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Extended finite element and meshfree methods: 2. Meshfree approximation and Partition of Unity Timon

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... and Structures, T. Belytschko, W.K. Liu and B. Moran, John Wiley and Sons, 2001 ... SPH method by Lucy and Monaghan [1977] Central particle. Neighbor ... – PowerPoint PPT presentation

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Title: Extended finite element and meshfree methods: 2. Meshfree approximation and Partition of Unity Timon


1
Extended finite element and meshfree methods2.
Meshfree approximation and Partition of Unity
Timon RabczukProf. Wolfgang Wall
2
Outline
  • Ergänzungsfach im Diplomstudiengang
  • Lecture notes will be done at the beginning of
    december
  • A written examen will be at the end of this
    semester. Only a non-programmable pocket
    calculator will be allowed at the examen. A
    detailed date will be announced later. It can
    also be found at our website
  • http//www.lnm.mw.tum.de/teaching/xfem/

3
Outline
  • Recommended literature
  • Classification and overview of meshfree methods,
    T.P Fries and H. Matthies, University Brunswick
    http//www.digibib.tu-bs.de/?docid00001418
  • Meshfree and Particle Methods, T. Belytschko and
    J.S. Chen, John Wiley and Sons, 2007
  • Nonlinear Finite Elements for Continua and
    Structures, T. Belytschko, W.K. Liu and B. Moran,
    John Wiley and Sons, 2001
  • Moes N., Dolbow J., Belytschko T A finite
    element method for crack growth without
    remeshing, Interational Journal for Numerical
    Methods in Engineering, 1999, 46(1), 133-150
  • Belytschko T., Moes N., Usui S., Parimi C.
    Arbitrary discontinuities in finite elements,
    Interational Journal for Numerical Methods in
    Engineering, 2001, 50(4), 993-1013
  • Advanced Fracture Mechanics, Lectures on
    Fundamentals of Elastic, Elastic-Plastic and
    Creep Fracture, Imperial College London,
    Department of Mechanical Engineering, 2002-2203

4
Outline
  • Meshfree approximation
  • Partition of Unity
  • Completeness, conservation, stability,
    convergence, continuity
  • Weighting function
  • Specific meshfree methods SPH, corrected SPH
    forms

5
Meshfree approximation
Central particle
FE
Meshfree
Neighbor particle
Meshfree approximation
Domain of influence (support)
6
Partition of unity
Partition of unity
Linear FEM
7
Partition of unity
Partition of unity
Quadratic FEM
1
2
3
8
Partition of unity
Partition of unity
The Kronecker-delta property is not
fulfilled in meshfree methods. This causes
difficulties in imposing Dirichlet BCs.
9
Partition of unity
10
Partition of unity
When the size of the domain of influence in
meshfree methods is decreased accordingly, then
the finite element shape functions are recovered.
11
Completeness
Completeness
Completeness is expressed in terms of the order
of the polynomial which must be represented
exactly. Completeness is often referred to
reproducing conditions. An approximation is
called complete of order n, if the approximation
is able to reproduce a polynomial of order n
exactly.
Completeness is important for the convergence of
a discretization.
12
Completeness
Completeness
The derivative reproducing conditions are also
important for several meshfree methods. In two
dimensions, the derivative reproducing conditions
for a linear field are
13
Completeness and conservation
An approximation that is of zeroth-order
completeness guarantees gallilean invariance.
An approximation that is of zeroth-order
completeness guarantees linear momentum.
Conservation of linear momentum requires that the
rate of change of linear momentum due to internal
forces is zero. Thus, in the absence of external
forces and body forces, conservation of linear
momentum requires that
14
Completeness and conservation
This requires
15
Completeness and conservation
An approximation that is linear complete
guarantees angular momentum. Conservation of
angular momentum requires that any change is
exclusively due to external forces. We will show
that the change in angular momentum in the
absence of external forces vanishes. The time
rate of change in angular momentum can be
expressed as
16
Completeness and conservation
17
Completeness, stability and convergence
A method is convergent if it is consistent and
stable, Lax-Richtmeyr. According to Strikwerda
(1989), a difference scheme Luf (L is the
differential operator, Lh the corresponding
difference operator) is consistent of order k for
any smooth function v if
In Galkerin methods, completeness takes the role
of consistency. Stability ensures that a small
defect stays small.
A method is convergent of order k (kgt0) if
18
Continuity
A method is considered to be n-th order
continuous (Cn) if their shape functions are n
times continuous differentiable.
19
Meshfree methods
Linear meshfree
Quadratic meshfree
20
Kernel function
Weighting/kernel/window functions
Cuartic B-Spline
21
Kernel function
h
h
22
Kernel function
Requirements usually imposed on the kernel
functions
23
Kernel function
Extension of the kernel function into higher
order dimensions Rectangular support
Circular support
24
Kernel function
25
Kernel function
The cubic B-Spline
26
Kernel function
The derivative of the Cubic B-spline
The cubic B-Spline WJ(X)
27
Kernel function
28
Kernel function
29
Kernel function
Lagrangian and Eulerian kernels
Eulerian kernels are usually applied for large
deformations. Eulerian kernels show a so-called
tensile instability, meaning methods based on
Eulerian kernels become instable when tensile
stresses occur. Methods based on Eulerian kernels
are generally not well-suited to model crack
initiation since such methods are usually not
capable of capturing the onset of fracture
properly. Therefore, we recommend the use of
Lagrangian kernels. When the deformations are too
large, then the Lagrangian kernels gets instable
when the domain of influence in the current
configuration is extremely distorted.
30
Stability of Lagrangian and Eulerian kernels
  • Instabilities due to (Belytschko et al. 2003)
  • Rank deficiency
  • Tensile instability (Swegle et al. 1993)
  • Material instability

Hyperelastic material law with strain softening
31
Meshfree Methods
Central particle
SPH method by Lucy and Monaghan 1977
Neighbor particle
Domain of influence
32
Meshfree Methods
Central particle
SPH method by Lucy and Monaghan 1977
Neighbor particle
Subtraction of
gives
If linear consistency is fulfilled, above is
guaranteed by the symmetry of the kernel
Domain of influence
33
Meshfree Methods
Central particle
SPH method by Lucy and Monaghan 1977
Neighbor particle
Domain of influence
34
Meshfree Methods
35
Meshfree Methods
36
Meshfree Methods
37
Meshfree Methods
Different ways to discretize a body
38
Meshfree Methods
Central particle
SPH method by Lucy and Monaghan 1977
Neighbor particle
Symmetrization
Domain of influence
39
Meshfree methods
Shepard functions
40
Meshfree methods
Krongauz-Belytschko correction
41
Meshfree methods
Randles-Libersky correction
Non-Symmetrized version
Symmetrized version
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