Numerical implementation of the integral-transform solution to Lamb's point-load problem

Abstract

The present work describes a procedure for the numerical evaluation of the classical integral-transform solution of the transient elastodynamic point-load (axisymmetric) Lamb's problem. This solution involves integrals of rapidly oscillatory functions over semi-in®nite intervals and inversion of one-sided (time) Laplace transforms. These features introduce dif®culties for a numerical treatment and constitute a challenging problem in trying to obtain results for quantities (e.g. displacements) in the interior of the half-space. To deal with the oscillatory integrands, which in addition may take very large values (pseudo-pole behavior) at certain points, we follow the concept of Longman's method but using as accelerator in the summation procedure a modi®ed Epsilon algorithm instead of the standard Euler's transformation. Also, an adaptive procedure using the Gauss 32-point rule is introduced to integrate in the vicinity of the pseudo-pole. The numerical Laplace-transform inversion is based on the robust Fourier-series technique of Dubner/AbateCrump-Durbin. Extensive results are given for sub-surface displacements, whereas the limit-case results for the surface displacements compare very favorably with previous exact results.

Surface Wave Effects on the Ballistic Response of Brittle Materials

“…Special care must be taken to isolate the branch points and pseudo pole and to perform further refinement during the numerical integration to ensure accuracy and stability of the method [15].…”

“…The displacements and stresses at a given point can be written in terms of potentials ϕ and ψ. The advantage of using potentials is that they satisfy the simple wave equations [14,15] …”

Performance comparison of numerical inversion methods for Laplace and Hankel integral transforms in engineering problems

“…This is due to instability of Epsilon table for large value of N [47]. However, we have removed this disadvantage and prepared a new code for Crump method, which allows any arbitrary value for N [see also 24]. The code is used for those test functions that cannot be evaluated by Mallet [39].…”

An Improvement to the Fourier Series Method for Inversion of Laplace Transforms Applied to Elastic and Viscoelastic Waves

“…Its implementation can be achieved in a computer program consisting of just a handful of lines. Its effectiveness can be measured by the frequency of its use, [Abate and Whitt (1995); Chen and Chou (1998); Frolov and Kitaev (1998); Georgiadis (1993); Georgiadis and Rigatios (1996); Georgiadis et al (1999)]. …”

“…Despite all its strengths, the DAC algorithm has one shortcoming within the context of wave propagation, Gibbs phenomenon, [Georgiadis et al (1999); Laverty (2003)]. The algorithm itself is a construction of an approximate Fourier Series based upon Laplace transform data.…”