A Natural Neighbours Method Based On Fraeijs De Veubeke Variational Principle

A Natural nEighbours Method (NEM) based on the FRAEIJS de VEUBEKE (FdV) variational principle is developed in the domain of 2D infinitesimal transformations.This method is firstly applied to linear elastic problems and then is extended to materially nonlinear problems and problems of linear elastic fracture mechanics (LEFM).In all these developments, thanks to the FdV variational principle, the displacement field, the stress field, the strain field and the support reaction field are discretized independently.In the spirit of the NEM, nodes are distributed in the domain and on its contour and the corresponding Voronoi cells are constructed.In linear elastic problems the following discretization hypotheses are used:The assumed displacements are interpolated between the nodes with Laplace functions.The assumed support reactions are constant over each edge of Voronoi cells on which displacements are imposed.The assumed stresses are constant over each Voronoi cell.The assumed strains are constant over each Voronoi cell.The degrees of freedom linked with the assumed stresses and strains can be eliminated at the level of the Voronoi cells so that the final equation system only involves the nodal displacements and the assumed support reactions.The support reactions can be further eliminated from the equation system if the imposed support conditions only involve constant imposed displacements (in particular displacements imposed to zero) on a part of the solid contour, finally leading to a system of equations of the same size as in a classical displacement-based method.For the extension to materially nonlinear problems, similar hypotheses are used. In particular, the velocities are interpolated by Laplace functions and the strain rates are assumed to be constant in each Voronoi cell.The final equations system only involves the nodal velocities. It can be solved step by step by time integration and Newton-Raphson iterations at the level of the different time steps.In the extension of this method for LEFM, a node is located on each crack tip. In the Voronoi cells containing the crack tip, the stress and the strain discretization includes not only a constant term but also additional terms corresponding to the solutions of LEFM for modes 1 and 2.In this approach, the stress intensity coefficients are obtained as primary variables of the solution. The final equations system only involves the nodal displacements and the stress intensity coefficients. Finally, an eXtended Natural nEighbours Method (XNEM) is proposed in which the crack is represented by a line that does not conform to the nodes or the edges of the cells.Based on the hypotheses used in linear elastic domain, the discretization of the displacement field is enriched with Heaviside functions allowing a displacement discontinuity at the level of the crack.In the cells containing a crack tip, the stress and strain fields are also enriched with additional terms corresponding to the solutions of LEFM for modes 1 and 2.The stress intensity coefficients are also obtained as primary variables of the solution.A set of applications are performed to evaluate these developments.The following conclusions can be drawn for all cases (linear elastic, nonlinear, fracture mechanics).In the absence of body forces, the numerical calculation of integrals over the area of the domain is avoided:only integrations on the edges of the Voronoi cells are required, for which classical Gauss numerical integration with 2 integration points is sufficient to pass the patch test.The derivatives of the nodal shape functions are not required in the resulting formulation.The patch test can be successfully passed.Problems involving nearly incompressible materials can be solved without incompressibility locking in all cases.The numerical applications show that the solutions provided by the present approach converge to the exact solutions and compare favourably with the classical finite element method.