Point-set manifold processing for computational mechanics

DANIEL MILLÁN; MARINO ARROYO

Vienna

Congreso; 6th European Congress on Computational methods in Applied Sciences and Engineering (ECCOMAS 2012); 2012

Vienna University of Technology

In some applications, one would like to perform calculations on smooth manifolds of dimension d embedded in high-dimensional spaces of dimension D, given by scattered points. Here, we consider Kirchhoff-Love thin shells (d=2, D=3), the description of the swimming stroke of micro-organisms from experimental videos (d=1, D=10000's), and the determination of collective variables for the con- formation dynamics of proteins (d=1...6, D=1000's).The point-set manifold is approximated by an over- lapping set of smooth parametric descriptions, which can be seen as meshfree macro-elements. This ap- proach avoids any global parameterization and hence is applicable to handle manifolds 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 manifold using smooth meshfree (here local maximum-entropy) approximants, and (3) patching together the local representations by means of a partition of unity.