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F inite Element Method. for readers of all backgrounds. G. R. Liu and S. S. Quek. CHAPTER 11:. MODELLING TECHNIQUES. CONTENTS. INTRODUCTION CPU TIME ESTIMATION GEOMETRY MODELLING MESHING Mesh density Element distortion MESH COMPATIBILITY Different order of elements - PowerPoint PPT Presentation
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1
FFinite Element Methodinite Element Method
MODELLING TECHNIQUES
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 11:
2Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS
INTRODUCTIONCPU TIME ESTIMATIONGEOMETRY MODELLINGMESHING
– Mesh density– Element distortion
MESH COMPATIBILITY– Different order of elements– Straddling elements
3Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS
USE OF SYMMETRY– Mirror symmetry– Axial symmetry– Cyclic symmetry– Repetitive symmetry
MODELLING OF OFFSETS– Creation of MPC equations for offsets
MODELLING OF SUPPORTSMODELLING OF JOINTS
4Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS
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
5Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
Ensure reliability and accuracy of results. Improve efficiency and accuracy.
0% 50% 100%
0% 50% 100%
EFFORT
ACCURACY
(ANALYSIS)
(RESULTS)
6Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
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.
7Finite Element Method by G. R. Liu and S. S. Quek
CPU TIME ESTIMATIONCPU TIME ESTIMATION
CPU doft n ( ranges from 2 – 3)
0
0
.sy
ndof
b 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.
8Finite Element Method by G. R. Liu and S. S. Quek
GEOMETRY MODELLINGGEOMETRY MODELLING Reduction of a complex geometry to a manageable one. 3D? 2D? 1D? Combination?
y
z
y
x
h
z
x
Shell Neutral surface
Neutral surface
x
fy1
z
fy2
Beam member
Bulky solids 3-D solid element mesh
2-D shell element mesh
1-D beam element mesh
Centroid
(Using 2D or 1D makes meshing much easier)
9Finite Element Method by G. R. Liu and S. S. Quek
GEOMETRY MODELLINGGEOMETRY 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.
10Finite Element Method by G. R. Liu and S. S. Quek
MESHINGMESHING
To minimize the number of DOFs, have fine mesh at important areas.
In FE packages, mesh density can be controlled by mesh seeds.
Mesh densityMesh density
(Image courtesy of Institute of High Performance Computing and Sunstar Logistics(s) Pte Ltd (s))
11Finite Element Method by G. R. Liu and S. S. Quek
Element distortionElement 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
12Finite Element Method by G. R. Liu and S. S. Quek
Element distortionElement distortion
Aspect ratio distortion
b
a
3 Stress analysis
10 Displacement analysis
b
a
Rule of thumb:
13Finite Element Method by G. R. Liu and S. S. Quek
Element distortionElement distortion
Angular distortion
skew Taper b a
b<5a
14Finite Element Method by G. R. Liu and S. S. Quek
Element distortionElement distortion
Curvature distortion
15Finite Element Method by G. R. Liu and S. S. Quek
Element distortionElement distortion
Volumetric distortion
1
2
3 4
1 2
3 4
x
y
1
1
1
1
Area outside distorted element maps into an internal area – negative volume integration
16Finite Element Method by G. R. Liu and S. S. Quek
Element distortionElement distortion
Volumetric distortion (Cont’d)
17Finite Element Method by G. R. Liu and S. S. Quek
Element distortionElement distortion
Mid-node position distortion
a
>b/4 b
>a/4
Shifting of nodes beyond limits can result in singular stress field (see crack tip elements)
18Finite Element Method by G. R. Liu and S. S. Quek
MESH COMPATIBILITYMESH COMPATIBILITY
Requirement of Hamilton’s principle – admissible displacement
The displacement field is continuous along all the edges between elements
19Finite Element Method by G. R. Liu and S. S. Quek
Different order of elementsDifferent order of elements
Quad
Linear
3
1
2 Quad
Linear
3
1
2
Crack like behaviour – incorrect results
20Finite Element Method by G. R. Liu and S. S. Quek
Different order of elementsDifferent order of elements Solution:
– Use same type of elements throughout
– Use transition elements
– Use MPC equations
Quad Quad
Linear
Linear
1 1
2
2
3
Quad Linear Transition
Transition Element
Vary quadratically along this edge Vary linearly along this edge
21Finite Element Method by G. R. Liu and S. S. Quek
Straddling elementsStraddling elements
Quad
Quad
3
1
2
Avoid straddling of elements in mesh
22Finite Element Method by G. R. Liu and S. S. Quek
USE OF SYMMETRYUSE OF SYMMETRY
Different types of symmetry:
Mirror symmetryAxial symmetry
Cyclic symmetry Repetitive symmetry
Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error.
23Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
Symmetry about a particular plane
Planes of symmetry
Modelling of quarter model is sufficient
24Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
Consider a 2D symmetric solid:
x
y
3
2
1
3
2
1
u1x = 0
u2x = 0
u3x = 0
Single point constraints (SPC)
25Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
y
x
P P
a b a
b
P
Deflection = FreeRotation = 0
Symmetric loading
26Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
Anti-symmetric loading y
x
P P
a b a
b P
Deflection = 0Rotation = Free
27Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
Plane of symmetry
u v w x y z
xy Free Free Fix Fix Fix Free
yz Fix Free Free Free Fix Fix
zx Free Fix Free Fix Free Fix
Symmetric
•No translational displacement normal to symmetry plane
•No rotational components w.r.t. axis parallel to symmetry plane
28Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetryAnti-symmetric
•No translational displacement parallel to symmetry plane
•No rotational components w.r.t. axis normal to symmetry plane
Plane of symmetry
u v w x y z
xy Fix Fix Free Free Free Fix
yz Free Fix Fix Fix Free Free
zx Fix Free Fix Free Fix Free
29Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
Any load can be decomposed to a symmetric and an anti-symmetric load
y
x
P/2
a b a
b
P/2
y
x
P/2
a b a
b
P/2
= +
y
x
P
a b a
b
Anti-Symmetric loading
Symmetric loading
Asymmetric loading
30Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
X
P
Y
Full frame structure
P/2
P/2
P/2
P/2 + =
Sym.
Anti-sym.
31Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
All nodes on this line fixed against vertical displacement.
2
P
X
All nodes on this line fixed against the horizontal displacement and rotation.
X
2
P
Y Y
Properties are halved for this member
32Finite Element Method by G. R. Liu and S. S. Quek
Mirror symmetryMirror symmetry
Dynamic problems (e.g. two half models to get full set of eigenmodes in eigenvalue analysis)
motion symmetric about this node
motion antisymmetric about this node
Rotation dof = 0 at this node
translational dof v = 0 at this node
33Finite Element Method by G. R. Liu and S. S. Quek
Axial symmetryAxial symmetry Use of 1D or 2D axisymmetric elements
– Formulation similar to 1D and 2D elements except the use of polar coordinates
z w2 w1
w = W sin
x
y
Cylindrical shell using 1D axisymmetric elements
3D structure using 2D axisymmetric elements
34Finite Element Method by G. R. Liu and S. S. Quek
Cyclic symmetryCyclic symmetry
uAn
uBt
uBn
F
F
F
F
Representative cell
uAt
Side B
F Side A
uAn = uBn
uAt = uBt
Multipoint constraints (MPC)
35Finite Element Method by G. R. Liu and S. S. Quek
Repetitive symmetryRepetitive symmetry
Representative cell
uBx
P
uAx
P
P
P
A
B
uAx = uBx
36Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF OFFSETSMODELLING OF OFFSETS Offset Length of beam l
Offset
Length of beam l
Corner Nodes
Joint point
, offset can be safely ignored l
100
l l
100 5 , offset needs to be modelled
l
5, ordinary beam, plate and shell elements should not be used.
Use 2D or 3D solid elements.
Guidelines:
37Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF OFFSETSMODELLING OF OFFSETS
Three methods:– Very stiff element– Rigid element– MPC equations
Very stiff element/ Rigid element
Corner Nodes
Rigid body connecting two corner nodes
Corner Nodes
38Finite Element Method by G. R. Liu and S. S. Quek
Creation of MPC equations for offsetsCreation of MPC equations for offsets
d1
q1
q2
q3
d2 d3
d5
d4
d6
2
1
3
1 1 3d q q
2 2d q
3 3d q
4 1d q
5 2 3d q q
6 3d q1 3 4
2 3 5
3 6
0
0
0
d d d
d d d
d d
Eliminate q1, q2, q3
DOFs at all nodes to be DOFs of the Equation of MPC connected by the rigid body rigid bodyN N N
39Finite Element Method by G. R. Liu and S. S. Quek
Creation of MPC equations for offsetsCreation of MPC equations for offsets
Rigid body
Node on neutral-surface of the plate
Node of the beam element
Neutral surface of the plate
40Finite Element Method by G. R. Liu and S. S. Quek
Creation of MPC equations for offfsetsCreation of MPC equations for offfsets
d1
d2 d3
d5
d4
d6 d9 d8 d7
x
z
y
A
B
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
41Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF SUPPORTSMODELLING OF SUPPORTS
Beam with “built-in end”
a) Full constraint only in the horizontal direction
c) Fully clamped support
b) Support provides full constraint only on the lower surface
42Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF SUPPORTSMODELLING OF SUPPORTS
b)
a)
c)
d)
(Prop support of beam)
43Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF JOINTSMODELLING OF JOINTS
turbine blade
turbine disc nodes at interface
(a) (b)
Perfect connection ensured here
44Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF JOINTSMODELLING OF JOINTS
u
v
u v
1
Mismatch between DOFs of beams and 2D solid – beam is free to rotate (rotation not transmitted to 2D solid)
1
Perfect connection by artificially extending beam into 2D solid (Additional mass)
45Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF JOINTSMODELLING OF JOINTS
Using MPC equations
d1
d2
d3
d4 d6
d5
d7
2a disc model
blade model
d5
ad7
d2 d6
d1
1 5d d
2 6 7d d ad
3 5d d
4 6 7d d ad
46Finite Element Method by G. R. Liu and S. S. Quek
MODELLING OF JOINTSMODELLING OF JOINTS
Mesh for plate
Mesh for Solid
Similar for plate connected to 3D solid
47Finite Element Method by G. R. Liu and S. S. Quek
OTHER APPLICATIONS OF OTHER APPLICATIONS OF MPC EQUATIONSMPC EQUATIONS
Modelling of symmetric boundary Modelling of symmetric boundary conditionsconditions
u
v
900
Axis of symmetry
n
Axis of symmetry x
y
1
2
3
1
2
3
dn = 0
ui cos + vi sin=0
or ui+vi tan =0
for i=1, 2, 3
48Finite Element Method by G. R. Liu and S. S. Quek
Enforcement of mesh compatibilityEnforcement of mesh compatibility
d1
Quad Linear
d2
d4
d5
d3 d6
0
-1
1
1
2
3
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
Use lower order shape function to interpolate
49Finite Element Method by G. R. Liu and S. S. Quek
Enforcement of mesh compatibilityEnforcement of mesh compatibility
d1
Quad Quad
d2 d6
d8
d5 d10
0
-1
1
d3
d4
d7
d9
-0.5
0.5
Use shape function of longer element to interpolate
dx = (1) d1 + (1+)(1) d3 + 0.5 (1+) d5
Substituting the values of for the two additional nodes
d2 = 0.251.5 d1 + 1.50.5 d3 0.250.5 d5
d4 = 0.250.5 d1 + 0.51.5 d3
+ 0.251.5 d5
50Finite Element Method by G. R. Liu and S. S. Quek
Enforcement of mesh compatibilityEnforcement of mesh compatibility
d1
Quad Quad
d2 d6
d8
d5 d10
0
-1
1
d3
d4
d7
d9
-0.5
0.5
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 d7+0.75 d8 0.125 d10 = 0
0.125 d6+0.75 d8 d9 + 0.375 d10 = 0
51Finite Element Method by G. R. Liu and S. S. Quek
Modelling of constraints by rigid body Modelling of constraints by rigid body attachmentattachment
l1
l3
l2
q2 q1
d1 d2 d3 d4
Rigid slab
d1 = q1
d2 = q1+q2 l1
d3=q1+q2 l2
d4=q1+q2 l3
(l2 /l1-1) d1 - ( l2 /l1) d2 + d3 = 0
(l3 /l1-1) d1 - ( l3 /l1) d2 + d4 = 0
Eliminate q1 and q2
(DOF in x direction not considered)
52Finite Element Method by G. R. Liu and S. S. Quek
IMPLEMENTATION OF MPC IMPLEMENTATION OF MPC EQUATIONSEQUATIONS
KD F
CD Q 0 (Matrix form of MPC equations)
(Global system equation)
Constant matrices
53Finite Element Method by G. R. Liu and S. S. Quek
Lagrange multiplier methodLagrange multiplier method
1 2 m
T
{ } 0T CD Q
p 1
2D KD D F CD QT T T { }
T
D FK C
λ QC 0
(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
54Finite Element Method by G. R. Liu and S. S. Quek
Lagrange multiplier methodLagrange 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 termsEfficiency of solving the system equations is
lower
55Finite Element Method by G. R. Liu and S. S. Quek
Penalty methodPenalty method
t CD Q (Constrain equations)
p 1
2
1
2D KD D F t tT T T
=1 2 ... m is a diagonal
matrix of ‘penalty numbers’
[ ]T T K C C D F C Q
Stationary condition of the modified functional requires the derivatives of p with respect to the Di to vanish
Penalty matrix
56Finite Element Method by G. R. Liu and S. S. Quek
Penalty methodPenalty method
[Zienkiewicz et al., 2000] :
= constant (1/h)p+1
Characteristic size of element
P is the order of element used
4 61.0 10 max (diagonal elements in the stiffness matrix)
5 81.0 10 or
Young’s modulus
57Finite Element Method by G. R. Liu and S. S. Quek
Penalty methodPenalty 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.