The α-shape Based Natural Element Method in Solid and Fluid Mechanics

David González (1, 2) , Icíar Alfaro (1, 2) , Elías Cueto (1, 2) , Manuel Doblaré (1, 2) , Francisco Chinesta (1, 2)

1 LMSP - Laboratoire de Mécanique des Systèmes et des Procédés
2 I3A - Aragón Institute of Engineering Research [Zaragoza]

The Natural Element method (also known as Natural Neighbour Galerkin method) is a Galerkin method based on the use of Natural Neighbour interpolation to construct the trial and test functions. Unlike many other meshless methods, it has some important characteristics, such as interpolant shape functions, easy imposition of essential boundary conditions and linear precision along convex boundaries. The natural neighbour interpolation scheme is based on the construction of a Delaunay triangulation of the given set of points. This geometrical link provides the NEM some other interesting properties. One of them is the ability of constructing models without any explicit (CAD) boundary description. Instead, by invoking the concept of α-shape of the cloud of points, the method is able to accurately extract the geometry described by the nodes as it evolves, thus avoiding complex geometrical checks in the formation of holes or waves in the domain, without any loss in mass conservation requirements. It has been also proved how the use of α-shapes ensures the strictly interpolant character of the shape functions along any type of boundary. In this work we review the main characteristics of the method in its application to Solid and Fluid Mechanics, including the study of mixed natural neighbour approximation, simulation of nearly incompressible media and some industrial applications.

Mots clés: Fluid Mechanics; Delaunay Triangulation; Voronoi Cell; Meshless Method; Strain Rate Tensor

Domaines: Mécanique [physics.med-ph]

David González, Icíar Alfaro, Elías Cueto, Manuel Doblaré, Francisco Chinesta. The α-shape Based Natural Element Method in Solid and Fluid Mechanics. Michael Griebel, Marc A. Schweitzer. Meshfree Methods for Partial Differential Equations II, 43, Springer, pp.55-69, 2005, Lecture Notes in Computational Science and Engineering, 978-3-540-23026-7. ⟨10.1007/3-540-27099-X_4⟩. ⟨hal-00021026⟩