ADVANCES IN SHAPE-FREE FINITE ELEMENT METHODS: A REVIEW

As the most important tool for simulation and computation, the finite element method has been widely applied in engineering and scientific problems. However, this powerful method still suffers from some inherent deficiencies, such as the sensitivity problem to mesh distortion and so on, which have not been completely solved yet. This paper systematacially introduces some research developments on new finite element methods, i.e., the shape-free finite element methods, including the hybrid stress-function elements for plane and 2D fracture problems, the hybrid displacement-function elements for Reissner-Mindlin plate bending problem, and new unsymmetric finite element for plane and 3D elasticity. Based on the existing hybrid stress and unsymmetric finite element methods, some advanced techniques, including the analytical trial function method, new natural coordinate method, generalized conforming technique, etc., were adopted in the aforementioned new approaches. The resulting new finite element models possess high precision and good robustness. In particular, they can keep their level of performance even in extremely distorted meshes. Furthermore, some historical challenges in the finite element method, such as the limitation defined by the MacNeal's theorem, the edge effect problem of the Resissner-Mindlin plate, were also successfully solved. At the end of this paper, some features of the new methods and further development are discussed.