SUMMARYContinued fraction absorbing boundary conditions (CFABCs) are highly effective boundary conditions for modelling wave absorption into unbounded domains. They are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries. In this paper, CFABCs are extended to the more general case of polygonal computational domains. The key to the current development is the surprising link found between the CFABCs and the complex co-ordinate stretching of perfectly matched layers (PMLs). This link facilitates the extension of CFABCs to oblique corners and, thus, to polygonal domains. It is shown that the proposed CFABCs are easy to implement, expected to perform better than PMLs, and are effective for general polygonal computational domains. In addition to the derivation of CFABCs, a novel explicit time-stepping scheme is developed for efficient numerical implementation. Numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain. Although this paper focuses on wave propagation, its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients.

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SUMMARYThis paper endows the recently-proposed granular element method (GEM) with the ability to perform 3D discrete element calculations. By using non-uniform rational B-Splines to accurately represent complex grain geometries, we proposed an alternative approach to clustering-based and polyhedra-based discrete element methods whereby the need for complicated and ad hoc approaches to construct 3D grain geometries is entirely bypassed. We demonstrate the ability of GEM in capturing arbitrary-shaped 3D grains with great ease, flexibility, and without excessive geometric information. Furthermore, the applicability of GEM is enhanced by its tight integration with existing non-uniform rational B-Splines modeling tools and ability to provide a seamless transition from binary images of real grain shapes (e.g., from 3D X-ray CT) to modeling and discrete mechanics computations.

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