effectiveness of tetrahedral finite elements in modeling tread patterns for rolling simulations
DESCRIPTION
Effectiveness of Tetrahedral Finite Elements in Modeling Tread Patterns for Rolling Simulations. Harish Surendranath. Overview. General Considerations Evolution of Contact Modeling Contact Discretization Constraint Enforcement Treaded Tire Model Conclusions. General Considerations. - PowerPoint PPT PresentationTRANSCRIPT
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Effectiveness of Tetrahedral Finite Elements in Modeling Tread Patterns for Rolling Simulations
Harish Surendranath
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Overview
General ConsiderationsEvolution of Contact ModelingContact DiscretizationConstraint EnforcementTreaded Tire ModelConclusions
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General Considerations
What is contact? Physically, contact involves interactions between bodies
that toucho Contact pressure resists penetrationo Frictional stress resists slidingo Electrical, thermal interactions
Fairly intuitive
Numerically challenging
• Numerically, contact is a severely discontinuous form of nonlinearity
• Inequality conditions
• Resist penetration (h≤0)
• Limited frictional stress (≤p)
• Contact status (open/closed, stick/slip)
• Conductance often has discontinuous dependence on contact status
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Evolution of Contact Modeling
Contact elements (e.g., GAPUNI):
v
h1
2
2 1 0h d n u u
Contact pairs: General contact:
Trends over time
Model all interactions between free surfaces
Many pairings for assemblies
User-defined element for each contact constraint
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Evolution of Contact Modeling
Flat approximation of master surface per slave node:
Master surface
Realistic representation of master surface:
Master surface
Trends over time
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Trends over time
Evolution of Contact Modeling
Slave surfaces treated as collection of discrete points:
Constraints based on integrals over slave surface:
Does not resist penetration at master nodes
Resists penetration at slave nodes
Good resolution of contact over the entire
interface
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Evolution of Contact Modeling
Goals: improve usability, accuracy, and performance More focus by user on physical aspects
o Less on idiosyncrasies of numerical algorithms Broad applicability Large models (assemblies)
General contact:
Model all interactions between free surfaces
Master surface
Realistic representation of master surface:
Constraints based on integrals over slave surface:
Good resolution of contact over the entire
interface
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Contact Discretization
Node-to-surface (N-to-S) contact discretization Traditional “point-against-surface” method Contact enforced between a node and surface facets local to the node
o Node referred to as a “slave” node; opposing surface called the “master” surface
slave
master
These nodes do not participate in contact
constraints
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Contact Discretization
Surface-to-surface (S-to-S) contact discretization Each contact constraint is formulated based on an integral over the region
surrounding a slave node
• Tends to involve more master nodes per constraint• Especially if master surface is more
refined than slave surface
slave
master
• Still best to have more-refined surface act as slave
• Better performance and accuracy
• Benefits of surface-to-surface approach
• Reduced likelihood of large localized penetrations
• Reduced sensitivity of results to master and slave roles
• More accurate contact stresses
• Inherent smoothing (better convergence)
• Also involves coupling among slave nodes
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Contact Discretization
S-to-S discretization often improves accuracy of contact stresses Related to better distribution of contact
forces among master nodes Example: Classical Hertz contact
problem:o Contact pressure contours much smoother
and peak contact stress in very close agreement with the analytical solution using surface-to-surface approach
Node-to-surface
Analytical CPRESSmax = 3.01e+05
Surface-to-surface
CPRESSmax = 3.425e+05
CPRESSmax = 3.008e+05
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Contact Discretization
S-to-S discretization fundamentally sound for situations in which quadratic elements underlie slave surfaceN-to-S struggles with some quadratic element types
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q pA
Zero force at corner nodes
q qq• Related to:
• Discrete treatment of slave surface• Consistent force distribution for element
• Workarounds (with pros and cons)• C3D10M, supplementary constraints, etc.
Slave: C3D10
Master: C3D8 Node-to-surface Surface-to-surface
Uniaxial pressure loading of 5.0
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Constraint Enforcement
Strict enforcement Intuitively desirable Can be achieved with Lagrange multiplier method in Abaqus/Standard Drawbacks:
o Can make it challenging for Newton iterations to convergeo Overlapping constraints problematic for equation solvero Lagrange multipliers add to solver cost
h < 0 h = 0
No penetration: no constraint required
Constraint enforced: positive contact pressure
h
p, contact pressure
Any pressure possible when in contact
No pressure
h, penetration
Physically “hard” pressure vs. penetration behavior
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Constraint Enforcement
Penalty method Penalty method is a stiff approximation of hard contact
p, contact pressure
Any pressure possible when in contact
No pressure
h, penetration
Strictly enforced hard contact
p, contact pressure
No pressure
h, penetration
Penalty method approximation of hard contact
k, penalty stiffness
K+Kp u f=
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Constraint Enforcement
Pros and cons of penalty method Advantages:
o Significantly improved convergence rateso Better equation solver performance
• No Lagrange multiplier degree of freedom unless contact stiffness is very high
o Good treatment of overlapping constraints
Disadvantages:o Small amount of penetration
• Typically insignificant
o May need to adjust penalty stiffness relative to default setting in some cases
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Treaded Tire Model
Tread pattern modeled using both hexahedral and tetrahedral elements Tread mesh density is varied
Non-axisymmetric tread pattern tied to the carcass using mesh independent tie constraintsTire rolling at low speed with 3300 N vertical load and 1000 N lateral loadFriction coefficient of 0.8 between the tread and the road
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Tread Pattern
Rolling Tire
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Contact Pressure Comparison
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Element Type – C3D8HElement Size – 6e-3 mmPeak Pressure – 1.084 MPa
Element Type – C3D10HElement Size – 12e-3 mmPeak Pressure – 1.378 MPa
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Contact Pressure Comparison
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Element Type – C3D8HElement Size – 3e-3 mmPeak Pressure – 1.860 MPa
Element Type – C3D10HElement Size – 6e-3 mmPeak Pressure – 2.514 MPa
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Contact Pressure Comparison
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Element Type – C3D8HElement Size – 1.5e-3 mmPeak Pressure – 4.418 MPa
Element Type – C3D10HElement Size – 3e-3 mmPeak Pressure – 4.786 MPa
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Conclusions
Tetrahedral elements provide an efficient way to represent the tread pattern geometriesResidual Aligning Torque results agree very well between the hexahedral and tetrahedral meshesContact pressure distribution as well as peak contact pressure show good agreement between hexahedral and tetrahedral meshes
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