A novel multi-grid based reanalysis approach for efficient prediction of fatigue crack propagation
Introduction
Engineering structures usually work under cyclic loads and fatigue failure becomes one of the major failure modes of structures [1], [2]. Microcrack will grow continuously during the service life until the final fracture occurs. In order to avoid serious safety accident, it is important to make proper predictions about the fatigue life of engineering structures [3], [4], [5]. With the ever increasing speed of computers [6], [7], [8], the prediction of fatigue crack propagation has become an efficient way to successfully assess the life of structures, which can be realized based on numerical methods, such as finite element method (FEM) [9], singular edge-based smoothed finite element method (ES-FEM) [10], [11], [12], singular node-based smoothed finite element method [13], [14], [15], etc. However, these methods are mainly derived from the FEM and hence, the mesh used in the computation must conform to the geometry of cracks in structures [16], [17], [18]. When crack propagation happens, the mesh around the crack must be updated continuously to trace the new crack path. It is obvious that computational effort spent on the generation of computational mesh will be huge especially when the number of simulation steps is large. In order to overcome the mesh-dependent difficulties in FEM, an improved numerical approach known as the extended finite element method (XFEM) has been proposed and further developed to simulate the crack propagation of engineering structures [19], [20], [21], [22]. In the XFEM, the modeling of crack can be greatly facilitated through the partition of unity enrichment of finite elements (PUM) [23]. The geometry of crack is independent of the finite element mesh and hence, there is no need to update the mesh when the crack grows. The computational effort related to the mesh regeneration can be saved during the simulation [24], [25], [26]. Up to now, the XFEM has become the most important tool for crack propagation simulation [27], [28].
Although XFEM can avoid the computational cost on the implementation of re-meshing process, the computational effort will still be tremendous especially when the finite element mesh is dense or the number of simulation steps becomes huge. This is mainly because that large scale of system equations will be solved repeatedly in these cases, until the fatigue crack propagation stops. In order to ensure the engineering practicality of simulation, it is crucial to further improve the computational efficiency.
For the prediction of crack propagation problems, the key feature is that only a small portion of the global stiffness matrix changes during the process of crack propagation. Therefore, only the stiffness matrix of newly added enriched elements needs to be updated. Some researches have been done recently focusing on this issue, which aim to deal with the system equations of crack propagation analysis without solving the complete set of system equations. Several methods have been developed, such as the incremental Cholesky factorization method [29], the combined approximations method [30]. These methods mainly belong to the reanalysis method families, which have been successfully utilized in the topology structural optimization problems [31], [32], [33], [34]. The crack propagation analysis and topology optimization analysis have some essential similarities [35], [36], [37]. The local change of global stiffness matrix also occurs in the topology optimization process and large numbers of system equations should also be solved repeatedly during the optimization analysis [38], [39], [40]. As there is no need to solve the complete set of system equations, the computational efficiency of crack propagation analysis can also be fully improved based on these reanalysis methods. Nevertheless, there are still some defects for these exiting methods. For example, the incremental Cholesky factorization method usually costs a large amount of storage especially for large scale analysis, which will hinder its applications in practice [41]. For the combined approximations method, it is hard to propose a general rule for the construction of basis vector space [42] and hence, an optimal tradeoff between the computational accuracy and efficiency should be determined, which is still a challenging task [43].
Different from the incremental Cholesky factorization and combined approximations methods, the multi-grid (MG) method, derived from the discretized partial differential equations [44], [45], [46], is much efficient for large scale algebraic system equations. For the MG analysis, algebraic system equations are solved through iterations between different scales of meshes and hence, the demand for storage is much lower. This feature is crucial especially for large scale problems [47], [48], [49], [50]. Since MG aims to improve the computation efficiency of algebraic system equations, it can be incorporated into XFEM to achieve accurate and efficient prediction of fatigue crack propagation. For example, Rannou presented a multi-grid extended finite element method for elastic crack growth simulation, which can greatly improve the computational efficiency. However, the enrichments for each scale of meshes need to be carried out and then, the specific multi-scale operators can be developed [51]. Berger-Vergiat provided another way to couple the XFEM with a multi-grid strategy, in which a multiplicative Schwarz preconditioner is formulated through special domain decomposition procedures. The problem domain needs to be divided into healthy part and cracked part that include all the XFEM-enriched degrees of freedom. Then, the MG analysis can be carried out in the healthy part of structure [52].
In order to avoid some pre-treatments to meshes or enriched degrees of freedom on each level of meshes, the multi-grid (MG) reanalysis method is introduced to the framework of XFEM in this work, which aims to fully improve the computational efficiency while achieving high accuracy for the prediction of fatigue crack propagation. The core idea of this MG reanalysis approach is that the intermediate data are calculated only once for the structure with no crack and will be reused during the crack propagation analysis. The result of initial MG analysis will also be used as an initial solution in the subsequent MG iterations. Then, the calculation and prediction of the fatigue crack propagation can be efficiently carried out by taking full advantage of MG and XFEM.
The contributions of this work that may add new insight to the prediction of fatigue crack propagation are described as the following two aspects.
(1) A novel multi-grid (MG) reanalysis method is developed under the framework of the extended finite element method (XFEM) to analyze the fatigue crack propagation problems.
(2) The presented MG-XFEM approach enables accurate prediction of the crack propagation with much less computational effort than traditional approach.
This paper is organized as follows. In Section 2, we briefly discussed the XFEM formulations and some fundamentals about fatigue crack propagation analysis. In Section 3, the algorithm for coupling MG reanalysis and XFEM is discussed in detail. In Section 4, several numerical examples are presented to fully investigate the accuracy and efficiency of the proposed algorithm. Some concluding remarks that close the paper are given in Section 5.
Section snippets
XFEM formulation
In this section, we briefly discuss the formulation of XFEM for fracture analysis. For the XFEM, the non-smooth behavior or field variables are described through a set of enrichment functions and this is the main difference between XFEM and standard FEM. These enrichment functions are chosen according to the local behavior of different problems. For linear fracture mechanics, the enriched displacement approximation is usually chosen as
MG formulations
The MG method can solve the algebraic equations based on different scales of meshes. In the analysis, the problem domains are usually divided into several levels of meshes (), as shown in Fig. 1. The iterative methods are used during the iteration process to reduce the iterative errors, such as Gauss–Seidel method, Jacobi iteration method, etc.
For the ith level of mesh , the equilibrium equation can be written as where, is the system stiffness matrix, denotes the
Results and discussion
For the fatigue crack propagation analysis, numerical examples are investigated to fully test the accuracy and efficiency of the proposed MG reanalysis algorithm. The Conjugate Gradient (CG) method has become one of the most powerful tools for large scale algebraic equations and hence, the CG is employed for the full analysis to verify the validity of the present reanalysis method [56]. For the MG V-cycle iteration procedure, 3 levels of meshes are mainly recommended for crack growth analysis
Conclusion
In this study, the MG based reanalysis algorithm is formulated and combined with XFEM for the fatigue crack propagation analysis. The overall formulations are straightforward and can be simply accomplished based on exiting XFEM codes. Several numerical examples are investigated to fully verify the accuracy and efficiency of the present MG reanalysis algorithm, when simulating the fatigue crack propagation. Based on these studies, some remarks can be drawn as follows:
- 1.
In the present algorithm,
Acknowledgments
This work is supported by National Key R&D Program Of China (2017YFB1301300), State Key Program of National Natural Science of China (11832011).
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