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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.


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.
History and recent development

One of the earlier methods without a mesh is smoothed particle hydrodynamics, presented in 1977.[1] Many methods listed in the next section are developed during the past 30 some years.

Recent advances on meshfree methods aim at the development of computational tools for automation in modeling and simulations. This is enabled by the so-called weakened weak (W2) formulation based on the G space theory.[2] The W2 formulation offers possibilities for formulate various (uniformly) "soft" models that works well with triangular meshes. Because triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence 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). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. Typical W2 models are the Smoothed Point Interpolation Methods (or S-PIM).[3] The S-PIM can be node-based (known as NS-PIM or LC-PIM),[4] edge-based (ES-PIM),[5] and cell-based (CS-PIM).[6] The NS-PIM was developed using the so-called SCNI technique.[7] It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.[8] The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments.

The W2 formulation has also led to the development of combination of meshfree techniques with the well-developed FEM techniques, and one can now use triangular mesh with excellent accuracy and desired softness. A typical such a formulation is the so-called Smoothed Finite Element Method (or S-FEM) [9] The S-FEM is the linear version of S-PIM, but with most of the properties of the S-PIM and much simpler.

It is a general perception that meshfree methods are much more expensive than the FEM counterparts. The recent study has found however, the S-PIM and S-FEM can be much faster than the FEM counterparts.[3][9]

The S-PIM and S-FEM works well for solid mechanics problems. For [CFD] problems, the formulation can be simpler, via strong formulation. A Gradient Smoothing Methods (GSM) has also be developed recently for [CFD] problems, implementing the gradient smoothing idea in strong form.[10][11] The GSM is similar to [FVM], but uses gradient smoothing operations exclusively in nested fashions, and is a general numerical method for PDEs.
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)
Dissipative particle dynamics (DPD) (1992)
Element-free Galerkin method (EFG / EFGM) (1994)
Reproducing kernel particle method (RKPM) (1995)
Finite pointset method (FPM) (1998)
Natural element method (NEM)
Material Point Method (MPM)
Meshless local Petrov Galerkin (MLPG)
Moving particle semi-implicit (MPS)
Generalized finite difference method (GFDM)
Particle-in-cell (PIC)
Moving particle finite element method (MPFEM)
Finite cloud method (FCM)
Boundary node method (BNM)
Meshfree moving Kriging interpolation method (MK)
Boundary cloud method (BCM)
Method of fundamental solution(MFS)
Method of particular solution (MPS)
Method of Finite Spheres (MFS)
Discrete Vortex Method (DVM)
Smoothed point interpolation method (S-PIM) (2005).[3]
Meshfree local radial point interpolation method (RPIM).[3]
Local Radial Basis Function Collocation Method (LRBFCM)[12]
Viscous vortex domains method (VVD)

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
Continuous blending method (enrichment and coupling of finite elements and meshless methods) – see Huerta & Fernández-Méndez (2000)
eXtended FEM, Generalized FEM (XFEM, GFEM) – variants of FEM (finite element method) combining some meshless aspects
Smoothed finite element method (S-FEM) (2007)
Gradient smoothing method (GSM) (2008)
Local maximum-entropy (LME) – see Arroyo & Ortiz (2006)
Space-Time Meshfree Collocation Method (STMCM) – see Netuzhylov (2008), Netuzhylov & Zilian (2009)

See also

Continuum mechanics
Smoothed finite element method[9]
G space[13]
Weakened weak form[2]
Boundary element method
Immersed Boundary Method
Stencil codes


^ Gingold RA, Monaghan JJ (1977). Smoothed particle hydrodynamics - theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389
^ a b G.R. Liu. A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory and Part II applications to solid mechanics problems. International Journal for Numerical Methods in Engineering, 81: 1093–1126, 2010
^ a b c d Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
^ Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY and Han X, A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, International Journal of Computational Methods, 2(4): 645–665, 2005.
^ G.R. Liu, G.R. Zhang. Edge-based Smoothed Point Interpolation Methods. International Journal of Computational Methods, 5(4): 621–646, 2008
^ G.R. Liu, G.R. Zhang. A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method. International Journal of Computational Methods, 6(1): 147–179, 2009
^ Chen, J. S., Wu, C. T., Yoon, S. and You, Y. (2001). A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Meth. Eng. 50: 435–466.
^ G. R. Liu and G. Y. Zhang. Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM). International Journal for Numerical Methods in Engineering, 74: 1128–1161, 2008.
^ a b c Liu, G.R., 2010 Smoothed Finite Element Methods, CRC Press, ISBN 978-1-4398-2027-8.
^ G. R. Liu, George X. Xu. A gradient smoothing method (GSM) for fluid dynamics problems. International Journal for Numerical Methods in Fluids, 58: 1101–1133, 2008.
^ J. Zhang, G. R. Liu, K.Y. Lam, H. Li, G. Xu. A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems. Finite Elements in Analysis and Design, 44: 889–909, 2008.
^ Sarler B, Vertnik R. Meshfree

Liu, G.R., Liu, M.B. (2003). Smoothed Particle Hydrodynamics, a meshfree and Particle Method, World Scientific, ISBN 981-238-456-1.
Atluri, S.N. (2004), "The Meshless Method (MLPG) for Domain & BIE Discretization", Tech Science Press. ISBN 0-9657001-8-6
Arroyo, M.; Ortiz, M. (2006), "Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods", International Journal for Numerical Methods in Engineering 65 (13): 2167–2202, Bibcode 2006IJNME..65.2167A, doi:10.1002/nme.1534.
Belytschko, T., Chen, J.S. (2007). Meshfree and Particle Methods, John Wiley and Sons Ltd. ISBN 0-470-84800-6
Belytschko, T.; Huerta, A.; Fernández-Méndez, S; Rabczuk, T. (2004), "Meshless methods", Encyclopedia of computational mechanics vol. 1 Chapter 10, John Wiley & Sons. ISBN 0-470-84699-2
Liu, G.R. 1st edn, 2002. Mesh Free Methods, CRC Press. ISBN 0-8493-1238-8.
Li, S., Liu, W.K. (2004). Meshfree Particle Methods, Berlin: Springer Verlag. ISBN 3-540-22256-1
Huerta, A.; Fernández-Méndez, S. (2000), "Enrichment and coupling of the finite element and meshless methods", International Journal for Numerical Methods in Engineering 11 (11): 1615–1636, Bibcode 2000IJNME..48.1615H, doi:10.1002/1097-0207(20000820)48:11<1615::AID-NME883>3.0.CO;2-S.
Netuzhylov, H. (2008), "A Space-Time Meshfree Collocation Method for Coupled Problems on Irregularly-Shaped Domains", Dissertation, TU Braunschweig, CSE – Computational Sciences in Engineering ISBN 978-3-00-026744-4, also as electronic ed..
Netuzhylov, H.; Zilian, A. (2009), "Space-time meshfree collocation method: methodology and application to initial-boundary value problems", International Journal for Numerical Methods in Engineering 80 (3): 355–380, Bibcode 2009IJNME..80..355N, doi:10.1002/nme.2638
Alhuri. Y, A. Naji, D. Ouazar and A. Taik. (2010). RBF Based Meshless Method for Large Scale Shallow Water Simulations: Experimental Validation, Math. Model. Nat. Phenom, Vol. 5, No. 7, 2010, pp. 4–10.

External links

The first available commercial fully automated meshfree structural analysis solution
The first available commercial meshfree CFD code
The USACM blog on Meshfree Methods

Mathematics Encyclopedia

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