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.

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 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

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)
• Dissipative particle dynamics (DPD) (1992)
• Element-free Galerkin method (EFG / EFGM) (1994)
• Reproducing kernel particle method (RKPM) (1995)
• Finite pointset method (FPM) (1998)
• hp-clouds
• 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)
• Boundary cloud method (BCM)
• Method of fundamental solution(MFS)
• Method of particular solution (MPS)
• Method of Finite Spheres (MFS)
• Discrete Vortex Method (DVM)

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
• Local maximum-entropy (LME) – see Arroyo & Ortiz (2006)
• Space-Time Meshfree Collocation Method (STMCM) – see Netuzhylov (2008), Netuzhylov & Zilian (2009)
• Mesh-free radial point interpolation method (RPIM)
• Meshfree moving Kriging interpolation method (MK)

See also

• Continuum mechanics
• Boundary element method
• Immersed Boundary Method

References

• 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, doi:10.1002/nme.1534.
• T. Belytschko, J. S. Chen (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
• G. R. Liu (2002). Mesh Free Methods, CRC Press. ISBN 0-8493-1238-8
• S. Li, W. K. Liu (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: 1615–1636, 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, doi:10.1002/nme.2638

External links

• The first available commercial meshfree CFD code
• The USACM blog on Meshfree Methods