Contact Analysis — CAE Glossary

Contact Analysis

This is called the Karush-Kuhn-Tucker (KKT) condition, or Hertz-Signorini-Moreau condition. The third equation is key: "either the gap is open (no contact force) or the gap is closed (contact force exists) — never both simultaneously."

Definition and Fundamentals

• Higher stiffness → master (e.g., mold is master, blank is slave)
• Coarser mesh → master (finer slave nodes detect contact more accurately)
• Larger area → master

However, modern solvers (Abaqus, Ansys) favor surface-to-surface formulations where master-slave assignment matters less. Still, following these rules is a safe bet when uncertain.

Contact Types (node-to-surface / surface-to-surface)

Node-to-Surface (NTS):

• Each slave node is projected onto the master surface
• Fast but contact pressure distribution is jagged
• Master surface can "skip through" slave nodes (zipping effect)
• Poor accuracy on coarse-mesh curves

Surface-to-Surface (STS):

• Integrates contact constraint over surfaces → smooth pressure
• Less sensitive to master-slave assignment (nearly identical results either way)
• Higher computational cost but better convergence

For example, a bolted flange contact analysis with NTS shows nodal chatter in contact pressure, while STS yields smooth distribution. Abaqus now defaults to STS.

Penalty and Augmented Lagrangian Methods

Penalty Method generates a restoring force proportional to penetration — like a virtual contact spring:

where $\epsilon_N$ is penalty stiffness and $\langle \cdot \rangle$ is the Macauley bracket (negative values clamped to zero). Advantages: fast, no extra unknowns. Disadvantages: $\epsilon_N$ must be tuned; too small allows penetration, too large degrades the condition number.

Updated each iteration:

This allows smaller $\epsilon_N$ while still achieving near-zero final penetration. Ansys Mechanical defaults to this method. It requires more iterations than penalty, but contact pressure accuracy and robustness are much better.

Decision guide:

• Penalty: Explicit impact analysis, large models where speed is critical
• Augmented Lagrangian: Static/implicit analysis where contact pressure accuracy matters
• Pure Lagrangian: Zero penetration is mandatory (but requires direct solvers due to zero diagonal terms)

Friction Models

The problem is the discontinuity at the transition, which damages convergence. In practice:

• Tangential Penalty Method: Allow small elastic slip in stick state
• Exponential Friction: Smooth $\mu$ variation with slip velocity
• Coulomb + Viscosity: $\tau = \mu p_N + c \dot{u}_T$ for smooth transition

Typical values: steel-on-steel $\mu = 0.15$–$0.3$, aluminum $0.1$–$0.35$, rubber-on-metal $0.5$–$1.0$. Measured data is best; surface roughness, lubricant, and temperature all affect friction significantly.

1. Stabilize contact first with $\mu = 0$. Confirm pure contact converges
2. Gradually ramp friction. Step 1: $\mu=0$, Step 2: $\mu=0.15$, etc.
3. Set tangential penalty to ~10% of normal penalty. Default values are often too high

For example, a press-forming simulation that fails to converge with $\mu=0.12$ from the start will often run smoothly if you form to mid-stage with $\mu=0$, then add friction. This is a proven trick in production CAE.

Convergence Challenges and Strategies

1. Inverted normal direction — Most common. Shell elements with flipped normals prevent contact detection. Check element orientation
2. Initial penetration — Mesh already intersects due to CAD fillets or clearance. Use *CONTACT CLEARANCE ASSIGNMENT in Abaqus
3. Inappropriate penalty stiffness — Too high: oscillation (chatter); too low: excessive penetration
4. Coarse load steps — Contact state cannot toggle rapidly. Refine step size
5. Rigid body motion — Free parts before contact lock the matrix. Use weak springs or ensure initial contact

Solutions:

• Contact Stabilization: Add damping to smooth state transitions. Abaqus: *CONTACT CONTROLS STABILIZE
• Automatic time-stepping: Reduce step size when chatter detected
• Over-closure tolerance: Allow small penetration before toggling contact state

In Ansys Mechanical: adjust CNOF and PINB parameters.

1. Contact is a boundary nonlinear problem — Do not treat it like linear analysis
2. Use surface-to-surface formulation — Node-to-surface is outdated and less accurate
3. Penalty is fast but penetrates; augmented Lagrangian is accurate but slower — Choose based on requirements
4. Stabilize contact without friction first, then add friction gradually
5. When convergence fails, suspect setup errors first (normal direction, initial penetration) before tweaking parameters

Contact analysis causes more CAE trouble than most disciplines, but these five principles solve 80% of real-world issues.

Related Terms

• Penalty Method — Constraint enforcement via penalty terms
• Lagrange Multiplier Method — Strict constraint enforcement using multipliers
• Friction — Sliding resistance at contact surfaces
• Boundary Nonlinearity — Nonlinearity from state-dependent boundary conditions
• Newton-Raphson Method — Standard iterative solver for nonlinear FEM
• Contact Stress — Surface pressure at contact
• Hertz Contact — Classical elastic contact theory
• Convergence — Iterative solution approaching true answer