Nonlinear manifold description from scattered points for meshfree thin shell analysis

DANIEL MILLÁN

Barcelona

Seminario; Cafes de CIMNE; 2010

International Center for Numerical Methods in Engineering

In spite of techniques developed over the last decade in computer graphics to represent and to manipulate a surface from a cloud of points, meshfree thin-shells analysis is still a challenging task. The fundamental difficulty is that, unlike surface meshes, a cloud of points does not describe well the geometry, nor provides the convenient local parameter spaces given by the reference elements. For this reason, unless a global parameterization of the surface is possible and the meshfree shape functions can be defined on this global 2D parametric domain (this is not the case for a sphere or other surfaces of complex topology), or unless a support mesh is used, there are no methods available. We present a method to process embedded smooth manifolds using sets of points alone. This method avoids any global parameterization and hence is applicable to surfaces of any genus. It combines three ingredients: (1) the automatic detection of the local geometric structure of the manifold 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. 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 approximants, this fourth-order partial differential equation is treated directly. We show the good performance of the method on the basis of the classical obstacle course. Additional calculations exemplify the flexibility of the proposed approach in treating surfaces of complex topology and geometry.