  We present OBMeshfree, an Optimization-Based Meshfree solver for compactly
supported nonlocal integro-differential equations (IDEs) that can describe
material heterogeneity and brittle fractures. OBMeshfree is developed based on
a quadrature rule calculated via an equality constrained least square problem
to reproduce exact integrals for polynomials. As such, a meshfree
discretization method is obtained, whose solution possesses the asymptotically
compatible convergence to the corresponding local solution. Moreover, when
fracture occurs, this meshfree formulation automatically provides a sharp
representation of the fracture surface by breaking bonds, avoiding the loss of
mass. As numerical examples, we consider the problem of modeling both
homogeneous and heterogeneous materials with nonlocal diffusion and
peridynamics models. Convergences to the analytical nonlocal solution and to
the local theory are demonstrated. Finally, we verify the applicability of the
approach to realistic problems by reproducing high-velocity impact results from
the Kalthoff-Winkler experiments. Discussions on possible immediate extensions
of the code to other nonlocal diffusion and peridynamics problems are provided.
OBMeshfree is freely available on GitHub.
