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Umbrella spherical integration: a stable meshless method for non‐linear solids
Author(s) -
Kucherov L.,
Tadmor E. B.,
Miller R. E.
Publication year - 2006
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1871
Subject(s) - meshfree methods , moving least squares , regularized meshless method , mathematics , bending , boundary value problem , grid , convergence (economics) , mathematical analysis , singular boundary method , finite element method , geometry , structural engineering , engineering , boundary element method , economics , economic growth
Abstract A stable meshless method for studying the finite deformation of non‐linear three‐dimensional (3D) solids is presented. The method is based on a variational framework with the necessary integrals evaluated through nodal integration. The method is truly meshless, requiring no 3D meshing or tessellation of any form. A local least‐squares approximation about each node is used to obtain necessary deformation gradients. The use of a local field approximation makes automatic grid refinement and the application of boundary conditions straightforward. Stabilization is achieved through the use of special ‘umbrella’ shape functions that have discontinuous derivatives at the nodes. Novel efficient algorithms for constructing the nodal stars and computing the nodal volumes are presented. The method is applied to four test problems: uniaxial tension, simple shear and bending of a bar, and cylindrical indentation. Convergence studies at infinitesimal strain show that the method is well‐behaved and converges with the number of nodes for both uniform and non‐uniform grids. Typical of meshless methods employing nodal integration, the total energy can be underestimated due to the approximate integration. At finite deformation the method reproduces known exact solutions. The bending example demonstrates an interesting example of torsional buckling resulting from the bending. Copyright © 2006 John Wiley & Sons, Ltd.