SUMMARYA spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an e cient tool for simulating elastic wave propagation in realistic geological structures in two-and three-dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, deÿned with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor-product of a Gauss -Lobatto -Legendre 1-D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an e cient 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=multicorrector format. Long term energy conservation and stability properties are illustrated as well as the e ciency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three-dimensional model. Very accurate modelling of Rayleigh wave propagation and surface di raction is obtained at a low computational cost. The method is shown to provide an e cient tool to study the di raction of elastic waves and the large ampliÿcation of ground motion caused by three-dimensional surface topographies.
S U M M A R YThe scattering of elastic waves by cracks is an old problem and various ways to solve it have been proposed in the last decades. One approach is using dual integral equations, another useful and common formulation is the Boundary Element Method (BEM). With the last one, the boundary conditions of the crack lead to hyper-singularities and particular care should be taken to regularize and solve the resulting integral equations.In this work, instead, the Indirect Boundary Element Method (IBEM) is applied to study problems of zero-thickness 2-D cracks. The IBEM yields the Crack Opening Displacement (COD) which is used to evaluate the solution away from the crack. We use a multiregional approach which consists of splitting a boundary S into two identical boundaries S + and S − chosen such that the cracks lie in the interface. The resulting integral equations are not hyper-singular and wave propagation within media that contain zero-thickness cracks can be rigorously solved.In order to validate the method, we deal with the scalar case, namely the scattering of antiplane SH waves by a 2-D crack. We compare results against a recently published analytic solution, obtaining an excellent agreement. This comparison gives us confidence to study cases where no analytic solutions exist. Some examples of incidence of P-or SV waves are depicted and the salient aspects of the method are also discussed.
Simple and accurate approaches to predict failure pressures in corroded pipelines are outlined in this work. It is shown that failure pressures for corroded pipelines can be predicted from the solution for undamaged pipelines using an equivalent wall thickness. Three different yield criteria (Tresca, ASSY (average shear stress yield), and von Mises) are reviewed in the light of reported experimental burst pressures. At first, failure pressures for cylindrical vessels with an infinitely long groove are studied by means of numerical simulations. The effect of groove size (depth and width) over the pipeline performance is quantified through a model. Finally, the scheme is extended to estimate the failure pressure of thin walled vessels with irregular finite defects.
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