Solution of 3D elasticity problems using meshless local equilibrated basis functions

A meshfree method is presented for 3D elasto-static problems in homogenous media using Equilibrated Basic functions. The method treats satisfaction of the Partial Differential Equation independent of the boundary conditions, using a weak weighted residual integration over a cubic fictitious domain embedding the main domain. All 3D integrals break into combination of 1D library integrals, resulting in the omission of the numerical integration. Chebyshev polynomials of the first kind are used to approximate the solution function, and exponential functions combined with polynomials are used as weight functions. The weights vanish over the boundaries of the cubic fictitious domain, removing the boundary integrals. The meshless method considers some nodes for definition of the Degrees of Freedom throughout the domain. Each node corresponds to a local sub-domain called cloud, including 98 other nodes than the main central one. The overlap between adjacent clouds ensures the continuity of both the displacement as well as stress components, an advantage with respect to the formulations. The approximation order within each cloud is 4. Boundary conditions are applied over a set of boundary points independent of the domain nodes, granting the method the ability of application for arbitrarily shaped domains without the drawback of irregularity in the nodal grid. Definition of curved boundary surfaces is easily done by inserting the coordinates of some boundary points located on it. Three numerical examples with various geometries and boundaries are presented to challenge the method. The results are compared with either the available exact solutions or the FEM.