Solving thermal and phase change problems with the eXtended finite element method

 The application of the eXtended finite element method (X-FEM) to thermal problems with moving heat sources and phase boundaries is presented. Of particular interest is the ability of the method to capture the highly localized, transient solution in the vicinity of a heat source or material interface. This is effected through the use of a time-dependent basis formed from the union of traditional shape functions with a set of evolving enrichment functions. The enrichment is constructed through the partition of unity framework, so that the system of equations remains sparse and the resulting approximation is conforming. In this manner, local solutions and arbitrary discontinuities that cannot be represented by the standard shape functions are captured with the enrichment functions. A standard time-projection algorithm is employed to account for the time-dependence of the enrichment, and an iterative strategy is adopted to satisfy local interface conditions. The separation of the approximation into classical shape functions that remain fixed in time and the evolving enrichment leads to a very efficient solution strategy. The robustness and utility of the method is demonstrated with several benchmark problems involving moving heat sources and phase transformations. Received 20 May 2001 / Accepted 19 December 2001     

An Abaqus implementation of the extended finite element method

In this paper, we introduce an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS^TM. User subroutine (UEL) in Abaqus is used to enable the incorporation of extended finite element capabilities. We provide details on the data input format together with the proposed user element subroutine, which constitutes the core of the finite element analysis; however, pre-processing tools that are necessary for an X-FEM implementation, but not directly related to Abaqus, are not provided. In addition to problems in linear elastic fracture mechanics, non-linear frictional contact analyses are also realized. Several numerical examples in fracture mechanics are presented to demonstrate the benefits of the proposed implementation.     

A multiscale extended finite element method for crack propagation

In this paper, we propose a multiscale strategy for crack propagation which enables one to use a refined mesh only in the crack’s vicinity where it is required. Two techniques are used in synergy: a multiscale strategy based on a domain decomposition method to account for the crack’s global and local effects efficiently, and a local enrichment technique (the X-FEM) to describe the geometry of the crack independently of the mesh. The focus of this study is the avoidance of meshing difficulties and the choice of an appropriate scale separation to make the strategy efficient. We show that the introduction of the crack’s discontinuity both on the microscale and on the macroscale is essential for the numerical scalability of the domain decomposition method to remain unaffected by the presence of a crack. Thus, the convergence rate of the iterative solver is the same throughout the crack’s propagation.     

Calculation of KII in crack face contacts using X-FEM. Application to fretting fatigue

In this work, the modeling of LEFM problems that imply crack face closure and contact using the extended finite element method (X-FEM) is presented aiming at its application to fretting fatigue problems. An assessment of the accuracy in the calculation of K_II is performed for two different techniques to model crack face contacts in X-FEM: one is based on the use of additional elements to establish the contact and the other on a segment-to-segment (or mortar) approach. It is concluded that only the segment-to-segment approach can lead to optimal convergence rates of the error in K_II. The crack face contact modeling has also been applied to a fretting fatigue problem, where the estimation of K_II under crack closure conditions plays an important role in the stage I of fatigue crack propagation. The effect of the crack face friction coefficient has been studied and its influence on the range of K_II has been ascertained during loading and unloading cycles.     

Crack analysis in orthotropic media using the extended finite element method

An extended finite element method has been proposed for modeling crack in orthotropic media. To achieve this aim a discontinuous function and two-dimensional asymptotic crack-tip displacement fields are used in a classical finite element approximation enriched with the framework of partition of unity. It allows modeling crack by standard finite element method without explicitly defining and re-meshing of surfaces of the crack. In this study, fracture properties of the models are defined by the mixed-mode stress intensity factors (SIFs), which are obtained by means of the domain form of the interaction integral (M-integral). Numerical simulations are performed to verify the approach, and the accuracy of the results is discussed by comparison with other numerical or (semi-) analytical methods.     

Using the X-FEM method to model the dynamic propagation and arrest of cleavage cracks in ferritic steel

Although initiation criteria have been the subject of many publications, the phenomena associated with the propagation and arrest of brittle cracks have not. To be able to predict the dynamic behaviour of cleavage cracks, we made a series of experiments and associated numerical studies. Tests of crack propagation and arrest were carried out on specimens of two different geometries (Compact Tension and compression ring) made of the 16MND5 ferritic steel of which French nuclear reactor vessels are constructed. The elastic-viscoplastic behaviour of this material was characterised and its nature was taken into account in the numerical simulations. The eXtended Finite Element Method (X-FEM) was developed in CAST3M finite element analysis software to enable fine, effective modelling of crack propagation. Propagation models based on principal stress were studied and it was found that critical cleavage stress depended on loading rate. The use of criteria calibrated for Compact Tension specimens gave excellent results in predictive calculations for similar specimens, and also for compression rings in both mode I and mixed-mode. The speed and path of crack predicted with the numerical simulations were in close agreement with the experimental results.     

Modeling stationary and moving cracks in shells by X-FEM with CB shell elements

The continuum-based (CB) shell theory is combined with the extended finite element method (X-FEM) in this paper to model crack propagation in shells under static and dynamic situations. Both jump function and asymptotic crack tip solution are adopted for describing the discontinuity and singularity of the crack in shells. Level set method (LSM) is used to represent the crack surface and define the enriched shape functions. Stress intensity factors (SIFs) are calculated by the displacement interpolation technique to prove the capability of the method and the maximum strain is applied for the fracture criterion. Also, an efficient integration scheme for the CB shell element with cracks is proposed.     

X-FEM implementation of VAMUCH: Application to active structural fiber multi-functional composite materials

In multiscale analysis of composite materials, there is usually a need to solve microstructures problems with complex geometries. The variational asymptotic method for unit cell homogenization (VAMUCH) is a recently developed variant of the asymptotic homogenization approach. In contrast to conventional asymptotic methods, VAMUCH carries out an asymptotic analysis of the variational statement, synthesizing the merits of both variational methods and asymptotic methods. This work gives an outline of the Extended Finite Element Method (X-FEM) implementation of VAMUCH for complex multi-material structures. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing an enrichment strategy. The X-FEM/VAMUCH approach is applied successfully to many examples reported in the VAMUCH literature. Numerical experiments on the periodic homogenization of complex unit cells demonstrate the accuracy and simplicity of the X-FEM/VAMUCH approach.     

X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation

This paper is devoted to the simulation of dynamic brittle crack propagation in an isotropic medium. It focuses on cases where the crack deviates from a straight-line trajectory and goes through stop-and-restart stages. Our argument is that standard methods such as element deletion or remeshing, although easy to use and implement, are not robust tools for this type of simulation essentially because they do not enable one to assess local energy conservation. Standard cohesive zone models behave much better when the crack’s path is known in advance, but are difficult to use when the crack’s path is unknown. The simplest method which consists in placing the cohesive segments along the sides of the finite elements leads to crack trajectories which are mesh-sensitive. The adaptive cohesive element formulation, which adds new cohesive elements when the crack propagates, is shown to have the proper energy conservation properties during remeshing. We show that the X-FEM is a good candidate for the simulation of complex dynamic crack propagation. A two-dimensional version of the proposed X-FEM approach is validated against dynamic experiments on a brittle isotropic plate.