Meshfree methods

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.

If the mesh becomes tangled or degenerate during simulation, the operators defined on it may no longer give correct values.

, 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

Smoothed-particle hydrodynamics (SPH), one of the oldest meshfree methods, solves this problem by treating data points as physical particles with mass and density that can move around over time, and carry some value

is a kernel function that operates on nearby data points and is chosen for smoothness and other useful qualities.

One disadvantage of SPH is that it requires extra programming to determine the nearest neighbors of a particle.

only returns nonzero results for nearby particles within twice the "smoothing length" (because we typically choose kernel functions with compact support), it would be a waste of effort to calculate the summations above over every particle in a large simulation.

So typically SPH simulators require some extra code to speed up this nearest neighbor calculation.

The main drawbacks of SPH are inaccurate results near boundaries and tension instability that was first investigated by Swegle.

This first method called the diffuse element method[4] (DEM), pioneered by Nayroles et al., utilized the MLS approximation in the Galerkin solution of partial differential equations, with approximate derivatives of the MLS function.

Thereafter Belytschko pioneered the Element Free Galerkin (EFG) method,[5] which employed MLS with Lagrange multipliers to enforce boundary conditions, higher order numerical quadrature in the weak form, and full derivatives of the MLS approximation which gave better accuracy.

Around the same time, the reproducing kernel particle method[6] (RKPM) emerged, the approximation motivated in part to correct the kernel estimate in SPH: to give accuracy near boundaries, in non-uniform discretizations, and higher-order accuracy in general.

[8] RKPM and other meshfree methods were extensively developed by Chen, Liu, and Li in the late 1990s for a variety of applications and various classes of problems.

[26] The common weak form requires strong enforcement of the essential boundary conditions, yet meshfree methods in general lack the Kronecker delta property.

This make essential boundary condition enforcement non-trivial, at least more difficult than the Finite element method, where they can be imposed directly.

However, it was later realized that low-energy modes were still present in SCNI, and additional stabilization methods have been developed.

[26] More recently, a framework has been developed to pass arbitrary-order patch tests, based on a Petrov–Galerkin method.

[29] One recent advance in meshfree methods aims at the development of computational tools for automation in modeling and simulations.

Because a triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence enables 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).

This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated.

[28] It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.

The recent study has found however, some meshfree methods such as the S-PIM and S-FEM can be much faster than the FEM counterparts.

[37][38] The GSM is similar to [FVM], but uses gradient smoothing operations exclusively in nested fashions, and is a general numerical method for PDEs.

Nodal integration has been proposed as a technique to use finite elements to emulate a meshfree behaviour.