The natural neighbour radial point interpolation method : Solid mechanics and mechanobiology application

This work presents and develops the Natural Neighbour Radial Point Interpolation Method (NNRPIM), a new truly meshless method. The NNRPIM is extended to several engineering fields, such as solid mechanics static and dynamic linear analysis and structural nonlinear analysis, and bone remodelling biomechanical analysis.;Within the NNRPIM the nodal connectivity is enforced using the Natural Neighbour concept. After the Voronoi diagram construction from the unstructured nodal mesh, which discretizes the problem domain, small cells are created, the "influence-cells". These cells are in fact influence-domains entirely nodal dependent. The Delaunay triangles are used to create a nodedepending background mesh used in the numerical integration of the NNRPIM interpolation functions. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed with the Radial Point Interpolators. In the construction of the NNRPIM interpolation functions no polynomial base is required, which is an innovation and the used Radial Basis Function (RBF) is the Multiquadric RBF. The NNRPIM interpolation functions posses the delta Kronecker property, which simplify the imposition of the natural and essential boundary conditions.;In the nonlinear analysis, large deformations and elastoplastic material behaviour are considered. The solution of the nonlinear equation system is obtained resorting to incremental/iterative methods, such as the Newton- Raphson method or the orthogonal actualized Ramm's method, this last permitting the analysis of structures that in some point evidence instability phenomenona such as the "snap-through" and the "snap-back". The material nonlinear behaviour is dealt considering for the elastoplastic model the Von Mises yield function and the efficient "forward-Euler" procedure is used in order to return the stress to the yield surface.;In the biomechanical analysis, a new mathematical model to obtain the bone material properties is proposed. This material law is based in experimental data, which support the idea that the law governing the mechanical behaviour of the bone tissue is the same for cortical bone and trabecular bone. A bone remodelling algorithm is proposed, adapted from Carter's remodelling algorithm. It is based on the assumption that the adaptation of bony tissue responds mainly to mechanical stimulus. A simple forward Euler scheme is implemented, resulting in an iterative remodelling process.;After the respective exposition, several benchmark solid mechanics static and dynamic linear examples are solved, and well-known demanding nonlinear reference examples are analysed. In biomechanics applications, bone remodelling validation tests are made and then two important bone structures in the human body are studied. The obtained results indicate that NNRPIM is an accurate, flexible and reliable meshless method that can be used in several fields of solid mechanics and biomechanical analysis.