Stochastic Meshfree Method for Computational Fracture Mechanics

Abstract

In the stochastic mechanics community, the need to account for uncertainty has long been recognized as key to achieving the reliable design of structural and mechanical systems. It is generally agreed that advanced computational tools must be employed to provide the necessary computational framework for describing structural response. A currently popular method is the stochastic finite element method (SFEM), which integrates probability theory with the standard finite element method (FEM). However, SFEM requires a structured mesh to perform the underlying finite element analysis. It is generally recognized that the creation of workable meshes for complex geometric configurations can be difficult, time consuming, and expensive. This discrepancy is further exacerbated when solving solid mechanics problems characterized by a continuous change in the geometry of the domain under analysis. The underlying structures of these methods, which rely on a mesh, are cumbersome in treating moving cracks or mesh distortion. Consequently, the only viable option when applying FEM is to remesh during each discrete step of the model’s evolution. This creates numerical difficulties, even for deterministic analysis, and often leads to degradation in solution accuracy, complexity in computer programming, and a computationally intensive environment. Consequently, there is considerable interest in eliminating or greatly simplifying the meshing task. In recent years, a class of Galerkin-based meshfree or meshless methods have been developed that do not require a structured mesh to discretize the problem, such as the element-free Galerkin method, and the reproducing kernel particle method. These methods employ a moving least-squares approximation method that allows resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes. Meshless discretization presents significant advantages for modeling fracture propagation. By sidestepping remeshing requirements, crack-propagation analysis can be dramatically simplified. Since mesh generation of complex structures can be far more time-consuming and costly than the solution of a discrete set of linear equations, the meshless method provides an attractive alternative to FEM. However, most of the development in meshless methods to date has focused on deterministic problems. Research into probabilistic meshless analysis has not been widespread and is only now gaining attention. Due to inherent uncertainties in loads, material properties and geometry, a probabilistic meshless model is ultimately necessary. Hence, there is considerable interest in developing stochastic meshless methods capable of addressing uncertainties in loads, material properties and geometry, and of predicting the probabilistic response of structures. This paper presents a new stochastic meshless method for predicting probabilistic structural response of cracked structures i.e., mean and variance of the fracture parameters.