Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis

Abstract

SUMMARYCalculations 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.

Phase-field modeling of fracture in linear thin shells

“…For more details, refer to [38,21] . With the Kirchhoff-Love and the small deformation hypothesis, the only remaining non-zero components of the Green-Lagrange strain tensor are…”

“…E is the Young's modulus, and ν is the Poisson's ratio [38,21]. The external potential energy expressed in the reference middle surface states as…”

“…Furthermore, many of the approaches are applied to simple geometries such as plates, or spherical and cylindrical geometries [18,19,12,20]. Towards a general, flexible and robust methodology to deal with fracture in Kirchhoff-Love shells, we propose here treating fracture with a phasefield model and discretizing the coupled thin-shell/phase-field equations with a recently proposed meshfree method for partial differential equations on manifolds of complex geometry and topology [21,22].…”

“…The higher order continuity of the meshfree basis functions has also been exploited for this purpose [14,15], but since meshfree basis functions are defined in physical space, these methods were applied to simple geometries with a single parametric patch. Recently, nonlinear manifold learning techniques have been exploited to parametrize 2D sub-domains of a point-set surface, which are then used as parametric patches and glued together with a partition of unity [38,21]. Here, we combine this methodology with local maximum-entropy (LME) meshfree approximants [39,40,5] because of their smoothness, robustness, and relative ease of quadrature compared with other meshfree approximants.…”

“…Section 2 describes the representation of general surfaces represented by a set of scattered points [21]. In Section 3, we review the Kirchhoff-Love theory of thin shells.…”