Mesh-free applications to fracture mechanics and an analysis of the Corrected Derivative Method.

다국어 초록 (Multilingual Abstract)

Mesh-free methods and extensions of the finite element method for the solution of stress analysis problems, with an emphasis on crack problems, are developed and analyzed. Application of radial weighting functions to the numerical solution of differential equations is considered. It is shown that traditional numerical schemes using radial weighting functions may not converge. The Corrected Derivative Method (CD) and the Element-free Galerkin Method (EFG) are considered and analyzed as corrections to the radial weighting functions to ensure convergence and good approximation.
A convergence theory for the Corrected Derivative Method is presented. This is a Petrov-Galerkin method using radial weighting functions to approximate the primary variable and corrects their derivatives to satisfy algebraic constraints. Under mild conditions relating nodal spacing and functional supports, this method converges with optimal order in the <italic>H</italic><super>1</super> norm. This method is used to model a two dimensional crack and compute stress intensity factors.
A coupled FE-EFG method for three dimensional fracture mechanics is described. In this method, the solution in the region near the crack front is approximated by intrinsically enriched EFG shape functions and linear finite elements are used in the rest of the domain. The plane strain asymptotic crack tip fields are used to enrich in a tubular region surrounding the crack front where they are well defined. The diffraction method and visibility criterion are generalized to three dimensional geometries. Example computations for stress intensity factors and crack growth illustrate the method.
An extension of the finite element method is developed for two dimensional crack problems. The crack is modeled variationally by using shape functions constructed so that they are discontinuous along the crack line. This method is applied to arbitrarily curved cracks. Examples determining stress intensity factors and crack growth are presented. More general extensions to three dimensional problems are described.