The pseudospectral method: Accurate representation of interfaces in elastic wave calculations
TL;DR: In this article, the pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy, and the mappings introduced in this paper also eliminate the other dominant error source.
Abstract: When finite‐difference methods are used to solve the elastic wave equation in a discontinuous medium, the error has two dominant components. Dispersive errors lead to artificial wave trains. Errors from interfaces lead to circular wavefronts emanating from each location where the interface appears “jagged” to the rectangular grid. The pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy. With this method, dispersive errors are essentially eliminated. The mappings introduced in this paper also eliminate the other dominant error source. Test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost. Although the mapping method is described here in connection with the pseudospectral method, it can also be used with high‐order finite‐difference approximations.
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TL;DR: The spectral element method as discussed by the authors is a high-order variational method for the spatial approximation of elastic-wave equations, which can be used to simulate elastic wave propagation in realistic geological structures involving complieated free surface topography and material interfaces for two- and three-dimensional geometries.
Abstract: We present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as Δ tC < O ( nel−1/nd N −2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.
1,183 citations
TL;DR: In this paper, a spectral element method for studying acoustic wave propagation in complex geological structures is presented, which shows more accurate results compared to the low-order finite element, the conventional finite difference and the pseudospectral methods.
262 citations
217 citations
TL;DR: In this paper, a new numerical method, named as Traction Image method, is proposed to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography.
Abstract: SUMMARY
In this study, we propose a new numerical method, named as Traction Image method, to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography. In this algorithm, the computational domain is discretized by boundary-conforming grids, in which the irregular surface is transformed into a ‘flat’ surface in computational space. Thus, the artefact of staircase approximation to arbitrarily irregular surface can be avoided. Such boundary-conforming gridding is equivalent to a curvilinear coordinate system, in which the first-order partial differential velocity-stress equations are numerically updated by an optimized high-order non-staggered finite difference scheme, that is, DRP/opt MacCormack scheme. To satisfy the free surface boundary conditions, we extend the Stress Image method for planar surface to Traction Image method for arbitrarily irregular surface by antisymmetrically setting the values of normal traction on the grid points above the free surface. This Traction Image method can be efficiently implemented. To validate this new method, we perform numerical tests to several complex models by comparing our results with those computed by other independent accurate methods. Although some of the testing examples have extremely sloped topography, all tested results show an excellent agreement between our results and those from the reference solutions, confirming the validity of our method for modelling seismic waves in the heterogeneous media with arbitrary shape topography. Numerical tests also demonstrate the efficiency of this method. We find about 10 grid points per shortest wavelength is enough to maintain the global accuracy of the simulation. Although the current study is for 2-D P-SV problem, it can be easily extended to 3-D problem.
216 citations
TL;DR: In this paper, an application of the local interaction simulation approach for wave propagation in metallic structures is reported, where the focus of the analysis is on damage detection applications, and the simulated results are validated experimentally.
Abstract: Lamb waves are the most widely used acousto-ultrasonic guided waves for damage detection. The method is generally complicated by the coexistence of at least two highly dispersive modes at any given frequency. Furthermore pure Lamb wave modes may generate a variety of other modes by interacting with defects and/or by crossing different boundaries. Knowledge and understanding of Lamb wave propagation is important for reliable damage detection. However, the theoretical analysis of guided wave scattering forms an extremely difficult problem. This paper reports an application of the local interaction simulation approach for wave propagation in metallic structures. The focus of the analysis is on damage detection applications. The study also involves wave propagation in a piezoceramic actuator/sensor diffusion bond model in which one of the piezoceramics generates the thickness mode vibration. The simulated results are validated experimentally. The results show the potential of the method for wave propagation analysis in damage detection applications.
204 citations
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Book•
01 Jan 1980
TL;DR: This work has here attempted to give a unified treatment of those methods of seismology that are currently used in interpreting actual data and develops the theory of seismic-wave propagation in realistic Earth models.
Abstract: In the past decade, seismology has matured as a quantitative science through an extensive interplay between theoretical and experimental workers. Several specialized journals have recorded this progress in thousands of pages of research papers, yet such a forum does not bring out key concepts systematically. Because many graduate students have expressed their need for a textbook on this subject and because many methods of seismogram analysis now used almost routinely by small groups of seismologists have never been adequately explained to the wider audience of scientists and engineers who work in the peripheral areas of seismology, we have here attempted to give a unified treatment of those methods of seismology th at are currently used in interpreting actual data. We develop the theory of seismic-wave propagation in realistic Earth models. We study specialized theories of fracture and rupture propagation as models of an earthquake, and we supplement these theoretical subjects with practical descriptions of how seismographs work and how data are analyzed and inverted. Our text is arranged in two volumes. Volume I gives a systematic development of the theory of seismic-wave propagation in classical Earth models, in which material properties vary only with depth. It concludes with a chapter on seismometry. This volume is intended to be used as a textbook in basic courses for advanced students of seismology. Volume II summarizes progress made in the major frontiers of seismology during the past decade. It covers a range of special subjects, including chapters on data analysis and inversion, on successful methods for quantifying wave propagation in media varying laterally (as well as with depth), and on the kinematic and dynamic aspects of motions near a fault plane undergoing rupture. The second volume may be used as a texbook in graduate courses on tectonophysics, earthquake mechanics, inverse problems in geophysics, and geophysical data processing.n
5,291 citations
TL;DR: In this paper, a nonreflecting boundary condition for the finite-difference method is proposed, which is based on gradual reduction of the amplitudes in a strip of nodes along the boundary of the mesh.
Abstract: I J ~ One of the nagging problems which arises in application of discrete solution methods for wave-propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh. These undesired events eventually over ride the actual seismic signals which propagate in the modeled region. The solution to avoiding boundary effects used to be to enlarge the numerical mesh, thus delaying the side reflections and wraparound longer than the range of times involved in the modeling. Obviously this solution considerably increases the expense of computation. More recently, nonreflecting bound ary conditions were introduced for the finite-difference method (Clayton and Enquist, 1977: Reynolds, (978). These boundary conditions are based on replacing the wave equation in the boundary region by one-way wave equations which do not permit energy to propagate from the boundaries into the nu merical mesh. This approach has been relatively successful, except that its effectiveness degrades for events which impinge on the boundaries at shallow angles. It is also not clear how to apply this type of boundary condition to global discrete meth ods such as the Fourier method for which all grid points are coupled. In this note we describe an alternative scheme for construct ing a nonreflecting boundary condition. It is based on gradual reduction of the amplitudes in a strip of nodes along the bound aries of the mesh. The method appears extremely simple and robust, and it can be applied to a wide variety of time dependent problems. Unlike other methods, the effectiveness of this boundary condition does not decrease for shallow angles of incidence.
949 citations
TL;DR: In this paper, the authors investigate more accurate difference methods and show that fourth order methods are optimal in some sense, and compare these methods with a variant of the Fourier technique.
Abstract: Historically, second order accurate difference methods have been used for computations in dynamic meteorology and oceanography. We investigate more accurate difference methods and show that fourth order methods are optimal in some sense. This method is then compared with a variant of the Fourier technique. DOI: 10.1111/j.2153-3490.1972.tb01547.x
628 citations
TL;DR: In this paper, a pseudospectral forward-modeling algorithm for solving the two-dimensional acoustic wave equation is presented, which utilizes a spatial numerical grid to calculate spatial derivatives by the fast Fourier transform.
Abstract: A Fourier or pseudospectral forward-modeling algorithm for solving the two-dimensional acoustic wave equation is presented. The method utilizes a spatial numerical grid to calculate spatial derivatives by the fast Fourier transform. time derivatives which appear in the wave equation are calculated by second-order differcncing. The scheme requires fewer grid points than finite-diffcrcnce methods to achieve the same accuracy. It is therefore believed that the Fourier method will prove more efficient than finitedifference methods. especially when dealing with threedimensional models. The Fourier forward-modeling method was tested against two problems, a single-layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling. The numerical results agreed with both the analytic and physical model results. Furthermore, the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account.
484 citations
TL;DR: In this paper, a finite difference for elastic waves is introduced and the model is based on the first order system of equations for the velocities and stresses of the elastic wave and is tested on a series of examples including the Lamb problem, scattering from plane interf aces and scattering from a fluid-elastic interface.
Abstract: A finite difference for elastic waves is introduced. The model is based on the first order system of equations for the velocities and stresses. The differencing is fourth order accurate on the spatial derivatives and second order accurate in time. The model is tested on a series of examples including the Lamb problem, scattering from plane interf aces and scattering from a fluid-elastic interface. The scheme is shown to be effective for these problems. The accuracy and stability is insensitive to the Poisson ratio. For the class of problems considered here it is found that the fourth order scheme requires for two-thirds to one-half the resolution of a typical second order scheme to give comparable accuracy.
182 citations