Point-set manifold processing for computational mechanics

DANIEL MILLÁN; MARINO ARROYO

Sao Paulo

Congreso; 10th World Congress on Computational Mechanics (WCCM 2012); 2012

International association for computational mechanics

In some applications, one would like to perform calculations on smooth manifolds of dimen- sion 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=10000s), and the determination of collective variables for the conformation dynamics of proteins (d=1...6, D=1000s).The point-set manifold is approximated by an overlapping set of smooth parametric descriptions, which can be seen as meshfree macro-elements. This approach 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.