Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part II: a 3D model

Abstract

SUMMARYThe displacement and stress Green's functions for a 3D triclinic half-space with embedded harmonic point load is considered. The resulting displacement and stress fields are expressed in terms of triple Fourier integrals. The first integral was evaluated using contour integration and the 3D Green's functions were obtained as a superposition of 2D results over the azimuthal angle. The resulting algorithm developed for evaluation of the Green's functions avoids repeated calculations of the same quantities and it utilizes the vectorized manipulation within MATLAB environment. The algorithm places no restriction on material properties, frequency and location of source and observation points. Extensive testing of the numerical results was performed for both displacement and stress. The tests confirm the accuracy of the numerical results.

Scattering of plane harmonic P, SV and Rayleigh waves by a completely embedded corrugated elastic inclusion

“…The advantage of using the full-space Green's functions lies in ease of their numerical evaluation when compared to the half-space Green's functions [23][24][25][26][27].…”

Simulation Methods for Guided Wave-Based Structural Health Monitoring: A Review

“…The problem that remains is to calculate the integral without knowing the poles' location. Commonly this computation is performed numerically with the help of adaptive quadrature formulae [31,32,149,218,274]. This method is closely related to the use of complex frequencies to obtain the response of the plate in the time domain using the exponential window method.…”

“…17is less problematic once the integration with respect to ξ has been completed. The most popular techniques used for solving this kind of integral are asymptotic methods like the stationary phase method [30,255] and the usual numerical quadrature formulas, as proposed in [32]. The first method is more convenient for cases where the results are needed at an observation pointx that is further away from the pointsx ′ where the external forces are applied and only the propagating modes have to be taken into account.…”

Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition

“…Spies applied Fourier transforms in space and time, but no methods to calculate the inverse transforms were proposed. More recently, in 2007, Chen and Dravinski [4,5] considered time-harmonic Green's functions for a triclinic half-space, using a double Fourier transform. The inversion was made by employing contour integration and Gauss-Legendre quadrature.…”