Elastodynamic Boundary Element Method Research Articles

Krauklis wave propagation within a complex fracture system: Modeling via a two-dimensional time-harmonic boundary element method.

Fluid-filled fractures involving kinks and branches result in complex interactions between Krauklis waves-highly dispersive and attenuating pressure waves within the fracture-and the body waves in the surrounding medium. For studying these interactions, we introduce an efficient 2D time-harmonic elastodynamic boundary element method. Instead of modeling the domain within a fracture as a finite-thickness fluid layer, this method employs zero-thickness, poroelastic Linear-Slip Interfaces to model the low-frequency, local fluid-solid interaction. Using this method, the scattering of Krauklis waves by a single kink along a straight fracture and the radiation of body waves generated by Krauklis waves within complex fracture systems are examined.

Laplace Domain Boundary Element Method for Structural Health Monitoring of Poly-Crystalline Materials at Micro-Scale

This paper describes, for the first time, the application of an Elastodynamic Boundary Element Method (BEM) in Laplace Domain for the Structural Health Monitoring (SHM) of poly-crystalline materials. The study focuses on Ultrasonic Guided Wave (UGW) propagation and investigates the wave–material interactions at micro-scale. The study aims to investigate the interaction of UGWs with assessing micro-structural features such as grain size, morphology, degradation, and flaws. Numerical simulations of the most common micro-structural features demonstrate the accuracy and validity of the proposed method. Particular attention is paid to the study of porosity and its influence on material macro-properties. Different crystal morphologies such as cubic, rhombic, and truncated octahedral are considered. The detection of voids based on the changes in the amplitude and Time of Arrival (ToA) of the backscattered signal is investigated. The study also considers inter-granular cracks, which cause laceration, and examines flaw position/orientation, length, and distance from a specific reference. Furthermore, a framework is proposed for generating Probability of Detection (PoD) curves using numerical simulations. Experimental tests in pristine conditions are shown to be in good agreement with the numerical simulations in terms of ToA, signal amplitude, and wave velocity. The numerical simulations provide insights into wave propagation and wave–material interactions, including different types of defects at the micro-scale. Overall, the BEM and UGW methods are shown to be effective tools for better understanding micro-structural features and their influence on the macro-structural properties of poly-crystalline materials.

A Direct Method for Solving Singular Integrals in Three-Dimensional Time-Domain Boundary Element Method for Elastodynamics

The analytically time integrable time-space domain (ATI-TSD) is discovered based on which the minimum time-space domain is identified for treatment on singularities in the three-dimensional time-domain boundary element method (3D TD-BEM) formulation. A direct method to solve singular integrals in the 3D TD-BEM formulation for elastodynamic problems is proposed. The wavefront singularity can be analytically eliminated in ATI-TSD, while the dual singularity can be treated by the direct method using Kutt’s quadrature in the identified minimum time-space domain. Three benchmark examples are presented to verify the correctness and the applicability of the direct method for solving the singular integrals in 3D TD-BEM.

A fast biem algorithm to evaluate seismic ground response in 2d elastic time domain

Since the topography and surface irregularities of the terrain have a significant effect on the seismic response of the earth's surface, the numerical analysis using boundary element method in elastodynamics, which leads to a significant increase in the number of DOF and stiffness matrix dimension, as well as the sparse and unsymmetrical structure of stiffness matrix, indicates the inefficiency of the conventional method. This paper presents a fast boundary element algorithm in seismic analysis of two-dimensional elastic media for the first time. In the proposed method, instead of the usual node-to-node, a method of cell-to-cell relation method is implemented by hierarchy tree structure. In addition, the Plane Wave Time Domain algorithm and the Iterative method has been used to solve a system of equations which increases the speed of network convergence, especially in high degrees of freedom that has not been used in the study of the seismic waves’ response yet. Nowadays, this new algorithm adopts for investigating the response wave fields of electrical, magnet [1], acoustic and thermal conductivity which are discussed in details in various articles and books.

Comparison of the convolution quadrature method and enhanced inverse FFT with application in elastodynamic boundary element method

Transient problems can often be solved with transformation methods, where the inverse transformation is usually performed numerically. Here, the discrete Fourier transform in combination with the exponential window method is compared with the convolution quadrature method formulated as inverse transformation. Both are inverse Laplace transforms, which are formally identical but use different complex frequencies. A numerical study is performed, first with simple convolution integrals and, second, with a boundary element method (BEM) for elastodynamics. Essentially, when combined with the BEM, the discrete Fourier transform needs less frequency calculations, but finer mesh compared to the convolution quadrature method to obtain the same level of accuracy. If further fast methods like the fast multipole method are used to accelerate the boundary element method the convolution quadrature method is better, because the iterative solver needs much less iterations to converge. This is caused by the larger real part of the complex frequencies necessary for the calculation, which improves the conditions of system matrix.

A kernel-independent fast multipole BEM for large-scale elastodynamic analysis

Purpose – The purpose of this paper is to develop an easy-to-implement and accurate fast boundary element method (BEM) for solving large-scale elastodynamic problems in frequency and time domains. Design/methodology/approach – A newly developed kernel-independent fast multipole method (KIFMM) is applied to accelerating the evaluation of displacements, strains and stresses in frequency domain elastodynamic BEM analysis, in which the far-field interactions are evaluated efficiently utilizing equivalent densities and check potentials. Although there are six boundary integrals with unique kernel functions, by using the elastic theory, the authors managed to accelerate these six boundary integrals by KIFMM with the same kind of equivalent densities and check potentials. The boundary integral equations are discretized by Nyström method with curved quadratic elements. The method is further used to conduct the time-domain analysis by using the frequency-domain approach. Findings – Numerical results show that by the fast BEM, high accuracy can be achieved and the computational complexity is brought down to linear. The performance of the present method is further demonstrated by large-scale simulations with more than two millions of unknowns in the frequency domain and one million of unknowns in the time domain. Besides, the method is applied to the topological derivatives for solving elastodynamic inverse problems. Originality/value – An efficient KIFMM is implemented in the acceleration of the elastodynamic BEM. Combining with the Nyström discretization based on quadratic elements and the frequency-domain approach, an accurate and highly efficient fast BEM is achieved for large-scale elastodynamic frequency domain analysis and time-domain analysis.

On Neumann ‐ Dirchlet coupling strategy of Finite Element and Boundary Element method in elastodynamics

AbstractThe Finite Element Method (FEM) and the Boundary Element Method (BEM) are the most used numerical tools for solid mechanics analysis. Each one of these methods has advantages and drawbacks in different cases. In order to take advantage of both methods, a nonoverlapping domain decomposition method FEM ‐ BEM in elastodynamics is presented. The domain is divided in two subdomains and each one of them is analyzed separately and only the interface information is exchanged. An iterative Neumann ‐ Dirchlet algorithm with relaxation is used, to get continuity and the equilibrium conditions at the interface. The FEM time integration is carried out using the Newmark's method and the BEM approach in time domain is based in the Convolution Quadrature Method developed by Lubich. Numerical examples are presented to show agreement with other available numerical results. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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

SUMMARYA 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.

Transient analysis of the DSIFs and dynamic T-stress for particulate composite materials—Numerical vs. experimental results

The fracture behavior of particulate composite materials when subjected to dynamic loading has been a great concern for many industrial applications as these materials are particularly susceptible to impact loading conditions. As a result, many numerical and experimental techniques have been developed to deal with this class of problems. In this work, the fracture behavior of particulate composites under impact loading conditions is numerically studied via the two most important fracture parameters: dynamic stress intensity factors (DSIFs) and dynamic T-stress (DTS), and the results are compared with the experimental data obtained in Refs. [1,2]. Here, micromechanics models (self-consistent, Mori–Tanaka, …) or experimental techniques need to be employed first to determine the effective material properties of particulate composites. Then, the symmetric-Galerkin boundary element method for elastodynamics in the Fourier-space frequency domain is used in conjunction with displacement correlation technique to evaluate the DSIFs and stress correlation technique to determine the DTS. To obtain transient responses of the fracture parameters, fast Fourier transform (FFT) and inverse FFT are subsequently used to convert the DSIFs and DTS from the frequency domain to the time domain. Test examples involving free–free beams made of particulate composites are considered in this study. The numerical results are found to agree very well with the experimental ones.

A new fast multi-domain BEM to model seismic wave propagation and amplification in 3-D geological structures

SUMMARY The analysis of seismic wave propagation and amplification in complex geological structures raises the need for efficient and accurate numerical methods. The solution of the elastodynamic equations using traditional boundary element methods (BEMs) is greatly hindered by the fully-populated nature of the matrix equations arising from the discretization. In a previous study limited to homogeneous media, the present authors have established that the fast multipole method (FMM) reduces the complexity of a 3-D elastodynamic BEM to N log N per GMRES iteration and demonstrated its effectiveness on 3-D canyon configurations. In this paper, the frequency-domain FM-BEM methodology is extented to 3-D elastic wave propagation in piecewise homogeneous domains in the form of a FM-accelerated multi-region BE–BE coupling approach. This new method considerably enhances the capability of the BEM for studying the propagation of seismic waves in 3-D alluvial basins of arbitrary geometry embedded in semi-infinite media. Several fully 3-D examples (oblique SV -waves) representative of such configurations validate and demonstrate the capabilities of the multi-domain FM approach. They include comparisons with available (low-frequency) results for various types of incident wavefields and time-domain results obtained by means of Fourier synthesis.

An indirect elastodynamic boundary element method with analytic bases

AbstractAn 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.

경계요소법을 이용한 결함의 초음파 산란장 해석

Numerical modeling of a nondestructive testing system plays an important role in many aspects of quantitative nondestructive evaluation (QNDE). The ultimate goal of a model is to predict test results for a specific flaw in a material. Thus, in ultrasonic testing, a system model should include the transducer, its radiation pattern, the beam reflection and propagation, and scattering from defects. In this paper attention is focused on the scattering model and the scattered fields by defects are observed by an elastodynamic boundary element method. Flaw types addressed are void-like and crack-like flaws. When transverse ultrasonic waves are obliquely incident on the flaw, the angular distribution of far-field scattered displacements are calculated and presented in the form of A-scan mode. The component signals obtained from each scattering problem are identified and their differences are addressed. The numerical results are also compared with those obtained by high frequency approximate solutions.

Identification of Plural Defects Using the Boundary Element Method.

In this paper, location, angle and size of plural defects in a structural component are identified by an elastodynamic boundary element method. Each defect is noncircular and represented by 5 parameters, a radius, an angle, x and y coordinates of the location and the degree of distortion of its shape. External force of harmonic excitation is applied to the component and the displacements at points on its boundary are chosen as additional information. The square sum of the differences in displacements between real values and inference from numerical solutions is minimized to identify the defects using the conjugate gradient method. Though the number of defects is assumed to be one in the beginning of the identification process, the assumed defect is divided into two parts when the value of the parameter representing its distortion satisfies a certain condition. To judge correctly whether it should be divided, it is assumed that defects are filled with imaginary material of which mass density and Young's modulus are considerably low in comparison with those of a structural component.

Simulation of multiple scattering of seismic waves by spatially distributed inclusions

A 2D elastodynamic boundary element method (BEM) is used to solve multiple scattering of elastic waves. The method is based on the integral representation of an elastic wave-field by assuming a fictitious source distribution on the scattering objects or inclusions, i.e. a mathematical description of Huygens’ principle, and the fictitious source distribution can be found by matching appropriate boundary conditions at the boundary of the inclusions. Numerical studies show that in the presence of cracks, spatial and scale-length distributions are important and different spatial arrangements of the same scatters lead to profound differences in scattering characteristics, in particular the frequency contents of the transmitted wave-fields. The frequency characteristics, such as the frequency of peak attenuation, can be related to spatial size parameters of the model.

"Causal" shape functions in the time domain boundary element method

Since failing to respect the causality condition has been identified as one of the main sources of inaccuracies in the time domain boundary element method for elastodynamics and scalar wave propagation problems, in this contribution new shape functions are investigated, which permit a more accurate simulation of the continuous propagation of wave fronts. The performance of these shape functions in 2D scalar wave propagation problems is tested both for the potential (displacement) and for the time gradient (velocity) equations. Analytical time integrations are developed and numerical results are presented.

Elastodynamic BEM modelling of multiple scattering of elastic waves by spatial distributions of inclusions

We use a 2D elastodynamic boundary integral equation or boundary element method (BEM) in this paper and apply it to solve crack scattering problems. The method is based on the integral representation of a scattered wavefield by assuming a fictitious source distribution on the scattering objects or inclusions (i.e. mathematical description of Huygens’ principle), and the fictitious source distribution can be found by matching appropriate boundary conditions at the boundary of the inclusions. We present two numerical examples to demonstrate the versatility of the BEM method. The first example shows that different spatial arrangements of the same scatters lead to profound differences in scattering characteristics, in particular the frequency contents of the transmitted wavefields using the method of time‐frequency analysis. The second example shows the effects of power‐law or fractal distribution of scalelengths on transmitted wavefields, and we conclude that frequency characteristics, such as the frequency of the peak attenuation, can be related to spatial size parameters of the model.

Semi-analytical integration of sub-parametric elements used in the BEM for three-dimensional elastodynamics

The steady-state elastodynamic boundary element method is an efficient method that can be used in the modelling of vibration systems. The ability to locate natural frequencies and predict displacements close to these frequencies requires the use of high order elements and accurate integration schemes. The possible unboundedness of the displacements at or close to a natural frequency highlights poor numerical conditioning of the algebraic equations resulting from the discretization process, thus making accurate integration schemes a necessity. This paper presents a semi-analytical integration scheme that can be applied to quadratic subparametric triangular elements. The scheme involves subdividing the triangular elements into four triangular subelements. The quadratic shape functions of the original element can then be represented in terms of the linear shape functions of each subelement. A semi-analytical scheme is applied to the integrals involving the linear shape functions of the subelements. Taylor expansions are utilized in the scheme presented to enable the formulation of the integrals into regular and singular parts. Standard numerical schemes are applied to the regular part. The singular part can be transformed into a line integral and evaluated numerically using Gauss–Legendre quadrature. The scheme can handle all integrals appearing in the steady-state elastodynamic BEM with good accuracy. In addition, the Cauchy principal value singular integrals can be dealt with without special treatment. The new scheme is tested by considering integration over two test elements and by application to simple test-problems for which analytical solutions are known.

A boundary element method in elastodynamics for geometrically axisymmetric problems

A direct boundary element method (BEM) in elastodynamics is developed for geometrically axisymmetric problems subjected to arbitrary external loads. Traction and displacement components are expressed in terms of Fourier series. Unlike classical BEMs, the kernel functions in the resulting integral governing equations in the present method are expanded as implicit functions of the difference of the polar angles at two field points. The new BEM is therefore capable of solving a wider variety of elastodynamic problems. Two numerical examples in foundation engineering show that the present BEM is also accurate and computationally efficient.

A Study on Scattered Fields Analysis of Ultrasonic SH-Wave from Multi-Defects by Boundary Element Method

Ultrasonic technique which is one of the most common nondestructive evaluation techniques has been applied to evaluate the integrity of structures by analyzing the characteristic of scattering sign al from internal defects. Therefore, a numerical analysis of ultrasonic scattering field due to defect profiles is absolutely needed for the accurate, quantitative estimation of internal defects. In this paper, the SH-wave scattering by multi-cavity defects and inclusion using Elastodynamic Boundary Element Method is studied. The effects of shape and distance of defects on transmitted and reflected fields are considered. The interaction of multi-cavity defects in SH-wave scattering is also investigated. Numerical calculations by the BEM have been carried out to predict near field solution of scattered fields of ultrasonic SH-wave. The presented results can be used to improve the detection sensitivity and pursue quantitative nondestructive evaluation for inverse problem.

Elastodynamic boundary element method for piecewise homogeneous media

A general higher-order formulation for the time domain elastodynamic direct boundary element method is presented for computing the transient displacements and stresses in multiply connected two-dimensional solids. The displacement and traction interpolation functions are linear in time and quadratic in space. All integrations are analytical, and are expressed in terms of twelve basic recurring integrals. Causality is ensured by integrating only over the dynamically active parts of each element, and the algorithm presented is time-marching and implicit. The use of analytical integrations allows both unbounded and bounded domain problems to be solved without having to introduce special enclosing elements. All of these improved features allow for a formulation that is very efficient and accurate. The stability and accuracy of the elastodynamic boundary element algorithm is demonstrated by solving several example problems and comparing the results with available analytical and numerical solutions. © 1998 John Wiley & Sons, Ltd.