Effectiveness of Tetrahedral Finite Elements in Modeling Tread Patterns for Rolling Simulations

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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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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


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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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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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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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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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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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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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


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Element Type – C3D8HElement Size – 1.5e-3 mmPeak Pressure – 4.418 MPa


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Residual Aligning Torque

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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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Thank You!

Documents

Effectiveness of Tetrahedral Finite Elements in Modeling Tread Patterns for Rolling Simulations