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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
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