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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 [email protected]

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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

    http://lastpage/
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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