Summary and conclusion
In this study, we introduce and discuss features and improvements of the stiffness matrix method developed in [1] that may enhance the applicability and flexibility of the method. Several numerical examples are provided to support the improvements and features. First, we introduce the acoustic layer stiffness matrices by means of three different formulations in terms of vertical displacement, velocity potential and nodal pressure, respectively. The latter two are easy for implementing air-gun source, while the first one is straightforward to apply a vertical disk load on seabed or within the water layer. Secondly, the soil stiffness matrices for the vertically transverse isotropic (VTI) layer are derived for both P-SV and SH wave modes. The structure of the VTI soil layer stiffness matrices is constructed in the same fashion as for the isotropic soil layer and such that the stiffness matrix for the isotropic case can be recovered by setting the anisotropy factors equal to 1. Thirdly, in order to simulate wave motions subjected to injected (air) volume or dislocation/slip at the interface of two layers, we formulate a technique where the displacement discontinuity can be implemented. Finally, by means of the continuum stiffness matrices we derive the key parameter of PML (thickness, hPML) that can be used in discrete numerical approaches (e.g. FEM, FDM, TLM). We also discuss how the continuum PML can be useful for the contour integration over the complex wavenumber domain, eliminating the need for the branch-cut integral, in the context of the wavenumber integration from the wavenumber to spatial domain.
