## 1 Introduction  
In welding applications, numerical simulation technology is a useful tool for quantitative investigation of various parameters, such as welding temperature, stress, deformations and crack effects. The mathematical model of the welding problem can be simplified as a heat conduction process of a moving point heat source applied on a medium. [PERSON] [1] developed a three-dimensional and transient model with a keyhole geometry-dependent heat source and arc pressure distribution for the plasma arc welding process. The dynamic variation of the temperature field and fluid flow in the weld pool were quantitatively analyzed as well as the keyhole shape and size. [PERSON] [2] investigated the effects of solid-state phase transformation on the residual stress in the welding of Q345 by numerical simulation. He [3] established the temperature field finite element numerical simulation model of twin arc movement; the loading form of the twin arc with a double ellipsoid heat source was discussed; and the laws of molten pool characteristics influenced by the welding speed, current and voltage of the twin-arc submerged arc welding parameters were analyzed. [PERSON] [4] conducted finite element simulation of thermal and residual welding stresses in the specific heat of a material. [PERSON] [5] developed a model of the heat transfer during welding by the Element Free Galerkin (EFG) method, and demonstrated the effectiveness and utilities of the EFG method for modeling and understanding the heat transfer processes in arc welding. [PERSON] [6] calculated the temperature distribution and thermal cycle in the workpiece by finite element analysis in dissimilar aluminum titanium laser welding, and he also carried out the finite element method (FEM) study of full penetration keyhole laser welding of Ti6 Al4V in butt configuration by varying the modeling strategy of the thermal source [7]. [PERSON] [8] investigated the underlying mechanism involved in the solidification crack in pulsed laser welding of 2024 aluminum alloy experimentally and numerically. He [9] established a three-dimensional welding thermal-mechanical coupling model of aluminum alloy sheet, and analyzed the welding solidification crack tendency by calculation of the welding temperature, stress and the crack driving force curves. The current numerical techniques of welding mainly focus on the non-crack homogeneous or nonhomogeneous structure. In fact, the presence of stress concentration in the material may result in the initiation and extension of a crack in the welding seam zone, which can affect the quality of the structure. However, only few existing models take into account the crack behavior under welding thermal loadings, which are critical to many welding applications, such as automobile, fuel cells and computer components fabrication, etc. In recent years, the extended finite element method (XFEM) has been the subject of considerable research as a powerful numerical procedure for analysis of the crack problems. Within the framework of thermo-elastic fracture by the XFEM, [PERSON] [10] investigated the static case of thermo-elastic fracture by XFEM, which considered thermal boundary conditions with different crack faces in both the 2D and 3D problems. [PERSON] [11] implemented the XFEM to model the effects of mechanical load and thermal shocks on a body with a stationary crack. [PERSON] [12] used the extended Element Free Galerkin method (XEFG) to model the crack growth in elastic materials, and investigated the effect of thermo-mechanical loading on the crack growth. [PERSON] [13] studied isotropic and orthotropic functionally graded materials (FGMs) under a combination of mechanical and thermal loadings using the XFEM. These implementations of the XFEM for modeling thermo-elastic fracture were successful in the analysis of the thermal crack mechanics, dynamic crack propagation and estimation of the dynamic stress intensity factors (SIFs).