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 u ( x , t ) {\displaystyle u(x,t)} , represented as a mesh of data values u i n {\displaystyle u_{i}^{n}} at points x i {\displaystyle x_{i}} , where
i = 0 , 1 , 2... {\displaystyle i=0,1,2...}
n = 0 , 1 , 2... {\displaystyle n=0,1,2...}
x i + 1 − x i = h ∀ i {\displaystyle x_{i+1}-x_{i}=h\ \forall i}
t n + 1 − t n = k ∀ n {\displaystyle t_{n+1}-t_{n}=k\ \forall n}
We can define the derivatives that occur in the equation being simulated using some finite difference formulae on this domain, for example
∂ u ∂ x = u i + 1 n − u i − 1 n 2 h {\displaystyle {\partial u \over \partial x}={u_{i+1}^{n}-u_{i-1}^{n} \over 2h}}
and
∂ u ∂ t = u i n + 1 − u i n k {\displaystyle {\partial u \over \partial t}={u_{i}^{n+1}-u_{i}^{n} \over k}}
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