  The Fragile Points Method (FPM) is an elementarily simple Galerkin meshless
method, employing Point-based discontinuous trial and test functions only,
without using element-based trial and test functions. In this study, the
algorithmic formulations of FPM for linear elasticity are given in detail, by
exploring the concepts of point stiffness matrices and numerical flux
corrections. Advantages of FPM for simulating the deformations of complex
structures, and for simulating complex crack propagations and rupture
developments, are also thoroughly discussed. Numerical examples of deformation
and stress analyses of benchmark problems, as well as of realistic structures
with complex geometries, demonstrate the accuracy, efficiency and robustness of
the proposed FPM. Simulations of crack initiation and propagations are also
given in this study, demonstrating the advantages of the present FPM in
modeling complex rupture and fracture phenomena. The crack and rupture
propagation modeling in FPM is achieved without remeshing or augmenting the
trial functions as in standard, extended or generalized FEM. The simulation of
impact, penetration and other extreme problems by FPM will be discussed in our
future papers.
