Post on 14-Apr-2018
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Universitt Stuttgart
Fakultt fr Bau und-Umweltingenieurwissenschaften Baustatik und Baudynamik
Modeling of Shells with
Three-dimensional Finite ElementsManfred Bischoff
Institute of Structural Mechanics
University of Stuttgartbischoff@ibb.uni-stuttgart.de
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Universitt Stuttgart
Fakultt fr Bau und-Umweltingenieurwissenschaften Baustatik und Baudynamik
acknowledgements
Ekkehard Ramm
Kai-Uwe
Bletzinger
Thomas Cichosz
Michael Gee
Stefan Hartmann
Wolfgang A. Wall
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Universitt Stuttgart
Fakultt fr Bau und-Umweltingenieurwissenschaften Baustatik und Baudynamik
outline
evolution of shell modelssolid-like shell or shell-like solid element?
locking and finite element technology
how three-dimensional are 3d-shells / continuum shells / solid shells?
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History
early attempts
Shell Theories / Shell Models
Leonhard Euler 1707 - 1783
ring models (Euler 1766)
lattice models (J. Bernoulli 1789)
continuous models (Germain,
Navier, Kirchhoff, 19th century)
Gustav Robert Kirchhoff
1824 - 1887
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History
August E.H. Love, 1888
membrane and bending action
inextensional deformations
first shell theory = Kirchhoff-Love theory
This paper is really an attempt to construct a theoryof the vibrations of bells
Lord Rayleigh (John W. Strutt)
Shell Theories / Shell Models
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All you need is Love?
August E.H. Love, 1888
first shell theory = Kirchhoff-Love theory
This paper is really an attempt to construct a theoryof the vibrations of bells
Shell Theories / Shell Models
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Evolution of Shell Models
fundamental assumptions
cross sections remain
- straight
- unstretched
- normal to midsurface
( )0,0 == zzzz
contradiction
requires modificationof material law
0
0
=
=
yz
xz
Shell Theories / Shell Models
Kirchhoff-Love
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Evolution of Shell Models
fundamental assumptions
cross sections remain
- straight
- unstretched
- normal to midsurface
( )0,0 == zzzz
contradiction
requires modificationof material law
0
0
yz
xz
Shell Theories / Shell Models
Reissner-Mindlin, Naghdi
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Evolution of Shell Models
fundamental assumptions
cross sections remain
- straight
- unstretched
- normal to midsurface
0,0 zzzz
contradiction
requires modificationof material law
0
0
yz
xz
Shell Theories / Shell Models
7-parameter formulation
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Evolution of Shell Models
fundamental assumptions
cross sections remain
- straight
- unstretched
- normal to midsurface
0,0 zzzz
contradiction
requires modificationof material law
0
0
yz
xz
Shell Theories / Shell Models
multi-layer, multi-director
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Evolution of Shell Models
from classical thin shell theories to 3d-shell models 1888: Kirchhoff-Love theory
membrane and bending effects
middle of 20th century: Reissner/Mindlin/Naghdi
+ transverse shear strains
1968: degenerated solid approach (Ahmad, Irons, Zienkiewicz)
shell theory = semi-discretization of 3d-continuum
1990+: 3d-shell finite elements, solid shells,surface oriented (continuum shell) elements
Schoop, Simo et al, Bchter and Ramm, Bischoff and Ramm,
Krtzig, Sansour, Betsch, Gruttmann and Stein, Miehe and Seifert,
Hauptmann and Schweizerhof, Brank et al., Wriggers and Eberlein,Klinkel, Gruttmann and Wagner, and many, many others
since ~40 years parallel development of theories and finite elements
Shell Theories / Shell Models
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Evolution of Shell Models
the degenerated solid approach Ahmad, Irons and Zienkiewicz (1968)
Shell Theories / Shell Models
1. take a three-dimensional finite element (brick)
2. assign a mid surface and a thickness direction
3. introduce shell assumptions and
refer all variables to mid surface quantities(displacements, rotations, curvatures, stress resultants)
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Derivation from 3d-continuum (Naghdi)
geometry of shell-like body
3d-shell Models
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Derivation from 3d-continuum (Naghdi)
deformation of shell-like body
3d-shell Models
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7-parameter Shell Model
displacements
+ 7th parameter for linear transverse normal strain distribution
geometry of shell-like body
3d-shell Models
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7-parameter Shell Model
linearized
strain tensor in three-dimensional space
approximation (semi-discretization)
strain components
+ linear part via 7th
parameter
3d-shell Models
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7-parameter Shell Model
in-plane
strain
components
membrane
bending
higher order effects
3d-shell Models
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Semi-discretization of Shell Continuum
straight cross sections: inherent to theory ordiscretization?
equivalence of shell theory and degenerated solid approach, Bchter and Ramm (1992)
discretization
(3-dim.)
linear shape functions
+ additional assumptions
discretization
(2-dim.)
dimensional reduction
Solid-like Shell or Shell-like Solid?
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Large Strains
metal forming, using 3d-shell elements (7-parameter model)
Solid-like Shell or Shell-like Solid?
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Large Strains
metal forming, using 3d-shell elements (7-parameter model)
3d stress state
contact
Solid-like Shell or Shell-like Solid?
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Large Strains
very thin shell (membrane), 3d-shell elements
Solid-like Shell or Shell-like Solid?
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Motivation
three-dimensional data from CAD
complex structures with stiffeners and intersections
connection of thin and thick regions,layered shells, damage and fracture,
why solid elements instead of 3d-shell elements?
Three-dimensional FEM for Shells
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Shell Analysis with Standard Solid Elements
a nave approach: take a commercial code and go!
maximum
displacement
pressure load
element formulation
include extra displacements
exclude extra displacements
Three-dimensional FEM for Shells
clamped
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Shell Analysis with Standard Solid Elements
a nave approach: take a commercial code and go!
0.0
1.0
2.0
3.0
0 100000 200000 300000
exclude extra displacements
= standard Galerkin elements
include extra displacements
= method of incompatible modes
Three-dimensional FEM for Shells
d.o.f.
displacement
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Shell Analysis with Standard Solid Elements
one layer of standard Galerkin
elements yields wrong results
0.0
1.0
2.0
3.0
0 100000 200000 300000
Three-dimensional FEM for Shells
exclude extra displacements
= standard Galerkin elements
include extra displacements
= method of incompatible modes
d.o.f.
displacement
shell elements
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Shell Analysis with Standard Solid Elements
refinement in transverse direction helps (but is too expensive!)
0.0
1.0
2.0
3.0
0 100000 200000 300000
Three-dimensional FEM for Shells
2 layers of
standard Galerkin elements
d.o.f.
displacement
shell elements
Poisson thickness locking
(volumetric locking)
7th
parameter in 3d-shell model = incompatible mode
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Three-dimensional Analysis of Shells
3d-shell (e.g. 7-parameter formulation)
two-dimensional meshdirector + difference vector
6 (+1) d.o.f. per nodestress resultants
continuum shell (solid shell)
three-dimensional mesh3 d.o.f. per node (+ internal d.o.f.)
stress resultants
3d-solid (brick)
three-dimensional mesh3 d.o.f. per node (+ internal d.o.f.)
3d stresses
there are (at least) three different strategies
3d-shell (e.g. 7-parameter formulation)
two-dimensional meshdirector + difference vector
6 (+1) d.o.f. per nodestress resultants
continuum shell (solid shell)
three-dimensional mesh3 d.o.f. per node (+ internal d.o.f.)
stress resultants
3d-solid (brick)
three-dimensional mesh3 d.o.f. per node (+ internal d.o.f.)
3d stresses
Three-dimensional FEM for Shells
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Surface Oriented Formulation
nodal displacements instead of difference vector
+ 7th parameter for linear transverse normal strain distribution
nodes on upper and lower shell surface
3d-shell Models
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Surface Oriented Formulation
membrane
and bending
strains
membrane
bending
higher ordereffects
continuum shell formulation
3d-shell Models
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Requirements
what we expect from finite elements for 3d-modeling of shells
Requirements
asymptotically correct (thickness 0)
numerically efficient for thin shells (locking-free)
consistent (patch test)
competitive to usual 3d-elements for 3d-problems
required for both 3d-shell elements and solid elements for shells
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A Hierarchy of Models
thin shell theory (Kirchhoff-Love, Koiter)
3-parameter model
Asymptotic Analysis
modification of material law required
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A Hierarchy of Models
first order shear deformation theory (Reissner/Mindlin, Naghdi)
5-parameter model
Asymptotic Analysis
modification of material law required
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A Hierarchy of Models
shear deformable shell + thickness change
6-parameter model
Asymptotic Analysis
asymptotically correct for membrane state
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A Hierarchy of Models
shear deformable shell + linear thickness change
7-parameter model
Asymptotic Analysis
asymptotically correct for membrane +bending
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Numerical Experiment (Two-dimensional)
a two-dimensional example: discretization of a beam with 2d-solids
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Numerical Experiment (Two-dimensional)
a two-dimensional example: discretization of a beam with 2d-solids
Asymptotic Analysis
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Requirements
what we expect from finite elements for 3d-modeling of shells
Requirements
asymptotically correct (thickness 0)
numerically efficient for thin shells (locking-free)
consistent (patch test)
competitive to usual 3d-elements for 3d-problems
required for both 3d-shell elements and solid elements for shells
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Locking Phenomena
3d-shell/continuum shell vs. solid
3d-shell / continuum shell 3d-solid
in-plane shear locking
transverse shear locking
membrane locking
Poisson thickness locking
curvature thickness locking
shear locking
(membrane locking)
volumetric locking
trapezoidal locking
Numerical Efficiency and Locking
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Comparison: Continuum Shell vs. 3d-solid
continuum shell 3d-solid
stress resultants
distinct thickness direction
linear in 3
stresses
all directions are equal
quadratic in 3
Numerical Efficiency and Locking
differences with respect to finite element technology
and underlying shell theory
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Trapezoidal Locking (Curvature Thickness locking)numerical example: pinched ring
Numerical Efficiency and Locking
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1,0E-01
1,0E+00
1,0E+01
1,0E+02
1,0E+03
1,0E+04
1 10 100 1000
1,0E-01
1,0E+00
1,0E+01
1,0E+02
1,0E+03
1,0E+04
1 10 100 1000
1,0E-01
1,0E+00
1,0E+01
1,0E+02
1,0E+03
1,0E+04
1 10 100 1000
Trapezoidal Locking (Curvature Thickness locking)numerical example: pinched ring
3d-solid
(standard Galerkin)
3d-solid with EAS
continuum shell, DSG method
Numerical Efficiency and Locking
d.o.f.
displacement
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Trapezoidal Locking (Curvature Thickness locking)origin of locking-phenomenon explained geometricallypure bending of an initially curved element
leads to artificial transverse normal strains and stresses
trapezoidal locking distortion sensitivity
Numerical Efficiency and Locking
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Cylindrical Shell Subject to External Pressure
slenderness
shell
elements
3d-solid elements
coarse mesh, 4608 d.o.f.
coarse mesh, 4608 d.o.f.
Numerical Efficiency and Locking
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Cylindrical Shell Subject to External Pressure
slenderness
shell
elements
3d-solid elements
coarse mesh, 4608 d.o.f.
coarse mesh, 4608 d.o.f.
fine mesh, 18816 d.o.f.
fine mesh, 18816 d.o.f.
Numerical Efficiency and Locking
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Cylindrical Shell Subject to External Pressure
slenderness
shell
elements
3d-solid elements
coarse mesh, 4608 d.o.f.
coarse mesh, 4608 d.o.f.
fine mesh, 18816 d.o.f.
fine mesh, 4608 d.o.f.
still 70% error!
factor 13!
due to trapezoidal locking
Numerical Efficiency and Locking
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Finite Element Technology: Summary
3d-shell, continuum shell, solid shell,
3d-solid (brick)
stress resultants allow separate treatment
of membrane and bending terms
anisotropic element technology
(trapezoidal locking)
no transverse direction
no distinction of membrane / bending
(usually) suffer from trapezoidal locking
general
effective methods for transverse shear locking available
membrane locking mild when (bi-) linear shape functions are used
Numerical Efficiency and Locking
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Finite Element Technology: Summary
triangles, tetrahedrons and wedges
tetrahedrons: hopeless
wedges: may be o.k.
in transverse direction
problem: meshing with hexahedrons
extremely demanding
A triangle!!
Numerical Efficiency and Locking
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Requirements
what we expect from finite elements for 3d-modeling of shells
Requirements
asymptotically correct (thickness 0)
numerically efficient for thin shells (locking-free)
consistent (patch test)
competitive to usual 3d-elements for 3d-problems
required for both 3d-shell elements and solid elements for shells
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Fundamental Requirement: The Patch Test
Consistency and the Patch Test
one layer of 3d-elements, x = const.
3d-solid continuum shell, DSG
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Fundamental Requirement: The Patch Test
one layer of 3d-elements, x = const., directors skewed
3d-solid continuum shell, DSG
Consistency and the Patch Test
T di i l M d l P bl
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Two-dimensional Model Problem
the fundamental dilemma of finite element technology
modeling constant stresses
Consistency and the Patch Test
T di i l M d l P bl
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Two-dimensional Model Problem
the fundamental dilemma of finite element technology
or pure bending?
Consistency and the Patch Test
F d t l R i t Th P t h T t
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Fundamental Requirement: The Patch Test
one layer of 3d-elements, x = const., directors skewed
continuum shell, no DSG
(trapezoidal locking in bending)
3d-solid continuum shell, DSG
Consistency and the Patch Test
F d t l R i t Th P t h T t
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Fundamental Requirement: The Patch Test
same computational results, different scales for visualization
continuum shell, DSG continuum shell, no DSG
(trapezoidal locking in bending)
avoiding trapezoidal locking
contradicts satisfaction of
patch test
(known since long,
e.g. R. McNeal text book)
much smaller error
originates fromshell assumptions!?
Consistency and the Patch Test
F d t l R i t Th P t h T t
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Fundamental Requirement: The Patch Test
curvilinear components of strain tensor
consistency: exactly represent
Consistency and the Patch Test
C
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Convergence
mesh refinement by subdivision
Consistency and the Patch Test
C
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0
100
200
300
400
500
600
700
800
900
1000
1100
0 100 200 300 400 500 600 700 800 900 1000
Convergence
mesh refinement
continuum shell, DSG
continuum shell, no DSG
3d-solid (reference)
Consistency and the Patch Test
Con ergence
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850
900
950
1000
1050
0 100 200 300 400 500 600 700 800 900 1000
Convergence
mesh refinement
effect from element technology
diminishes with mesh refinement
effect from theory
remains
Consistency and the Patch Test
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Requirements
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Requirements
what we expect from finite elements for 3d-modeling of shells
Requirements
asymptotically correct (thickness 0)
numerically efficient for thin shells (locking-free)
consistent (patch test)
competitive to usual 3d-elements for 3d-problems
required for both 3d-shell elements and solid elements for shells
Panel with Skew Hole
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Panel with Skew Hole
3d-problems
distorted elements, skew directors
Panel with Skew Hole
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Panel with Skew Hole
continuum shell elements
continuum shell
no DSG
continuum shell
DSG
3d-problems
Panel with Skew Hole
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Panel with Skew Hole
continuum shell elements
continuum shell
no DSG
continuum shell
DSG
3d-problems
Panel with Skew Hole
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Panel with Skew Hole
continuum shell elements
continuum shell
no DSG
continuum shell
DSG
3d-problems
Panel with Skew Hole
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Panel with Skew Hole
comparison to brick elements
continuum shell
DSG
3d-solid (brick)
3d-problems
Cylinder with Skew Hole
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Cylinder with Skew Hole
distorted and curved elements, skew directors
3d-problems
Cylinder with Skew Hole
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Cylinder with Skew Hole
continuum shell elements
continuum shell
no DSGcontinuum shell
DSG
3d-problems
Cylinder with Skew Hole
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Cylinder with Skew Hole
continuum shell elements
continuum shell
no DSGcontinuum shell
DSG
3d-problems
Cylinder with Skew Hole
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Cylinder with Skew Hole
continuum shell elements
continuum shell
no DSGcontinuum shell
DSG
3d-problems
Cylinder with Skew Hole
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Cylinder with Skew Hole
comparison to brick elements
continuum shell
DSG3d-solid elements
(bricks)
3d-problems
Cylinder with Skew Hole
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Cylinder with Skew Hole
comparison to brick elements
continuum shell
standard Galerkin3d-solid elements
(bricks)
3d-problems
The Conditioning Problem
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spectral condition norm
thin shells worse than thick shells
3d-shell elements ( ) worse than standard shell elements ( )
The Conditioning Problem
The Conditioning Problem
condition numbers for classical shell and 3d-shell elements
Wall, Gee and Ramm (2000)
classical shell:
3d-shell:
Significance of Condition Number
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Significance of Condition Number
The Conditioning Problem
error evolution in iterative solvers
error of solution vectorx afterkth iteration
estimated number of iterations (CG solver)
comparison of three different concepts
Wall, Gee, Ramm, The challenge of a three-dimensional shell formulation the conditioning
problem, Proc. IASS-IACM, Chania, Crete (2000)
Eigenvalue Spectrum
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Eigenvalue Spectrum
The Conditioning Problem
shell, 3d-shell and brick
Eigenvectors (Deformation Modes)
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Eigenvectors (Deformation Modes)
The Conditioning Problem
Scaled Director Conditioning
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Scaled Director Conditioning
The Conditioning Problem
scaling of director
da =3
21,aa
*
3 da =c
21,aa
Scaled Director Conditioning
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Scaled Director Conditioning
The Conditioning Problem
scaling of director
linear scaling ofw
does not influence results
acts like a preconditioner
Scaled Director Conditioning
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Scaled Director Conditioning
The Conditioning Problem
reference configuration
current configuration
Improved Eigenvalue
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Improved Eigenvalue Spectrum
The Conditioning Problem
numerical example
5116 d.o.f.
BiCGstab solver
ILUT preconditioning
(fill-in 30%)
400 load steps
no scaling
scaled director
Improved Eigenvalue
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p o ed ge a ue Spectrum
The Conditioning Problem
numerical example
5116 d.o.f.
BiCGstab solver
ILUT preconditioning
(fill-in 30%)
400 load steps
Conclusions
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3d-solids
3d-shell and continuum shell (solid shell)
mechanical ingredients identical
stress resultants
flexible and most efficient finite element technology
neglecting higher order terms bad for 3d-applications
best for 3d-analysis of real shells
usually suffer from trapezoidal locking
(curvature thickness locking)
pass all patch tests (consistent) higher order terms naturally included
best for thick-thin combinations