ECIV 720 A Advanced Structural

Mechanics and Analysis

Lecture 16 & 17: Higher Order Elements (review)3-D Volume ElementsConvergence RequirementsElement Quality

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

N1,2,3,4 Graphical Representation

N5,6,7,8 Graphical Representation

N9 Graphical Representation

Polynomials & the Pascal Triangle n

nxaxyayaxaayx 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

3-D Stress state

3-D Stress State

AssumptionSmall Deformations

Strain Displacement Relationships

xy

xz

yz

z

y

x

xy

xz

yz

z

y

x

Material Matrix

3-D Finite Element Analysis

Simplest Element (Lowest Order)

Tetrahedral Element

Solution Domain is VOLUME

12

3

45

6

78

91011

12

3-D Tetrahedral Element

Parent

(Master)

1 (1,0,0)

2 (0,1,0)

4 (0,0,0)

3 (0,0,1)

Can be thought of an extension of the 2D CST

3-D Tetrahedral

1

2

3

4

Shape Functions

1N

2N

3N

14N

Volume Coordinates

Geometry – Isoparametric Formulation

44332211 xNxNxNxNx

44332211 yNyNyNyNy

44332211 zNzNzNzNz In view of shape functions

3424144 xxxxx 3424144 yyyyy

3424144 zzzzz

jiij xxx

jiij yyy

jiij zzz

Jacobian of Transformation

343434

242424

141414

zyx

zyx

zyx

zyx

zyx

zyx

J

JJ det6

1det

1

0

1

0

1

0

dddVe

Strain-Displacement Matrix

B is CONSTANT

x

N

z

Ny

N

z

Nx

N

y

Nz

Ny

Nx

N

B

ii

ii

ii

i

i

i

i

0

0

0

00

00

00

Stiffness Matrix

ee V

T

V

Te dVdVU Dεεσε

2

1

2

1

Element Strain Energy

ee V

TT

V

TT dVdV DBqBqDBqBq2

1

2

1

eeTe

Te

TV qkqDBqBq2

1

2

1

Force Terms

Body Forces

ddddVWP TT

V

Tf

e JfNqfu det

Tzzyxzyxe

e fffffffV

)112(4 f

Element Forces

Surface Traction

1

2

3

4Applied on FACE of element

ee A

TT

A

TT dAdAWP TNqTu

Tzyxzyxzyxe TTTTTTTTTA

)112(123 0004

T

eg on face 123

Stress Calculations

= DB qe

Stress Tensor

Constant

Stress Invariants

zyxI 1

2222 xyxzyzxzzyyxI

222

3 2

xyzxzyyzx

xyxzyzzyxI

Stress Calculations

Principal Stresses

cos31

1 cI

3

2cos

31

2

cI

3

4cos

31

3

cI

32 ac

ac

b3cos

3

1 1

2

21

3I

Ia 3

21

3

1

332 I

IIIb

Other Low Order Elements

1

2

3

4

5

6

18 dof 5-hedral

1

2

3

5

6

7

8

24 dof 6-hedral

Degenerate Elements

1

2

3

5

6

7

8

1

2

4

5

6

8

,3

,7

Still has 24 dof

Degenerate

1

2

3

5

6

7

8

1

2,3

4

5,6,7,8

Still has 24 dof

Higher Order Elements 10-node 4-hedral

2

Z

X

Y

1

2

3

4

5 6

7

8

9103

4

6

810

1

9

7

1314

15

5

N L L i

N L L N L L N L L

N L L N L L N L L

i i i

2 1 1 3 4

4 4 4

4 4 45 1 2 6 2 3 7 3 1

8 1 4 9 2 4 10 3 4

,2, ,

15-node 5-hedral

Z

X

Y

L1

L3

L2

1

2

3

4

5

6

7 8

9

10 11

12

13

14

15

3

2

4

6

8

10

1112

13

14

15

19

7

1314

15

5

15-node 5-hedral Shape Functions

N L L L i

N L L L i p

N L L i p q

N L L i p q

N L i p

i i i i

i j j j

i p q

i p q

i p

1

22 1 1 1 1 3

1

22 1 1 1 4 5 6 1 3

2 1 7 8 9 1 3 2 31

2 1 101112 1 3 2 31

2 1 131415 1 3

2

2

2

,2,

, , ,2,

, , ,2, , ,

, , ,2, , ,

, , ,2,

20-node 6-hedral

Z

X

Y

1

2

3

4

5

6

7

910

1112

13 14

1516

17

18

19

20

24

23

22

8

20-node 6-hedral Shape Functions

20,19,18,171114

1

16,14,12,101114

1

15,13,11,91114

1

8,...,2,121118

1

2

2

2

iN

iN

iN

iN

iii

iii

iii

iiiiiii

Convergence Considerations

For monotonic convergence of solution

Elements (mesh) must be compatible

Elements must be complete

Requirements

Monotonic Convergence

Exact Solution

FEM Solution

No of Elements

For monotonic convergence the elements must beFor monotonic convergence the elements must be

complete and the mesh must be compatiblecomplete and the mesh must be compatible

Mixed Order Elements

Consider the following Mesh

8-node

4-node

Incompatible Elements…

Mixed Order Elements

We can derive a mixed order element for grading

8-node 4-node

By blending shape functions appropriately

7-node

Convergence Considerations

For monotonic convergence of solution

Elements (mesh) must be compatible

Elements must be complete

Requirements

Element Completeness

For an element to be complete

Assumption for displacement field

nnxaxyayaxaayx 3210, u

•RIGID BODY MOTION

•CONSTANT STRAIN STATE

must accommodate

Element Completeness

Consider

yaxaayxu 321,

This is not a complete polynomial

However,

Element Completeness

yaxaayxu 321, The Computed nodal displacement corresponding to this field

iii yaxaau 321 i=1,…,#of nodes

Test for ELEMENT completeness

Isoparametric Formulation

iiuNuNuNuNu 332211

Assume displacement field

Element Completeness

iiiii yNaxNaNau 321

Isoparametric Formulation

iixNx ii yNy

Thus, computed displacement field

yaxaNau i 321

Element Completeness

yaxaayxu 321, Assumed

yaxaNau i 321 Computed

In order for the computed displacements to be the

assumed ones we must satisfy

1 iN Condition for element completeness

Effects of Element Distortion

Loss of predictive capability of isoparametric element

No Distortion

1

x y

x2 xy y2

x2y xy2

Behavior accurately predicted

Effects of Element Distortion

Angular Distortion

1

x y

x2 xy y2

x2y xy2

Predictability is lost for all quadratic terms

Effects of Element Distortion

Quadratic Curved Edge Distortion

1

x y

x2 xy y2

x2y xy2

Predictability is lost for all quadratic terms

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

Effects of Element Distortion

Loss of predictive capability of isoparametric element

No Distortion

1

x y

x2 xy y2

x2y xy2

Behavior accurately predicted

x2y2

9-node

Effects of Element Distortion

Angular Distortion

1

x y

x2 xy y2

x2y xy2

9-node

Behavior predicted better than 8-node

Effects of Element Distortion

Quadratic Curved Edge Distortion

1

x y

x2 xy y2

x2y xy2

Predictability is lost for high order terms

9-node

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

Polynomial Element Predictability

Tests of Element Quality

Eigenvalue TestIdentify Element Deficiencies

Patch TestConvergence of Solutions

Eigenvalue Test

1

2

3

4

Apply loads –{r} in proportion to displacements

dr

rdk

Eigenvalue Test

dr

rdk 0dIk

Eigenproblem

As many eigenvalues as dof

For each there is a solution for {d}

Displacement Modes & Stiffness Matrix

nn

n

n

nn d

d

d

d

d

d

d

d

d

2

1

2

22

12

1

21

11

D

n

2

1

Λ

DΛKD

For all eigenvalues and modes

dKd

Eigenvalue Test

Scale {d} so that

1dd T

U

TT

2 dddkd

then

Eigenvalue Test

U

TT

2 dddkd

Rigid Body Motion => System is not strained => U=0

System is strained => U=0

Rigid Body Motion

Rigid Body Modes

Total Number of Element Displacement Modes

(=number of degrees of freedom)

Element Straining Modes+

Displacement Modes & Stiffness Matrix

Consider the 2-node axial element

11

11

L

AEK

Identify all possible modes of displacement

Displacement Modes & Stiffness Matrix

t=1

E=1

v=0.3

8 degrees of freedom 8 modes

1

1

Consider the 4-node plane stress element

Solve Eigenproblem

Displacement Modes & Stiffness Matrix

01

Rigid Body Mode 02

Rigid Body Mode

Displacement Modes & Stiffness Matrix

03

Rigid Body Mode

Displacement Modes & Stiffness Matrix

495.05

Flexural Mode

495.04 Flexural Mode

Displacement Modes & Stiffness Matrix

769.06 Shear Mode

Displacement Modes & Stiffness Matrix

769.07 Stretching Mode

43.18 Uniform Extension Mode

(breathing)

Displacement Modes & Stiffness Matrix

The eigenvalues of the stiffness matrix display directly how stiff the element is in the corresponding displacement mode

U2

Patch Test

Objective

Examine solution convergence for displacements, stresses and strains in a particular element type with mesh refinement

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

Patch Test - Procedure

HtF x2

1

Compute results of FE patch

If

(computed x) = (assumed x)

test passed