ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order Elements
Dec 18, 2015
ECIV 720 A Advanced Structural
Mechanics and Analysis
Lecture 15: Quadrilateral Isoparametric Elements
(cont’d)Force VectorsModeling Issues
Higher Order Elements
Integration of Stiffness Matrix
1
1
1
1
det dJdt Te DBBk
B (3x8)
D (3x3)
BT(8x3)
ke (8x8)
Integration of Stiffness Matrix
Each term kij in ke is expressed as
1
1
1
1
1
1
1
1
3
1
3
1
,
,det
ddgt
ddJBDBtkm l
ljmlTimij
Linear Shape Functions is each Direction
Gaussian Quadrature is accurate if
We use 2 Points in each direction
Integration of Stiffness Matrix
311 312
311
311
2222212112121111 ,,,, gwwgwwgwwgww
1
1
1
1
, ddgt
11 w
12 w
22211211 ,,,, gggg
Choices in Numerical Integration
• Numerical Integration cannot produce exact results
• Accuracy of Integration is increased by using more integration points.
• Accuracy of computed FE solution DOES NOT necessarily increase by using more integration points.
FULL Integration
• A quadrature rule of sufficient accuracy to exactly integrate all stiffness coefficients kij
• e.g. 2-point Gauss ruleexact for
polynomials
up to 2nd order
311 312
311
311
311 312
311
311
Reduced Integration, Underintegration
Use of an integration rule of less than full order
Advantages
• Reduced Computation Times
• May improve accuracy of FE results
• Stabilization
Disadvantages
Spurious Modes (No resistance to nodal loads that tend to activate the mode)
Spurious Modes
t=1
E=1
v=0.3
8 degrees of freedom 8 modes
1
1
Consider the 4-node plane stress element
Solve Eigenproblem
Spurious Modes
01
Rigid Body Mode 02
Rigid Body Mode
Spurious Modes
03
Rigid Body Mode
Spurious Modes
495.05
Flexural Mode
495.04 Flexural Mode
Spurious Modes
769.06 Shear Mode
Spurious Modes
769.07 Stretching Mode
43.18 Uniform Extension Mode
(breathing)
Element Body Forces
ii
Ti
el
T
eA
T
eA
T
e
e
e
tdl
tdA
tdA
Pu
Tu
fu
Dεε2
1
ii
Ti
eA
T
eV
T
eV
T
e
e
e
dA
dV
dV
Pφ
Tφ
fφ
φεσ 0
Total Potential Galerkin
Body Forces
eA
T dAd detfu
Integral of the form
8
1
4321
4321
0000
0000
q
q
NNNN
NNNN
v
u
8
1
4321
4321
0000
0000
NNNN
NNNN
y
x
eA
T dAd detfφ
Body Forces
In both approaches
e
e
e
e
Ay
Ax
Ay
Ax
dAdNf
dAdNf
dAdNf
dAdNf
qqWP
det
det
det
det
4
2
1
1
81
Linear Shape Functions
Use same quadrature as stiffness maitrx
Element Traction
ii
Ti
el
T
eA
T
eA
T
e
e
e
tdl
tdA
tdA
Pu
Tu
fu
Dεε2
1
ii
Ti
eA
T
eV
T
eV
T
e
e
e
dA
dV
dV
Pφ
Tφ
fφ
φεσ 0
Total Potential Galerkin
Element Traction
Similarly to triangles, traction is applied along sides of element
4
2
3
Tx
Ty
u
v
1
4
0
12
1
12
1
0
4
3
2
1
N
N
N
N
el
TT tdlWP Tu
32
Traction
8
1
32
32
000000
000000
q
q
NN
NN
v
u
0
0
0
0
232
8132
y
x
eT T
Tlt
qqWP
For constant traction along side 2-3
Traction
components
along 2-3
Stresses
311 312
311
311
DBqσ
12221121
1121
1222
00
00
det
1
JJJJ
JJ
JJ
JA
10101010
10101010
01010101
01010101
4
1G
More Accurate at
Integration points
Stresses are calculated at any
Modeling Issues: Nodal Forces
ii
Ti
el
T
eA
T
eA
T
e
e
e
tdl
tdA
tdA
Pu
Tu
fu
Dεε2
1
A node should be
placed at the location
of nodal forces
In view of…
Or virtual potential energy
Modeling Issues: Element Shape
Square : Optimum Shape
Not always possible to use
Rectangles:
Rule of Thumb
Ratio of sides <2
Angular Distortion
Internal Angle < 180o
Larger ratios
may be used
with caution
Modeling Issues: Degenerate Quadrilaterals
Coincident Corner Nodes
1
2
3
4
32
1
4
xx
xx
x
xx
x
Integration Bias
Less accurate
Modeling Issues: Degenerate Quadrilaterals
Three nodes collinear
1
2
3
4
xx
xx
1
2
3
4 x
xx
x
Less accurate
Integration Bias
Modeling Issues: Degenerate Quadrilaterals
Use only as necessary to improve representation of geometry
2 nodes
Do not use in place of triangular elements
A NoNo Situation
3
4
1 2
x
y
(3,2) (9,2)
(7,9)
(6,4)
Parent
All interior angles < 180
J singular
Another NoNo Situation
x, y
not uniquely
defined
FEM at a glance
It should be clear by now that the cornerstone in FEM procedures is the interpolation of the displacement field from discrete values
i
m
ni zyxzyx uNu
1
,,,,
Where m is the number of nodes that define the interpolation and the finite element and N is a set of Shape Functions
FEM at a glance
1=-1 2=1
m=2
1=-1
1
3
2=1
2m=3
FEM at a glance
12
3q6
q5
q4
q3
q2
q1
vu
m=3
4
1
2
3
m=4
FEM at a glance
In order to derive the shape functions it was assumed that the displacement field is a polynomial of any degree, for all cases considered
nn xaxaxaax 2
210u
..., 3210 xyayaxaayxu
Coefficients ai represent generalized coordinates
1-D
2-D
FEM at a glance
For the assumed displacement field to be admissible we must enforce as many boundary conditions as the number of polynomial coefficients
1=-1
1
3
2=1
2e.g.
12121101 uxaxaaxx u
22222102 uxaxaaxx u
32323103 uxaxaaxx u
FEM at a glance
This yields a system of as many equations as the number of generalized displacements
nn u
u
a
a
zyx
Matrix
tCeofficien
10
),,(
nn u
u
C
a
a
1
10
that can be solved for ai
FEM at a glance
nn xaxaxaax 2
210u
Substituting ai in the assumed displacement field
and rearranging terms…
i
m
ni uxNxu
1
FEM at a glance
113N
12
11N 1
2
12N
u(-1)=a0 -a1 +a2 =u1
u(1)=a0 +a1 +a2 =u2
u(0)=a0 =u3
33
22
11
uN
uN
uNu
…
u()=a0+a1 +a2 2
1=-1
1
2=1
3 2
Let’s go through the exercise
x1
1
x2
2
220 xaaxu
Assume an incomplete form of quadratic variation
Incomplete form of quadratic variation
We must satisfy
12
1201 uxaaxu
22
2202 uxaaxu
2
1
1
0
22
21
1
1
u
u
a
a
x
x
x1
1
x2
2
Incomplete form of quadratic variation
2
1
1
0
22
21
1
1
u
u
a
a
x
x
And thus,
2
121
22
21
221
0
11
1
u
uxx
xxa
a
Incomplete form of quadratic variation
21
22
2211
22
0 xx
uxuxa
21
22
211 xx
uua
220 xaaxu
And substituting in
221
22
2121
22
2211
22 x
xx
uu
xx
uxuxxu
Incomplete form of quadratic variation
221
22
2121
22
2211
22 x
xx
uu
xx
uxuxxu
2
1
21
22
221
21
22
222
u
u
xx
xx
xx
xxxu
Which can be cast in matrix form as
Isoparametric Formulation
The shape functions derived for the interpolation of the displacement field are used to interpolate geometry
2
1
21
22
221
21
22
222
x
x
xx
xx
xx
xxx
x1
1
x2
2
Intrinsic Coordinate Systems
Intrinsic coordinate systems are introduced to eliminate dependency of Shape functions from geometry
1 (-1,-1) 2 (1,-1)
4 (-1,1) 3 (1,1)
The price?
Jacobian of transformation
iiiN 114
1
Great Advantage for the money!
Field Variables in Discrete Form
nNxx
Geometry
Displacement
nNuu
= DB un
Stress Tensor
= B un
Strain Tensor
)(intrinsic
)(cartesianJ
FEM at a glance
eeTeV
Te
e
dVU ukuσDε2
1
2
1
Element Strain Energy
ee V
TTeV
Tf dVdVW fNufu
Work Potential of Body Force
ee S
TTeS
Tf dSdSW TNuTu
Work Potential of Surface Traction
etc
Higher Order Elements
Quadrilateral Elements
Recall the 4-node
4321, aaaau
Complete Polynomial
4 generalized displacements ai
4 Boundary Conditions for admissible displacements
Higher Order Elements
Quadrilateral Elements
29
28
227
26
25
432
1,
aaaaa
aaa
au
Assume Complete Quadratic Polynomial
9 generalized displacements ai
9 BC for admissible displacements
9-node quadrilateral
9-nodes x 2dof/node = 18 dof
BT18x3 D3x3 B3x18
ke 18x18
9-node element Shape Functions
Following the standard procedure the shape functions are derived as
1 2
34
4,3,2,14
1 iN iii
Corner Nodes
5
6
7
8
8,7,6,5
11
12
1 22
i
N
iiii
iii
Mid-Side Nodes9
Middle Node
911 22 iN i
9-node element – Shape Functions
113N
12
11N 1
2
12N
Can also be derived from the 3-node axial element
1=-1
1
2=1
3 2
Construction of Lagrange Shape Functions
(1,)(1,1)
1 (-1,-1)
12
11N
12
11N
1
2
11
2
1, 111 NNN
4,3,2,14
1 iN iii
N1,2,3,4 Graphical Representation
N5,6,7,8 Graphical Representation
N9 Graphical Representation
Polynomials & the Pascal Triangle n
n xaxyayaxaayx 3210, u
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
Degree
1
2
3
4
5
0
Pascal Triangle
Polynomials & the Pascal Triangle
To construct a complete polynomial
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
etc
Q1
xyayaxaayx 3210, u
4-node QuadQ2
39
28
27
36
254
23
21
01
,
yaxyayxaxa
yaxyaxa
yaxa
a
yx
u
9-node Quad
Incomplete Polynomials
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
yaxaayx 210, u
3-node triangular
Incomplete Polynomials
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
27
26
254
23
21
01
,
xyayxa
yaxyaxa
yaxa
a
yx
u
8-node quadrilateral
Assume interpolation
1 2
34
5
6
7
8
27
26
254
23
21
01
,
xyayxa
yaxyaxa
yaxa
a
yx
u
8 coefficients to determine for admissible displ.
8-node quadrilateral
8-nodes x 2dof/node = 16 dof
BT16x3 D3x3 B3x16
ke 16x16
8-node element Shape Functions
Following the standard procedure the shape functions are derived as
1 2
34
4,3,2,1
1114
1
i
N iiiii
Corner Nodes
5
6
7
8
8,7,6,5
112
1 22
i
N iiiii
Mid-Side Nodes
N1,2,3,4 Graphical Representation
N5,6,7,8 Graphical Representation
Incomplete Polynomials
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
254
23
21
01
,
yaxyaxa
yaxa
a
yx
u
6-node Triangular
Assume interpolation
1 2
3
4
56
254
23
21
01
,
yaxyaxa
yaxa
a
yx
u
6 coefficients to determine for admissible displ.
6-node triangular
6-nodes x 2dof/node = 12 dof
BT12x3 D3x3 B3x12
ke 12x12
1 2
3
4
56
6-node element Shape Functions
Following the standard procedure the shape functions are derived as
3,2,112 iLLN iii
Corner Nodes
1 2
3
214 4 LLN
Mid-Side Nodes
4
56
325 4 LLN
136 4 LLN Li:Area coordinates
Other Higher Order Elements
1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
…….
x5 x4y x3y2 x2y3 xy4 y5
12-node quad
1 2
34
Other Higher Order Elements
x5 x4y x3y2 x2y3 xy4 y5
16-node quad1
x y
x2 xy y2
x3 x2y xy2 y3
x4 x3y x2y2 xy3 y4
……. x3y21 2
34