A novel X-FEM based fast computational method for crack propagation
📝 Abstract
This study suggests a fast computational method for crack propagation, which is based on the extended finite element method (X-FEM). It is well known that the X-FEM might be the most popular numerical method for crack propagation. However, with the increase of complexity of the given problem, the size of FE model and the number of iterative steps are increased correspondingly. To improve the efficiency of X-FEM, an efficient computational method termed decomposed updating reanalysis (DUR) method is suggested. For most of X-FEM simulation procedures, the change of each iterative step is small and it will only lead a local change of stiffness matrix. Therefore, the DUR method is proposed to predict the modified response by only calculating the changed part of equilibrium equations. Compared with other fast computational methods, the distinctive characteristic of the proposed method is to update the modified stiffness matrix with a local updating strategy, which only the changed part of stiffness matrix needs to be updated. To verify the performance of the DUR method, several typical numerical examples have been analyzed and the results demonstrate that this method is a highly efficient method with high accuracy.
💡 Analysis
This study suggests a fast computational method for crack propagation, which is based on the extended finite element method (X-FEM). It is well known that the X-FEM might be the most popular numerical method for crack propagation. However, with the increase of complexity of the given problem, the size of FE model and the number of iterative steps are increased correspondingly. To improve the efficiency of X-FEM, an efficient computational method termed decomposed updating reanalysis (DUR) method is suggested. For most of X-FEM simulation procedures, the change of each iterative step is small and it will only lead a local change of stiffness matrix. Therefore, the DUR method is proposed to predict the modified response by only calculating the changed part of equilibrium equations. Compared with other fast computational methods, the distinctive characteristic of the proposed method is to update the modified stiffness matrix with a local updating strategy, which only the changed part of stiffness matrix needs to be updated. To verify the performance of the DUR method, several typical numerical examples have been analyzed and the results demonstrate that this method is a highly efficient method with high accuracy.
📄 Content
1
A novel X-FEM based fast computational method for crack propagation
Zhenxing Chenga, Hu Wanga* a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, P.R. China
Abstract This study suggests a fast computational method for crack propagation, which is based on the extended finite element method (X-FEM). It is well known that the X- FEM might be the most popular numerical method for crack propagation. However, with the increase of complexity of the given problem, the size of FE model and the number of iterative steps are increased correspondingly. To improve the efficiency of X-FEM, an efficient computational method termed decomposed updating reanalysis (DUR) method is suggested. For most of X-FEM simulation procedures, the change of each iterative step is small and it will only lead a local change of stiffness matrix. Therefore, the DUR method is proposed to predict the modified response by only calculating the changed part of equilibrium equations. Compared with other fast computational methods, the distinctive characteristic of the proposed method is to update the modified stiffness matrix with a local updating strategy, which only the changed part of stiffness matrix needs to be updated. To verify the performance of the DUR method, several typical numerical examples have been analyzed and the results demonstrate that this method is a highly efficient method with high accuracy. Keywords Crack propagation, Fast computational method, Decomposed updating reanalysis method, Extended finite element method
- Corresponding author Tel: +86 0731 88655012; fax: +86 0731 88822051. E-mail address: wanghu@hnu.edu.cn (Hu Wang) 2
1 Introduction A great amount of engineering practice indicates that the quality and stability of engineering structures are closely related to the internal crack propagation[1, 2]. Therefore, prediction of the path of crack propagation and analysis of the stability of crack are significant for estimating the safety and the reliability of engineering structures. There are many numerical methods have been developed to simulate crack propagation process, such as finite element method (FEM) [3], boundary element method (BEM) [4], meshless method [5, 6], edge-based finite element method (ES- FEM) [7, 8], numerical manifold method (NMM) [9, 10], extended finite element method (X-FEM) [11-13] and so on[14]. Compared with above methods, the X-FEM might be the most popular numerical method for the crack propagation simulation due to its superiority of modeling both strong and weak discontinuities within a standard FE framework. The X-FEM was first developed by Belytschko and Black [15]. They analyzed the crack propagation problem with minimal re-meshing. Then, Dolbow et al [16]. and Moës et al. [17] improved this method by using a Heaviside function to enrichment function, and it also has been extended to 3D static crack modeling by Sukumar et al.[18]. Sequentially, the X-FEM was significantly improved by coupling with the level set method (LSM) which is used to track both the crack position and tips [19]. Moreover, the X-FEM has been applied to multiple engineering fields, such as dynamic crack propagation or branching [20, 21], crack propagation in composites [22] or shells [23-25], multi-field problems [26], multi-material problems [27, 28], solidification [29], shear bands [30], dislocations [31] and so on. More details of the development of X-FEM can be found in Refs [12, 32-34]. Generally, in order to improve the accuracy of simulation, a very refined mesh with a very small increment of crack propagation or fatigue cycles should be engaged. Correspondingly, the computational cost is expensive. Therefore, reanalysis algorithm is used to improve the efficiency. 3
Reanalysis, as a fast computational method, is used to predict the response of modified structures efficiently without full analysis, and reanalysis method can be divided into two categories: direct methods (DMs) and approximate methods. DMs can update the inverse of modified stiffness matrix quickly by Sherman-Morrison-Woodbury lemma [35, 36] and obtain the exact response of the modified structure, but usually it can only solve the problems of local or low-rank modifications. In recent decades, many DMs have been suggested. Such as, Song et al. suggested a novel direct reanalysis algorithm based on the binary tree characteristic to update the triangular factorization in sparse matrix solution [37]. Liu et al. applied Cholesky factorization to structural reanalysis [38]. Huang and Wang suggested an independent coefficient (IC) method for large-scale problems with local modification [39]. Compared with DMs, approximate methods can solve the high-rank modifications, but the exact response usually cannot be obtained. The approximate methods mainly include loca
This content is AI-processed based on ArXiv data.