Computing the elastic scattering from inclusions using the multiple multipoles method in three dimensions

Abstract

S U M M A R YThe paper extends the multiple multipole method (MMP) to compute the multiple scattering effects from inclusions in a homogeneous full-space from two dimensions to three dimensions. MMP methods are based on a scattered wavefield model where the unknown weighting coefficients are determined from satisfying the boundary conditions at discrete matching points. The wavefield model is most commonly constructed from multiple spherical wavefunctions with different expansion centres, but any solution or approximation to the wave equation may also be used. Such an expansion is non-orthogonal and requires use of overdetermined matrix systems, but this system can be constructed rapidly because simple point matching without costly integration is sufficient. Furthermore, irregularities of the boundary are resolved locally from the nearest centre of expansion, which decouples different parts of the boundary, and thus, promotes rapid conversion with small numbers of expansion functions. The resulting algorithm is a very general tool to solve relatively large and complex 3-D scattering problems.

Elastic wave field computation in multilayered nonplanar solid structures: A mesh-free semianalytical approach

“…one can clearly see that the shear stress is zero at the top and bottom boundaries and continuous across the interface ͑x 2 =12 mm͒. Similar plots ͑not shown͒ shows discontinuities in 11 at the interface but no such jump at the interface is observed in 22 plot. The displacement components u 1 and u 2 show continuity across the interface and nonzero values at the plate boundaries.…”

“…3-5 it is not clear if the continuity conditions ͑continuity of u 1 , u 2 , 22 , and 12 ͒ across the interface and the boundary condition ͑ 12 =0͒ at the top and bottom boundaries are satisfied. These conditions can be verified from the line plot in Figs.…”

“…In the bygone years several researchers have developed various techniques to solve the above-mentioned problems. Some of the most popular methods are finite element method ͑FEM͒, 11 boundary element method ͑BEM͒, 12,13 multiGaussian beam model ͑MGBM͒, [14][15][16][17][18] charge simulation technique, 19 multiple multipole program ͑MMP͒, [20][21][22] etc. Although the MGBM technique has some computational advantage over other techniques mentioned here, it also has a number of limitations similar to those of other paraxial models.…”

Ultrasonic field modeling in plates immersed in fluid

“…MMP was first proposed for electromagnetic wave problems by Hafner (1985). Later Imhof (1996Imhof ( , 2004 and Rokhlin (1990) extended the method to acoustics and elastic wave scattering problems in isotropic unbounded media. But with best of authors' knowledge the method has never been used for ultrasonic nondestructive problems where elastic waves are generated by finite size sources.…”

“…In MMP the error is minimized and coefficients of basis vectors are obtained by enforcing boundary and continuity conditions. Bessel functions and Hankel functions are used as basis functions in spherical wave function expansion (Imhof, 1996). If the center of expansion is located inside the domain of interest, Bessel solutions are used corresponding to standing waves.…”