Analysis of dynamic stress concentration problems employing spline-based wavelet Galerkin method

Abstract

Two-dimensional (2D) dynamic stress concentration problems are analyzed using the wavelet Galerkin method (WGM). Linear B-spline scaling/wavelet functions are employed. We introduce enrichment functions for the X-FEM to represent a crack geometry. In the WGM, low-resolution scaling functions are periodically located across the entire analysis domain to approximate deformations of a body. High-resolution wavelet functions and enrichment functions including crack tip singular fields are superposed on the scaling functions to represent the severe stress concentration around holes or crack tips. Heaviside functions are also enriched to treat the displacement discontinuity of the crack face. Multiresolution analysis of the wavelet basis functions plays an important role in the WGM. To simulate the transients, the wavelet Galerkin formulation is discretized using a Newmark-β time integration scheme. A path independent J-integral is adopted to evaluate the dynamic stress intensity factor (DSIF). We solve dynamic stress concentration problems and evaluate DSIF of 2D cracked solids. The accuracy and effectiveness of the proposed method are discussed through the numerical examples.

Introduction

In recent years, various new methodologies for solving solid and structural problems have been proposed. A number of these are meshfree methods, such as the element-free Galerkin method [1], reproducing kernel particle method [2], and meshless local Petrov–Galerkin method [3]. Because there are no finite elements (FEs) in the discretization, the nodes need only be relocated to represent severe stress concentration near holes or cracks. Applications for static and dynamic problems using meshfree methods are discussed in [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. In addition, the extended finite element method (X-FEM) [17], [18] has been adopted for analyzing dynamic crack problems. X-FEM is well suited to the treatment of crack problems. Crack tip singular fields and displacement discontinuities are introduced in the displacement functions based on partition of unity method [19], [20]. Because the crack shape can be modeled independent of the FE meshes, crack propagation simulations are carried out without remeshing. Applications of dynamic crack problems using X-FEM have been presented in [21], [22], [23], [24], [25]. In addition, the combination of meshfree methods and X-FEM for dynamic crack problems was proposed in [26], [27].

Wavelet-based methods [28], [29], [30], [31] have also been adopted to solve science and engineering problems. Among them, we focus on the wavelet Galerkin methods (WGMs). Some of these methods are discretized based on a fixed-grid (voxel-based approach) [32], [33]. Scaling/wavelet functions are adopted as the basis functions, and are periodically located on the analysis domain, and equally spaced structured cells are adopted for numerical integration of the stiffness matrix. The wavelet basis functions permit a multiresolution analysis (MRA), so the spatial resolution of the wavelet Galerkin (WG) model can be easily controlled. Therefore, the approach is considered as an enhanced voxel FEM. To date, WGMs have been adopted to solve several boundary value problems [34], [35], [36], [37], [38], [39], [40]. Other related wavelet-based numerical methods are summarized in [41].

In the present study, dynamic problems including holes or cracks are analyzed using the WGM. Because the WG discretization is based on fixed-grid, model generation is relatively easy in the elastodynamic problems as well as the meshfree methods [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and boundary element methods (BEMs) [42], [43], [44]. As it is easy to control the spatial resolution near the holes or crack tips, a stress concentration can effectively be evaluated with fewer degrees of freedom (DOFs). Although the multilevel wavelet functions are fixed in the elastodynamic analyses in this study, spatial mesh adaptation will be enabled with an error estimator, e.g. [45], [46], because relocation of the wavelet functions is easy. When carrying out the fracture mechanics analysis in the WGM, it was difficult to introduce displacement discontinuities such as crack faces, because the displacement functions in most WGMs are assumed to be continuous. In a previous study by some of the present authors [47], [48], [49], X-FEM enrichment functions were introduced to represent the crack geometry and severe stress concentration near the crack tip. To carry out transient simulations, a Newmark-β time integration scheme is adopted. A path-independent J-integral is employed to evaluate the dynamic stress intensity factor (DSIF). We consider certain dynamic WG problems, and examine the discretization and accuracy of the solutions.

This paper is organized as follows. Section 2 presents the governing equations for dynamic problems for 2D elastic solids. In Section 3, WG formulation and the discretization for the dynamic problems is presented. Section 4 explains how we evaluate the DSIF by employing a path-independent J-integral. Section 5 describes the numerical results from various examples. Section 6 summarizes our conclusions.