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ECIV 720 A Advanced Structural Mechanics and Analysis

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ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 16 & 17: Higher Order Elements (review) 3-D Volume Elements Convergence Requirements – PowerPoint PPT presentation

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Title: ECIV 720 A Advanced Structural Mechanics and Analysis


1
ECIV 720 A Advanced Structural Mechanics and
Analysis
  • Lecture 16 17
  • Higher Order Elements (review)
  • 3-D Volume Elements
  • Convergence Requirements
  • Element Quality

2
Higher Order Elements
Quadrilateral Elements
Recall the 4-node
4 generalized displacements ai
3
Higher Order Elements
Quadrilateral Elements
Assume Complete Quadratic Polynomial
9 generalized displacements ai
9 BC for admissible displacements
4
9-node quadrilateral
9-nodes x 2dof/node 18 dof
5
9-node element Shape Functions
Following the standard procedure the shape
functions are derived as
6
N1,2,3,4 Graphical Representation
7
N5,6,7,8 Graphical Representation
8
N9 Graphical Representation
9
Polynomials the Pascal Triangle
Degree
Pascal Triangle
1
.
10
Polynomials the Pascal Triangle
To construct a complete polynomial
etc
11
Incomplete Polynomials
3-node triangular
12
Incomplete Polynomials
13
8-node quadrilateral
Assume interpolation
8 coefficients to determine for admissible displ.
14
8-node quadrilateral
8-nodes x 2dof/node 16 dof
15
8-node element Shape Functions
Following the standard procedure the shape
functions are derived as
h
x
16
N1,2,3,4 Graphical Representation
17
N5,6,7,8 Graphical Representation
18
Incomplete Polynomials
19
6-node Triangular
Assume interpolation
6 coefficients to determine for admissible displ.
20
6-node triangular
6-nodes x 2dof/node 12 dof
21
6-node element Shape Functions
Following the standard procedure the shape
functions are derived as
LiArea coordinates
22
Other Higher Order Elements
12-node quad
23
Other Higher Order Elements
16-node quad
24
3-D Stress state
25
3-D Stress State
Assumption Small Deformations
26
Strain Displacement Relationships
27
3-D Finite Element Analysis
Solution Domain is VOLUME
Simplest Element (Lowest Order) Tetrahedral
Element
28
3-D Tetrahedral Element
Can be thought of an extension of the 2D CST
29
3-D Tetrahedral
Shape Functions
Volume Coordinates
30
Geometry Isoparametric Formulation
In view of shape functions
31
Jacobian of Transformation
32
Strain-Displacement Matrix
B is CONSTANT
33
Stiffness Matrix
Element Strain Energy
34
Force Terms
Body Forces
35
Element Forces
Surface Traction
Applied on FACE of element
eg on face 123
36
Stress Calculations
Stress Invariants
37
Stress Calculations
Principal Stresses
38
Other Low Order Elements
39
Degenerate Elements
Still has 24 dof
40
Degenerate
Still has 24 dof
41
Higher Order Elements 10-node 4-hedral
Z
2

Y
X

42
15-node 5-hedral
Z
Y
X
43
15-node 5-hedral Shape Functions
44
20-node 6-hedral
z
24
23
8
16
15
22
5
20
h
7
x
13
14
6
17
4
19
12
11
18
1
3
Z
9
10
2
Y
X
45
20-node 6-hedral Shape Functions
46
Convergence Considerations
For monotonic convergence of solution
Requirements
Elements (mesh) must be compatible
Elements must be complete
47
Monotonic Convergence
No of Elements
For monotonic convergence the elements must
be complete and the mesh must be compatible
48
Mixed Order Elements
Consider the following Mesh
4-node
8-node
Incompatible Elements
49
Mixed Order Elements
We can derive a mixed order element for grading
4-node
8-node
7-node
By blending shape functions appropriately
50
Convergence Considerations
For monotonic convergence of solution
Requirements
Elements (mesh) must be compatible
Elements must be complete
51
Element Completeness
For an element to be complete
Assumption for displacement field
must accommodate
  • RIGID BODY MOTION
  • CONSTANT STRAIN STATE

52
Element Completeness
Consider
This is not a complete polynomial
However,
53
Element Completeness
Assume displacement field
Test for ELEMENT completeness
54
Element Completeness
55
Element Completeness
In order for the computed displacements to be
the assumed ones we must satisfy
Condition for element completeness
56
Effects of Element Distortion
Loss of predictive capability of isoparametric
element
No Distortion
1
x2y
xy2
Behavior accurately predicted
57
Effects of Element Distortion
Angular Distortion
1
x2y
xy2
Predictability is lost for all quadratic terms
58
Effects of Element Distortion
Quadratic Curved Edge Distortion
1
x2y
xy2
Predictability is lost for all quadratic terms
59
Effects of Element Distortion
The advantage (reduced of dof) of using 8-node
higher order element based on an incomplete
polynomial is lost when high element distortions
are present
60
Effects of Element Distortion
Loss of predictive capability of isoparametric
element
No Distortion
9-node
1
x2y
xy2
x2y2
Behavior accurately predicted
61
Effects of Element Distortion
9-node
Angular Distortion
Behavior predicted better than 8-node
62
Effects of Element Distortion
Quadratic Curved Edge Distortion
9-node
Predictability is lost for high order terms
63
Effects of Element Distortion
The advantage (reduced of dof) of using higher
order element based on an incomplete polynomial
is lost when high element distortions are present
For angular distortion 9-node element shows
better behavior
For Curved edge distortion all elements give low
order prediction
64
Polynomial Element Predictability
65
Tests of Element Quality
Eigenvalue Test Identify Element
Deficiencies Patch Test Convergence of Solutions
66
Eigenvalue Test
67
Eigenvalue Test
Eigenproblem
As many eigenvalues l as dof
For each l there is a solution for d
68
Displacement Modes Stiffness Matrix
For all eigenvalues and modes
69
Eigenvalue Test
Scale d so that
70
Eigenvalue Test
Rigid Body Motion gt System is not strained gt U0
71
Rigid Body Motion
Rigid Body Modes
72
Displacement Modes Stiffness Matrix
Consider the 2-node axial element
Identify all possible modes of displacement
73
Displacement Modes Stiffness Matrix
Consider the 4-node plane stress element
1
t1 E1 v0.3
1
Solve Eigenproblem
74
Displacement Modes Stiffness Matrix
75
Displacement Modes Stiffness Matrix
76
Displacement Modes Stiffness Matrix
77
Displacement Modes Stiffness Matrix
78
Displacement Modes Stiffness Matrix
79
Displacement Modes Stiffness Matrix
The eigenvalues of the stiffness matrix display
directly how stiff the element is in the
corresponding displacement mode
80
Patch Test
Objective
Examine solution convergence for displacements,
stresses and strains in a particular element type
with mesh refinement
81
Patch Test - Procedure
Build a simple FE model
Consists of a Patch of Elements
At least one internal node
Load by nodal equivalent forces consistent with
state of constant stress
Internal Node is unloaded and unsupported
82
Patch Test - Procedure
Compute results of FE patch
If (computed sx) (assumed sx) test passed
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