Moreover, these results are obtained using less degree of freedom without any additional internal points that are commonly used to improve the accuracy of the results.

An Asymptotic Approach for the Elastodynamic Problem of a Plate under Impact Loading

An approach is presented for analyzing the transient elastodynamic problem of a plate under an impact loading. The plate is considered to be in the form of a long strip under plane strain conditions. The loading is taken as a concentrated line force applied normal to the plate surface. It is assumed that this line force is suddenly applied and maintained thereafter (i.e., it is a Heaviside step function of time). Inertia effects are taken into consideration and the problem is treated exactly within the framework of elastodynamic theory. The approach is based on multiple Laplace transforms and on certain asymptotic arguments. In particular, the one‐sided Laplace transform is applied to suppress time dependence and the two‐sided Laplace transform to suppress the dependence upon a spatial variable (along the extent of the infinite strip). Exact inversions are then followed by invoking the asymptotic Tauber theorem and the Cagniard‐deHoop technique. Various extensions of this basic analysis are also discussed.

Evaluation of the Stress Intensity Factors for Cracks in Continuously Nonhomogeneous Solids, Part II: Meshless Method

A meshless method based on the local Petrov-Galerkin approach is proposed for crack analysis in two-dimensional (2-D), orthotropic and linear elastic solids with continuously varying material properties. Both quasi-static thermal and transient elastodynamic problems are considered. For time-dependent problems, the Laplace-transform technique is utilized. A unit step function is used as the test functions in the local weak-form. It leads to local boundary integral equations (LBIEs) involving only a domain-integral in the case of transient dynamic problems.The analyzed domain is divided into small subdomains with a circular shape. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the mode-I stress intensity factors is discussed by comparison with available analytical or numerical solutions.

A coupled numerical scheme of dual reciprocity BEM with DQM for the transient elastodynamic problems

AbstractThe two‐dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first‐ and the second‐order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second‐order ordinary differential equations in time. This system is then discretized by the polynomial‐based DQM with respect to time, which gives a system of linear algebraic equations after the imposition of both the boundary and the initial conditions. Therefore, the solution is obtained at any required time level at one stroke without the use of an iterative scheme and without the need of very small step size in time direction. The numerical results are visualized in terms of graphics. Copyright © 2008 John Wiley & Sons, Ltd.

A meshless local boundary integral equation method for solving transient elastodynamic problems

A new local boundary integral equation (LBIE) method for solving two dimensional transient elastodynamic problems is proposed. The method utilizes, for its meshless implementation, nodal points spread over the analyzed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the global boundary, displacements and tractions are treated as independent variables. The local integral representation of displacements at each nodal point contains both surface and volume integrals, since it employs the simple elastostatic fundamental solution and considers the acceleration term as a body force. On the local boundaries, tractions are avoided with the aid of the elastostatic companion solution. The collocation of the local boundary/volume integral equations at all the interior and boundary nodes leads to a final system of ordinary differential equations, which is solved stepwise by the θ-Wilson finite difference scheme. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.

Dual reciprocity BEM analysis of 2D transient elastodynamic problems by time-discontinuous Galerkin FEM

Dual reciprocity BEM analysis of 2D transient elastodynamic problems by time-discontinuous Galerkin FEM