A Hierarchical Finite Element Approach to Modeling ...

A Hierarchical Finite Element Approach to Modeling Process-Induced Deformations in Composite Structures presented at

Composites Testing and Model Identification Conference (CompTest 2006) April 10-12, 2006, Porto, Portugal

A. Arafath, R. Vaziri and A. Poursartip Composites Group Civil Engineering / Materials Engineering The University of British Columbia Vancouver, Canada

Outline • Introduction and Background • Using Standard Solid Elements to Model Shell-Like Structures • Hierarchical Element Approach • Summary and Conclusions

Acknowledgements • Natural Sciences and Engineering Research Council of Canada (NSERC) • University of British Columbia Graduate Fellowship (Patrick David Campbell Endowment) • The Boeing Company

Process Modelling is for Large/Complex Structures • From raw material to finished structure in one step • Realistic structures are large and complex • Mesh creation time, run-time, and ability to easily make changes are all important

• UBC process model family, COMPRO, used heavily in the Boeing 787 program • Has provided significant insight into what is needed for the next round of improvements

Using Solid Elements to Model Shell-Like Structures Current approach Large aspect ratio

Locking Problems

Possible Remedies

• Shear locking

• Reduced integration method

• Volumetric locking

• Assumed natural strain method

• Trapezoidal locking

• Higher order shape functions

• ……………….

24-node Solid Element Thickness direction 15

12

11

23 20

16 24

22 6 13 10

9

21

17

18 5

1

2

y x

3

14

7

4

8

z

19

Why our own element? • Every code uses its own patches for locking ¾ No cross-platform uniformity • Flexibility to add features for easy modelling of composite structures

Main Features of the Element • Selective Reduced Integration 7

4

3 hn

3

η

Layer n

4

8

6

ξ

ζ h2 Layer 2

1

2 h1 Layer 1

1

5

2

Gauss Quadrature (in-plane)

Simpson’s Rule (through-thickness)

Variable layer thickness t13

t14

t11

Layer 1

Layer thicknesses are defined at 4 corners of the element and interpolated linearly to the integration points Additional layer rotation w.r.t. axis - 3'

t12

Layer orientation w.r.t. element material orientation

3' 2'

[0] [90]

1'

3' 2' 1' [0] [90]

Honeycomb

z

[90] [0]

[90] [0] 3

3

2

2 Global coordinate x

y

1

1 Element material orientation w.r.t. global coordinates

Verification Example Cantilever Beam under end shear Rectangular shaped elements P

t = 10 mm L = 1000 mm

Trapezoidal shaped elements

Parallelogram shaped elements

E = 200 GPa ν = 0.0

PL3 wmax = − 3EI

Sensitivity of aspect ratio Normalized Maximum Warpage Element Type

Mesh size (Length*Depth) / (aspect ratio) 1*1 / (100)

2*1 / (50)

4*1 / (25)

8*1 / (12.5)

10*1 / (10)

20-node-F

0.750

0.938

0.985

0.996

0.998

27-node-F

0.750

0.938

0.985

0.996

0.998

20-node-R

1.000

1.000

1.000

1.000

1.000

24-node-R

1.000

1.000

1.000

1.000

1.000

27-node-R

0.908

0.995

1.000

1.000

1.000

Skew Sensitivity F – Full integration R – Reduced Integration 20-node and 27-node elements are ABAQUS built-in elements

Normalized Maximum Warpage Element Type

Parallelogram

Trapezoid

Skew angle (Deg)

Skew angle (Deg)

0º

45º

75º

0º

45º

75º

20-node-F

0.998

0.988

0.510

0.998

0.917

0.502

27-node-F

0.998

0.992

0.857

0.998

0.994

0.829

20-node-R

1.000

0.997

0.726

1.000

0.984

0.356

24-node-R

1.000

1.000

0.978

1.000

1.000

0.878

27-node-R

1.000

1.000

0.978

1.000

1.000

0.878

Warpage Phenomenon Autoclave Pressure Part Tool

Low CTE part is stretched by higher CTE tooling due to interfacial shear stress

Autoclave Pressure Part Tool

Shear deformation within the part results in non-uniform stress distribution through part thickness. Upon curing this is locked in a residual stress. Residual stress distribution causes a net bending moment ⇒ warped part

Warpage of a Flat Part Interface

y

t2

Part

x

Tool

t1

2L

y

y

x

x L

Boundary conditions before tool removal

Boundary conditions after tool removal

Temperature (ºC)

180

Cure temperature cycle

20 80

220

300

320

Time (min)

n elements

Finite element model

m elements

Dimensions

Material Part = T300/3900-2 Tool = Aluminum

Interface condition

Fully bonded Length = 1200 mm Part thickness = 1.6 mm (8 layers) Lay-up = Unidirectional Tool thickness = 5.0 mm

Mesh Size

Warpage (mm)

Time (min)

10*1

73.77

1.46

20*1

73.71

2.93

30*1

73.70

4.27

30*2

47.16

5.87

30*4

37.56

9.73

30*8

35.52

18.7

30*10

35.38

23.43

30*12

35.32

25.43

30*16

35.29

35.38

60*16

35.29

64.13

300*8

35.52

170.37

Not very sensitive to number of elements in in-plane direction

Very sensitive to number of elements in thickness direction

Need large number of elements in thickness direction even though the part thickness is very small (1.6 mm)

• How to decide on the number of elements – Mesh refining – Costly and time consuming, particularly for realistic structures

• Need to understand the physics of the problem so that the number of elements can be selected prior to the analysis

Closed-Form Solution τt

y u =0

t σ =0

x

τb

l Exponential function

σ x = F ( x) G ( β , y ) β=

πt 2l

Ex Gx y Material dependent Geometry dependent

Material Dependency 6

6 150

Resin modulus (GPa)

4.5

Temperature 100

3

Resin modulus 50

1.5

5 4 1

3

2

0

0 0

50

100

150

200

250

300

350

Time (min) 1.6

4 Thickness coord. (mm)

Temperature (ºC)

200

cure stage - 6

3

cure stage - 5 cure stage - 4

5,6

cure stage - 3 cure stage - 2

0.8

2

cure stage - 1

1

0 -50

0

50

100

150

200

250

300

Axial stress (MPa)

350

400

450

500

• Stress gradient in the thickness direction reduces with the curing of the material ¾ Large number of elements may be necessary only at the beginning of the cure cycle

• Adaptive finite element techniques may be the best option to change the number of elements dynamically based on the material properties ¾ h – method ¾ p – method (or hierarchical – method)

h- vs. p-Refinement

e r h

nt e em n i f

p-r e

fin em e

nt

Hierarchical Finite Element Formulation Thickness direction Top surface

z

Mid surface y

x

Bottom surface

• Similar to the 24-node element introduced earlier • The element has three surfaces • The in-plane shape functions for each surface are similar to the standard elements

7

4

3

η 8

1

ξ

5

6

2

• Only the through-thickness interpolation function is changed hierarchically ζ ζ=+1 R2 R3 ζ=0 ξ, η

1 (1 − ζ ) R1 = 2 1 (1 + ζ ) R2 = 2 ⎧ π ⎛ ⎞ ( ) Cos n ζ − 2 ⎜ ⎟ ⎪⎪ 2 ⎝ ⎠ Rn = ⎨ ⎪ Sin ⎛⎜ (n − 2 ) π ζ ⎞⎟ ⎪⎩ 2 ⎝ ⎠

R1

ζ=-1 R4

for n is odd n = 3 , 4 ...... for n is even

Standard vs. Hierarchical Shape Functions N3

N1 1

ξ

3

ξ = −1

1 2

ξ = +1

ξ =0

(

)

(

)

1 2 ξ −ξ 2 1 2 N2 = ξ +ξ 2 N3 = 1− ξ 2 N1 =

1 1

N1

N2

(

)

Standard

N2

1

1

ξ

1

ξ = −1

2

ξ = +1

1 (1 − ξ ) 2 1 N 2 = (1 + ξ ) 2 N1 =

N3

N1

N2

1

1

1 1

ξ = −1

ξ

3

ξ =0

Hierarchical

1 (1 − ξ ) 2 1 N 2 = (1 + ξ ) 2 N3 = 1−ξ 2 N1 =

2

ξ = +1

(

)

• Additional degrees of freedom are introduced at the mid-surface nodes u =

( u1 v1 w1 " u8 v8 w8 ) (u9 v9 w9 " u16 v16 w16 ) (u17 v17 w17 " u24 v24 w24 )  

   

R1

R2

Basic 24-node element

R3  

mid plane nodes

Interpolation function is increased by an order

u =

( u1 v1 w1 "u8 v8 w8 )(u9 v9 w9 "u16 v16 w16 )(u17 v17 w17 "u24 v24 w24 )(u25 v25 "u32 v32 ) "   

   

 

R1

R2

R4 3 R    

mid plane nodes

How to select the number of terms? • From the closed-form solution:

u ≈ eβ ζ ⎛E π t ⎜ β= 2l ⎜⎝ G

⎞ ⎟ ⎟ 13 ⎠

11

• By Taylor series expansion:

βζ β ζ u ≈ 1+ + ⎛⎜ ⎝

⎞⎟ ⎠

2 +

⎛⎜ ⎝

βζ

3

⎞⎟ ⎠

+"

1! 2! 3! ≈ 1+ C ζ + C ζ 2 + C ζ 3 +" 1 2 3

• The basic element has a quadratic shape function in the thickness direction • The additional order of the shape function is decided based on the ratio of (Cn+1/Cn) • If this ratio is less than a certain threshold value, then the order of the shape function is n • Currently the threshold value is set to 1.0

(Cn+1/Cn) ratio for the flat composite part at the initial stages of cure Symmetric line

1.6 mm Part Tool

5 mm

300 mm

4

3

2

11 terms

1

0

C3 C2

C4 C3

C5 C4

C6 C5

C7 C6

C8 C7

C9 C8

C 10 C 11 C 9 C 10

C 12 C 13 C 11 C 12

C 14 C 13

C 15 C 16 C 14 C 15

Axial stress variation in the thickness direction – standard vs. hierarchical method 1.6

8 Elements Thickness coord (mm)

2 Terms 4 Terms 10 Terms 0.8

0 -100

0

100

200

300

Axial stress (MPa)

400

500

Comparison of warpage prediction – standard vs. hierarchical

Normalized warpage

2.5

2

1.5

1

0.5

0 0

2

4

6

Number of terms

8

10

12

Implementation in a FE code • The FE code needs to allow the degrees of freedom at a node to change dynamically • This can also be done by constraining the unnecessary degrees of freedom dynamically

200

6

150

4.5

100

3

Temperature Resin modulus

Resin modulus (GPa)

Temperature (ºC)

• Currently this cannot be done in ABAQUS, which is our current FE platform

1.5

50

12 Terms

2 Terms 0

0 0

50

100

150

200

Time (min)

250

300

350

Warpage of a Flat Composite Part Dimensions

Material

Interface condition

Length = 1200 mm Part thickness = 1.6 mm (8 layers) Lay-up = unidirectional Tool thickness = 5.0 mm

Part = T300/3900-2 Tool = Aluminum

Fully bonded

A work-around to achieve reasonable run-time efficiency is to run the problem in two steps, with more terms early in the cure cycle Warpage (mm)

Run Time (min)

Standard

Mesh (60*16)

35.29

46.0

Hierarchical (12 Terms)

Fixed

35.29

119.0

Changing

35.29

53.0

Thus one can get an accurate answer, with equivalent efficiency even with an un-optimized FE platform, with the benefit of:

- much more efficient meshing - no requirement for mesh density evaluation

Summary and Conclusions • From an industrial perspective there is a need for a process model that is both accurate and efficient • Understanding the physics of the problem helps make the numerical model more efficient • In this study, a hierarchical finite element method is developed to meet this need