On the minimization of wave reflection at the interface of a discrete system and a dispersively similar continuum

The minimization of wave reflection is considered in this paper at the interface between a regular lattice and a corresponding continuum. This problem is of importance for the multi-scale and hybrid numerical modelling of materials. It is known that the classical continuum is capable of non-reflecting the long waves falling on the interface from the lattice provided that their wavelength is much larger than the period of the lattice. However, the shorter the incident wave, the higher the reflection coefficient. In this paper a new idea is formulated that a gradient continuum can be used instead of the classical one to minimize the wave reflection at a wide frequency range. It is shown that in the one-dimensional case a second-gradient continuum can serve as a perfect non-reflecting boundary that provides no reflection at the complete frequency band in which waves can propagate in the lattice. It is remarkable that the dispersive properties of such a continuum differ from those of the lattice even at low frequencies. This raises a question as to whether the non-reflective coupling is possible at the interface of a lattice and a gradient continuum whose dispersive properties reproduce those of the lattice at a wide frequency range. It is shown in this paper that this is possible provided that a special interface cell is introduced whose elastic and inertial properties are dissimilar to those of the inner cells of the lattice. It is noted that such a dissimilar cell helps minimize the wave reflection even from the classical continuum, which is another new finding of this paper.