A CG-FFT approach to the solution of a stress-velocity formulation of three-dimensional elastic scattering problems

Inverse scattering analysis for an elastic half space based on a fast volume integral equation method

“…The back ground Lamé constants and mass density for the model are λ 0 =4.0 GPa, μ 0 =2.0 GPa and ρ 0 =2.0 g/cm 3 , respectively. Namely, the P wave velocity for the back ground structure of the wave field is 2 km/s and the S wave velocity for that is 1 km/s.…”

“…As a result, a number of examples of applications of the volume integral equation are increasing recently in spite of its deficiency, that is the requirement of a huge scale and dense matrix for numerical analyses. Several methods for the volume integral equation presented recently are aimed to resolve the deficiency of the volume integral equation (for example, [2][3][4]). Even in this situation, application of the volume integral equation to inverse scattering analysis of an elastic half space is still a task for the future.…”

A Volume Integral Equation Method for the Direct/Inverse Problem in Elastic Wave Scattering Phenomena

“…For example, Hudson and Heritage (1981) used the Born approximation of the solution of the volume integral equation obtained from the background structure of the wave field for the seismic scattering problem. Recently, Zaeytijd, Bogaert, and Franchois (2008) presented the MLFMA-FFT method for analyzing electro-magnetic waves, and Yang, Abubaker, van den Berg et al (2008) used a CG-FFT approach to solve elastic scattering problems. These methods were used to establish a fast algorithm to solve the volume integral equation via a Fast Fourier transform, which is used for efficient calculation of the convolution integral.…”

“…The starting point of the analysis is the volume integral equation in the wavenumber domain, which includes the operators of the Fourier integral and its inverse transforms. This starting point yields a different method from previous approaches (for example, Yang et al, 2008). By replacing these operators with discrete Fourier transforms, the volume integral equation in the wavenumber domain can be treated as a Fredholm equation of the 2nd kind with a nonHermitian operator on a finite dimensional vector space, which is to be solved by the Krylov subspace iterative scheme .…”

Volume integral equation method for the analysis of scattered waves in an elastic half space

“…Nevertheless, a number of applications of the volume integral equation to scattering problems are increasing. The application fields are extended to elastic wave as well as electro-magnetic wave propagations (for example [4,5]), in which methods to overcome the deficiency of the volume integral equation are formulated.…”