Thin shell analysis from scattered points with maximum‐entropy approximants

Abstract

SUMMARYWe present a method to process embedded smooth manifolds using sets of points alone. This method avoids any global parameterization and hence is applicable to surfaces 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 surface using smooth meshfree (here maximum-entropy) approximants; and (3) patching together the local representations by means of a partition of unity. Mesh-based methods can deal with surfaces of complex topology, since they rely on the element-level parameterizations, but cannot handle high-dimensional manifolds, whereas previous meshfree methods for thin shells consider a global parametric domain, which seriously limits the kinds of surfaces that can be treated. We present the implementation of the method in the context of Kirchhoff-Love shells, but it is applicable to other calculations on manifolds in any dimension. With the smooth approximants, this fourth-order partial differential equation is treated directly. We show the good performance of the method on the basis of the classical obstacle course. Additional calculations exemplify the flexibility of the proposed approach in treating surfaces of complex topology and geometry.

Phase-field modeling of fracture in linear thin shells

“…We consider here LME basis functions. See [39,40,38] for the LME formulation, properties, and the evaluation of the basis functions and their derivatives. Then, let p a (ξ) denote the LME approximants associated to the point-set Ξ κ on a domain A κ ⊂ R 2 , a subset of the convex hull of the reduced node set conv Ξ κ .…”

“…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…”

“…Note that M a , B a ∈ R 3×3 , see [38] for a detailed description. The force contribution of the a-th node is…”

“…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.…”