Natural Neighbor Interpolation

Consider an interpolation scheme for a function : , in the form:

where ( ) are the function values at the natural neighbors, and are the weights (shape functions in FE) associated with each node. In the context of natural neighbor interpolation, the weights are taken as the n-n coordinates of the point in the plane. Natural-neighbor coordinates were introduced by [1,2] and may be defined in any number of dimensions. To illustrate, we use an example indicated in [9]. In Figure 1a, five nodes and the sides of the Voronoi cells are shown. If a point is added in cell 3, then a new Voronoi cell can be placed around it (see Figure 1b). The n-n coordinates of with respect to a neighbor is defined as the ratio of the area of their overlapping Voronoi cells to the total area of the Voronoi cell about . For instance, in Figure 1b, is given by

(3) |

The five overlapping regions in Figure 1b are known as

(4) |

Now, if were to coincide with any node, say node for instance, then it is readily seen that and . Therefore the n-n coordinates satisfy the Kronecker-delta property:

(5) |

and hence Eq. (2) is an interpolation: .

Since
is only non-zero in the union
of the circles that pass through the vertices of the
Delaunay triangles about node , the n-n interpolation
is a local (interpolating) scheme. In addition, the
are continuosly differentiable
() everywhere except at the nodes,
, where all the derivatives are
discontinuous [1]. The proof of
the above in one-dimension is trivial. In 1D,
it is readily derived that the are precisely
linear FE shape functions:
,
, where
. Hence
FEM is realized (by the n-n interpolation) in 1D,
where it is well-known that the displacement derivative
is discontinuous at nodes (**element
boundaries** in 1D).

The natural neighbor coordinates also satisfies an important property known as the Local Coordinate Property (LCP) -- geometrical coordinates are interpolated exactly [1]:

(6) |

Since the interpolant satisfies linear consistency, in the context of elastostatics, the approximation (trial function) can exactly represent rigid-body motion and linear displacement fields.