Nonlinear analysis of tape springs:

Comparison of two geometrically

exact finite element formulations

Florence Dewalque, Valentin Sonneville and Olivier Brüls

Dept. of Aerospace and Mechanical Engineering, University of Liège, Belgium

11th World Congress on Computational Mechanics

5th European Conference on Computational Mechanics

Barcelona, July 22, 2014

Outline

� Tape springs

� Motivations for a formulation of shells on the

special Euclidean group

� Comparison with a classical formulation

� Test cases

� Tape springs

� Conclusions

Tape springs – Main features

Definition: Thin plate curved along its

width used as a compliant mechanism

characterised by its elastic deformation

General characteristics:� Elastic energy

� Deformation

� No external energy sources

� Space applications

0 2 4 6 8 10 12-300

-200

-100

0

100

200

300

400

500

Bending angle [deg]

B

endi

ng m

omen

t [N

mm

]Tape springs – Mechanical behaviour

� Nonlinear behaviour

� Buckling, hysteresis and self-locking

phenomena

� Senses of bending

Opposite sense

of bending

Equal sense

of bending

Formulation of shells on SE(3)

Motivations:

� Framework based on the Lie group theory where rotations and

translations are treated in a unified and frame invariant way

� Equilibrium equations formulated in a parameterization-free

way

� Singularities due to rotation parameterization naturally avoided

� Significant reduction of the geometrical nonlinearities

� Locking-free and coupled nonlinear interpolation field for

translations and rotations

� No need to update the tangent stiffness matrix at each

iteration/time step

100

101

1.6634

1.6638

1.6642

1.6646

1.6649

1.6653

1.6657

1.6661

Mesh size [mm]

Tip

dis

plac

emen

t [m

m]

SamcefLie

Comparison with a classical formulation

Classical formulation = use of the commercial software SAMCEF

in which shells are based on the Mindlin-Reissner model

Flat plate with a lumped mass submitted to bending (10 Nm)

100

101

0

2

4

6

8

10

Mesh size [mm]

Mea

n nu

mbe

r of N

-R it

erat

ions

[-]

SamcefLie (non constant tan. stiff. matrix)Lie (constant tan. stiff. matrix)

101

102

103

10-2

10-1

100

Nbr. of elements [-]

Rel

ativ

e er

ror i

n S

E(3

) [%

]

SE(3)Unit line

Comparison with a classical formulation

Classical formulation = use of the commercial software SAMCEF

in which shells are based on the Mindlin-Reissner model

Flat plate with a lumped mass submitted to bending (10 Nm)

100

101

78.4152

78.4513

78.4874

78.5236

78.5597

Mesh size [mm]

Tip

dis

plac

emen

t [m

m]

SamcefLie

Comparison with a classical formulation

Flat plate with a lumped mass submitted to bending (500 Nm)

100

101

78.4152

78.4513

78.4874

78.5236

78.5597

Mesh size [mm]

Tip

dis

plac

emen

t [m

m]

SamcefLie

Comparison with a classical formulation

Flat plate with a lumped mass submitted to bending (500 Nm)

100

101

2

3

4

5

6

Mesh size [mm]

Mea

n nu

mbe

r of N

-R it

erat

ions

SamcefLie (non constant tan. stiff. matrix)

100

101

31.842

31.9154

Mesh size [mm]

Tip

dis

plac

emen

t [m

m]

SamcefLie

Comparison with a classical formulation

Flat plate submitted to a surface force (1000 N/m²)

100

0

1

2

3

4

5

Mesh size [mm]

Mea

n nu

mbe

r of N

-R it

erat

ions

SamcefLie (non constant tan. stiff. matrix)

10101

0.022909

0.023014

0.023121

0.023227

0.023335

Mesh size [mm]

Cen

tral d

ispl

acem

ent [

mm

]

SamcefSE(3)Theory

Comparison with a classical formulation

Circular plate submitted to a surface force (100 N/m²)

100

101

1

1.5

2

2.5

3

3.5

4

Mesh size [mm]

Mea

n nu

mbe

r of N

-R it

erat

ions

SamcefLie (non constant tan. stiff. matrix)Lie (constant tan. stiff. matrix)

100

101

0.016596

0.016982

0.017378

0.017783

0.018197

0.018621

0.019055

Mesh size [mm]

Cen

tral d

ispl

acem

ent [

mm

]

SamcefLie

Comparison with a classical formulation

Square plate submitted to a surface force (1000 N/m²)

100

101

0

1

2

3

4

5

Mesh size [mm]

Mea

n nu

mbe

r of N

-R it

erat

ions

[-]

SamcefLie (non constant tan. stiff. matrix)Lie (constant tan. stiff. matrix)

100

101

0.039811

0.050119

Mesh size [mm]

Cen

tral d

ispl

acem

ent [

mm

]

SamcefLieTheory

Comparison with a classical formulation

Rhombic plate submitted to a surface force (1 N/m²)

100

101

0

1

2

3

4

5

Mesh size [mm]

Mea

n nu

mbe

r of N

-R it

erat

ions

[-]

SamcefLie (non constant tan. stiff. matrix)Lie (constant tan. stiff. matrix)

101

102

103

104

1

10

100

Nbr. of elements [-]

Ver

tical

dis

plac

emen

t of B

[mm

]

SamcefLie

Comparison with a classical formulation

Barrel roof submitted to a surface

force (6250 N/m²)

101

102

103

104

1

Nbr. of elements [-]

Ver

tical

dis

plac

emen

t of C

[mm

]

SamcefLie

Comparison with a classical formulation

Barrel roof submitted to a surface

force (6250 N/m²)

101

102

103

1

2

3

4

5

6

7

Nbr. of elements [-]

Mea

n nu

mbe

r of N

-R it

erat

ions

[-]

SamcefLie (non constant tan. stiff. matrix)Lie (constant tan. stiff. matrix)

Comparison with a classical formulation

Tape spring submitted to a surface force (10 000 N/m²)

101

102

103

104

4.7534

4.7643

4.7753

4.7863

4.7973

4.8084

4.8195

Nbr. of elements [-]

Tip

dis

plac

emen

t [m

m]

SamcefLie

101

102

103

104

0

1

2

3

4

Nbr. of elements [-]

Mea

n nu

mbe

r of N

-R it

erat

ions

SamcefLie (non constant tan. stiff. matrix)

Conclusions

� Convergence to the same results for the formulation on SE(3) and

the classical formulation (Samcef)

� No need to always update the tangent stiffness matrix

� Reduction of the amount of geometric nonlinearities

� Good representation of nonlinear behaviours

� Good representation of structures with an initial curvature (tape

springs)

Next developments:

� Improvement of the convergence rate

� Dynamic formulation

� Add a continuation method to model the buckling in tape springs

Nonlinear analysis of tape springs: Comparison of

two geometrically exact finite element

formulations

Florence Dewalque, Valentin Sonneville and Olivier Brüls

Dept. of Aerospace and Mechanical Engineering, University of Liège, Belgium

11th World Congress on Computational Mechanics

5th European Conference on Computational Mechanics

Barcelona, July 22, 2014

Thank you for your attention