The Research Of Two-dimensional Smoothed Extended Finite Element Method

The Extended Finite Element Method and The Smoothed Finite Element Methodare both based on the general finite element method, while inheriting all of itsadvantages. The Extended Finite Element Method is a calculating method of solvingthe discontinuous problems, such as interface, inclusion and crack, which mainlyconducts its analysis through PUM and The Level Sets Method. Smoothed finiteelement method mainly adopts strain smoothing function and the divergence theoremtheorem to simplify the integration to easier line integral (two-dimensional problem),allowing the elements to exist with any shapes, without any isoparametric, therebysolving the distortion of elements and shear locking. The Smoothed finite elementmethod can be as precise as the eneral finite element method without spending muchcalculating time, which can be used to deal with more complex models, such as largedeformation and high-speed impact. Because of the many advantages of XFEM andS-FEM, the extended finite element method with strain smoothing (Sm-XFEM) can beformed when they are coupled together.First of all, this thesis studied the two-dimensional S-FEM and calculates infiniteplate with a circular hole and a cantilever beam model and through two-dimensionalS-FEM. The effect of irregular elements for the solving accuracy has been discussed.Secondly, this thesis researched XFEM and exploited the code it adoptstwo-dimensional XFEM to solve the discontinuous problems such as crack andinclusion.Moreover, it based on the S-FEM and XFEM, developed it to the entended finiteelement method with strain smoothing. The strain smoothing is applied to theextended finite element method, the regional integral will be converted to theboundary integral by the divergence theory. In this paper, we introduce thecontrruction of the smoothing domain, and by the numerical experiments of inclusionand crack to verify the effectiveness of the method.Finally, we will develop the XFEM to the axisymmetric model, and bring in thestrain smoothing, deduce the basic theory of the Sm-XFEM for the axisymmetricmodel. Exploit the code by using Matlab, it will be used to solving the axisymmetricmodel with interface, such as spherical inclusion and cylindrical inclusion. The effectof irregular elements for the solving accuracy will be discussed. It proved that theSm-XFEM is very effective for the discontinuous problem, and illustrates the precision is not sensitive to the irregular elements.