The Best of Both Worlds: Coupling FE & Meshless Methods.

Meshless methods offer the ability to simulate large strain problems without the issues associated with mesh deformation that can hamper the application of FE to high speed dynamic events. These methods are typically formulated in a similar way to FE, in as much as they characterise the solution as a weighted sum of basis functions with each basis function being associated with a node, but as the name suggests, they dispense with the mesh used to link the solution values at the nodes in FE. The basis functions and the method of linking the solution parameters (which are not necessarily nodal solution values) varies from method to method, but many methods effectively assign a "domain of influence" to each node and formulate equations linking all the nodes whose domains of influence overlap the domain of influence of a given node (conceptually this is similar to the FE approach, with the domain of influence of a node being all of the elements of which that node is a member). As an example, figure 1 shows a sketch of some nodes in a domain connected in two ways: the upper sketch shows a possible finite element mesh over part of the domain, and the lower sketch shows each node as having a circular domain of influence, with the overlapping regions shown as a darker shade.