A New Discontinuous Galerkin Formulation for Kirchhoff-Love Shells
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University of Liège R. Radovitzky Department of Aeronautics and Astronautics Massachusetts Institute of Technology Cambridge, MA A New Discontinuous Galerkin Formulation for Kirchhoff-Love Shells 9th US National Congress on Computational Mechanics San Francisco, California, USA, July 23-26, 2007 L. Noels Aerospace and Mechanical Engineering Department University of Liège Belgium
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A New Discontinuous Galerkin Formulation for Kirchhoff-Love Shells. L. Noels Aerospace and Mechanical Engineering Department University of Liège Belgium. R. Radovitzky Department of Aeronautics and Astronautics Massachusetts Institute of Technology Cambridge, MA. - PowerPoint PPT Presentation
Transcript of A New Discontinuous Galerkin Formulation for Kirchhoff-Love Shells
University of Liège
R. RadovitzkyDepartment of Aeronautics and Astronautics
Massachusetts Institute of TechnologyCambridge, MA
A New Discontinuous Galerkin Formulation for
Kirchhoff-Love Shells
9th US National Congress on Computational MechanicsSan Francisco, California, USA, July 23-26, 2007
L. NoelsAerospace and Mechanical Engineering Department
University of LiègeBelgium
Slide 2University of Liège
Introduction
• Discontinuous Galerkin methods– Finite-element discretization
allowing for inter-elements discontinuities
– Weak enforcement of compatibility equations and continuity (C0 or C1, …) through interelement integrals called numerical fluxes
– Stability is ensured with quadratic interelement integrals
• Applications of DG to solid mechanics– Allowing weak enforcement of C0 continuity:
• Non-linear mechanics (Noels and Radovitzky 2006; Ten Eyck and Lew 2006)• Reduction of locking for shells (Güzey et al. 2006)• Beams and plates (Arnold et al. 2005, Celiker and Cockburn 2007)
– Allowing weak enforcement of C1 continuity (strong enforcement of C0):• Beams and plates (Engel et al. 2002)• Strain gradient continuity (Molari et al. 2006)
Slide 3University of Liège
Introduction
• Purpose of the presentation: to develop a DG formulation– for Kirchhoff-Love shells,– which is a C0 displacement formulation,– without addition of degrees of freedom,– where C1 continuity is enforced by DG interface terms,– which leads to an easy implementation of the shell elements in the reduced
coordinates,– without locking in bending
• Scope of the presentation– Kirchhoff-Love shells– DG formulation– Numerical properties– Implementation– Numerical examples
Slide 4University of Liège
Kirchhoff-Love shells
• Kinematics of the shell– Shearing is neglected
– Small displacements formulation
– Resultant linear and angular equilibrium equations
and ,
in terms of the resultant stress components and , with
, and
and are the resultant applied tension and torque
E1
E2
E3
12
A
2=cst
1 =c
st
0=0t0
t0
1=cst
2=cst
t
S
S0
°0-1
=t
Slide 5University of Liège
• Constitutive behavior and BC– Resultant strain components
high order– Linear constitutive relations
– Boundary conditions and
Kirchhoff-Love shells
Slide 6University of Liège
Discontinuous Galerkin formulation
• Hu-Washizu-de Veubeke functional
– Polynomial approximation uhPk ⊂ C0
– New inter-elements term accounting for discontinuities in the derivatives
E1
E2
E3
A e-
A h
dNA h
dUA e-
A e+
dNA e+
S0
-
S0e- S0e+
dIAe-
dUA h
dIA h
0=0t0
Slide 7University of Liège
Discontinuous Galerkin formulation
• Minimization of the functional (1/2)– With respect to the resultant strains and
and
– With respect to the resultant stresses and
Discontinuities result in new terms (lifting operators)
and in the introduction of a stabilization parameter .
– With respect to the displacement field balance equation (next slide)
Slide 8University of Liège
Discontinuous Galerkin formulation
• Minimization of the functional (2/2)– With respect to the displacement field uh
– Reduction to a one-field formulation
Mesh size
with
Slide 9University of Liège
• Consistency– Exact solution u satisfies the DG formulation
• Definition of an energy norm
• Stability
• Convergence rate of the error in the mesh size hs
– Energy norm:
– LL2 norm:
Numerical properties
General Pure
bending
Pure
membrane
k-1 k-1 k
≥ k-1 k+1 (if k>2) k+1 (if k>0)
Motivates the use of
quadratic elements
with C>0 if > Ck, Ck depends only on k.
Slide 10University of Liège
Implementation of 8-node bi-quadratic quadrangles
• Membrane equations
– Solved in (1, 2) system– 3 X 3 Gauss points with EAS method or
2 X 2 Gauss points
• Bending equations
– Solved in (1, 2) system– 3 X 3 or 2 X 2 Gauss points – Locking taken care of by the
DG formulation
Straightforward implementation of the equations
E1
E2
E3
A e
1
2
3
4
5
6
7
8AB
C D
12
Se
EF G
HI
0=0t0
Slide 11University of Liège
Implementation of 8-node bi-quadratic quadrangles
• Interface equations
– Interface element s solved in 1 system– 3 or 2 Gauss points
– Neighboring elements Se– and Se+ evaluate values (t, t, , , Hm)
on the interface Gauss points andsend them to the interface element s
– Local frame (,1, ,2, t0) of interface
element s is the average of the neighboringelements’ frames
E1
E2
E3
s
1
2
3
4
5
6
7
8A
B
C
1
Se-
Se+
9
13
12
11
10
-
0=0t0
Slide 12University of Liège
Implementation of 16-node bi-cubic quadrangles
• Membrane equations– Solved in (1, 2) system
– 4 X 4 Gauss points
(without EAS method)
• Bending equations– Solved in (1, 2) system
– 4 X 4 Gauss points
• Interface equations– Interface element s
solved in 1 system
– 4 Gauss points
E1
E2
E3
A e
1
2
3
4
5
7
9
11
AB
CD
1 2
Se
E
FGH
I
0=0t0
6
8 10
12
1314
15
16J
KL
M
NO
P
E1
E2
E3
s
1
2
3
4
5
6
7
8
A
BC
1
Se-
Se+
9
1312
11
10 -
0=0t0
14
1516
17
18
192021
22
2324
25
26
27
28
D
Slide 13University of Liège
• 8-node bi-quadratic quadrangles Membrane test Bending test
– Bending test:• Instability if ≤ 10 and locking if > 1000• Convergence rate k-1 in the energy-norm and k+1 in the L2-norm
Numerical example: Cantilever beam (L/t = 10)
0 0.05 0.10
0.5
1
1.5
hs/ L
Nor
mal
ized
10-2
10-1
100
10-2
10-1
100
11
1
2
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
Membrane
10-2
10-1
100
10-3
10-2
10-1
100
1
3
hs/ L
Err
or o
n
= 10 = 102
= 104
= 106
Membrane
10-2
10-1
100
10-2
10-1
100
11
1
2
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
Membrane
10-2
10-1
100
10-2
10-1
100
11
1
2
hs/ L
|||e|
|| = 10 = 102
= 104
= 106
Membrane
10-2
10-1
100
10-2
10-1
100
11
1
2
hs/ L|||
e|||
= 10 = 102
= 104
= 106
Membrane
Slide 14University of Liège
• 8-node bi-quadratic quadranglesClamped/Clamped Supported/Clamped Supported/Supported
– Instability if ≤ 10 and locking if > 1000
Numerical example: Plate bending (L/t = 100)
Sym.
Sym
. Sym.
Sym
.
Sym.
Sym
.
0 0.05 0.1 0.15 0.20
0.5
1
1.5
hs/ L
Nor
mal
ized
z
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
10-3
10-2
10-1
100
10-3
10-2
10-1
100
1
3
hs/ L
Err
or o
n z
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L|||
e|||
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
Slide 15University of Liège
• 8-node bi-quadratic quadrangles
Numerical example: Pinched ring (R/t = 10)
0 20 40 60 80-0.5
0
0.5
[°]
Mzz
/PR
hs/L = 1
hs/L = 1/2
hs/L = 1/8Exact
= 100
0 20 40 60 80-0.5
0
0.5
[°]
Mzz
/PR
hs/L = 1
hs/L = 1/2
hs/L = 1/8Exact
= 100
10-1
100
10-3
10-2
10-1
100
1
2
hs/ L
Err
or o
n
y
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
0.2 0.4 0.6 0.8 10.4
0.6
0.8
1
hs/ L
No
rma
lized
m
ax
= 1.e1 = 1.e2 = 1.e4 = 1.e6 = 1.e8
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 1.e1 = 1.e2 = 1.e4 = 1.e6 = 1.e8
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
Bending and membrane coupling• Instability if ≤ 10• Convergence: k-1 in the energy-
norm and k in the L2-norm
Slide 16University of Liège
8-node bi-quad. 16-node bi-cub.
Numerical example: Pinched open-hemisphere (R/t = 250)
0 0.05 0.1 0.15 0.20
0.5
1
1.5
hs/ R
Nor
mal
ized
x
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-3
10-2
10-1
100
1
3
hs/ R
Err
or o
n x
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108 Double curvature• Instability if ≤ 10• Locking if > 1000 (quad.)
and if > 100000 (cubic)• Convergence in L2 norm: k+1
0 0.05 0.1 0.15 0.20
0.5
1
1.5
hs/ R
Nor
mal
ized
x
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
|| = 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-4
10-3
10-2
10-1
100
1
4
hs/ R
Err
or o
n x
= 10 = 102
= 104
= 106
= 108
10-2
10-1
100
10-2
10-1
100
1
1
hs/ L
|||e|
||
= 10 = 102
= 104
= 106
= 108
Slide 18University of Liège
8-node bi-quad. 16-node bi-cub.
Numerical example: Pinched cylinder (R/t = 100)
0 0.5 10
0.5
1
1.5
hs/ L
Nor
mal
ize
d y
= 10 = 102
= 104
= 106
0 0.5 10
0.5
1
1.5
hs/ L
Nor
mal
ized
y
= 10 = 102
= 104
= 106
10-2
10-1
100
10-3
10-2
10-1
100
1
2
hs/ L
Err
or o
n y
= 10 = 102
= 104
= 106
0 0.5 10
0.5
1
1.5
hs/ L
Nor
mal
ized
y
= 10 = 102
= 104
= 106
Complex membrane state• Instability if ≤ 10• Locking if > 10000 (quad.)
and if > 100000 (cubic)• Convergence in L2-norm: k
0 0.5 10
0.5
1
1.5
hs/ L
Nor
mal
ized
y = 10
= 102
= 104
= 106
0 0.5 10
0.5
1
1.5
hs/ L
Nor
mal
ized
y
= 10 = 102
= 104
= 106
10-2
10-1
100
10-3
10-2
10-1
100
1
3
hs/ L
Err
or o
n y = 10
= 102
= 104
= 106
0 0.5 10
0.5
1
1.5
hs/ L
Nor
mal
ized
y
= 10 = 102
= 104
= 106
Slide 20University of Liège
Conclusions
• Development of a discontinuous Galerkin framework for Kirchhoff-Love shells:
– Displacement formulation (no additional degree of freedom)– Strong enforcement of C0 continuity– Weak enforcement of C1 continuity
• Quadratic elements:– Method is stable if ≥ 100– Bending locking avoided if ≤ 1000– Membrane equations integrated with EAS or Reduced integration
• Cubic elements:– Method is stable if ≥ 100– Bending locking avoided if ≤ 100000– Full Gauss integration
• Convergence rate:– k-1 in the energy norm– k or k+1 in the L2-norm
