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

L. Kucherov, E. B. Tadmor, R. E. Miller

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2807-2847
Number of pages41
JournalInternational Journal for Numerical Methods in Engineering
Issue number13
StatePublished - Mar 26 2007


  • Computational mechanics
  • Finite deformation
  • Meshless methods
  • Multiscale modelling
  • Nodal integration
  • Stability


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