Fast variable density 3-D node generation

Abstract

Mesh-free solvers for partial differential equations perform best on scattered quasi-uniform nodes. Computational efficiency can be improved by using nodes with greater spacing in regions of less activity. We present an advancing front type method to generate variable density nodes in 2-D and 3-D with clear generalization to higher dimensions. The exhibited cost of generating a node set of size N in 2-D and 3-D with the present method is O(N ).

An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

“…It now remains to insert the estimate (28) into (27) and then also combine this with consistency estimates (22), (24) to arrive at the final error estimate:…”

“…The interpolation and evaluation points are -conceptually speaking -constructed in the same way as in the two-dimensional cases. We start by using a surface point-cloud of the diaphragm [23] in place of the boundary evaluation points, which are placed in a three-dimensional box that contains interpolation points computed using an algorithm from [24]. Then we enforce q evaluation points (again computed by an algorithm from [24]) around each interpolation point and after that remove those evaluation points which are placed outside of the diaphragm.…”

“…We start by using a surface point-cloud of the diaphragm [23] in place of the boundary evaluation points, which are placed in a three-dimensional box that contains interpolation points computed using an algorithm from [24]. Then we enforce q evaluation points (again computed by an algorithm from [24]) around each interpolation point and after that remove those evaluation points which are placed outside of the diaphragm. At last we remove the interpolation points according to the criterion given in Section 4.…”