Elastodynamic Boundary 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 log-linear time algorithm for the elastodynamic boundary integral equation method

We present a fast and memory-efficient algorithm for transient space–time-domain elastodynamic boundary-integral analysis. Associated data-sparse approximations and operations are named fast domain partitioning hierarchical matrices (FDP=H-matrices). The fast domain partitioning method (the FDPM) solves a known problem of hierarchical matrices (H-matrices) in compressing discretized elastodynamic kernel functions. A novel set of plane-wave approximations unites the FDPM and H-matrices in an accurate analytic manner. Memory usage is O(NlogN) and computation time O(NMlogN) in our algorithm for a single run with N boundary elements and M time steps. Consequent cost reduction is remarkable, considering the O(N2M) memory usage and O(N2M2) computational time to run the orthodox time-marching implementation. Numerical experiments verify FDP=H-matrices realize O(NM/logN) times smaller memory and computation time with ensuring the accuracy of integral analyses.

An adaptive cellular automata approach with the use of radial basis functions for the simulation of elastic wave propagation

A physics-based adaptive approach constructed on cellular automata is proposed for the analysis of two-dimensional (2D) elastodynamic initial boundary value problems. Although any 2D cell may be used in this approach, because of easy construction of arbitrary geometry, triangular plane cells are used as primary reference. The analysis of the problem includes two principal parts. In the first part, after discretization of the problem domain, a physics-based configuration considering the conservation of mass and the balance of momentum is developed as reference for cellular automata used in the next step. In the second part, radial basis functions (RBFs) are employed for an adaptive solution. For this purpose, after analysing the problem based on an arbitrary coarse mesh in the first step as reference, the problem domain may be discretized by iterative schemes, where RBFs indicate the lack and redundancy of computational points of the domain, from which some computational points may be added or eliminated for the next analysis. This process may be repeated to get the most appropriate mesh with the most accurate results. The principle of causality is taken into account in the proposed method, meaning that the unexcited parts of the domain are not in the solution domain. Similarly, those parts of the domain where the wave has already passed from would be deducted from the solution domain. The results of the proposed approach are compared with numerical and analytical results available in the literature. The comparison illustrates very good agreements between the results of the proposed approach and the results of those already reported in the literature.

Identification of delamination based on the Lamb wave scattering resonance frequencies

The present paper is aimed to test the idea of using scattering resonance frequencies, which are sensitive to the crack's size and depth variation, for the express non-destructive estimation of its severity. The width and depth of artificial strip-like delaminations are estimated by matching theoretical eigenfrequencies, calculated in the framework of a corresponding 2D elastodynamic boundary value problem, with the experimentally obtained resonance frequencies. The high accuracy of the delamination characteristics obtained by this way indicates its potential utility for practice.

Interplay between Process Zone and Material Heterogeneities for Dynamic Cracks

Using an elastodynamic boundary integral formulation coupled with a cohesive model, we study the problem of a dynamic rupture front propagating along an heterogeneous plane. We show that small-scale heterogeneities facilitate the supershear transition of a mode-II crack. The elastic pulses radiated during front accelerations explain how microscopic variations of fracture toughness change the macroscopic rupture dynamics. Perturbations of dynamic fronts are then systematically studied with different microstructures and loading conditions. The process zone size is the intrinsic length scale controlling heterogeneous dynamic rupture. The ratio of this length scale to asperity size is proposed as an indicator to transition from quasihomogeneous to heterogeneous fracture. Moreover, we discuss how the shortening of the process zone size with increasing crack speed brings the front to interact with smaller details of the microstructure. This study shines new light on recent experiments reporting perturbations of dynamic rupture fronts, which intensify with crack propagation speed.

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 low frequency elastodynamic fast multipole boundary element method in three dimensions

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the `low frequency' regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer---inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to $$60\,\%$$60%. This combination of strategies converges 2---2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.

Acoustic coupled fluid-structure interactions using a unified fast multipole boundary element method.

This paper presents a numerical model for the acoustic coupled fluid-structure interaction (FSI) of a submerged finite elastic body using the fast multipole boundary element method (FMBEM). The Helmholtz and elastodynamic boundary integral equations (BIEs) are, respectively, employed to model the exterior fluid and interior solid domains, and the pressure and displacement unknowns are coupled between conforming meshes at the shared boundary interface to achieve the acoustic FSI. The low frequency FMBEM is applied to both BIEs to reduce the algorithmic complexity of the iterative solution from O(N(2)) to O(N(1.5)) operations per matrix-vector product for N boundary unknowns. Numerical examples are presented to demonstrate the algorithmic and memory complexity of the method, which are shown to be in good agreement with the theoretical estimates, while the solution accuracy is comparable to that achieved by a conventional finite element-boundary element FSI model.

Spectral formulation of the elastodynamic boundary integral equations for bi-material interfaces

A spectral formulation of the plane-strain boundary integral equations for an interface between dissimilar elastic solids is presented. The boundary integral equations can be written in two equivalent forms: (a) the tractions can be written as a space–time convolution of the displacement continuities at the interface (Budiansky and Rice, 1979) (b) the displacement discontinuities can be written as a space–time convolution of the tractions at the interface (Kostrov, 1966). Prior work on spectral formulation of the boundary integral equations has adopted the former as the starting point. The present work has for its basis the latter form based on a space–time convolution of the tractions. Tractions and displacement components are given a spectral representation in the spatial coordinate along the interface. The radiation damping term is then explicitly extracted to avoid singularities in the convolution kernels. With the spectral forms introduced, the space–time convolutions reduce to convolutions in time for each Fourier mode. Due to continuity of tractions at the interface, this leads to a simpler formulation and form of the convolution kernels in comparison to the formulation involving convolutions over the slip and opening history at the bi-material interface. The convolution kernels are validated by studying some model problems to which analytical solutions are known. When coupled with a cohesive law or a friction law at the interface, the formulation proposed here is of wide applicability for studying spontaneous rupture propagation.

A study of frictional contact in dynamic fracture along bimaterial interfaces

We investigate numerically the dynamic in-plane propagation of a centered crack along bimaterial interfaces using a spectral formulation of the elastodynamic boundary integral equations. Particular attention is given to the effect of contact zones at the subsonic/intersonic transition. In a single set-up, we simulate and describe the different phenomenon observed experimentally (distinct natures of contact zones, unfavorable velocity range, asymmetric crack propagation). We show that different behaviors are observed as function of the crack propagation direction, i.e., with respect to the particle displacements of the compliant material. When the crack propagates in the same direction, the propagation velocities between $$c_\mathrm{R}$$ and $$c_\mathrm{s}$$ are forbidden and the subsonic/intersonic transition occurs with the nucleation of a daughter crack in front of the main rupture. The intersonic stress field at the crack front is compressive due to the material mismatch and a contact zone appears behind the tip. In the opposite direction, a smooth subsonic/intersonic transition occurs although crack face closure (in normal direction) is observed for speeds between $$c_\mathrm{s}$$ and $$\sqrt{2}c_\mathrm{s}$$ . In this regime, a Rayleigh disturbance is generated at the crack surface causing a contact zone which detaches from the tip. Using a contact model governed by a regularized Coulomb law, we provide a quantitative evaluation of the influence of friction on the effective fracture toughness. Finally, we show the applicability of our analysis to the description of different bimaterial situations as well as the single-material set-up.

Efficient wave scattering analysis for damaged cylindrical waveguides

Abstract This paper addresses scattering effects at arbitrarily shaped defects in waveguides with finite cross-sections. The main subject is to predict scattered wavefields induced by an incident wavefield in order to localize and characterize damages. The numerical approach involves the solution of a boundary value problem in combination with a decomposition method of scattered wavefields. In order to investigate mode conversion phenomena at defects, a boundary element model of the damaged waveguide section is built up. Implementation of the Boundary Element Method based on the elastodynamic boundary integral equation and the Waveguide Finite Element Method allows for a numerically efficient calculation of scattering coefficients. For defect characterization and mode sensitivity analysis, various types of surface opening defects as well as shear fractures are considered. Numerical results are presented for cylindrical waveguides and are verified experimentally.

Discussion on “Two-dimensional elastodynamics by the time-domain boundary element method: Lagrange interpolation strategy in time integration” by Carrer J A M et al.

Abstract The purpose is to discuss the potential flaws in the elastodynamic boundary integral equation with the authors to the mentioned paper.

Homogenization of microstructures in dynamics using periodic boundary conditions

AbstractA homogenization method is used to calculate effective material properties of periodic microstructures subjected to small, time‐harmonic strain fields. The method is based on the equivalence of averaged micro‐field quantities in dispersive heterogeneous media and those in effective homogeneous media with frequency dependent elastic constants. Solutions to the elastodynamic boundary value problem (BVP) are obtained from a boundary element method where periodic boundary conditions on the unit‐cell are incorporated using Lagrangian multipliers. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Importance of footing–soil separation on dynamic stiffness of piled embedded footings

AbstractDifferent phenomena such as soil consolidation, erosion, and scour beneath an embedded footing supported on piles may lead to loss of contact between soil and the pile cap underside. The importance of this separation on the dynamic stiffness and damping of the foundation is assessed in this work. To this end, a numerical parametric analysis in the frequency domain is performed using a rigorous three‐dimensional elastodynamic boundary element–finite element coupling scheme. Dimensionless plots relating dynamic stiffness functions computed with and without separation effects are presented for different pile–soil configurations. Vertical, horizontal and rocking modes of oscillation are analyzed for a wide range of dimensionless frequencies. It is shown that the importance of separation is negligible for frequencies below those for which dynamic pile group effects start to become apparent. Redistribution of stiffness contributions between piles and footing is also addressed. Copyright © 2009 John Wiley & Sons, Ltd.

Adaptive Cross Approximation in an elastodynamic Boundary Element formulation

AbstractElastodynamic phenomena can be effectively analyzed by using the Boundary Element Method (BEM), especially in unbounded media. However, for the simulation of such problems, beside others, two difficulties restrict the BEM to rather small or medium–sized problems. Firstly, one has to deal with dense matrices and secondly the treatment of the kernel functions is very costly. Several approaches have been developed to overcome these drawbacks. Approaches, such as Fast Multipole and Panel Clustering etc. gain their efficiency basically from an analytic kernel approximation. The main difficulty of these methods is that the so called degenerate kernel has to be known explicitly.Hence, the present work focuses on a purely algebraic approach, the adaptive cross approximation (ACA). By means of a geometrical clustering and a reliable admissibility condition, first, a so called hierarchical matrix structure is set up. Then each admissible block can be represented by a low–rank approximation. The advantage of the ACA is based on the fact that only a few of the original matrix entries have to be generated.As will be shown numerically, the presented approach is suitable for an efficient simulation of elastodynamic problems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Seismic radiation from dynamic coalescence, and the reconstruction of dynamic source parameters on a planar fault

SUMMARY The dynamic coalescence of two mode II cracks on a planar fault is simulated here using the elastodynamic boundary integral equation method. We focus on the complexity of the resultant slip rate and seismic radiation in the crack coalescence model (CCM) and on the reconstruction of a single crack model (SCM) that can reproduce the CCM waveforms from heterogeneous source parameters rather than coalescence. Simulation results reveal that localized higher slip rates are generated by coalescence as a result of stress interaction between the approaching crack tips. The synthesized seismic radiation exhibits a distinct coalescence phase that has striking similarities to stopping phases in the radiation and propagation properties. The corresponding SCM yields a singular increase in the stress drop distribution, which is accompanied by a sudden decrease in it across the point of coalescence in the CCM. This implies that the generation of high-frequency radiation is more efficient from coalescence than from stopping, although both phenomena exhibit the same strong ω −2 -type displacement spectra.

Elastodynamic boundary element formulation employing discontinuous curved elements

A boundary element formulation having discontinuous curved quadratic elements is presented for 2D elastodynamics. The first fundamental solution for static case is subtracted from and added to the first fundamental solution for dynamic case. As both kernels have the same order of singularity, the integral involving the regular expression arising from the subtraction can be calculated. H ̲ matrix is calculated by employing the well-known rigid-body motion technique. The formulation is performed in Fourier transform space. Based on the formulation presented in this study, a general purpose computer program is developed for elastic or visco-elastic 2D elastodynamic problems. The program performs the analysis in Fourier transform space and can also be used for static analysis by assigning a very small value close to zero for the frequency. The results of some elastodynamic and dynamic soil–structure interaction problems obtained using the present study are compared with those in the literature.

Mode-3 spontaneous crack propagation in unsymmetric functionally graded materials

The effects of the inhomogeneous material property variation on the mode-3 crack propagation characteristics have been analyzed. The spectral form of the elastodynamic boundary integral equations are derived for a functionally graded material, which is used as the numerical tool in our analysis. The material property gradient is assumed to vary unsymmetrically in the direction normal to the fracture plane. A parametric study has been performed by systematically varying the inhomogeneity length scale. In comparison with a homogeneous material, an unsymmetric functionally graded material offers lesser fracture resistance. The fracture resistance progressively decreases with increase in inhomogeneity, quantified by the increase in crack sliding displacement jumps, crack tip velocities and accelerations. The material property inhomogeneity affects only the transient crack propagation velocities, while the quasi-steady-state velocity remains unaltered.

レーザ超音波計測による波形データを用いたクラック決定解析

This work considers the determination of cracks with the time domain elastodynamic boundary integral equation method (BIEM) in laser-ultrasonic non-destructive evaluation problems. We carry out an inverse problem to determine the thermally expanded source from the measured velocimetory waveform data using Green's function of heat conduction problems and that of elastodynamics. With the thermally expanded source thus determined, we can calculate the laser generated elastic wave field numerically using the time domain BIEM. We then represent the unknown crack in the material with shape parameters and compute the normal velocities on the surface of the test piece with BIEM. The shape parameters of the crack are determined by a comparison between the computed waveforms and the corresponding measured data at the measuring points. We show that we can determine the shape and the position of the surface crack in the test piece made of aluminum alloy accurately using time domain BIEM and real data obtained in a laser-ultrasonics measurement.