  Exceptional points (EPs) are complex singularities of parametric linear
operators where two or more eigenvalues and eigenvectors coalesce. EPs are
attracting increasing interest in mechanical metamaterials due to their strong
potentials for wave filtering, cloaking, and sensing applications. This work
studies the band topology and scattering behaviors near EPs, using discrete
models of metamaterial (MM) systems. The questions of existence of EPs and
their physical manifestations will be addressed with particular focus on
symmetry considerations and scattering behavior. Discrete mass-spring models
with adjustable parameters are used here to elucidate the EP-related phenomena
in a fundamental form. The transfer and scattering matrices are analyzed to
provide practical insights on the restrictions associated with reciprocity and
fundamental symmetries. By including complex stiffness in frequency domain as a
representation of non-conservative mechanical loss or gain, the MM arrays can
be tuned to achieve bi-directional transparency or one-way reflection when
operating at the EPs. This analytical study will contribute to the
understandings of EPs in mechanical context and the design of micro-structured
media for novel applications.
