Applications in Computational Mechanics

In the preceding sections, a concise description of natural neighbor interpolation and its implementation in the context of a Galerkin procedure to solve PDEs was delineated. Having at least given a flavor (I hope) of the interpolation scheme, one can be undoubtedly ambitious in suggesting its possible applications within the field of computational mechanics:

1. Patch test: The first step in the testing of a numerical method for the elliptic operator. The patch test is a sufficient condition to prove the convergence of a numerical method [22].
2. Smooth boundary-value problems in elastostatics (2D) -- plate with a circular/elliptical hole, beam in bending, etc.
3. Benchmark crack problems in 2D: The edge-crack, center-crack, and shear crack problems, where symmetric boundary conditions can be imposed. The case of inclined cracks (no planes of symmetry) might pose a problem, since the whole domain has to be modeled -- non-convex domain!
4. The higher order smoothness (or regularity) of the NEM interpolant in $\Omega \backslash \mathbf{x}_I$ allows one to consider higher order PDEs [10]. The FE interpolant, $u_{FE}^h(\mathbf{x}) \in C^0$, since $u_{FE}^h(\mathbf{x})$ is continuous but its gradients are discontinuous across element boundaries (edges of the Delaunay triangle). The approach used by [23] to successfully avoid locking in beams and plates by using cardinal splines on regular grids can be explored using NEM in the context of irregular nodal arrangements. The jumps in the displacement gradients at the nodes do not pose a problem since the numerical integration is carried out inside the triangle.
5. Finite deformation: Mesh distortion and mesh entanglement which are the bane of FE are not issues in NEM. The density of the nodes dictates the quality of the n-n interpolation and not the Delaunay triangles.
6. Fluid-structure interaction: See [10].
7. Moving interfaces $\ldots$