History and Recent Development
One of the earlier methods without a mesh is smoothed particle hydrodynamics, presented in 1977. Many methods listed in the next section are developed during the past 30 some years.
Recent advances on meshfree methods aim at the development of computational tools for automation in modeling and simulations. This is enabled by the so-called weakened weak (W2) formulation based on the G space theory. The W2 formulation offers possibilities for formulate various (uniformly) "soft" models that works well with triangular meshes. Because triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence automation in modeling and simulation. In addition, W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the Smoothed Point Interpolation Methods (or S-PIM). The S-PIM can be node-based (known as NS-PIM or LC-PIM), edge-based (ES-PIM), and cell-based (CS-PIM). The NS-PIM was developed using the so-called SCNI technique. It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free. The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments.
The W2 formulation has also led to the development of combination of meshfree techniques with the well-developed FEM techniques, and one can now use triangular mesh with excellent accuracy and desired softness. A typical such a formulation is the so-called Smoothed Finite Element Method (or S-FEM) The S-FEM is the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler.
It is a general perception that meshfree methods are much more expensive than the FEM counterparts. The recent study has found however, the S-PIM and S-FEM can be much faster than the FEM counterparts.
The S-PIM and S-FEM works well for solid mechanics problems. For problems, the formulation can be simpler, via strong formulation. A Gradient Smoothing Methods (GSM) has also be developed recently for problems, implementing the gradient smoothing idea in strong form. The GSM is similar to, but uses gradient smoothing operations exclusively in nested fashions, and is a general numerical method for PDEs.
Read more about this topic: Meshfree Methods
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