Extended finite element and meshfree methods: 11' EXtended finite elements Timon Rabczuk Prof' Wolfg

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Extended finite element and meshfree methods11.
EXtended finite elements Timon RabczukProf.
Wolfgang Wall
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Outline

• Tracking the crack path
• Embedded Elements
• Interface elements
• Extended Meshfree Methods
• Introductions into cohesive zone models
• Crack initiation/propagation
• Problems

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Tracking the crack path

• Global methods
• Local methods
• The level set method

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Embedded elements

• Statical optimal symmetric elements traction
• continuity is enforced but wrong kinematics
  (rigid
• motion of two element parts not guaranteed)
• Kinematical optimal symmetric elements correct
• kinematic but no traction continuity
• Kinematicstatic optimal non-symmetric elements
• correct kinematicenforcement of traction
  continuity
• but non symmetric

SOS shows stress locking-gt mixed formulation
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Embedded vs. XFEM
Embedded
XFEM
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Embedded elements
Elemental enrichment
S
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Embedded elements
Stiffness matrices
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Embedded elements
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Interface separation models
Advantages

• Simple implementation

Drawbacks

• Sensitive with respect to element size, form and
  arrangement
• Meshdependent results

Ortiz et al., Needleman et al.
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Discontinuities in meshfree methods
XFEM (Belytschko et al. 1999)
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Discontinuities in meshfree methods
Two solutions to model a meshfree crack
without the branch enrichment
Domain decrease method
Lagrange multiplier method
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Discontinuities in meshfree methods

• Tracing the crack paths
• Evolution law for the level set
• Difficulties
• Crack branching/intersections
• Crack propagation

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Discontinuities in meshfree methods
An alternative approach to treat
discontinuities with meshfree methods
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The cracking particle approach
Methods that do not enforce crack path continuity
Continuous crack
model
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The cracking particle approach
Decomposition of the displacements
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Discretization

• Difference to methods before
• crack goes directly through a particle
• crack is modeled by a set of
• cracked particles
• no representation of the crack path is necessary

crack
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Drawback of FE
Continuous crack path
Crack segments
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Application to shear bands
Decomposition of the displacements
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Discretization
Shear band
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Cracks in thin shells

• Meshfree thin shell
• Combination of KL shell (kinematical assumption)
• and continuum based shell

Kinematics
Extension to cracks
Discretization
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Cohesive zone model
COHESIVE ZONE MODELS A BRIEF INTRODUCTION
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Introduction into cohesive cracks
load
Strain softening
displacement

• Problem IBVP is ill-posed, mesh-dependent
  results (localization
• in a set of measure zero,
  zero-energy dissipation with mesh
• refinement)

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Regularization
Regularization techniques for unstable materials

• Gradient models
• Integral or nonlocal models
• Viscous (rate dependent) models

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Outline
Gradient model
Integral model
Viscous model
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Outline
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Cohesive cracks

• Important
• Traction continuity
• Complementary problem

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Cohesive cracks
wmax
w
Prozesszone
ft
Rissoberflaeche
tf(w)
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Cohesive cracks
Starke Form
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Cohesive cracks
Variational formulation
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Smeared cracks

• Fixed single cracks
• Fixed orthorgonal cracks
• Fixed multiaxial cracks
• Rotating cracks

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Smeared cracks
Advantages

• Easy to implement

Drawbacks

• Wrong crack kinematics
• Stress locking
• diffuses crack pattern
• mesh-dependent results (mesh alignment
  sensitivity)
• Problems, when process-zone larger than element
  size
• Fixed cracks too stiff
• Rotating cracks too soft

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Crack initiation/propagation
Crack initiation

• When is a crack initiated
• Crack orientation
• Crack length

Crack propagation

• Basically same as initiation but simpler

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Crack onset
Stress based Rankine
Strain based
Others Damage
Loss of material stability (ellipticity/hyperbolic
ity)
Acoustic tensor

• Difficulty
• 2 angles obtained in the
• localization analysis

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Discontinuous bifurcation analysis
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Crack orientation
Mode I
Mode II
h
h
n
Mixed Mode
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Crack orientation/length

• Crack orientation
• Rankine perpendicular to principal tensile
  stress
• Strain based perpendicular to principal tensile
  strain
• Others ???
• Loss of Material Stability obtained from
  bifurcation analysis

• Crack length
• through the entire element
• Control crack length
• e0

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Problems

• Cohesive models
• Unloading/reloading
• fmax cannot be exceeded
• Convergence difficulties with rigid cohesve
  models (unloading)
• Dynamic fracture energy?

• Bifurcation analysis
• Bifurcation analysis for problems with
  constraints (shells) (shear bands)
• Ambiguous normal
• PDE does not loose ellipticity for large time
  steps

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The end
Thank you very much for your attention