Yuanbing Miao

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Meshless Natural Neighbour Method and Its
Application in Elasto-plastic Problems

• Yuanbing Miao
• Graduate Student, Department of Geotechnical
  Engineering,
• Tongji University, Shanghai, P.R. China
• 2004.11.6

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Overview

• Introduction
• Theory of Meshless Natural Neighbour Method
• Numerical Examples
• Conclusions

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Introduction

• Finite Element Method(FEM) and Meshless Methods
• Classification of Meshless Methods
• Advantages of Meshless Natural Neighbour Method
  (MNNM)

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FEM and Meshless Methods

Element Mesh by FEM
Nodal points by Meshless Methods
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Difficulties In FEM

• Finite element method are difficult to analyze
  problems involving
• Moving discontinuities, e.g., crack propagation.
• High gradients, e.g., shock waves.
• Large material distortion, e.g., manufacturing
  processes.
• Multiple-scale phenomenon, e.g., strain
  localization.

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Advantages of Meshless Methods

• It requires neither domain nor surface
  discretization.
• Better handles large material distortion.
• Customized shape functions for desired
  smoothness.
• High solution accuracy and rate of convergence.

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Classification of Meshless Method
SPH
Kernel function approximation
RKPM,
EFG
Least square approximation
FPM
Hp Clouds,
NEM
Natural neighbour interpolation
NNM
MNNM,
Others
Radial basis function,
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• Notice that the first two classes are base on
  approximation method
• Difficult to impose essential boundary
  conditions.
• Difficult to treat discontinuities.
• Methods based on Natural neighbour interpolation
  such as NEM and NNM
• Essential boundary conditions can be imposed as
  easily as FEM.
• But the Delaunay triangulation of the whole
  analysis region is needed to construct the
  natural neighbour shape function .

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Advantages of MNNM

• Its shape function takes full advantages of
  natural neighbour shape function.
• Adopts the means similar to EFG to seek the
  natural neighbour points of the intergal points
  and the Delaunay triangulation of the whole
  region are avoided.

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Theory of MNNM
Search for natural neighbours
The algorithm for the neighbour-search in MNNM is
based on the local Delaunay triangles

• Set up a set of distinct nodes at the arbitrary
  geometry shape of domain.
• Let the initial influence nodes of the point P be
  confined within the dashed lines of the square as
  shown in Fig.1.

Fig.1 Discrete model of regionO and its arbitrary
integrate point p(x,y)
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Search for natural neighbours

• Find the node 1 which is the nearest to sample
  point P.
• Starting with edge P-1, use the empty
  circumcircle criterion to find node 2, then form
  a Delaunay triangle ?P12.
• In the same way, we can form a set of locally
  defined triangles.
• The nodes 1-6 are just the natural neighbours of
  the point P.

Delaunay triangle
Fig.2 natural neighbour of point p
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Laplace interpolation
The voronoi cell of the point P
Laplace shape function for node i is defined as
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From the definition of Laplace interpolation, the
following properties can be obtained easily
These properties make the Laplace interpolant the
only meshless data interpolation which can
exactly satisfy essential boundary conditions.
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After defining of the shape function, The global
forms of displacement approximations of point
can be written as
Using the same procedure as FEM, we have applied
the MNNM to the analysis of two-dimensional
elasto-plastic problems.
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Numerical Examples
Example I Thick Cylinder
The material is perfectly elasto-plastic with E
85570Kpa, µ 0.3. The Von-Mises yield criterion
is adopted and ss 10Kpa.
A thick cylinder subjected to internal pressure
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Comparison of normal stress
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Comparison of shear stress
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Example II Cantilever Beam
A rectangle cantilever beam subjected to
concentrated load at the free end
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Comparison of sx
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Conclusions

• The MNNM takes full advantages of NNM and avoid
  the Delaunay triangulation of the whole region.
• By combining with incremental initial stress
  method, the MNNM has been applied into
  elasto-plastic analysis successfully.
• It is expected that the proposed method can be
  used to solve some more complicated geotechnical
  engineering, as well as large deformation
  analysis such as landsliding and pile
  penetration.

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• Thank you for your attention!