Meshfree methods

Meshfree methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. Traditional simulation algorithms relied on a grid or a mesh, meshfree methods in contrast use the geometry of the simulated object directly for calculations. Meshfree methods exist for fluid dynamics as well as for solid mechanics. Some methods are able to handle both cases.

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Description

Meshfree methods eliminate some or all of the traditional mesh-based view of the computational domain and rely on a particle (either Lagrangian or Eulerian) view of the field problem.

A goal of meshfree methods is to facilitate the simulation of increasingly demanding problems that require the ability to treat large deformations, advanced materials, complex geometry, nonlinear material behavior, discontinuities and singularities. For example the melting of a solid or the freezing process can be simulated using meshfree methods.

There is also an additional 'sales' oriented aspect of this name. Meshfree (or 'meshless' as this is also used) methods seem attractive as alternative to finite elements (FEM) for the general engineering community, which consider the process of generating finite element meshes as more difficult and expensive than the remainder of analysis process.

History

One of the earlier methods without a mesh is smoothed particle hydrodynamics, presented in 1977.

List of methods and acronyms

The following numerical methods are generally considered to fall within the general class of "meshfree" methods. Acronyms are provided in parentheses.

• Smoothed particle hydrodynamics (SPH) (1977)
• Diffuse element method (DEM) (1992)
• Element-free Galerkin method (EFG / EFGM) (1994)
• Reproducing kernel particle method (RKPM) (1995)
• hp-clouds
• Natural element method (NEM)
• Material point method (MPM)
• Meshless local Petrov Galerkin (MLPG)
• Generalized finite difference method (GFDM)
• Particle-in-cell (PIC)
• Moving particle finite element method (MPFEM)
• Finite cloud method (FCM)
• Boundary node method (BNM)
• Boundary cloud method (BCM)
• Method of Finite Spheres (MFS)

Related methods:

• Moving least squares (MLS) - provide general approximation method for arbitrary set of nodes
• Partition of unity methods (PoUM) - provide general approximation formulation used in some meshfree methods
• eXtended FEM, Generalized FEM (XFEM, GFEM) - variants of FEM (finite element method) combining some meshless aspects

See also

• Finite element method
• Continuum mechanics
• Boundary element method

References

• S. Li, W. K. Liu (2004). Meshfree Particle Methods, Berlin: Springer Verlag. ISBN 3-540-22256-1
• G. R. Liu, M. E. Gershwin, A. M. Lucas (2002). Mesh Free Methods, CRC Press. ISBN 0-8493-1238-8
• T. Belytschko, J. S. Chen (2007). Meshfree and Particle Methods, John Wiley and Sons Ltd. ISBN 0-470-84800-6