A 3D meshless computational procedure for nonlinear analysis of structures

The purpose of the research presented in this dissertation is to demonstrate that a meshless computational procedure can be used to solve nonlinear-dynamic solid deformation problems that are usually solved using the finite element method. The various meshless methods that have been presented in the literature are summarized and explained here, and the relationships between them are examined. It is shown that each meshless method is part of a larger framework of approximate methods for solving partial differential equations, and that each differs only by differing choices of formulation, interpolation and spatial integration schemes—the three ingredients of a meshless method. As such, it is shown that the finite element method is part of the same framework by which all meshless methods are constructed.; The Voronoi diagram is used in this research as a tool by which to determine spatial relationships among nodes; specifically, it is used to specify the size and sometimes the shape of the influence domains of interpolation functions. The dual of the Voronoi diagram, the Delauney tessellation, is used as an integration grid over which the spatial integrals of the weak form of the momentum equations are approximated. The difficulties encountered in constructing the 3D Voronoi diagram are described, and methods of overcoming these difficulties are discussed. The essential algorithms used to construct a Voronoi diagram in a computer program are also presented.; A nonlinear-dynamic meshless computational procedure was created by building on lessons learned from first creating both linear-static and nonlinear-static meshless computer programs. Results are obtained that agree with simple analytical solutions and with more complicated solutions obtained using commercial finite element programs. It was found that, while this research shows meshless methods to be capable of reproducing finite element solutions, the full potential of meshless methods will not be realized until a technique can be found to store state variables without the need to map them to a new set of points (thus introducing error). Finding such a technique should be the objective of future research.