Meshfree Code.

I think he is talking about that in the same way as in finite elements, you can do a double loop to compute the meshfree basis functions (and therefore their derivatives). The first one over the elements and the second one over the Gauss points such that you cover all the Gauss points in the domain. In the double loop, you evaluate the meshfree basis functions (and their derivatives) at current Gauss point and you have to find the connectivity for that evaluation. Once you have the connectivity (usually called the neighbor list), you have to do the assembly of the stiffness matrix for the current Gauss point evaluation using the neighbor list. Finding the neighbor list might be the more expensive part of the meshfree process. To speed up the computation, you may use a kd-Tree algorithm to find the neighbor list.

Instead of computing the basis functions (and their derivatives) inside the double loop, you can construct a matrix at the beginning with the evaluation of the basis function derivatives at Gauss points of the entire domain (using the kd-tree algorithm to find the neighbors, i.e, the non-zero entries, will speed up this procedure). Rows being the basis function derivatives (let's think in 1d so we only have one derivative per basis function) and columns the Gauss points of the entire domain. This way there will be a lot of zero entries in the matrix (that is why I think he mentioned the sparse matrix). For example in the first row, there will be non-zero entries (the neighbor list) only where the basis function of the first row has non-vanishing values. If you construct such a matrix, then you can assembly the global stiffness matrix with tensor products of rows of the basis function derivative matrix, which is clearly faster than the double loop method. You can also construct a matrix for basis functions so that the global force vector can be assembled in a similar fashion (don't need to do the tensor product in this case, though).