Seismic Waves in a Quarter and Three-Quarter Plane

Alterman, Z.; Loewenthal, Dan

doi:10.1111/j.1365-246x.1970.tb06058.x

Cited by 111 publications

(39 citation statements)

References 9 publications

(4 reference statements)

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“…Based on von Neumann stability analysis for timedependent problems, allowable time-steps were derived for the two ®nite element discretizations considered in the paper. The allowable time step for the ®nite difference scheme was originally derived [2]. These are listed in Table 2.…”

Section: Discussionmentioning

confidence: 99%

“…(16) and (17) is an explicit scheme, the time step cannot be prescribed arbitrarily and a stability analysis is typically used to derive the allowable time step for a given spatial mesh. By using a von Neumann stability analysis, Altermann and Loewenthal [2] obtained the allowable time step Dt for the difference equations (16) and (17) as…”

Section: Stability Conditionmentioning

confidence: 99%

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Dispersion analysis of numerical approximations to plane wave motions in an isotropic elastic solid

Cherukuri

2000

Computational Mechanics

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It is well known that isotropic, nondispersive continuous hyperbolic problems become dispersive and anisotropic upon discretization. The purpose of this paper is to conduct a dispersion analysis of the nondissipative numerical approximations to plane wave motions in isotropic elastic solids. The discrete formulations considered are: an explicit, second-order accurate ®nite difference scheme, a consistent mass matrix formulation with linear quadrilateral elements and the corresponding lumped mass matrix formulation. Dispersion relation is derived for each of these formulations. In the context of the ®nite difference scheme, expressions for group velocity for both the shear and longitudinal waves are derived and the effect of using meshes of unequal size in x and y directions is studied. Results from numerical experiments con®rming the predictions of analysis are also presented. IntroductionDispersion relation is the relation between the frequency and wavenumber. The ratio of the frequency and wavenumber is the phase speed whereas the gradient of the frequency with respect to the wavenumber is the group speed. For a nondispersive problem, the group speed does not depend on the frequency and is identical to the phase speed. For a dispersive problem on the other hand, the group speed depends on the frequency and in general is different from the phase speed.It is well known that when a nondispersive continuous problem is discretized, the discrete model becomes dispersive. In the case of one-dimensional problems, dispersion leads to waves of different lengths propagating at different speeds. In the case of two and three dimensions, not only do different wavelengths travel at different speeds but they also travel in wrong directions. That is, the discrete models become anisotropic even if the continuous problem is isotropic.A survey of the literature shows that a large body of work has been done on analyzing the dispersion errors induced due to numerical approximations. Much of the work related to one-dimensional and two-dimensional scalar wave equations can be found in the monograph by Vichnevetsky [10]. Dispersion analysis of a three-dimensional scalar wave equation was conducted by Abboud and Pinsky [1]. The excellent review article by Trefethen [9] surveys the role of group velocity in ®nite difference approximations to one-dimensional and two-dimensional scalar wave equations. Recently, Monk and Parrott [8] studied the dispersion errors introduced in various ®nite element approximations to Maxwell's equations.However, surprisingly, there appears to be very little work done on the dispersion analysis of numerical approximations to the wave propagation problems in isotropic, elastic solids in two and three dimensions. Only the one-dimensional elastic wave propagation problem appears to have received the attention of several researchers previously (see [3,7] and the references cited therein). In addition, most of this work appears to consider only one type of wave propagating through the body.In the case of two and th...

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Section: Discussionmentioning

confidence: 99%

Section: Stability Conditionmentioning

confidence: 99%

Dispersion analysis of numerical approximations to plane wave motions in an isotropic elastic solid

Cherukuri

2000

Computational Mechanics

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“…The motion of longitudinal and shear waves in the medium can be obtained, as long as the stability requirement for the finite difference equations is satisfied. The time step ∆t is then chosen according to the von Neumann stability criterion (15) , ∆t ≤ ε(v…”

Section: Numerical Evaluation Of Focusing Abilitymentioning

confidence: 99%

In-Situ Observation of Alumina Particles in Molten Aluminum Using a Focused Ultrasonic Sensor

Ihara

Aso²,

Burhan

2004

JSME Int. J., Ser. A

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Ultrasonic in-situ observation of alumina particles in molten aluminum at temperature up to 800• C is presented. A focused ultrasonic sensor is employed for the high temperature measurement with a high spatial resolution. The sensor mainly consists of a conventional piezoelectric transducer, a taper-shaped clad buffer rod as a waveguide and a cooling system. Martensitic stainless steel is used as a material for the buffer rod and an acoustic lens is fabricated at the probe end of the rod. In order to examine the focusing ability of the acoustic lens, the acoustic field near the focusing zone is numerically evaluated by a finite difference method. Using the developed focused ultrasonic sensor, backscattered echoes from alumina particles of 160 µm suspended in the molten aluminum at 800• C have been observed clearly in pulse echo mode at 10 MHz. The effect of the size and the aggregation condition of the particle on the backscattered echoes has also been examined.

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“…The finite difference algorithms are obtained by discretizing the equations of motion, (1) and (2), and the boundary conditions, (7) and (8), for both the temporal and spatial derivatives [6,7].…”

Section: Reduction To Particle Displacementsmentioning

confidence: 99%