On the Efficient Implementation of the Integral Equation Method in Elastodynamics

Abstract

The authors deal with low-cost computer implementation of the integral equation method in application to wave excitation, propagation and diffraction phenomena in solids. Typically these are frequency domain problems of seismic and physical acoustics and of nondestructive testing. Fast codes are obtained basing on analytical extracting and taking account of singular and slowly converging components of the solution beforehand. The present paper gives ideas of several such approaches, which have already proved their efficiency in practical applications. More details including numerical examples and discussion of similar methods known from the literature can be found in the surveyed papers.1–12

Interface damage modeled by spring boundary conditions for in-plane elastic waves

“…of the derivation of Green's matrices for the 3D case has been given in [17,16]. In the problem under consideration only the 2D Green's matrix is used…”

“…At first a single interface crack between two half-spaces is considered for normal incidence of a plane longitudinal or transverse wave. The solution is obtained using a type of analytical boundary integral equation method [15,16]. Then the reflection and transmission coefficients for normal incidence for a random and a periodic distribution of equally sized cracks at the interface between two half-spaces are calculated.…”

Lamb Wave Scattering Analysis for Interface Damage Detection between a Surface-Mounted Block and Elastic Plate

“…Here, and are the Fourier transform of the Green’s matrix and the arbitrary surface load function (more details can be found in [ 31 ]).…”

“…To extract wave-fields related to scattered Lamb waves with a specific wavenumber corresponding to a specific Lamb wave, relations similar to Equation ( 13 ) are employed (for more details, see [ 17 , 31 ]). It should be mentioned that the equality is valid for energy distribution coefficients.…”

“…The standard FEM was applied for the simulation of the transient problems, whereas the SAHA, which was in a good agreement with the FEM, was used to compute eigenfrequencies and to analyze mode conversion/reflection by obstacles. The SAHA is based on the boundary integral equation method [ 31 ] and the spectral element method (SEM) [ 32 ]; these two methods meet in the contact area, where the traction vector is unknown. One can see that the SAHA is based on an idea similar to the so-called global–local approach, where a semi-analytical technique is employed to discretize a semi-infinite or elongated area and the FEM discretizes the “local” portion, which contains scatterers of various kinds, including stiffeners [ 33 , 34 , 35 ].…”