 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.
Contents
Description
Meshfree methods eliminate some or all of the traditional meshbased 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 the finite element method (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 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 socalled 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 remeshing 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 forcedriving 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 SPIM) ^{[3]}. The SPIM can be nodebased (known as NSPIM or LCPIM) ^{[4]}, edgebased (ESPIM) ^{[5]}, and cellbased (CSPIM) ^{[6]}. The NSPIM was developed using the socalled SCNI technique ^{[7]}. It was then discovered that NSPIM is capable of producing upper bound solution and volumetric locking free ^{[8]}. The ESPIM is found superior in accuracy, and CSPIM behaves in between the NSPIM and ESPIM. 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 welldeveloped FEM techniques, and one can now use triangular mesh with excellent accuracy and desired softness. A typical such a formulation is the socalled Smoothed Finite Element Method (or SFEM) ^{[9]} The SFEM is the linear version of SPIM, but with most of the properties of the SPIM 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 SPIM and SFEM can be much faster than the FEM counterparts ^{[3]}^{[9]}.
The SPIM and SFEM 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)
 Elementfree Galerkin method (EFG / EFGM) (1994)
 Reproducing kernel particle method (RKPM) (1995)
 Finite pointset method (FPM) (1998)
 hpclouds
 Natural element method (NEM)
 Material Point Method (MPM)
 Meshless local Petrov Galerkin (MLPG)
 Moving particle semiimplicit (MPS)
 Generalized finite difference method (GFDM)
 Particleincell (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 (SPIM) (2005)^{[3]}.
 Meshfree local radial point interpolation method (RPIM)^{[3]}.
 Local Radial Basis Function Collocation Method (LRBFCM)^{[12]}
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 & FernándezMéndez (2000)
 eXtended FEM, Generalized FEM (XFEM, GFEM) – variants of FEM (finite element method) combining some meshless aspects
 Smoothed finite element method (SFEM) (2007)
 Gradient smoothing method (GSM) (2008)
 Local maximumentropy (LME) – see Arroyo & Ortiz (2006)
 SpaceTime 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
References
 ^ Gingold RA, Monaghan JJ (1977). Smoothed particle hydrodynamics  theory and application to nonspherical 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: 10931126, 2010
 ^ ^{a} ^{b} ^{c} ^{d} Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 9781420082099
 ^ Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY and Han X, A linearly conforming point interpolation method (LCPIM) for 2D solid mechanics problems, International Journal of Computational Methods, 2(4): 645665, 2005.
 ^ G.R. Liu, G.R. Zhang. Edgebased Smoothed Point Interpolation Methods. International Journal of Computational Methods, 5(4): 621646, 2008
 ^ G.R. Liu, G.R. Zhang. A normed G space and weakened weak (W2) formulation of a cellbased Smoothed Point Interpolation Method. International Journal of Computational Methods, 6(1): 147179, 2009
 ^ Chen, J. S., Wu, C. T., Yoon, S. and You, Y. (2001). A stabilized conforming nodal integration for Galerkin meshfree 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 (LCPIM). International Journal for Numerical Methods in Engineering, 74: 11281161, 2008.
 ^ ^{a} ^{b} ^{c} Liu, G.R., 2010 Smoothed Finite Element Methods, CRC Press, ISBN: 9781439820278.
 ^ G. R. Liu, George X. Xu. A gradient smoothing method (GSM) for fluid dynamics problems. International Journal for Numerical Methods in Fluids, 58: 11011133, 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: 889909, 2008.
 ^ Sarler B, Vertnik R. Meshfree
 ^ Liu GR, ON G SPACE THEORY, INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS, Vol. 6 Issue: 2,257289, 2009
 Liu MB, Liu GR, Zong Z, AN OVERVIEW ON SMOOTHED PARTICLE HYDRODYNAMICS, INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS Vol. 5 Issue: 1, 135188, 2008.
 Liu, G.R., Liu, M.B. (2003). Smoothed Particle Hydrodynamics, a meshfree and Particle Method, World Scientific, ISBN 9812384561.
 Atluri, S.N. (2004), "The Meshless Method (MLPG) for Domain & BIE Discretization", Tech Science Press. ISBN 0965700186
 Arroyo, M.; Ortiz, M. (2006), "Local maximumentropy approximation schemes: a seamless bridge between finite elements and meshfree methods", International Journal for Numerical Methods in Engineering 65 (13): 2167–2202, doi:10.1002/nme.1534.
 Belytschko, T., Chen, J.S. (2007). Meshfree and Particle Methods, John Wiley and Sons Ltd. ISBN 0470848006
 Belytschko, T.; Huerta, A.; FernándezMéndez, S; Rabczuk, T. (2004), "Meshless methods", Encyclopedia of computational mechanics vol. 1 Chapter 10, John Wiley & Sons. ISBN 0470846992
 Liu, G.R. 1st edn, 2002. Mesh Free Methods, CRC Press. ISBN 0849312388.
 Li, S., Liu, W.K. (2004). Meshfree Particle Methods, Berlin: Springer Verlag. ISBN 3540222561
 Huerta, A.; FernándezMéndez, S. (2000), "Enrichment and coupling of the finite element and meshless methods", International Journal for Numerical Methods in Engineering 11 (11): 1615–1636, doi:10.1002/10970207(20000820)48:11<1615::AIDNME883>3.0.CO;2S.
 Netuzhylov, H. (2008), "A SpaceTime Meshfree Collocation Method for Coupled Problems on IrregularlyShaped Domains", Dissertation, TU Braunschweig, CSE  Computational Sciences in Engineering ISBN 9783000267444, also as electronic ed..
 Netuzhylov, H.; Zilian, A. (2009), "Spacetime meshfree collocation method: methodology and application to initialboundary value problems", International Journal for Numerical Methods in Engineering 80 (3): 355–380, 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. 410.
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
Categories: Numerical analysis
 Numerical differential equations
 Computational fluid dynamics
Wikimedia Foundation. 2010.
Look at other dictionaries:
Domain decomposition methods — Domain dec … Wikipedia
Mortar methods — are discretization methods for partial differential equations, which use separate finite element discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced … Wikipedia
Neumann–Neumann methods — In mathematics, Neumann–Neumann methods are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains.[1] Just like all domain decomposition methods … Wikipedia
Material Point Method — The Material Point Method (MPM), is an extension of the Particle in cell (PIC) Method in computational fluid dynamics to computational solid dynamics, and is a Finite element method (FEM) based particle method. It is primarily used for multiphase … Wikipedia
Numerical partial differential equations — is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). Numerical techniques for solving PDEs include the following: The finite difference method, in which functions are represented by… … Wikipedia
Computer simulation — This article is about computer model within a scientific context. For artistic usage, see 3d modeling. For simulating a computer on a computer, see emulator. A 48 hour computer simulation of Typhoon Mawar using the Weather Research and… … Wikipedia
List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra … Wikipedia
Diffuse element method — The diffuse element method (DEM) is a computer simulation technique used in engineering analysis. It is a meshfree method. The diffuse element method was developed by B. Nayroles, G. Touzot and Pierre Villon at the Universite de Technologie de… … Wikipedia
Computational fluid dynamics — Computational physics Numerical analysis … Wikipedia
Multigrid method — Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in (but not… … Wikipedia