Shape-Free Finite Element Method: Another Way between Mesh and Mesh-Free Methods

Performances of the conventional finite elements are closely related to the mesh quality. Once distorted elements are used, the accuracy of the numerical results may be very poor, or even the calculations have to stop due to various numerical problems. Recently, the author and his colleagues developed two kinds of finite element methods, named hybrid stress-function (HSF) and improved unsymmetric methods, respectively. The resulting plane element models possess excellent precision in both regular and severely distorted meshes and even perform very well under the situations in which other elements cannot work. So, they are called shape-free finite elements since their performances are independent to element shapes. These methods may open new ways for developing novel high-performance finite elements. Here, the thoughts, theories, and formulae of above shape-free finite element methods were introduced, and the possibilities and difficulties for further developments were also discussed.