Meshfree methods
Encyclopedia
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.
) 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.
, presented in 1977. 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 . 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) . The S-PIM can be node-based (known as NS-PIM or LC-PIM) , edge-based (ES-PIM) , and cell-based (CS-PIM) . The NS-PIM was developed using the so-called SCNI technique . It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free . 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) 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 .
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. The GSM is similar to [FVM], but uses gradient smoothing operations exclusively in nested fashions, and is a general numerical method for PDEs.
Related methods:
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
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
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
as well as for solid mechanics
Solid mechanics
Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...
. 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 EulerianLagrangian and Eulerian coordinates
In fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
) 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
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
(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 hydrodynamicsSmoothed particle hydrodynamics
Smoothed-particle hydrodynamics is a computational method used for simulating fluid flows. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography...
, presented in 1977. 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
G space
G space is a functional space used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solve PDEs in particular solid mechanics as well as fluid dynamics problems.-Description:For simplicity we...
theory . 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) . The S-PIM can be node-based (known as NS-PIM or LC-PIM) , edge-based (ES-PIM) , and cell-based (CS-PIM) . The NS-PIM was developed using the so-called SCNI technique . It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free . 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) 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 .
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. 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 hydrodynamicsSmoothed particle hydrodynamicsSmoothed-particle hydrodynamics is a computational method used for simulating fluid flows. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography...
(SPH) (1977) - Diffuse element methodDiffuse element methodThe diffuse element method 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 Compiegne, in 1992.It is in concept rather similar to the...
(DEM) (1992) - Dissipative particle dynamicsDissipative particle dynamicsDissipative particle dynamics is a stochastic simulation technique for simulating the dynamic and rheological properties of simple and complex fluids. It was initially devised by Hoogerbrugge and Koelman to avoid the lattice artifacts of the so-called lattice gas automata and to tackle...
(DPD) (1992) - Element-free Galerkin method (EFG / EFGM) (1994)
- Reproducing kernel particle method (RKPM) (1995)
- Finite pointset methodFinite pointset methodIn applied mathematics, the Finite Pointset Method is a method for the solution of the equations governing viscous fluid flows, including the effects of heat and mass transfer. FPM models problems in continuum mechanics. The method solves not only fluid flows, but also problems with elastic or...
(FPM) (1998) - hp-clouds
- Natural element method (NEM)
- Material Point MethodMaterial Point MethodThe Material Point Method , is an extension of the Particle-in-cell Method in computational fluid dynamics to computational solid dynamics, and is a Finite element method -based particle method. It is primarily used for multiphase simulations, because of the ease of detecting contact without...
(MPM) - Meshless local Petrov Galerkin (MLPG)
- Moving particle semi-implicit (MPS)
- Generalized finite difference method (GFDM)
- Particle-in-cellParticle-in-cellThe Particle-in-Cell method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution such as densities and currents are computed...
(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).
- Meshfree local radial point interpolation method (RPIM).
- Local Radial Basis Function Collocation Method (LRBFCM)
Related methods:
- Moving least squaresMoving least squaresMoving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the reconstructed value is requested....
(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
- eXtended FEMExtended finite element methodThe extended finite element method , also known as generalized finite element method or partition of unity method is a numerical technique that extends the classical finite element method approach by enriching the solution space for solutions to differential equations with discontinuous...
, Generalized FEM (XFEM, GFEM) – variants of FEM (finite element method) combining some meshless aspects - Smoothed finite element methodSmoothed finite element methodSmoothed Finite Element methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method...
(S-FEM) (2007) - Gradient smoothing method (GSM) (2008)
- Local maximum-entropy (LME) – see
- Space-Time Meshfree Collocation Method (STMCM) – see ,
See also
- Continuum mechanicsContinuum mechanicsContinuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
- Smoothed finite element methodSmoothed finite element methodSmoothed Finite Element methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method...
- G spaceG spaceG space is a functional space used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solve PDEs in particular solid mechanics as well as fluid dynamics problems.-Description:For simplicity we...
- Weakened weak formWeakened weak formWeakened weak form is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems....
- Boundary element methodBoundary element methodThe boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations . It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture...
- Immersed Boundary MethodImmersed boundary methodThe immersed boundary method is an approach – in computational fluid dynamics – to model and simulate mechanical systems in which elastic structures interact with fluid flows...
- Stencil codesStencil codesStencil codes are a class of iterative kernelswhich update array elements according to some fixed pattern, called stencil.They are most commonly found in the codes of computer simulations, e.g...