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What areParticle and Meshless Methods?
NAFEMS [email protected]
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What are Particle and Meshless Methods?
Where Can I Learn More?
• S. Li and W. K. Liu. Meshfree particle methods. Springer,
2004. ISBN 978-3-430-22256-9.
• L. Wright. Hybrid Methods: Review of particle-based numerical
methods and their couplingto other continuum methods, NPL Report
MS10.
• Blumrich, R., Grün, N. and Schuetz, T. C. (2016) Numerical
Methods. In: Schuetz, T. C. (Ed)Aerodynamics of Road Vehicles, 5th
Edition. Warrendale: SAE International.
• B. Peters and A. Donoso. Why Do Discrete Element Analysis?,
NAFEMS 2018 ISBN 978-1-910643-43-3
Figure 3: Simulating the discharge of a hopper froma plate using
the Discrete Element Method
WT07 - What are Particle Methods?_NAFEMS 07/08/2018 09:23 Page
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What are Particle and Meshless Methods? www.nafems.org
What are Particle and Meshless Methods?
Why Do We Need New Methods?
Finite element methods are a very powerful tool for numerical
solution of engineeringproblems. They approximate the solution to
the partial differential equations that governphysics by:• defining
a set of points and a mesh that connects them (discretisation),
• specifying a set of shape functions or basis functions, each
associated with a point andonly non-zero in some neighbourhood of
that point, that can be used to write thesolution in terms of a
finite number of unknown parameters, typically the values of
thesolution at the points, (local approximation), and
• applying a form of the partial differential equation to the
approximate solution tocreate a large matrix of simultaneous
equations that can be solved to obtain theunknown parameter
values
There are still some problems that FE methods do not solve
efficiently or accurately. Inparticular, FE methods are not ideally
suited for solution of dynamic problems involving a largeamount of
deformation because an accurate solution requires regularly-shaped
elements andthe movement of the nodes can lead to mesh distortion
and badly-shaped elements. Oneapproach to avoiding the problems
associated with mesh deformation is remeshing, where anew mesh of
regularly shaped elements is created during the solution process.
This approachis computationally expensive and has potential loss of
accuracy as it requires the mapping ofthe solution from one mesh to
the next.
Are There Disadvantages to Using Particle and Meshless
Methods?Yes. The methods can be computationally expensive, due to
both the need to update the listof which groups of points are
linked via equations and the need to use numerical
quadraturetechniques for accurate evaluation of complicated
integrals.
A further complication is that the imposition of boundary
conditions is not always simple.Because the unknown parameters in
FE are the values of the quantity of interest at the nodalpoints,
imposition of (say) fixed displacement conditions just involves
fixing a set of parametervalues, but the parameters in most
meshless methods are not related to the values of thequantity of
interest so simply. Imposing these conditions (for instance through
Lagrangemultipliers) can generate extra equations and further
computational cost.
One way to reduce the effects of these disadvantages is to only
use a meshless method whereit is needed, to use FE in areas of low
deformation, and to couple the methods together overan interface
region in a way that ensures continuity of forces and
displacements. Techniquesfor coupling to FE have been developed for
many meshless methods.
The methods typically feature a length parameter that defines
whether two points are linkedvia equations. It is not always simple
to choose a value for this parameter that balancescomputational
efficiency and numerical stability.
What Are The Advantages of Particle and Meshless
Methods?Meshless methods avoid the problems associated with mesh
deformation by not having amesh. The methods still use local
approximations, centred on a set of points, anddiscretisation to
calculate an approximate solution to the governing equations, but
thedomains over which each local approximation is applied do not
(necessarily) deform with thematerial. The approximation function
associated with each point has a fixed ``domain ofinfluence'' on
which they are non-zero, and the connectivity of a given point
depends onwhich other points lie within that point's domain of
influence at any given time.
FE methods are also not always suitable for problems involving
rapid changes in value of avariable, such as shock waves or other
discontinuities, because the mesh required to describesuch rapid
changes accurately can be too computationally expensive. Meshless
methodsgenerally use local approximations that are better able to
cope with sharp changes. In somecases (similar to the extended
finite element method, XFEM) extra functions and
associatedparameters are introduced to describe behaviour
accurately close to a discontinuity.
Some meshless methods are designed to model problems that
involve many interactingparticles or lumps, such as powders,
granular flow, and rock mechanics. Some of thesemethods can
simulate deformation of individual particles as well as the
interaction betweenthem.
Which Methods Might I Find Useful? Smoothed particle
hydrodynamics (SPH) is probably the most “mature” meshless method
and isalready available in several commercial packages. It is used
for solid mechanics that involve a lotof distortion, including
fragmentation, and CFD problems such as sloshing. Reproducing
kernelparticle methods are a family of methods related to SPH that
address some of the weaknessesof SPH.
Material point methods represent the material as a set of points
that move through a fixed grid.Each point has a set of values
carried with it as it moves, such as mass, velocity, and stress,
thatare updated as the analysis progresses. The methods have been
successfully applied toproblems in geotechnics and particulate
flow.
Discrete element methods are used to model particulate
interaction, and can be combined withtraditional FE (FDEM) to
simulate interaction of a large collection of deformable objects
such asmasonry structures and rock systems.
Element-free Galerkin methods are a family of meshless methods
that use the same formulationapproach as FE and are therefore
well-suited to coupling with FE in a consistent way.
Lattice Boltzmann methods simulate unsteady fluid flows using a
finite number of velocitydistribution functions to define particle
densities on a regular static lattice of points. These areupdated
by considering both the propagation of particles along fixed
vectors and collisionsbetween them. The approach may be combined
with turbulence-closure models, wallfunctions, porous media and
dispersed phase and multi-phase flow models.
Figure 1: Meshless method: Element-free Galerkin Figure 2: An
example of a 3D Lattice Boltzmann lattice unit. Arrowsindicate the
various velocity directions that a particle could take,from the
centre of the cell at 1 to the corners and the face centres
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