The Extended finite element method (X-FEM) is a numerical method for modeling discontinuties, such as cracks, within the standard finite element framework. In X-FEM, special functions are added to the finite element approximation. For crack modeling in linear elasticity, appropriate functions are used for modeling discontinuties along the crack length and simulating the singularity in the crack tip element. As a result, the degrees of freedom (D.O.F.) for the nodes arround the crack tip and the crack length are increased, the so-called node-enrichment scheme. This virtual crack modeling, which is mesh independent, avoids the usage of refined mesh and singular elements arround the crack tip, and does not require remeshing during crack growth simulation. In this paper the principles of the X-FEM are described. A new method, based on the local orthogonal coordinate system, is proposed for the node-enrichment scheme. The development of a special-purpose computer code for modeling 2D cracks using the X-FEM and the new method is presented. The code and the new method are verified through the analyses of different standard cracked geometries.

The finite element method (FEM), and other numerical methods, in recent years, is widely used in modeling of the fracture problem. Remeshing requirements and mesh sensitivity are the major disadvantages in analyzing crack growth using conventional FEM methods. Recently, advanced FEM methods, such as the Extended finite element method (X-FEM), have been proposed to model discontinuities through the elements. The advantage of these methods is that remeshing is not required in the crack growth process. The cohesive crack method is a simplified field model to simulate the complicated behavior of the crack growth in quasi-brittle materials. In this paper, we use the advantages of the X-FEM and crack length control method for modeling of the cohesive crack growth.

one of the simplest numerical integration method which provides a large saving in computational efforts, is the well known one-point Gauss quadrature which is widely used for 4 nodes quadrilateral elements. On the other hand, the biggest disadvantage to one-point integration is the need to control the zero energy modes, called hourglassing modes, which arise. The efficiency of four different anti-hourglassing approaches, Flanagan (elastic approach), Dyna3d, Hansbo and Liu have been investigated. The first two approaches have been used in 2 and 3-D explicit codes and the latter have been employed in 2-D implicit codes. For 2-D explicit codes, the computational time was reduced by 55% and 60% for elastic and Dyna3d, respectively. However, for 3-D codes the reduction was dependent on the number of elements and was obtained between 50% and 70%. Also, the error due to the application of elastic methods was less than that for Dyna3d when the results were compared with those obtained from 2-points Gauss quadrature. Nevertheless, the convergence occurred more rapidly and the oscillations were damped out more quickly for Dyna3d approach. For implicit codes, the anti-hourglassing methods had no effect on the computations and therefore a 2-points Gauss quadrature is recommended for implicit codes as it provide the results more accurately

In this study, the Extended finite element method was used for modeling dynamic fracture in Kirchhoff plate and shell problems. A new set of tip functions was extracted from analytical solutions of Kirchhoff plates. The semi-discrete method was used to simulate the dynamic behavior. An unconditionally stable implicit Newmark scheme was used for temporal discretization. The performance of the code in simulation of dynamic behaviors was proved by solving several benchmark problems and comparing the obtained results with other numerical and analytical solutions. Also, the problem of cracked thin tubes under gaseous detonation loading was simulated by the dynamic XFEM code. The results were compared with analytical and other numerical solutions and the obtained results showed that the method has good capability for simulation of these problems.

In this paper, the Extended finite element method (XFEM) is employed to investigate the statics and vibration problems of cracked isotropic bars and beams. Three kinds of elements namely the standard, the blended and the enriched elements are utilized to discretize the structure and model cracks. Two techniques referred as the increase of the number of Gauss integration points and the rectangle sub-grid are applied to refine the integration within the blended and enriched elements of the beam in which the priority of the developed rectangle sub-grid technique is identified. The stiffness and the mass matrices of the beam are Extended by considering the Heaviside and the crack tip functions. In a plane stress analysis, the effects of various crack positions and depths, different boundary conditions and other geometric parameters on the displacement and the stress contours are detected. Moreover, in a free vibration analysis, changes of the natural frequencies and the mode shapes due to the aforementioned effects are determined.

The Newmark method is an effective method for numerical time integration in dynamic problems. The results of Newmark method are function of its parameters (β , γ and Δ t). In this paper, a stationary mode I dynamic crack problem is coded in Extended finite element method )XFEM( framework in Matlab software and results are verified with analytical solution. This paper focuses on effects of main parameters in Newmark method for dynamic XFEM problems. Also use of the response surface method (RSM) a regression model is presented for estimating error of dynamic stress intensity factors (DSIF) with high validity according to results of analysis of variance (ANOVA). This work enables one to understand the effect of Newmark parameters on error of DSIFs and to find optimum β and γ for a determined number of time steps (N). This procedure is highly effective in order to manage the computational cost and enhance the accuracy at the desired domain. The effect of the considered parameters on error, is investigated using RSM in Minitab software and optimum state for minimization of errors is illustrated.

Initiation and progression of cracks in a saturated porous media is an important topic which has attracted considerable attention from researchers in the recent years. Extended finite element method (EFEM) is a contemporary technique removing the necessity of consecutive meshing of the problem in the analysis process. In the EFEM by enriching the elements whose discontinuity there exists, there is no need for re-meshing at each step of the analysis. . In this paper, EFEM is used to evaluate progression of cohesive crack in a two phase saturated porous media. To analyze the saturated porous media, at the first, the equations of mass conservation, momentum conservation, and energy conservation are established to consider simultaneous effects of displacement, pressure, and temperature on the crack progression. The cohesive model is used to simulate crack progression. Heavy-side functions are used to enrich finite elements and the resulting system of equations are solved by Newton Raphson method. Finally, the two numerical models were analyzed by other researchers is considered to evaluate the derived relationships. Numerical result show that maximum variation by other researchers is 5%.

Damage detection of structures is an important issue for maintaining structural safety and integrity. In order to evaluate the health condition of structures, many structural health monitoring (SHM) techniques have been proposed over the last decades. Major approaches of SHM are non-destructive in nature and are widely used for damage detection in engineering structures. The existence of damage in a structure may be traced by comparing the response of time-domain wave traveling in the structure at its present state with a base-line response. Thus, presence of damage in a structure is detected by inspecting at the wave parameters affected by the damage. The commonly used wave parameters are those representing attenuation, reflection and mode conversion of waves due to damage. Although detection of flaws is extremely important for many industrial applications, current approaches are severely restricted to specific flaws, simple geometries and homogeneous materials. In addition, the computational burden is very large due to the inverse nature of the problems where one solves many forward and backward problems. For instance, conventional ultrasonic methods measure the time difference of returning waves reflected from a crack; however, for laminated composite plates, the ultrasonic wave would be partially reflected at the interface of two layers where no crack actually exists, and partially continues to propagate further where it eventually is reflected back by the true crack. Numerical methods employed in crack detection algorithms require the solution of inverse problems in which the spatial problem is often discretized in space using finite elements in association with an optimization scheme. The solution of these problems is not unique, and sometimes the optimization algorithm may converge to local minima which are not the real optimal solution. Moreover, they often require hundreds of iterations to converge considering the algorithm used in the process. On the other hand, an accurate detection of cracks requires the re-meshing of the finite element domain at each iteration of the optimization. This is a severe limitation to any numerical approach when the conventional finite element method is employed for crack modeling, as the re-meshing of a domain is often not a trivial task. This paper investigates crack detection of two-dimensional (2D) structures using the Extended finite element method (XFEM) along with particle swarm optimization (PSO) algorithm. The XFEM is utilized to model the cracked structure as a forward problem, while the PSO is employed for finding crack location as an optimization scheme. The XFEM is a robust tool for analysis of structures having discontinuities without re-meshing. Therefore, it is an efficient tool for an iterative process. Also, the PSO is a well-known non-gradient based method which is suitable for this inverse problem. The problem is formulated such that the PSO algorithm searches crack coordinates in order to detect the existing crack by minimizing an error function based upon sensor measurements. This problem is a non-destructive evaluation of a structure. Three benchmark numerical examples are solved to demonstrate capability and accuracy of the XFEM and the PSO for crack detection of 2D domains.

The present paper investigates and compares the crack propagation in concrete gravity dams using two models of linear fracture mechanics and plasticity damage concrete. The first model is based on linear concrete behavior using the Extended finite element method without considering the effect of strain softening on the crack tip while the second model is based on the nonlinear concrete behavior and the strain softening in tension with damage parameter. According to two different algorithms and based on two models, several benchmark examples are reviewed and the results compared with those reported in the literature. Then, path of the crack growth in Koyna gravity dam due to a seismic excitation of Koyna earthquake in 1962 has been performed by considering the dam-reservoir interaction. The results show that due to low compressive stresses during analysis of concrete gravity dams, consideration of compressive nonlinear behavior has no effect on crack initiation and almost is the same for two models. However because of crack opening and closing with tapping the crack faces together in Extended finite element model, the compressive stress will be more than the allowable stress of concrete. Crack initiation at downstream and upstream face occurred at angle of 90 and zero degrees respectively, which in both models, the numerical results are in agreement with the experimental model. The crack in the Extended finite element model grows faster such that the crest block of dam in this model is separated from the dam body, earlier than the concrete plastic damage model. Also the values of dam crest displacement and hydrodynamic pressure in the reservoir in Extended finite element model with linear elastic fracture mechanic are more than the other model, which can be attributed to the linear and nonlinear behavior of concrete in Extended finite element and concrete plastic damage model respectively. In the Extended finite element model, due to using linear fracture mechanic, the maximum principal stress in the cracked elements reaches the values greater than the maximum tensile strength, but in the concrete plastic damage model as soon as the stress reaches a tension limit value, elements are damaged and the stress is reduced. In both models, the crack located at the slope change area, propagates with the downward slope from downstream dam face and connects to the crack at upstream face which is growth horizontally. Because of laboratory sample dimension and boundary condition of dam-reservoir compared with actual manner, neither of two crack profiles covered the experimental model, accurately. But it is shown that the crack profiles in the Extended finite element model are more consistent with experimental results. Finally, the results show that the crack profile are slightly different in the two models because of quasi brittle behavior of the dam concrete, which can be attributed to the small fracture process zone of the crack tip in comparison with the dimension of the concrete gravity dams such that by removing strain softness part, the error in the amount of additional computation can be neglected.