On a Finite Element Method with Variable Element Topology

Rashid, M. M., & Gullett, P. (). On a Finite Element Method with Variable Element Topology. Computer Methods in Applied Mechanics and Engineering. Elsevier. 190(11), 1509-1527.

A new finite-element-like approximation method for problems in solid mechanics, here called the variable-element-topology finite element method (VETFEM), is presented. The displacement-based variational basis of the conventional finite element method (FEM) is retained in the VETFEM, as is the discretization of the problem into elements. However, VETFEM elements are not subject to any of the geometric or topological restrictions of conventional elements: they may contain any number of nodes in any arrangement. The polynomial VETFEM shape functions emerge from a constrained minimization process on each element, instead of from an isoparametric transformation from a parent element as in the conventional FEM. All the powerful features normally associated with the conventional FEM are exhibited by the new method. In addition, because of the absence of geometric and/or topological restrictions on the elements, automatic mesh generation is enormously simplified. For this reason, the VETFEM is thought to be particularly useful for problems involving very complex geometry, adaptive remeshing, or crack extension.