  Nonlocality brings many challenges to the implementation of finite element
methods (FEM) for nonlocal problems, such as large number of queries and invoke
operations on the meshes. Besides, the interactions are usually limited to
Euclidean balls, so direct numerical integrals often introduce numerical
errors. The issues of interactions between the ball and finite elements have to
be carefully dealt with, such as using ball approximation strategies. In this
paper, an efficient representation and construction methods for approximate
balls are presented based on combinatorial map, and an efficient parallel
algorithm is also designed for assembly of nonlocal linear systems.
Specifically, a new ball approximation method based on Monte Carlo integrals,
i.e., the fullcaps method, is also proposed to compute numerical integrals over
the intersection region of an element with the ball.
