Nonlinear Shell Finite Elements - intales · Nonlinear Shell Finite Elements Robert Winkler 1st...

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Nonlinear Shell Finite Elements Robert Winkler 1 st Workshop on 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

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