Meshfree methods
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field. Motivation Numerical methods such as the finite difference method, finite-volume method, and finite element method were originally defined on meshes of data points. In such a mesh, each point has a fixed number of predefined neighbors, and this connectivity between neighbors can be used to define mathematical operators like the derivative. These operators are then used to construct the equations to simulate—such as the Euler equations or the Navier–Stokes equations. But in simulations where the material being simulated can move around (as in computational fluid dynamics) or where large deformations of the material can occur (as in simulations of plastic materials), the connectivity of the mesh can be difficult to maintain without introducing error into the simulation. If the mesh becomes tangled or degenerate during simulation, the operators defined on it may no longer give correct values. The mesh may be recreated during simulation (a process called remeshing), but this can also introduce error, since all the existing data points must be mapped onto a new and different set of data points. Meshfree methods are intended to remedy these problems. Meshfree methods are also useful for: Simulations where creating a useful mesh from the geometry of a complex 3D object may be especially difficult or require human assistance Simulations where nodes may be created or destroyed, such as in cracking simulations Simulations where the problem geometry may move out of alignment with a fixed mesh, such as in bending simulations Simulations containing nonlinear material behavior, discontinuities or singularities Example In a traditional finite difference simulation, the domain of a one-dimensional simulation would be some function , represented as a mesh of data values at points , where We can define the derivatives that occur in the equation being simulated using some finite difference formulae on this domain, for example and Then we can use these definitions of and its spatial and temporal derivatives to write the equation being simulated in finite difference form, then simulate the equation with one of many finite difference methods. In this simple example, the steps (here the spatial step and timestep ) are constant along all the mesh, and the left and right mesh neighbors of the data value at are the values at and , respectively. Generally in finite differences one can allow very simply for steps variable along the mesh, but all the original nodes should be preserved and they