Research Article

Umbrella spherical integration: a stable meshless method for non-linear solids

  1. L. Kucherov1,
  2. E. B. Tadmor1,*,
  3. R. E. Miller2

Article first published online: 5 SEP 2006

DOI: 10.1002/nme.1871

Issue

International Journal for Numerical Methods in Engineering

International Journal for Numerical Methods in Engineering

Volume 69, Issue 13, pages 2807–2847, 26 March 2007

Additional Information

How to Cite

Kucherov, L., Tadmor, E. B. and Miller, R. E. (2007), Umbrella spherical integration: a stable meshless method for non-linear solids. International Journal for Numerical Methods in Engineering, 69: 2807–2847. doi: 10.1002/nme.1871

Author Information

  1. 1

    Department of Mechanical Engineering, Technion, Israel Institute of Technology, 32000 Haifa, Israel

  2. 2

    Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Ont., Canada K1S 5B6

*Department of Mechanical Engineering, Technion, Israel Institute of Technology, 32000 Haifa, Israel

Publication History

  1. Issue published online: 2 MAR 2007
  2. Article first published online: 5 SEP 2006
  3. Manuscript Accepted: 13 JUL 2006
  4. Manuscript Revised: 27 JUN 2006
  5. Manuscript Received: 23 APR 2006

Funded by

  • Israel Science Foundation (ISF). Grant Number: 208/00

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Keywords:

  • meshless methods;
  • nodal integration;
  • stability;
  • finite deformation;
  • computational mechanics;
  • multiscale modelling

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.

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