  In this paper, a new method, named the Fragile Points Method (FPM), is
developed for computer modeling in engineering and sciences. In the FPM,
simple, local, polynomial, discontinuous and Point-based trial and test
functions are proposed based on randomly scattered points in the problem
domain. The local discontinuous polynomial trial and test functions are
postulated by using the Generalized Finite Difference method. These functions
are only piece-wise continuous over the global domain. By implementing the
Point-based trial and test functions into the Galerkin weak form, we define the
concept of Point Stiffnesses as the contribution of each Point in the problem
domain to the global stiffness matrix. However, due to the discontinuity of
trial and test functions in the domain, directly using the Galerkin weak form
leads to inconsistency. To resolve this, Numerical Flux Corrections, which are
frequently used in Discontinuous Galerkin methods are further employed in the
FPM. The resulting global stiffness matrix is symmetric and sparse, which is
advantageous for large-scale engineering computations. Several numerical
examples of 1D and 2D Poisson equations are given in this paper to demonstrate
the high accuracy, robustness and convergence of the FPM. Because of the
locality and discontinuity of the Point-based trial and test functions, this
method can be easily extended to model extreme problems in mechanics, such as
fragility, rupture, fracture, damage, and fragmentation. These extreme problems
will be discussed in our future studies.
