NUMERICAL INTEGRATION OF GREEN’S FUNCTIONS FOR LAYERED MEDIA: A CASE STUDY

Resumo

Classical Green’s functions for transversely isotropic media are typically expressed in terms
of improper integrals containing a number of singularities and a decaying tail that oscillates indefinitely.
Currently, there are no known numerical methods capable of dealing precisely with both characteristics
of these integrands simultaneously. In this work, Green’s functions for layered media are presented in
terms of an exact stiffness matrix scheme, in which a stiffness matrix for the medium is assembled from
the stiffness matrices of each layer. The integrand in such cases is characterized by an infinite number

of singularities, corresponding to the propagation and reflection waves in the medium. The oscillatory-
decaying tail presents more than one frequency of oscillation, which makes them difficult to integrate

by classical extrapolation methods. This work presents strategies with which to evaluate such integrals
numerically. We have shown that the singularities can be located within the integration interval at points
that correspond to physical wavenumbers of each layer, which are then integrated through a appropriate
contour deformation paths. For the oscillatory-decaying part, we use a combination of strategies. The
first is to use Fast Fourier Transforms to break down the oscillation into its component frequencies.
The fundamental frequency is used to yield a sequence of partial sums, from which the integral can be
obtained by extrapolation though the -algorithm. As a case study, the scheme is used to evaluate the
displacement of layered, transversely isotropic soil medium under time-harmonic external excitations.
The results are compared with classical adaptive quadrature integration schemes.