Meshless Simulation Methods

It has been well known since the early days of finite elements that mesh distortion leads to a significant lack of accuracy in simulations. Mesh generation is certainly the most expensive task in building a simulation model, so avoiding the need for meshing, or even remeshing, has been seen as an attractive goal to achieve, one that seemed out of reach for decades.

Meshless methods were born precisely to counter this lack of accuracy resulting from mesh distortion, and to try and minimize the tedious step of mesh construction. In essence, meshless methods are simulation techniques (based either on collocation or Galerkin-type approaches, although we'll be concentrating mainly on Galerkin-type methods) that don't need a mesh to approximate the problem fields (displacement field for solid mechanics, velocity field for fluid mechanics, temperature field for thermal problems...).

After a few years of intense activity, punctuated by numerous successes, the research community thought that meshless methods would one day replace finite elements in our commercial codes. Today, we know that this will not be the case (unless a completely unexpected revolution emerges in the field), and finite elements are set to endure, despite the fact that meshless methods have enabled unprecedented simulations to be carried out in fields such as free-surface flow, fracture and fragmentation, processes involving very large transformations, and so on.

In addition, meshless methods have greatly helped to understand the mathematical foundations of finite elements, and to build better approximations with less user intervention.

In this article, we review the most relevant aspects of meshless methods and, while the existing bibliography is rather substantial, we provide the most important references for the interested reader, inclined to continue reading and learning in this fascinating field.