Time discontinuous linear traction approximation in time‐domain BEM: 2‐D elastodynamics

This paper presents a further improvement of the standard time‐domain BEM formulation for 2‐D elastodynamics. Linear‐time interpolation is assumed for both boundary displacements and tractions.As this assumption implies time continuity for the variables, a procedure to consider tractions time discontinuities must be worked out. The initial step of this procedure consists of adding to the basic BEM integral equation the integral equation for velocities: after discretization is accomplished, this is done only for nodes with prescribed values of displacements. Additional equations are then incorporated to the final system of algebraic equations, providing the means for the determination of the extra unknowns (represented by the tractions at the begining of each time interval). The integral equations are presented by employing the concept of finite part of integrals, and the kernels are integrated analitically on time: complete time‐integrated expressions are included in an appendix. Three numerical examples are presented in order to assess the accuracy of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Rayleigh wave correction for the BEM analysis of two‐dimensional elastodynamic problems in a half‐space

A simple, elegant approach is proposed to correct the error introduced by the truncation of the infinite boundary in the BEM modelling of two‐dimensional wave propagation problems in elastic half‐spaces. The proposed method exploits the knowledge of the far‐field asymptotic behaviour of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution involving the same boundary integrals that form the original BEM system. The method is formulated for a two‐dimensional homogeneous, isotropic, linearly elastic half‐space and its implementation in a frequency domain boundary element scheme is discussed in some detail. The formulation is then validated for a free Rayleigh pulse travelling on a half‐space and successfully tested for a benchmark problem with a known approximation to the analytical solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Precorrected FFT accelerated BEM for large‐scale transient elastodynamic analysis using frequency‐domain approach

A precorrected fast Fourier transform (pFFT) accelerated boundary element method (BEM) for large‐scale transient elastodynamic analysis is developed and described in this paper. The frequency‐domain approach is used. To overcome the ‘wrap‐around’ problem associated with the discrete Fourier transform, the exponential window method (EWM) is employed and incorporated in the frequency‐domain BEM. An improved implementation scheme of the pFFT method based on polynomial interpolation technique is developed and applied to accelerate the elastodynamic BEM. This new scheme reduces the memory required to save the convolution matrix by a factor of 8. To further improve the efficiency of the code, a newly developed linear system solver based on the induced dimension reduction method is employed. Its performance is investigated and compared with that of the well‐known GMRES. The accuracy and computational efficiency of the method are evaluated and demonstrated by three examples: a classical benchmark, a plate subject to an impact loading and a porous cube with nearly half million DOFs subject to a step traction loading. Both analytical and experimental results are employed to validate the method. It has been found that the EWM can effectively resolve the wrap‐around problem and accurate time responses for an arbitrarily chosen time period can be obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

Influence matrix diagonal block elements for two‐dimensional elastodynamic problems

A method is described in this article to calculate the diagonal block elements of the influence matrices used in the boundary element method for two‐dimensional elastodynamic problems. Currently, a method that is used in the calculation of diagonal block elements is a combination of rigid‐body translation and Gaussian quadrature for the adjoining elements to the singularity point. This paper proposes to show an alternative method for the calculation of diagonal block elements of the traction influence matrix in elastodynamic problems where linear elements are used. In addition, the method is applied for use in the displacement influence matrix. The accuracy of the method is demonstrated in an example problem. Copyright © 1999 John Wiley & Sons, Ltd.     

A 2‐D time‐domain boundary element method with damping

A new material damping model which is convenient for use in the time‐domain boundary element method (TDBEM) is presented and implemented in a proposed procedure. Since only fundamental solutions for linear elastic material are employed, the procedure has high efficiency and is easy to be integrated into current TDBEM codes. Analytical and numerical results for benchmark problems demonstrate that the accuracy of the proposed method is high. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐domain BEM for three‐dimensional site response analysis of topographic structures

This paper presents a formulation of a time‐domain three‐dimensional boundary element method for site response analysis of topographic structures. The boundary element algorithm that uses the presented time‐convoluted traction kernels is applied to site response analyses of topographic structures. The seismic responses of canyon and ridge subjected to incident P and S waves are analyzed to demonstrate the accuracy of the kernels and the applicability of the presented boundary element algorithm for site response analysis of topographic structures. Seismic response analyses of three‐dimensional Gaussian‐shaped ridges show that the three‐dimensional axisymmetric ridge has a more amplification potential compared with three‐dimensional non‐axisymmetric elongated and two‐dimensional ridges, if the ridge is impinged by incident waves with wavelength of about the ridge's width. Copyright© 2009 John Wiley & Sons, Ltd.     

Boundary element‐free method (BEFM) for two‐dimensional elastodynamic analysis using Laplace transform

In this paper, we present a direct meshless method of boundary integral equation (BIE), known as the boundary element‐free method (BEFM), for two‐dimensional (2D) elastodynamic problems that combines the BIE method for 2D elastodynamics in the Laplace‐transformed domain and the improved moving least‐squares (IMLS) approximation. The formulae for the BEFM for 2D elastodynamic problems are given, and the numerical procedures are also shown. The BEFM is a direct numerical method, in which the basic unknown quantities are the real solutions of the nodal variables, and the boundary conditions can be implemented directly and easily that leads to a greater computational precision. For the purpose of demonstration, some selected numerical examples are solved using the BEFM. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast hierarchical dual boundary element method for three‐dimensional elastodynamic crack problems

In this work a fast solver for large‐scale three‐dimensional elastodynamic crack problems is presented, implemented, and tested. The dual boundary element method in the Laplace transform domain is used for the accurate dynamic analysis of cracked bodies. The fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the Laplace parameter. An ACA (adaptive cross approximation) algorithm is used for the population of the low rank blocks and its performance at varying Laplace parameters is investigated. A preconditioned GMRES is used for the solution of the resulting algebraic system of equations. The preconditioners are built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. An original strategy, based on the computation of some local preconditioners only, is presented and tested to further speed up the overall analysis. The reported numerical results demonstrate the effectiveness of the technique for both uncracked and cracked solids and show significant reductions in terms of both memory storage and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

A two‐and‐a‐half‐dimensional displacement‐based PML for elastodynamic wave propagation

This paper presents a perfectly matched layer (PML) technique for the numerical simulation of three‐dimensional linear elastodynamic problems, where the geometry is invariant in the longitudinal direction. Examples include transportation infrastructure, dams, lifelines, and alluvial valleys. For longitudinally invariant geometries, a computationally efficient two‐and‐a‐half‐dimensional (2.5D) approach can be applied, where the Fourier transform from the longitudinal coordinate to the wavenumber domain allows for the representation of the three‐dimensional radiated wave field on a two‐dimensional mesh. In this 2.5D framework, the equilibrium equations of a PML continuum are formulated in a weak form for an isotropic elastodynamic medium and discretized using a Galerkin approach. The 2.5D PML methodology is validated by computing the Green's displacements of a homogeneous halfspace, demonstrating that the 2.5D PML absorbs all propagating waves for different angles of incidence. Furthermore, the dynamic stiffness of a rigid strip foundation and the efficiency of a vibration isolating screen are computed. The examples demonstrate that the PML methodology is computationally efficient, especially when only the response of the structure or the near field response is of interest.Copyright © 2011 John Wiley & Sons, Ltd.     

A two‐dimensional BEM/FEM coupling applied to viscoelastic analysis of composite domains

A coupling between the boundary‐element and finite‐element methods is studied for the viscoelastic analysis of reinforced media. The viscous behaviour of the composed body is taken into account by an alternative BEM methodology developed for the Boltzmann model. This methodology is based on differential constitutive relations for viscoelasticity. The reinforcements are modelled by finite elements and are considered elastic. The coupling is based on the sub‐region technique due to its generality and easy implementation. The resulting time‐marching process is able to represent both the instantaneous and the time‐dependent behaviour of a body subjected to general boundary conditions. The method is validated by an experimental result and its accuracy tested by comparing numerical results with analytical solutions. The generality of the method is proved by an infinite domain application. Copyright © 2003 John Wiley & Sons, Ltd.     

Time‐domain soil–structure interaction analysis in two‐dimensional medium based on analytical frequency‐dependent infinite elements

A direct method for soil–structure interaction analysis in two‐dimensional medium is presented in time domain, which is based on the transformation of the analytical frequency‐dependent dynamic stiffness matrix. The present dynamic stiffness matrix for the far‐field region is constructed by assembling stiffness matrices of the analytical frequency‐dependent dynamic infinite elements, so that the equation of motion can be analytically transformed into the time‐domain equation. An efficient procedure is devised to evaluate the dynamic responses in time domain. Verification of the present formulation is carried out by comparing the compliances for a strip foundation on a homogeneous and layered half‐spaces with those obtained by other methods. Numerical analyses are also carried out for the transient responses of an elastic block and tunnel in a homogeneous and a layered half‐space. The comparisons with those by other approaches indicate that the proposed time‐domain method for soil–structure interaction analysis gives good solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Scalar wave equation by the boundary element method: A D‐BEM approach with constant time‐weighting functions

A D‐BEM approach, based on time‐weighting residuals, is developed for the solution of two‐dimensional scalar wave propagation problems. The modified basic equation of the D‐BEM formulation is generated by weighting, with respect to time, the basic D‐BEM equation, under the assumption of linear and cubic time variation for the potential and for the flux. A constant time‐weighting function is adopted. The time integration reduces the order of the time‐derivative that appears in the domain integral; as a consequence, the initial conditions are directly taken into account. An assessment of the potentialities of the proposed formulation is provided by the examples included at the end of the work. Copyright © 2009 John Wiley & Sons, Ltd.     

Fast time‐domain simulation for large‐order linear time‐invariant state space systems

Time‐domain simulation is essential for both analysis and design of complex systems. Unfortunately, high model fidelity leads to large system size and bandwidths, often causing excessive computation and memory saturation. In response we develop an efficient scheme for large‐order linear time‐invariant systems. First, the A matrix is block diagonalized. Then, subsystems of manageable dimensions and bandwidth are formed, allowing multiple sampling rates. Each subsystem is then discretized using a O(n_s) scheme, where n_s is the number of states. Subsequently, a sparse matrix O(n_s) discrete‐time system solver is employed to compute the history of the state and output. Finally, the response of the original system is obtained by superposition. In practical engineering applications, closing feedback loops and cascading filters can hinder the efficient use of the simulation scheme. Solutions to these problems are addressed in the paper. The simulation scheme, implemented as a MATLAB function fastlsim, is benchmarked against the standard LTI system simulator lsim and is shown to be superior for medium to large systems. The algorithm scales close to O(n) for a set of benchmarked systems. Simulation of a high‐fidelity model of (n_s ≈ 2200) the Space Interferometry Mission spacecraft illustrates real world application of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

Stokes flow around cylinders in a bounded two‐dimensional domain using multipole‐accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N²) (iterative solvers) or O(N³) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well‐posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill‐conditioned systems of equations. The model system here is a two‐dimensional wide‐gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well‐conditioned to allow the use of fast multipole methods. The direct approach results in severe ill‐conditioning, to a point where application of the multipole method leads to non‐convergence of the solution iteration. Copyright © 1999 John Wiley & Sons, Ltd.     

Efficient non‐linear solid–fluid interaction analysis by an iterative BEM/FEM coupling

An iterative coupling of finite element and boundary element methods for the time domain modelling of coupled fluid–solid systems is presented. While finite elements are used to model the solid, the adjacent fluid is represented by boundary elements. In order to perform the coupling of the two numerical methods, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. In the case of local non‐linearities within the finite element subdomain, it is straightforward to perform the iterative coupling together with the iterations needed to solve the non‐linear system. In particular a more efficient and a more stable performance of the new coupling procedure is achieved by a special formulation that allows to use different time steps in each subdomain. Copyright © 2005 John Wiley & Sons, Ltd.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

Frequency‐domain analysis of time‐integration methods for semidiscrete finite element equations—part I: Parabolic problems

Time‐integration methods for semidiscrete equations emanating from parabolic differential equations are analysed in the frequency domain. The discrete‐time transfer functions of three popular methods are derived, and subsequently the forced response characteristics of single modes are studied in the frequency domain. To enable consistent comparison of the frequency responses of different algorithms, three characteristic numbers are identified. Frequency responses and L₂‐norms of the phase and magnitude errors are compared for the three time‐integration algorithms. The examples demonstrate that frequency‐domain analysis provides substantial insight into the time‐domain properties of time‐integration algorithms. Copyright © 2001 John Wiley & Sons, Ltd.     

A Green's function time‐domain boundary element method for the elastodynamic half‐plane

The transient Green's function of the 2‐D Lamb's problem for the general case where point source and receiver are situated beneath the traction‐free surface is derived. The derivations are based on Laplace‐transform methods, utilizing the Cagniard–de Hoop inversion. The Green's function is purely algebraic without any integrals and is presented in a numerically applicable form for the first time. It is used to develop a Green's function BEM in which surface discretizations on the traction‐free boundary can be saved. The time convolution is performed numerically in an abstract complex plane. Hence, the respective integrals are regularized and only a few evaluations of the Green's function are required. This fast procedure has been applied for the first time. The Green's function BEM developed proved to be very accurate and efficient in comparison with analogue BEMs that employ the fundamental solution. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element shakedown analysis of two‐dimensional structures

In the present paper, the formulation proposed by Casciaro and Garcea (Comput. Meth. Appl. Mech. Eng., 2002; 191 :5761–5792) and applied to the shakedown analysis of plane frames, is extended to the analysis of two‐dimensional flat structures in both the cases of plane‐stress and plane‐strain. The discrete formulation is obtained using a mixed finite element in which both stress and displacement fields are interpolated. The material is assumed to be elasto‐plastic and a linearization of the elastic domain is performed. The result is a versatile iterative scheme well suited to implementation in general purpose FEM codes. An extensive series of numerical tests is presented showing the reliability of the proposed formulation. Copyright © 2005 John Wiley & Sons, Ltd.