ABSTRACT
Creation of MFree shape functions is the central and most important issue in MFree methods. The challenge is how to create shape functions using only nodes scattered arbitrarily in a domain without any predefined mesh to provide connectivity of the nodes. Development of more effective methods for constructing shape functions is thus one of the hottest areas of research in the area of MFree methods. A good method of shape function construction should satisfy the following basic requirements:
1. The nodal distribution can be arbitrary within reason, at least more flexible than that in the finite element method (FEM) (
arbitrary nodal distribution
). 2. The algorithm must be stable (
stability
). 3. The shape function constructed should satisfy a certain order of consistency
(
consistency
). 4. The domain for field variable approximation/interpolation (termed the support
domain or influence domain or smoothing domain) should be small compared with the entire problem domain (
compact support
). 5. The algorithm should be computationally efficient. It should be of the same order
of complexity as that of FEM (
efficiency
). 6. Ideally, the shape function should possess the Kronecker delta function property
(
delta function property
). 7. Ideally, the field approximation using the shape function should be compatible
throughout the problem domain (
compatibility
).
