Nonlinear manifold learning for meshfree finite deformation thin shell analysis

Author (s): Millán, D; Rosolen, A and Arroyo, M.
Journal: International Journal for Numerical Methods in Engineering

Volume: 93, Issue 7
Pages: 685 – 713
Date: 2013

Calculations on general point-set surfaces are attractive due to their flexibility and simplicity in the preprocessing, but present important challenges. The absence of a mesh makes it nontrivial to decide if two neighboring points in the three-dimensional embedding are nearby or rather far apart on the manifold. Furthermore, the topology of surfaces is generally not that of an open two-dimensional set, ruling out global parametrizations. We propose a general and simple numerical method analogous to the mathematical theory of manifolds, in which the point-set surface is described by a set of overlapping charts forming a complete atlas. We proceed in four steps: (1) partitioning of the node set into sub-regions of trivial topology, (2) automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods, (3) parametrization of the surface using smooth meshfree (here maximum-entropy) approximants, and (4) gluing together the patch representations by means of a partition of unity. Each patch may be viewed as a meshfree macroelement.
We exemplify the generality, flexibility, and accuracy of the proposed approach by numerically approximating the geometrically nonlinear Kirchho?-Love theory of thin shells. We analyze standard benchmark tests as well as point-set surfaces of complex geometry and topology.



@article {NME:NME4403,
author = {Millán, Daniel and Rosolen, Adrian and Arroyo, Marino},
title = {Nonlinear manifold learning for meshfree finite deformation thin-shell analysis},
journal = {International Journal for Numerical Methods in Engineering},
volume = {93},
number = {7},
publisher = {John Wiley & Sons, Ltd},
issn = {1097-0207},
url = {},
doi = {10.1002/nme.4403},
pages = {685--713},
keywords = {shells, meshfree methods, partition of unity, point-set surfaces, maximum-entropy approximants, nonlinear dimensionality reduction},
year = {2013},