  In this work, the fast-convolving reproducing kernel particle method
(FC-RKPM) is introduced. This method is hundreds to millions of times faster
than the traditional RKPM for 3D meshfree simulations. In this approach, the
meshfree discretizations with RK approximation are expressed in terms of
convolution sums. Fast Fourier transform (FFT) is then used to efficiently
compute the convolutions. Certain modifications to the domain and shape
functions are considered to maintain generality for complex geometries and
arbitrary boundary conditions. The new method does not need to identify, store,
and loop over the neighbors which is one of the bottleneck of the traditional
meshfree methods. As a result, the run-times and memory allocations are
independent of the number of neighbors and the shape function support size. As
a model problem, the method is laid out for a Galerkin weak form of the Poisson
problem with the RK approximation, and is verified in 1D, 2D, and 3D. Tables
with run-times and allocated memory are presented to compare the performance of
FC-RKPM with the traditional method in 3D. The performance is studied for
various node numbers, support size, and approximation degree. All the
implementation details and the roadmap for software development are also
provided. Application of the new method to nonlinear and explicit problems are
briefly discussed as well.
