What is Meshless Method?
by Yi Zhang
Let’s change the question: what is not meshless method? Well, a method with mesh, of course. That includes the FEM, FDM, FVM. Those methods essentially require the existence of a mesh, which leads to a major difficulty in solving dynamical problem, mesh generation. In highly nonlinear case, mesh must be regenerated frequently, which is very costly in computation. The meshless methods are designed to alleviate this difficulty. So, in this paper by Idelsohn et al defines a meshless method (MM) as one with shape function dependent only on node position.
However, the time consuming mesh generation is transited to same order time consuming task of determining the connectivity of nodes. In exchange of easier/no problem of conforming elements, distorted/degenerated elements, MMs induce difficulties of meeting boundary condition, anisotropic/inhomogeneous field variable. Just like a successful FEM code must have a efficient mesh generator, one MM code must have an efficient connectivity generation subroutine to be useful. At this moment there are so many such methods that bear the name of “meshless”:
Smooth Particle Hydrodynamics (SPH);
Element Free Galerkin (EFG);
Partition of Unity Finite Element Method (PUFEM); (this one is actually having much more mathematical elegance than many others in an encompass way.)
Meshless Finite Element Method (MFEM);
Natural Element Method (NEM);
Particle Finite Element Method (PFEM);