1 © Dassault Systèmes Ι Confidential Information Effectiveness of Tetrahedral Finite Elements in Modeling Tread Patterns for Rolling Simulations Harish Surendranath

2 © Dassault Systèmes Ι Confidential Information Overview General Considerations Evolution of Contact Modeling Contact Discretization Constraint Enforcement Treaded Tire Model Conclusions 2

3 © Dassault Systèmes Ι Confidential Information General Considerations What is contact?  Physically, contact involves interactions between bodies that touch o Contact pressure resists penetration o Frictional stress resists sliding o 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

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

5 © Dassault Systèmes Ι Confidential Information Evolution of Contact Modeling Flat approximation of master surface per slave node: Master surface Realistic representation of master surface: Master surface Trends over time

6 © Dassault Systèmes Ι Confidential Information 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

7 © Dassault Systèmes Ι Confidential Information 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

8 © Dassault Systèmes Ι Confidential Information 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

9 © Dassault Systèmes Ι Confidential Information 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

10 © Dassault Systèmes Ι Confidential Information 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 CPRESS max = 3.01e+05 Surface-to-surface CPRESS max = 3.425e+05 CPRESS max = 3.008e+05

11 © Dassault Systèmes Ι Confidential Information 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 Zero force at corner nodes q q q 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-surfaceSurface-to-surface Uniaxial pressure loading of 5.0

12 © Dassault Systèmes Ι Confidential Information 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 converge o Overlapping constraints problematic for equation solver o Lagrange multipliers add to solver cost h < 0h = 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

13 © Dassault Systèmes Ι Confidential Information 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+K p u f =

14 © Dassault Systèmes Ι Confidential Information Constraint Enforcement Pros and cons of penalty method  Advantages: o Significantly improved convergence rates o 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

15 © Dassault Systèmes Ι Confidential Information 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 15 Tread Pattern Rolling Tire

16 © Dassault Systèmes Ι Confidential Information Contact Pressure Comparison 16 Element Type – C3D8H Element Size – 6e-3 mm Peak Pressure – MPa Element Type – C3D10H Element Size – 12e-3 mm Peak Pressure – MPa

17 © Dassault Systèmes Ι Confidential Information Contact Pressure Comparison 17 Element Type – C3D8H Element Size – 3e-3 mm Peak Pressure – MPa Element Type – C3D10H Element Size – 6e-3 mm Peak Pressure – MPa

18 © Dassault Systèmes Ι Confidential Information Contact Pressure Comparison 18 Element Type – C3D8H Element Size – 1.5e-3 mm Peak Pressure – MPa Element Type – C3D10H Element Size – 3e-3 mm Peak Pressure – MPa

19 © Dassault Systèmes Ι Confidential Information Residual Aligning Torque 19

20 © Dassault Systèmes Ι Confidential Information 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 20

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