Nonlinear Shell Finite Elements - intales · Nonlinear Shell Finite Elements Robert Winkler 1st...
Transcript of Nonlinear Shell Finite Elements - intales · Nonlinear Shell Finite Elements Robert Winkler 1st...
Nonlinear Shell Finite Elements
Robert Winkler
1st Workshopon Nonlinear Analysis of Shell Structures
INTALES GmbH Engineering Solutions
University of Innsbruck, Faculty of Civil Engineering
University of Innsbruck, Faculty of Mathematics, Informatics and Physics
Natters/Tyrol, 15/06/2010
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 1 / 20
Introduction
f77 subdivision
Franz-Josef Falkner Stability
Roland Traxl Material
Stefan Schett Solid shell elements
Outline
The role of the element formulation
Current developments and ...
their impact on (near future) applications
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 2 / 20
Introduction
f77 subdivision
Franz-Josef Falkner Stability
Roland Traxl Material
Stefan Schett Solid shell elements
Outline
The role of the element formulation
Current developments and ...
their impact on (near future) applications
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 2 / 20
Challenge
Ill-posed continuous problem due to extreme slenderness
→ numerical difficulties treating the discretized structure
Membrane behavior
Shear lockingParallelogram lockingTrapezoidal lockingPoisson failureNormal strain failure
Bending behavior
Transverse shear lockingMembrane locking
Additionally (solid shell elements only)
Volumetric lockingPinching locking (‘Curvature thickness locking’)Transverse Poisson failure (‘Poisson thickness locking’)Transverse normal strain failure
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 3 / 20
Challenge
Ill-posed continuous problem due to extreme slenderness→ numerical difficulties treating the discretized structure
Membrane behavior
Shear lockingParallelogram lockingTrapezoidal lockingPoisson failureNormal strain failure
Bending behavior
Transverse shear lockingMembrane locking
Additionally (solid shell elements only)
Volumetric lockingPinching locking (‘Curvature thickness locking’)Transverse Poisson failure (‘Poisson thickness locking’)Transverse normal strain failure
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 3 / 20
Challenge
Ill-posed continuous problem due to extreme slenderness→ numerical difficulties treating the discretized structure
Membrane behavior
Shear lockingParallelogram lockingTrapezoidal lockingPoisson failureNormal strain failure
Bending behavior
Transverse shear lockingMembrane locking
Additionally (solid shell elements only)
Volumetric lockingPinching locking (‘Curvature thickness locking’)Transverse Poisson failure (‘Poisson thickness locking’)Transverse normal strain failure
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 3 / 20
State-of-the-art
Abaqus S4R/S4 large strain shell elements (e.g.)late 1990’sResultant based (Simo & Fox 1989)Penalized drill rotation (Fox & Simo 1992)S4R: Reduced integration (Belytschko et al. 1992)S4: Assumed enhanced membrane strains (Betsch et al. 1996)
General limits of perfectibilityTrapezoidal locking in bilinear (4 node) elementsMembrane locking in quadratic (8/9 node) elements
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 4 / 20
State-of-the-art
Abaqus S4R/S4 large strain shell elements (e.g.)late 1990’sResultant based (Simo & Fox 1989)Penalized drill rotation (Fox & Simo 1992)S4R: Reduced integration (Belytschko et al. 1992)S4: Assumed enhanced membrane strains (Betsch et al. 1996)
General limits of perfectibilityTrapezoidal locking in bilinear (4 node) elementsMembrane locking in quadratic (8/9 node) elements
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 4 / 20
Current Developments
ProblemsS4R: 3 algorithmic parameter*HOURGLASS STIFFNESS .,.,.,.
S4: sightly less robust, still 1 algo. parameter
For thin to moderately thick, smooth, and homogeneous shellsthere are no substantial improvements to be expected...
Even though, what can we do better?Getting rid of any algo. parameter (done...)Improved treatment of shell intersections (stiffened structures)Improved transverse shear distribution (layered structures)
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 5 / 20
Current Developments
ProblemsS4R: 3 algorithmic parameter*HOURGLASS STIFFNESS .,.,.,.
S4: sightly less robust, still 1 algo. parameter
For thin to moderately thick, smooth, and homogeneous shellsthere are no substantial improvements to be expected...
Even though, what can we do better?Getting rid of any algo. parameter (done...)Improved treatment of shell intersections (stiffened structures)Improved transverse shear distribution (layered structures)
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 5 / 20
Conventional vs. Solid Shell Elements
Shell theory attempts the impossible: to provide a two-dimensionalrepresentation of an intrinsically three-dimensional phenomenon(Koiter & Simmonds 1972)
4 node quadrilateral shell element
8 node hexahedral solid shell element
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 20
Conventional vs. Solid Shell Elements
Shell theory attempts the impossible: to provide a two-dimensionalrepresentation of an intrinsically three-dimensional phenomenon(Koiter & Simmonds 1972)
4 node quadrilateral shell element
8 node hexahedral solid shell element
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 20
Conventional vs. Solid Shell Elements
Shell theory attempts the impossible: to provide a two-dimensionalrepresentation of an intrinsically three-dimensional phenomenon(Koiter & Simmonds 1972)
4 node quadrilateral shell element
8 node hexahedral solid shell element
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 20
Solid Shell Elements
BasicsNodes on bottom and top surfaces, 3 D.O.F.s, each3D stress state, explicit thickness change
ProsArbitrary 3D constitutive lawsUnlimited rotationsLarge strains
ConsMesh depends on thicknessIncreased number of overall D.O.F.s by factor (e.g.)
(1+ 1n )(1+ 1
m )∼ 1.56
Extra ill-conditioning (dynamics, iterative solvers)
cond K e ∼ ( ht )
4
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 7 / 20
Shell Intersections
How to treat the joint section?Conventional solid elementsSolid shell elements (which is the thickness direction?)Solid beam elements
Increased efforts for mesh generation...
Use AAR elements!Arbitrary aspect ratiosInadequate for layered structuresIncreased stress oscillations
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 8 / 20
Shell Intersections
How to treat the joint section?Conventional solid elementsSolid shell elements (which is the thickness direction?)Solid beam elements
Increased efforts for mesh generation...
Use AAR elements!Arbitrary aspect ratiosInadequate for layered structuresIncreased stress oscillations
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 8 / 20
ACOSTA Work Package (selected items)
4 node quadrilateral shell elements QS4A8E5..11Prototype: Bischoff & Ramm (1997)Matlab, Abaqus (UEL), DianaResearch tool: RI, SRI, WRI, ACMSBifurcation analysis (analytical tangent)Limited applicability (stiffened structures)
8 node hexahedral shell elements HS8A8E8..Prototype: Klinkel et al. (1999)Passes in-plane and bending patch test’Optimal’ solid shell element for layered structuresMatlab, Abaqus (UEL) → ’small launcher’ (RT)
8 node hexahedral continuum elements HC8E18..21Prototype: Alves de Souza et al. (2003)Significantly improved versionPasses in-plane and volumetric patch test, pinching locking!Matlab, Abaqus (UEL) → aniso. large strain plasticity (RT)
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 9 / 20
ACOSTA Work Package (selected items)
4 node quadrilateral shell elements QS4A8E5..11Prototype: Bischoff & Ramm (1997)Matlab, Abaqus (UEL), DianaResearch tool: RI, SRI, WRI, ACMSBifurcation analysis (analytical tangent)Limited applicability (stiffened structures)
8 node hexahedral shell elements HS8A8E8..Prototype: Klinkel et al. (1999)Passes in-plane and bending patch test’Optimal’ solid shell element for layered structuresMatlab, Abaqus (UEL) → ’small launcher’ (RT)
8 node hexahedral continuum elements HC8E18..21Prototype: Alves de Souza et al. (2003)Significantly improved versionPasses in-plane and volumetric patch test, pinching locking!Matlab, Abaqus (UEL) → aniso. large strain plasticity (RT)
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 9 / 20
ACOSTA Work Package (selected items)
4 node quadrilateral shell elements QS4A8E5..11Prototype: Bischoff & Ramm (1997)Matlab, Abaqus (UEL), DianaResearch tool: RI, SRI, WRI, ACMSBifurcation analysis (analytical tangent)Limited applicability (stiffened structures)
8 node hexahedral shell elements HS8A8E8..Prototype: Klinkel et al. (1999)Passes in-plane and bending patch test’Optimal’ solid shell element for layered structuresMatlab, Abaqus (UEL) → ’small launcher’ (RT)
8 node hexahedral continuum elements HC8E18..21Prototype: Alves de Souza et al. (2003)Significantly improved versionPasses in-plane and volumetric patch test, pinching locking!Matlab, Abaqus (UEL) → aniso. large strain plasticity (RT)
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 9 / 20
ACOSTA Work Package (cont.)
8 node hexahedral continuum element HC8A18E9Arbitrary Aspect Ratios (AAR)’Optimal’ formulation wrt. locking and shape sensitivityFails patch tests (increased stress oscillations)Matlab, Abaqus (UEL) → ’small launcher’ (RT)
8 node hexahedral beam elements HB8A4E7..11’Solid beam’ elementsModeling joint sections, solid-beam transitions, etc.Matlab, Abaqus (UEL)
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 10 / 20
ACOSTA Work Package (cont.)
8 node hexahedral continuum element HC8A18E9Arbitrary Aspect Ratios (AAR)’Optimal’ formulation wrt. locking and shape sensitivityFails patch tests (increased stress oscillations)Matlab, Abaqus (UEL) → ’small launcher’ (RT)
8 node hexahedral beam elements HB8A4E7..11’Solid beam’ elementsModeling joint sections, solid-beam transitions, etc.Matlab, Abaqus (UEL)
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 10 / 20
Selected Results
Standard benchmark ‘Pinched hemisphere’
material
E = 6.825 ·107
ν = 0.3
geometry
R = 10
t = 0.04
F = 200
Abgebildet ist nur ein Viertel der oben o�enen Halbkugel. Das Loch entsteht durch eine
um 18�von der x3-Achse geneigten Gerade, welche um die x3-Achse rotiert. Zweck der
Ö�nung ist es, eine ungünstige Netzeinteilung am Pol der Halbkugel zu verhindern. An
den Punkten wo die Halbkugel die ξ1-Achse berührt sind diese Punkte in ξ2- und ξ3-
Richtung festgehalten und an den Punkten wo die ξ2-Achse die Hlabkugel berührt sind
diese in ξ1- und ξ3-Richtung festgehalten, somit ist die Lagerbedingung de�niert. An die-
sen Punkten (exemplarisch dargestellt in Abb. ?? mit K1 und K2) greift die Belastung in
die dort mögliche Verscheiungsrichtung an, Einzelkräfte entlag der ξ2-Achse richten sich
zum Mittelpunkt und die Kräfte entlang der ξ1-Achse zeigen vom Mittelpunkt weg. Zum
unterschied zur Schlenformulierung wird hier die Kraft nicht in der Schalenmittel�äceh
aufgebracht, sondern an der Schalen Innen- und Auÿen�äch, vgl. [?, S. 113f]. Die Kraft
F wird zu gleichen Teile auf die 8 Belastungspunkte aufgeteilt.
Je in Abb. (??) dargestellter Kante wird die Struktur in N-Elemente unterteilt. Zunächst
wird im linearem Bereich die Konvergenz bei Netzverfeinerung überprüft und das Ver-
halten von Membran- und Schublocking untersucht. In Tab. (??) sind als Ergebnis die
Radialverschiebungen am Belastungepunkt K2 je Netzeinteilung aufgelistet.
1
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 11 / 20
Selected Results
0
15
30
45
60
75
90
0 1 2 3 4 5 6
HC8A18E9(u1)
HS8A8E9(u1)
HC8E18(u1)
HC8E21(u1)
SC8R(u1)
S4R(u1)
QS4A8E9(u1)
HC8A18E9(u2)
HS8A8E9(u2)
HC8E18(u2)
HC8E21(u2)
SC8R(u2)
S4R(u2)
QS4A8E9(u2)
displacement
load
leve
l
1
Pinched hemisphere, d=0.04
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 12 / 20
Selected Results
0
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
0 1 2 3 4 5 6
HC8A18E9(u1)
HS8A8E9(u1)
HC8E18(u1)
HC8E21(u1)
SC8R(u1)
HC8A18E9(u2)
HS8A8E9(u2)
HC8E18(u2)
HC8E21(u2)
SC8R(u2)
displacement
load
leve
l
1
Pinched hemisphere, d=0.01
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 13 / 20
Structure
C-section cantilever beammaterial
E = 1 ·107
ν = 0.3333
geometry
L = 36
t = 0.05
b = 2.025
h = 6.05
Um die E�zients der Elemente im Bezug auf das Beispiel besser zu untersuchen ist eine
Netzverfeinerung vorgenommen worden. Netz 1 beinhaltet je nach Querschnittsdiskreti-
sierung nach Abb. (??) 10(a) oder 8(b) Elemente über den Querschnitt und 36 Elemente
über die Trägerlänge, somit ergeben sich eine Vernetzung beim Netz 1 von 360(10x36)
Elemente für (a) und 288(8x36) Elemente für (b). Netz 2 wurden in Bezug auf Netz 1
um die Hälfte verfeinert und so ergeben sich für Netz 2 1440(20x72) Elemente für (a)
und 1296(18x72) Elemente für (b). Als Ergebnis wird die Verschiebung des Belastungs-
punktes in die Belastungsrichtung je nach Belastungröÿe in Abb. (??) für Netz 1 und
Abb. (??) für Netz 2 dargestellt.
1
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 14 / 20
Selected Results
0
20
40
60
80
100
120
140
0 0.25 0.50 0.75 1.00 1.25 1.50
QS4A8E9S4R
SC8RC
SC8R
HC8E21C
HC8E21
HC8E18C
HC8E18
HS8A8E9C
HS8A8E9
HC8A18E9C
HC8A18E9
absolute displacement of the loaded node
exte
rnal
load
leve
l
1
C-section cantilever beam, coarse mesh
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 15 / 20
Selected Results
0
20
40
60
80
100
120
140
0 0.25 0.50 0.75 1.00 1.25 1.50
QS4A8E9S4R
SC8RC
SC8R
HC8E21C
HC8E21
HC8E18C
HC8E18
HS8A8E9C
HS8A8E9
HC8A18E9C
HC8A18E9
absolute displacement of the loaded node
exte
rnal
load
leve
l
1
C-section cantilever beam, fine mesh
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 16 / 20
Structure
I-section cantilever beammaterial
E = 2 ·106
ν = 0.30
geometry
L = 48
t = 0.25
b = 3
h = 3
load
F =λ · 1Fh = F/1000
Abbildung 0.1: Diskretisierung Kragträger I-Pro�l
1
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 17 / 20
Results
0
900
1800
2700
3600
4500
5400
6300
0 5 10 15 20 25 30
HC8A18E9− C
HC8A18E9−B
HC8A18E9−A
HS8A8E9− C
HS8A8E9− B
HS8A8E9− A
SC8R−A
SC8R−B
SC8R− C
HC8E18− A
HC8E18− B
HC8E18− C
HC8E21− A
HC8E21− B
HC8E21− C
C3D8− C
C3D8−D
absolute vertical displacement of the loaded node
exte
rnal
load
leve
l
1
I-section cantilever beam
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 18 / 20
Lessons learned
Solid shell elements are a valuable tool for...
strength analyses (shell-solid transitions etc.)large strain and/or contact problems (metal forming etc.)delamination...Conditioning problem in implicit dynamics
Conventional shell elements should be preferred for
stability analyseslarge-scale applications... whenever feasible!
Rigid joints in multi-shell structures
Feasible approximation in most stability analysesDeteriorating in the large displacement/strain regime
Mesh quality cannot be judged utilizing linear results!
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 19 / 20
Public Relations
Winkler “Three dimensional shell elements for industrialapplications” CEAS Berlin 2007
Winkler & Falkner “Numerical stability analyses of shellstructures, revisited” SSTA Jurata 2009
Winkler “Two comments on membrane locking” GAMMKarlsruhe 2010
Winkler, Traxl, Schett “Large strain continuum elements witharbitrary aspect ratios” ECCM Paris 2010
Winkler, Traxl, Schett “Low-order continuum shell elementsat finite strains, their limits of perfectibility and applications”SolMech Warshaw 2010
Robert Winkler (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 20 / 20