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.     

An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher‐order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency‐domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap‐around errors brought by the discrete Fourier transform. The frequency‐domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Frequency domain analysis by the exponential window method and SGBEM for elastodynamics

Dynamic analysis of a system can be carried out either in the time or frequency domain. Time responses/ histories of this system may be directly obtained using time-domain analysis. In case of frequency domain analysis in the Fourier space, the inverse fast Fourier transform (inverse FFT) would naturally be an appropriate choice for converting frequency solutions to the desired time responses. However, the standard FFT can not be applied to undamped systems as the free-vibration terms of these systems never decay which violates the periodic nature of the standard FFT algorithm. In addition, the FFT may be computationally expensive for lightly damped systems. An alternative to overcome the above limitations is the so-called exponential window method (EWM) commonly used in digital signal processing. This paper presents a combination of the EWM and the symmetric-Galerkin boundary element method for 2-D elastodynamic analysis in the frequency domain of undamped and lightly damped systems. Several numerical examples, including fracture problems, are given to illustrate the efficiency and accuracy of the proposed frequency domain analysis.     

Compression of time‐generated matrices in two‐dimensional time‐domain elastodynamic BEM analysis

This paper describes a new scheme to improve the efficiency of time‐domain BEM algorithms. The discussion is focused on the two‐dimensional elastodynamic formulation, however, the ideas presented apply equally to any step‐by‐step convolution based algorithm whose kernels decay with time increase. The algorithm presented interpolates the time‐domain matrices generated along the time‐stepping process, for time‐steps sufficiently far from the current time. Two interpolation procedures are considered here (a large number of alternative approaches is possible): Chebyshev–Lagrange polynomials and linear. A criterion to indicate the discrete time at which interpolation should start is proposed. Two numerical examples and conclusions are presented at the end of the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

An indirect elastodynamic boundary element method with analytic bases

An indirect time‐domain boundary element method (BEM) is presented here for the treatment of 2D elastodynamic problems. The approximated solution in this method is formulated as a linear combination of a set of particular solutions, which are called bases. The displacement and stress fields of a basis are analytically derived by means of solving Lame's displacement potentials. A semi‐collocation method is proposed to be the time‐stepping algorithm. This method is equivalent to a displacement discontinuity method with piecewise linear discontinuities in both space and time. The resulting time‐stepping scheme is explicit. The BEM is implemented to solve three numerical examples, Lamb's problem, half‐plane with a buried crack and Selberg's problem. Though Lamb's problem is considered a difficult problem for numerical methods, the current numerical results for the surface displacements show accurately the characteristics of the Rayleigh wave. This method is efficient and accurate. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses.     

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.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Precise and fast phase wraps reduction in fringe projection profilometry

Phase wraps in a 2D wrapped phase map can be completely eliminated or greatly reduced by frequency shifting. But it usually cannot be optimally reduced using conventional fast Fourier transform (FFT) because the spectrum can be shifted only by a integer number in the frequency domain. In order to achieve a significant phase wrap reduction, we propose a fast and precise two-step method for phase wraps reduction in this paper, which is based on the iterative local discrete Fourier transform (DFT). Firstly, initial estimate of the frequency peak is obtained by FFT. Then sub-pixel spectral peak with high resolution is determined by iteratively upsampling the local DFT around the initial peak location. Finally, frequency shifting algorithm that operates in the spatial domain is used to eliminate phase wraps. Simulations and experiments are conducted to demonstrate the superb computing efficiency and overall performance of the proposed method.     

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.     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.     

An accelerated surface discretization‐based BEM approach for non‐homogeneous linear problems in 3‐D complex domains

For non‐homogeneous or non‐linear problems, a major difficulty in applying the boundary element method (BEM) is the treatment of the volume integrals that arise. An accurate scheme that requires no volume discretization is highly desirable. In this paper, we describe an efficient approach, based on the precorrected‐FFT technique, for the evaluation of volume integrals resulting from non‐homogeneous linear problems. In this approach, the 3‐D uniform grid constructed initially to accelerate surface integration is used as the baseline mesh for the evaluation of volume integrals. As such, no volume discretization of the interior problem domain is necessary. Moreover, with the uniform 3‐D grid, the matrix sparsification techniques (such as the precorrected‐FFT technique used in this work) can be extended to accelerate volume integration in addition to surface integration, thus greatly reducing the computational time. The accuracy and efficiency of our approach are demonstrated through several examples. A 3‐D accelerated BEM solver for Poisson equations has been developed and has been applied to a 3‐D multiply‐connected problem with complex geometries. Good agreement between simulation results and analytical solutions has been obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic stresses created by the sudden appearance of a transverse crack in periodically layered composites

The time-dependent stress field generated by the sudden appearance of a transverse crack in a periodically layered composite that is subjected to a remote loading is determined. The resulting two-dimensional elastodynamic problem is solved by combining two approaches. In the first one, the representative cell method, which has been presently generalized to dynamic problems, is employed for the construction of the time-dependent Green’s functions generated by the displacement jumps along the crack line. This is performed in conjunction with the application of the double finite discrete Fourier transform. Thus the original problem for the cracked periodic composite is reduced to the problem of a domain with a single period in the transform space. The second approach is based on a wave propagation in composites theory which has been presently generalized to admit arbitrary types of loading. This theory is based on the elastodynamic continuum equations where the transformed time-dependent displacement vector is expressed by a second-order expansion, and the equations of motion and the various interfacial and boundary conditions are imposed in the average (integral) sense. The time-dependent field in any observation point in the plane can be obtained by the application of the inverse transform. This field is valid as long as no reflected waves from external boundaries have been arrived. Results along the crack line as well as the full field are given for cracks of various lengths for Mode I, II and III deformations. In particular the dynamic magnification with respect to the static case is determined at the interface within the first unbroken stiff layer.     

Weight Function for a Crack in a Two-dimensional Orthotropic Medium Under Impact Shear Loading

The paper examines the elastodynamic response of an infinite two-dimensional orthotr- opic medium containing a central crack under impact shear loading. Laplace and Fourier integral transforms are employed to reduce the problem to a pair of dual integral equations in the Laplace transformed plane. These equations are reduced to integral differential equations, which have been solved in the low frequency domain by iterations. To determine time dependence, these equations are inverted to yield the dynamic stress intensity factor (SIF) for shear point force loading that corresponds to the weight function for the crack under shear loading. It is used to derive SIF for polynomial loading.     

Axial symmetric elasticity analysis in non‐homogeneous bodies under gravitational load by triple‐reciprocity boundary element method

In general, internal cells are required to solve elasticity problems by involving a gravitational load in non‐homogeneous bodies with variable mass density when using a conventional boundary element method (BEM). Then, the effect of mesh reduction is not achieved and one of the main merits of the BEM, which is the simplicity of data preparation, is lost. In this study, it is shown that the domain cells can be avoided by using the triple‐reciprocity BEM formulation, where the density of domain integral is expressed in terms of other fields that are represented by boundary densities and/or source densities at isolated interior points. Utilizing the rotational symmetry, the triple‐reciprocity BEM formulation is developed for axially symmetric elasticity problems in non‐homogeneous bodies under gravitational force. A new computer program was developed and applied to solve several test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Transient dynamic analysis of interface cracks in layered anisotropic solids under impact loading

Transient elastodynamic crack analysis in two-dimensional (2D), layered, anisotropic and linear elastic solids is presented in this paper. A time-domain boundary element method (BEM) in conjunction with a multi-domain technique is developed for this purpose. Time-domain elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are applied in the present time-domain BEM. The spatial discretization of the boundary integral equations is performed by a Galerkin-method, while a collocation method is adopted for the temporal discretization of the arising convolution integrals. An explicit time-stepping scheme is developed to compute the unknown boundary data and the crack-opening-displacements (CODs). To show the effects of the crack configuration, the material anisotropy, the layer combination and the dynamic loading on the dynamic stress intensity factors and the scattered elastic wave fields, several numerical examples are presented and discussed.     

A NUMERICAL SCHEME FOR MODE III DYNAMIC FRACTURE PROBLEMS

This paper summarizes the formulation and numerical implementation of a spectral scheme specially designed for dynamic anti-plane shear (mode III) fracture problems. The scheme allows for a wide variety of simulations ranging from the dynamic loading of stationary cracks to the spontaneous propagation of faults. The method is based on a spectral form of the elastodynamic relation between the shearing tractions acting on the fracture surface and the resulting slip velocity response for a planar two-dimensional crack in an infinite linearly elastic medium. The formulation is expressed in the Fourier domain and involves a convolution over the past slip or slip rate history. Conversion between spectral and real domains is performed through the fast Fourier transform algorithm. The time-integration scheme is explicit and a variety of constitutive laws can be used to express the strength of the material on the fault plane. The stability and accuracy of the numerical scheme are discussed through comparison with existing analytical solutions involving non-propagating and propagating cracks. The extraction of the dynamic stress intensity factor from the computed slip history is described. © 1997 by John Wiley & Sons, Ltd.     

A time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.