Finite Deformation thin Shell Analysis from Scattered Points with Maximum-Entropy Approximants

Abstract

We present a method to process embedded smooth surfaces using sets of points alone. The goal is to perform calculations on general point-set surfaces avoiding any global parameterization. We achieve this aim by approximating the point-set surface as an overlapping set of smooth local parametric descriptions. We combine three ingredients: (1) the automatic detection of the nonlinear local geometric structure of the surface by statistical learning methods, (2) the local parameterization of the surface using smooth meshfree (here maximum-entropy) approximants, and (3) patching together the local representations by means of a partition of unity.
Mesh-based methods can deal with surfaces of complex topology, since they rely on the element-level parameterizations, but cannot handle high-dimensional manifolds, whereas previous meshfree methods for thin shells consider a global parametric domain, which seriously limits the kinds of surfaces that can be treated.
We present the implementation of the method in the context of Kirchhoff-Love shells, but it is applicable to other calculations on manifolds in any dimension. With the smooth maximum-entropy approximants, this fourth-order partial differential equation is treated directly. We exemplify the flexibility of the proposed approach dealing with large deformations and surfaces of complex geometry.