Elastic Wave Modeling Research Articles

Modal and characteristics-based approaches for modeling elastic waves induced by time-dependent boundary conditions

In this paper, we present a characteristics-based approach for solving elastic wave problems with time-dependent traction boundary conditions. A generalized mathematical model for this important class of problems is expressed as a set of first-order, linear, hyperbolic partial differential equations. We analyze the mathematical structure of this first-order linear system, verify its hyperbolicity, derive its characteristic form, and deduce its eigenvalues, eigenvectors, and Riemann invariants. The eigenvalues correspond to the wave speeds, while the Riemann invariants are used to construct a solution by the method of characteristics. We benchmark the method of characteristics against several popular modal approaches. Two of these, which we refer to as the concentrated body force method (CBFM) and the homogeneous eigenfunction expansion method (HEEM), were developed to simplify the well-established but tedious Mindlin–Goodman method. To homogenize the boundary conditions and enable modal analysis, the CBFM and HEEM forgo the usual formalism of linear transformations (a la Mindlin–Goodman) in favor of intuitive modeling assumptions and postulated solution structures. We find, however, that these approaches introduce an artificial stress discontinuity at the forced boundary in their reformulated problems. When these reformulated problems are solved by modal analysis, spurious oscillations and significant overshoot, similar to the Gibbs phenomenon, emerge in the stress profile at the artificial discontinuity. We demonstrate that these oscillations and overshoot are physical manifestations of a series solution for stress, obtained from term-by-term differentiation, that is not uniformly convergent, as required by the formalism of mathematical analysis. The method of characteristics solution, on the other hand, is exact to within machine precision, yielding no artificial discontinuities, spurious oscillations, or unphysical overshoot. Unlike the modal approaches, the method of characteristics solves the first-order problem with time-dependent boundary conditions ‘as is’ without any reformulation, restructuring, or postulated solution structures. Further, its solutions require no post-processing: no a posteriori solution treatment like l'Hopital's rule to accommodate resonance (resonant behavior is inherently captured without the emergence of singularities), no term-by-term differentiation to deduce stress (stress is primitive in a first-order velocity–stress formulation of elastodynamics), and no convergence tests.

Kidney stone fracture by surface waves generated with focused ultrasound tone bursts

Previous studies have provided insight into the physical mechanisms of stone fracture in shock wave lithotripsy. Broadly focused shocks efficiently generate shear waves in the stone leading to internal tensile stresses, which in concert with cavitation at the stone surface, cause cracks to form and propagate. Here, we propose a separate mechanism by which stones may fragment from sinusoidal ultrasound bursts without shocks. A numerical elastic wave model was used to simulate propagation of tone bursts through a cylindrical stone at a frequency between 0.15 and 2 MHz. Results suggest that bursts undergo mode conversion into surface waves on the stone that continually create significant stresses well after the exposure is terminated. Experimental exposures of artificial cylindrical stones to focused burst waves in vitro produced periodic fractures along the stone surface. The fracture spacing and resulting fragment sizes corresponded well with the spacing of stresses caused by surface waves in simulation at different frequencies. These results indicate surface waves may be an important factor in fragmentation of stones by focused tone bursts and suggest that the resulting stone fragment sizes may be controlled by ultrasound frequency. [Work supported by NIH 2T32DK007779-11A1, R01 EB007643, P01 DK043881, R01 DK092197, NSBRI through NASA NCC 9-58.]

A hybrid absorbing boundary condition for frequency-domain finite-difference modelling

Liu and Sen (2010 Geophysics 75 A1?6; 2012 Geophys. Prospect. 60 1114?32) proposed an efficient hybrid scheme to significantly absorb boundary reflections for acoustic and elastic wave modelling in the time domain. In this paper, we extend the hybrid absorbing boundary condition (ABC) into the frequency domain and develop specific strategies for regular-grid and staggered-grid modelling, respectively. Numerical modelling tests of acoustic, visco-acoustic, elastic and vertically transversely isotropic (VTI) equations show significant absorptions for frequency-domain modelling. The modelling results of the Marmousi model and the salt model also demonstrate the effectiveness of the hybrid ABC. For elastic modelling, the hybrid Higdon ABC and the hybrid Clayton and Engquist (CE) ABC are implemented, respectively. Numerical simulations show that the hybrid Higdon ABC gets better absorption than the hybrid CE ABC, especially for S-waves. We further compare the hybrid ABC with the classical perfectly matched layer (PML). Results show that the two ABCs cost the same computation time and memory space for the same absorption width. However, the hybrid ABC is more effective than the PML for the same small absorption width and the absorption effects of the two ABCs gradually become similar when the absorption width is increased.

A 21-point finite difference scheme for 2D frequency-domain elastic wave modelling

The 21-point finite difference scheme for the frequency-space domain elastic wave forward modelling is designed through optimising the impedance matrix, especially calculating the spatial derivative terms and the mass acceleration terms of the elastic wave displacement equation as accurately as possible. Comparative tests show that the 21-point finite difference scheme is much better in grid dispersion, memory requirement, and computation time than the 9-point scheme and slightly better than the 25-point scheme. The 21-point finite difference scheme is ~15% lower in memory consumption and computing time than the 25-point scheme. The numerical examples show that the 21-point finite difference scheme is valid in the sense of the numerical simulation of ideal models.

2.5-D forward and inverse modelling of full-waveform elastic seismic survey

SUMMARY We present a two-and-half-dimensional (2.5-D) forward and inversion algorithm for the interpretation of surface seismic elastic full-waveform data. The 2-D modelling approach for elastic waves might not be sufficiently accurate because of its line-source assumption. On the other hand, a full 3-D modelling of elastic waves is still computationally very expensive for seismic exploration applications. By employing the 2.5-D modelling approach, we assume that the elastic medium is 2-D while the source is a 3-D point source. We solve the 2.5-D problem by first transforming the elastic wave equation in the spatial domain into the wavenumber domain. Then, for each wavenumber we solve a 2-D problem using the finite difference method with staggered grids. After that an inverse wavenumber transform is performed to compute the 3-D field distribution. To handle the inverse transform, we develop an efficient numerical integration scheme by subdividing the integration domain and applying different integration rules to each subinterval. We show that this approach works well for surface seismic applications. The 2.5-D approach offers a more realistic modelling of the elastic wave data, hence it produces more accurate inversion results than the 2-D inversion approach. Finally, we validate this approach by using a numerical test. In this test we used our 2.5-D full-waveform inversion algorithm to invert the synthetic data generated by a 3-D finite-difference time-domain simulation.

A truncated implicit high-order finite-difference scheme combined with boundary conditions

In this paper, first we calculate finite-difference coefficients of implicit finitedifference methods (IFDM) for the first- and second-order derivatives on normal grids and firstorder derivatives on staggered grids and find that small coefficients of high-order IFDMs exist. Dispersion analysis demonstrates that omitting these small coefficients can retain approximately the same order accuracy but greatly reduce computational costs. Then, we introduce a mirrorimage symmetric boundary condition to improve IFDMs accuracy and stability and adopt the hybrid absorbing boundary condition (ABC) to reduce unwanted reflections from the model boundary. Last, we give elastic wave modeling examples for homogeneous and heterogeneous models to demonstrate the advantages of the proposed scheme.

21216 一次元弾性波モデルを用いたレーザ割断における熱応力挙動の基礎検討(OS4-4【機械工学が支援する微細加工技術(4)】)

21216 一次元弾性波モデルを用いたレーザ割断における熱応力挙動の基礎検討(OS4-4【機械工学が支援する微細加工技術(4)】)

주파수영역 탄성파모델링에 대한 CFS-PML경계조건의 적용 및 개선

탄성파의 수치 모델링은 유한한 경계에서 발생하는 인공적인 반사파의 제거를 위한 경계조건을 필요로 한다. 이 연구에서는 주파수영역의 탄성파 수치 모델링에 CFS-PML (Complex Frequency Shifted-Perfectly Matched Layer) 경계조건을 적용하였다. 수치모델링 검증을 위해 Lamb's Problem의 해석해와 수치모델링 결과를 비교한 결과 일치하였다. 모형 내의 운동에너지, 최대크기오차, 그리고 스펙트럼오차를 통하여 CFS-PML경계조건이 기존의 흡수경계조건들 보다 유한경계에서 발생한 인공적인 반사파를 효과적으로 제거할 수 있음을 확인하였다. CFS-PML경계조건의 변수 <TEX>${\kappa}_{max}$</TEX>와 <TEX>${\alpha}_{max}$</TEX>의 최적값은 운동에너지를 이용하여 산정할 수 있었다. 또한, 주파수에 따른 함수로 정의된 <TEX>${\alpha}_{max}$</TEX>를 변수로 갖는 변형된 CFS-PML경계조건을 제안하여 기존 PML경계조건, CFS-PML경계조건, 그리고 변형된 CFS-PML경계조건의 성능을 운동에너지, 최대크기오차, 스펙트럼 오차로 비교하였다. 기존 PML경계조건에서 나타난 스쳐가는 입사각에 대한 반사파 문제가 CFS-PML경계조건, 그리고 변형된 CFS-PML경계조건에서는 개선되었다. Absorbing boundary conditions are used to mitigate undesired reflections that can arise at the model's truncation boundaries. We apply a complex frequency shifted perfectly matched layer (CFS-PML) to elastic wave modeling in the frequency domain. Modeling results show that the performance of our implementation is superior to other absorbing boundaries. We consider the coefficients of CFS-PML to be optimal when the kinetic energy becomes to the minimum, and propose the modified CFS-PML that has the CFS-PML coefficient <TEX>${\alpha}_{max}$</TEX> defined as a function of frequency. Results with CFS-PML and modified CFS-PML are significantly improved compared with those of the classical PML technique suffering from large spurious reflections at grazing incidence.

Staggered Grid with Collocated Grid Points of Velocities for Modeling Propagation of Elastic Waves in Anisotropic Solids by Finite-Difference Time Domain Method

A staggered grid for velocity–stress formulation is presented for modeling elastic waves in anisotropic solids by the finite-difference time domain method. To simply impose boundary conditions on numerical models, our grid is derived by applying a finite integration technique to a single control volume satisfying Newton's law instead of interleaved control volumes for conventional staggered grids. Computed results for the numerical dispersions of the new grid for propagating vertically polarized shear and longitudinal waves in an isotropic solid show that the numerical dispersions of the new grid can be suppressed to the same levels as those of the conventional staggered grids by using a third-degree bi-polynomial interpolation.

Staggered Grid with Collocated Grid Points of Velocities for Modeling Propagation of Elastic Waves in Anisotropic Solids by Finite-Difference Time Domain Method

A staggered grid for velocity–stress formulation is presented for modeling elastic waves in anisotropic solids by the finite-difference time domain method. To simply impose boundary conditions on numerical models, our grid is derived by applying a finite integration technique to a single control volume satisfying Newton's law instead of interleaved control volumes for conventional staggered grids. Computed results for the numerical dispersions of the new grid for propagating vertically polarized shear and longitudinal waves in an isotropic solid show that the numerical dispersions of the new grid can be suppressed to the same levels as those of the conventional staggered grids by using a third-degree bi-polynomial interpolation.

On elastic-electromagnetic mathematical equivalences

Elastic-electromagnetic mathematical equivalences, which here consist of recasting Maxwell′s equations and the constitutive relations for an electromagnetic medium into a form that is mathematically identical with that of the basic equations for elastic waves, are presented. These equivalences allow us, for example, to use a finite-difference modelling code developed for elastic wave propagation to model electromagnetic wave propagation. The reverse process, which consists of using a finite-difference modelling code developed for electromagnetic wave propagation to model elastic wave propagation, is valid only for particular forms of elastic wave equations, such as acoustic wave equations in a 2-D medium (i.e. an infinitely long line source in the cross-line direction and acoustic properties invariant in the cross-line direction).

A hybrid absorbing boundary condition for elastic staggered‐grid modelling

ABSTRACTWe recently proposed an efficient hybrid scheme to absorb boundary reflections for acoustic wave modelling that could attain nearly perfect absorptions. This scheme uses weighted averaging of wavefields in a transition area, between the inner area and the model boundaries. In this paper we report on the extension of this scheme to 2D elastic wave modelling with displacement‐stress formulations on staggered grids using explicit finite‐difference, pseudo‐implicit finite‐difference and pseudo‐spectral methods. Numerical modelling results of elastic wave equations with hybrid absorbing boundary conditions show great improvement for modelling stability and significant absorption for boundary reflections, compared with the conventional Higdon absorbing boundary conditions, demonstrating the effectiveness of this scheme for elastic wave modelling. The modelling results also show that the hybrid scheme works well in 2D rotated staggered‐grid modelling for isotropic medium, 2D staggered‐grid modelling for vertically transversely isotropic medium and 2D rotated staggered‐grid modelling for tilted transversely isotropic medium.

Structure-preserving modelling of elastic waves: a symplectic discrete singular convolution differentiator method

SUMMARY In this paper, we introduce the so-called symplectic discrete singular convolution differentiator (SDSCD) method for structure-preserving modelling of elastic waves. In the method presented, physical space is discretized by the DSCD, whereas an explicit third-order symplectic scheme is used for the time discretization. This approach uses optimization and truncation to form a localized operator. This preserves the fine structure of the wavefield in complex media and avoids non-causal interaction when parameter discontinuities are present in the medium. Theoretically, the approach presented is a structure-preserving algorithm. Also, some numerical experiments are shown in this paper. Elastic wavefield modelling experiments on a laterally heterogeneous medium with high parameter contrasts demonstrate the superior performance of the SDSCD for suppression of numerical dispersion. Long-term computational experiments exhibit the remarkable capability of the approach presented for long-time simulations. Promising numerical results suggest the SDSCD is suitable for high-precision and long-time numerical simulations, as it has structure-preserving property and it can suppress effectively numerical dispersion when coarse grids are used.

Anisotropic finite-difference algorithm for modeling elastic wave propagation in fractured coalbeds

The simulation of wave propagations in coalbeds is challenged by two major issues: (1) strong anisotropy resulting from high-density cracks/fractures in coalbeds and (2) numerical dispersion resulting from high-frequency content (the dominant frequency can be higher than 100 Hz). We present a staggered-grid high-order finite-difference (FD) method with arbitrary even-order ([Formula: see text]) accuracy to overcome the two difficulties stated above. First, we derive the formulae based on the standard Taylor series expansion but given in a neat and explicit form. We also provide an alternative way to calculate the FD coefficients. The detailed implementations are shown and the stability condition for anisotropic FD modeling is examined by the eigenvalue analysis method. Then, we apply the staggered-grid FD method to 2D and 3D coalbed models with dry and water-saturated fractures to study the characteristics of the 2D/3C elastic wave propagation in anisotropic media. Several factors, like density and direction of vertical cracks, are investigated. Several phenomena, like S-wave splitting and waveguides, are observed and are consistent with those observed in a real data set. Numerical results show that our formulae can correlate the amplitude and traveltime anisotropies with the coal seam fractures.

Structural dynamic and stress wave models for the analysis of shotcrete on rock exposed to blasting

During blasting in tunnels and mines, the interaction between shotcrete (sprayed concrete) and rock is influenced by propagating stress waves. Shotcrete support in hard rock tunnels is studied here through numerical analysis using three different modelling approaches. The stress response in the shotcrete closest to the rock when exposed to P-waves striking perpendicularly to the shotcrete–rock interface is simulated. The first model tested is a structural dynamic model that consists of masses and spring elements. The second is a model built up with finite element beam elements interconnected with springs. The third is a one-dimensional elastic stress wave model. The models give comparable results, although the definition of the dynamic loads is different. The analysis results can be used to estimate whether the shotcrete will fail or not for a prescribed distance to detonating explosives inside the rock.

Generalized optical theorems for the reconstruction of Green’s function of an inhomogeneous elastic medium

This paper investigates the reconstruction of elastic Green's function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation-the resolvent formula-Green's function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell T-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell T-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The T-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green's function reconstruction in a scattering medium is discussed.

Application of the Nearly Perfectly Matched Layer to Seismic-Wave Propagation Modeling in Elastic Anisotropic Media

Seismic anisotropy has become a main research topic in both seismic data processing and interpretation (e.g., migration and impedance inversion). As an important part of seismic anisotropy, numerical modeling can help us better understand seismic responses when seismic waves propagate through the interior of the Earth. For the numerical implementation of finite-difference time-domain algorithms, boundary conditions are needed to suppress the spurious reflections from the artificially truncated edges of the discrete model. Here, the nearly perfectly matched layer (NPML) technique is applied to a numerical simulation of seismic-wave propagation in a 2D elastic anisotropic medium, which has been proven to be a very effective absorbing boundary condition in electromagnetic, acoustic, and elastic isotropic media. The equivalence of the wave absorbing performance between the NPML and the standard perfectly matched layer (PML) for elastic wave modeling is theoretically proven in this paper. To check the absorbing ability of the NPML in the cases of grazing incidence and longer simulations (up to 200 s), we choose an elongated anisotropic model in our test. Test results indicate that the NPML has essentially the same computational accuracy as the standard PML but with higher computational savings.

SH wave scattering from 2-D fractures using boundary element method with linear slip boundary condition

SUMMARY A boundary element method (BEM) combined with a linear slip boundary condition is proposed to calculate SH wave scattering from fractures in the frequency domain. The linear slip boundary condition was proposed by Schoenberg to model elastic wave propagation through an imperfectly bonded interface, where the traction cross the interface is continuous and displacement is discontinuous. Here, we demonstrate how to simulate SH wave scattering from fractures by applying the BEM and this linear slip boundary condition. We compare the displacement discontinuity across a traction-free crack calculated using the BEM and an analytical model to show the validity and accuracy of our approach. Comparisons between results obtained using our model with those obtained using a finite difference method are also performed and very good matches between these two methods are found. An example of SH wave scattering from three curved, crossing fractures is also given.

엇격자 유한차분법을 이용한 극지해역 지진파 모델링

이 연구에서는 기존의 탄성파 모델링 알고리즘에 지진 송신원을 적용하고, 음향-탄성파 결합 매질을 구현하여 남극대륙 주변과 같은 극지해역에서 발생할 수 있는 지진파의 거동을 모사한다. 기존의 변위근사 유한차분법과 달리 속도-응력 식으로 구성되는 엇격자 유한차분법의 경우 다양한 송신원을 구현하는데 적합하므로 변위-속도-응력 식에 기초하여 개발된 3차원 엇격자 유한차분법 알고리즘과 이중 우력(Double Couple Forces)을 이용하여 구현한 지진 송신원을 접목시켜 지진파의 거동을 모사한다. 좌수향 주향이동단층, 정단층, 역단층 형태의 지진 송신원에 대해서 개발된 알고리즘을 검증한 결과 이론적으로 예측되는 P파의 초동을 정확히 모사할 수 있었고, 섭입대 모델에 대한 수치모형실험 결과 섭입대에서 역단층에 의해 발생된 후 대륙지각, 해양지각 및 해양에서 전파되는 지진파의 거동양상이 정확하게 모사되는 것을 확인할 수 있었다. We simulate the propagation of earthquake waves in the continental margin of Antarctica using the elastic wave modeling algorithm, which is modified to be suitable for acoustic-elastic coupled media and earthquake source. To simulate the various types of earthquake source, the staggered-grid finite-difference method, which is composed of velocity-stress formulae, can be more appropriate to use than the conventional, displacement-based, finite-difference method. We simulate the elastic wave propagation generated by earthquakes combining 3D staggered-grid finite-difference algorithm composed of displacement-velocity-stress formulae with double couple mechanisms for earthquake source. Through numerical tests for left-lateral strike-slip fault, normal fault and reverse fault, we could confirm that the first arrival of P waves at the surface is in a good agreement with the theoretically-predicted results based on the focal mechanism of an earthquake. Numerical results for a model made after the subduction zone in the continental margin of Antarctica showed that earthquake waves, generated by the reverse fault and propagating through the continental crust, the oceanic crust and the ocean, are accurately described.

The Parameter Averaging Technique in Finite-Difference Modeling of Elastic Waves in Combined Structures with Solid, Fluid and Porous Subregions

AbstractTo finite-difference model elastic wave propagation in a combined structure with solid, fluid and porous subregions, a set of modified Biot’s equations are used, which can be reduced to the governing equations in solids, fluids as well as fluid-saturated porous media. Based on the modified Biot’s equations, the field quantities are finite-difference discretized into unified forms in the whole structure, including those on any interface between the solid, fluid and porous subregions. For the discrete equations on interfaces, however, the harmonic mean of shear modulus and the arithmetic mean of the other parameters on both sides of the interfaces are used. These parameter averaging equations are validated by deriving from the continuity conditions on the interfaces. As an example of using the parameter averaging technique, a 2-D finite-difference scheme with a velocity-stress staggered grid in cylindrical coordinates is implemented to simulate the acoustic logs in porous formations. The finite-difference simulations of the acoustic logging in a homogeneous formation agree well with those obtained by the analytical method. The acoustic logs with mud cakes clinging to the borehole well are simulated for investigating the effect of mud cake on the acoustic logs. The acoustic logs with a varying radius borehole embedded in a horizontally stratified formation are also simulated by using the proposed finite-difference scheme.