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

1. Effectiveness of Tetrahedral Finite Elements in Modeling Tread Patterns for Rolling Simulations Harish Surendranath

2. Overview • General Considerations • Evolution of Contact Modeling • Contact Discretization • Constraint Enforcement • Treaded Tire Model • Conclusions

3. General Considerations • What is contact? • Physically, contact involves interactions between bodies that touch • Contact pressure resists penetration • Frictional stress resists sliding • Electrical, thermal interactions Fairly intuitive • Numerically, contact is a severely discontinuous form of nonlinearity • Inequality conditions • Resist penetration (h≤0) • Limited frictional stress (t≤mp) • Contact status (open/closed, stick/slip) • Conductance often has discontinuous dependence on contact status Numerically challenging

4. Evolution of Contact Modeling Contact elements (e.g., GAPUNI): General contact: Contact pairs: 2 n v h 1 Model all interactions between free surfaces User-defined element for each contact constraint Many pairings for assemblies Trends over time

5. Evolution of Contact Modeling Flat approximation of master surface per slave node: Realistic representation of master surface: Master surface Master surface Trends over time

6. Good resolution of contact over the entire interface 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 Trends over time

7. Good resolution of contact over the entire interface Evolution of Contact Modeling Constraints based on integrals over slave surface: General contact: • Goals: improve usability, accuracy, and performance • More focus by user on physical aspects • Less on idiosyncrasies of numerical algorithms • Broad applicability • Large models (assemblies) Realistic representation of master surface: Master surface Model all interactions between free surfaces

8. 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 • Node referred to as a “slave” node; opposing surface called the “master” surface slave master These nodes do not participate in contact constraints

9. slave master • Tends to involve more master nodes per constraint • Especially if master surface is more refined than slave surface 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 • Also involves coupling among slave nodes • 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)

10. 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: • Contact pressure contours much smoother and peak contact stress in very close agreement with the analytical solution using surface-to-surface approach Analytical CPRESSmax = 3.01e+05 CPRESSmax = 3.008e+05 CPRESSmax = 3.425e+05 Surface-to-surface Node-to-surface

11. Contact Discretization • S-to-S discretization fundamentally sound for situations in which quadratic elements underlie slave surface • N-to-S struggles with some quadratic element types q • Related to: • Discrete treatment of slave surface • Consistent force distribution for element • Workarounds (with pros and cons) • C3D10M, supplementary constraints, etc. q q Zero force at corner nodes Uniaxial pressure loading of 5.0 Slave: C3D10 Master: C3D8 Node-to-surface Surface-to-surface

12. p, contact pressure Any pressure possible when in contact No pressure Physically “hard” pressure vs. penetration behavior h, penetration - h h < 0 h = 0 No penetration: no constraint required Constraint enforced: positive contact pressure Constraint Enforcement • Strict enforcement • Intuitively desirable • Can be achieved with Lagrange multiplier method in Abaqus/Standard • Drawbacks: • Can make it challenging for Newton iterations to converge • Overlapping constraints problematic for equation solver • Lagrange multipliers add to solver cost

13. p, contact pressure Any pressure possible when in contact No pressure h, penetration Constraint Enforcement • Penalty method • Penalty method is a stiff approximation of hard contact p, contact pressure  k, penalty stiffness No pressure h, penetration Strictly enforced hard contact Penalty method approximation of hard contact = u K+Kp f

14. Constraint Enforcement • Pros and cons of penalty method • Advantages: • Significantly improved convergence rates • Better equation solver performance • No Lagrange multiplier degree of freedom unless contact stiffness is very high • Good treatment of overlapping constraints • Disadvantages: • Small amount of penetration • Typically insignificant • May need to adjust penalty stiffness relative to default setting in some cases

15. 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 constraints Tire rolling at low speed with 3300 N vertical load and 1000 N lateral load Friction coefficient of 0.8 between the tread and the road Tread Pattern Rolling Tire

16. Contact Pressure Comparison Element Type – C3D8H Element Size – 6e-3 mm Peak Pressure – 1.084 MPa Element Type – C3D10H Element Size – 12e-3 mm Peak Pressure – 1.378 MPa

17. Contact Pressure Comparison Element Type – C3D8H Element Size – 3e-3 mm Peak Pressure – 1.860 MPa Element Type – C3D10H Element Size – 6e-3 mm Peak Pressure – 2.514 MPa

18. Contact Pressure Comparison Element Type – C3D8H Element Size – 1.5e-3 mm Peak Pressure – 4.418 MPa Element Type – C3D10H Element Size – 3e-3 mm Peak Pressure – 4.786 MPa

19. Residual Aligning Torque

20. Conclusions Tetrahedral elements provide an efficient way to represent the tread pattern geometries Residual Aligning Torque results agree very well between the hexahedral and tetrahedral meshes Contact pressure distribution as well as peak contact pressure show good agreement between hexahedral and tetrahedral meshes

21. Thank You!