Nonlinear manifold learning for meshfree finite deformation thin-shell analysis

SUMMARY Calculations on general point‐set surfaces are attractive because of 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 f...

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Published in:	International journal for numerical methods in engineering Vol. 93; no. 7; pp. 685 - 713
Main Authors:	Millán, Daniel, Rosolen, Adrian, Arroyo, Marino
Format:	Journal Article Publication
Language:	English
Published:	Chichester, UK John Wiley & Sons, Ltd 17.02.2013 Wiley Subscription Services, Inc John Wiley & Sons
Subjects:	74 Mechanics of deformable solids 74S Numerical methods Anàlisi numèrica Classificació AMS Manifolds Matemàtiques i estadística Mathematical analysis Mathematical models maximum-entropy approximants meshfree methods Meshless methods Mètodes numèrics nonlinear dimensionality reduction Nonlinearity Numerical analysis Numerical methods and algorithms Parametrization partition of unity point-set surfaces Resistència de materials shells Topology Àrees temàtiques de la UPC
ISSN:	0029-5981, 1097-0207
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Summary:	SUMMARY Calculations on general point‐set surfaces are attractive because of 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 subregions 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 macro‐element. We exemplify the generality, flexibility, and accuracy of the proposed approach by numerically approximating the geometrically nonlinear Kirchhoff–Love theory of thin‐shells. We analyze standard benchmark tests as well as point‐set surfaces of complex geometry and topology. Copyright © 2012 John Wiley & Sons, Ltd.
Bibliography:	istex:3F0994DFDA90659E01859465EC041F76569A3F0E ArticleID:NME4403 ark:/67375/WNG-KM9QQ4FC-T ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23 ObjectType-Feature-2
ISSN:	0029-5981 1097-0207
DOI:	10.1002/nme.4403