The Extend Finite Element Method Based On Hamilton Theory

Finite element method is the most effective numerical method of solving engineering problem. However, for some special problems, especially the discontinuous problems, the finite element method exists some shortcomings. Extended finite element method has all the advantages of the conventional finite element method and also has the unique advantages in addressing the problem of discontinuity. The shape functions of currently existing extended finite element methods are extended on the basis of the displacement interpolation method.The Hamilton methods appeared in the 1980 s. The development of Hamilton system theory research has achieved unprecedented growth in numerical aspects. In the end of the20 th century, the Hamilton system was introduced to the elastic mechanics and brought a new thought of solving mechanics problems.This paper studies the extended finite element method based on the Hamilton theory system. By using the same interpolation mode as the displacement interpolation in the extended finite element to calculate stress and innovatively combining them together, the stress intensity factors at the crack tip are calculated. The results are quite accurate even under sparse mesh density, which enables the present method greatly simplify the preprocessing.Firstly, the research status at home and abroad and the basic theory of the extended finite element method(XFEM) were introduced;Secondly, according to the basic theory of Hamilton and the basic principle of XFEM,the extended finite method under the Hamilton system was established and the corresponding formulation and discrete control equations were derived.Finally, several methods of stress intensity factor calculation were introduced. Through the numerical analysis of classical plane fracture models and the comparison of the present results and the analytical solutions, the accuracy and reliability of the present method were verified. Besides, the comparison of multiple sets of data also indicated the parameters affecting the stress intensity factor. The appropriate parameter values were decided to improve the calculation accuracy. This work provides foundation for further research of crack propagation.The work of this paper provides a new method for the crack extended problem, and enrichs the engineering application of the Hamilton theoretical system.