Finite Element Method

1
Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 11

• MODELLING TECHNIQUES

2
CONTENTS

• INTRODUCTION
• CPU TIME ESTIMATION
• GEOMETRY MODELLING
• MESHING
• Mesh density
• Element distortion
• MESH COMPATIBILITY
• Different order of elements
• Straddling elements

3
CONTENTS

• USE OF SYMMETRY
• Mirror symmetry
• Axial symmetry
• Cyclic symmetry
• Repetitive symmetry
• MODELLING OF OFFSETS
• Creation of MPC equations for offsets
• MODELLING OF SUPPORTS
• MODELLING OF JOINTS

4
CONTENTS

• OTHER APPLICATIONS OF MPC EQUATIONS
• Modelling of symmetric boundary conditions
• Enforcement of mesh compatibility
• Modelling of constraints by rigid body attachment
• IMPLEMENTATION OF MPC EQUATIONS
• Lagrange multiplier method
• Penalty method

5
INTRODUCTION

• Ensure reliability and accuracy of results.
• Improve efficiency and accuracy.

6
INTRODUCTION

• Considerations
• Computational and manpower resources that limit
  the scale of the FEM model.
• Requirement on results that defines the purpose
  and hence the methods of the analysis.
• Mechanical characteristics of the geometry of the
  problem domain that determine the types of
  elements to use.
• Boundary conditions.
• Loading and initial conditions.

7
CPU TIME ESTIMATION
(? ranges from 2 3)
Bandwidth, b, affects ?
- minimize bandwidth
Aim

• To create a FEM model with minimum DOFs by using
  elements of as low dimension as possible, and
• To use as coarse a mesh as possible, and use fine
  meshes only for important areas.

8
GEOMETRY MODELLING

• Reduction of a complex geometry to a manageable
  one.
• 3D? 2D? 1D? Combination?

(Using 2D or 1D makes meshing much easier)
9
GEOMETRY MODELLING

• Detailed modelling of areas where critical
  results are expected.
• Use of CAD software to aid modelling.
• Can be imported into FE software for meshing.

10
MESHING
Mesh density

• To minimize the number of DOFs, have fine mesh at
  important areas.

• In FE packages, mesh density can be controlled by
  mesh seeds.

(Image courtesy of Institute of High Performance
Computing and Sunstar Logistics(s) Pte Ltd (s))
11
Element distortion

• Use of distorted elements in irregular and
  complex geometry is common but there are some
  limits to the distortion.
• The distortions are measured against the basic
  shape of the element
• Square ? Quadrilateral elements
• Isosceles triangle ? Triangle elements
• Cube ? Hexahedron elements
• Isosceles tetrahedron ? Tetrahedron elements

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Element distortion

• Aspect ratio distortion

Rule of thumb
13
Element distortion

• Angular distortion

14
Element distortion

• Curvature distortion

15
Element distortion

• Volumetric distortion

Area outside distorted element maps into an
internal area negative volume integration
16
Element distortion

• Volumetric distortion (Contd)

17
Element distortion

• Mid-node position distortion

Shifting of nodes beyond limits can result in
singular stress field (see crack tip elements)
18
MESH COMPATIBILITY

• Requirement of Hamiltons principle admissible
  displacement
• The displacement field is continuous along all
  the edges between elements

19
Different order of elements
Crack like behaviour incorrect results
20
Different order of elements

• Solution
• Use same type of elements throughout
• Use transition elements
• Use MPC equations

21
Straddling elements
Avoid straddling of elements in mesh
22
USE OF SYMMETRY

• Different types of symmetry

Use of symmetry reduces number of DOFs and hence
computational time. Also reduces numerical error.
Mirror symmetry
Axial symmetry
Cyclic symmetry
Repetitive symmetry
23
Mirror symmetry

• Symmetry about a particular plane

24
Mirror symmetry
Consider a 2D symmetric solid
u1x 0
u2x 0
u3x 0
Single point constraints (SPC)
25
Mirror symmetry
Symmetric loading
Deflection Free Rotation 0
26
Mirror symmetry
Anti-symmetric loading
Deflection 0 Rotation Free
27
Mirror symmetry

• Symmetric
• No translational displacement normal to symmetry
  plane
• No rotational components w.r.t. axis parallel to
  symmetry plane

28
Mirror symmetry

• Anti-symmetric
• No translational displacement parallel to
  symmetry plane
• No rotational components w.r.t. axis normal to
  symmetry plane

29
Mirror symmetry

• Any load can be decomposed to a symmetric and an
  anti-symmetric load

30
Mirror symmetry
31
Mirror symmetry
32
Mirror symmetry

• Dynamic problems (e.g. two half models to get
  full set of eigenmodes in eigenvalue analysis)

33
Axial symmetry

• Use of 1D or 2D axisymmetric elements
• Formulation similar to 1D and 2D elements except
  the use of polar coordinates

Cylindrical shell using 1D axisymmetric elements
3D structure using 2D axisymmetric elements
34
Cyclic symmetry
uAn uBn
uAt uBt
Multipoint constraints (MPC)
35
Repetitive symmetry
uAx uBx
36
MODELLING OF OFFSETS
Guidelines
, offset can be safely ignored
, offset needs to be modelled
, ordinary beam, plate and shell elements should
not be used. Use 2D or 3D solid elements.
37
MODELLING OF OFFSETS

• Three methods
• Very stiff element
• Rigid element
• MPC equations

38
Creation of MPC equations for offsets
Eliminate q1, q2, q3
39
Creation of MPC equations for offsets
40
Creation of MPC equations for offfsets
d6 d1 ? d5 or d1 ? d5 - d6 0 d7
d2 - ? d4 or d2 - ? d4 - d7 0 d8 d3
or d3 - d8 0 d9 d5
or d5 - d9 0
41
MODELLING OF SUPPORTS
42
MODELLING OF SUPPORTS
(Prop support of beam)
43
MODELLING OF JOINTS
Perfect connection ensured here
44
MODELLING OF JOINTS
Mismatch between DOFs of beams and 2D solid
beam is free to rotate (rotation not transmitted
to 2D solid)
Perfect connection by artificially extending beam
into 2D solid (Additional mass)
45
MODELLING OF JOINTS

• Using MPC equations

46
MODELLING OF JOINTS
Similar for plate connected to 3D solid
47
OTHER APPLICATIONS OF MPC EQUATIONS
Modelling of symmetric boundary conditions
dn 0
ui cos? vi sin?0 or uivi tan? 0
for i1, 2, 3
48
Enforcement of mesh compatibility
Use lower order shape function to interpolate
dx 0.5(1-?) d1 0.5(1?) d3
dy 0.5(1-?) d4 0.5(1?) d6
Substitute value of ? at node 3
0.5 d1 - d2 0.5 d3 0
0.5 d4 - d5 0.5 d6 0
49
Enforcement of mesh compatibility
Use shape function of longer element to
interpolate
dx -0.5? (1-?) d1 (1?)(1-?) d3 0.5? (1?)
d5
Substituting the values of ? for the two
additional nodes
d2 0.25?1.5 d1 1.5?0.5 d3 - 0.25?0.5 d5
d4 -0.25?0.5 d1 0.5?1.5 d3 0.25?1.5 d5
50
Enforcement of mesh compatibility
In x direction,
0.375 d1 - d2 0.75 d3 - 0.125 d5 0
-0.125 d1 0.75 d3 - d4 0.375 d5 0
In y direction,
0.375 d6- d70.75 d8- 0.125 d10 0
-0.125 d60.75 d8 - d9 0.375 d10 0
51
Modelling of constraints by rigid body attachment
d1 q1 d2 q1q2 l1 d3q1q2 l2 d4q1q2 l3
Eliminate q1 and q2
(l2 /l1-1) d1 - ( l2 /l1) d2 d3 0 (l3 /l1-1)
d1 - ( l3 /l1) d2 d4 0
(DOF in x direction not considered)
52
IMPLEMENTATION OF MPC EQUATIONS
(Global system equation)
(Matrix form of MPC equations)
Constant matrices
53
Lagrange multiplier method
(Lagrange multipliers)
Multiplied to MPC equations
Added to functional
The stationary condition requires the derivatives
of ?p with respect to the Di and ?i to vanish.
?
Matrix equation is solved
54
Lagrange multiplier method

• Constraint equations are satisfied exactly
• Total number of unknowns is increased
• Expanded stiffness matrix is non-positive
  definite due to the presence of zero diagonal
  terms
• Efficiency of solving the system equations is
  lower

55
Penalty method
(Constrain equations)
???1 ?2 ... ?m? is a diagonal matrix of
penalty numbers
Stationary condition of the modified functional
requires the derivatives of ?p with respect to
the Di to vanish
Penalty matrix
56
Penalty method
Zienkiewicz et al., 2000
? constant (1/h)p1
P is the order of element used
Characteristic size of element
max (diagonal elements in the stiffness matrix)
or
Youngs modulus
57
Penalty method

• The total number of unknowns is not changed.
• System equations generally behave well.
• The constraint equations can only be satisfied
  approximately.
• Right choice of ? may be ambiguous.