Nonpenetration

Nonpenetration is a principle used in mechanics, mathematics, and numerical simulation to ensure that solid bodies do not occupy the same space at the same time. It is central to rigid-body dynamics and contact mechanics, where interactions change abruptly when bodies come into contact. The concept distinguishes between free motion, where bodies are separated, and contact, where a contact surface resists interpenetration.

Mathematically, nonpenetration is often described with a gap function g that measures the shortest distance between two surfaces along their contact normal. Nonpenetration requires g(t) ≥ 0 at all times. When g(t) = 0, the bodies are in contact and a contact force acts along the normal direction. This relationship is commonly expressed with a complementarity condition: either there is no contact force and the gap is positive, or there is a nonnegative contact force and the gap is zero. In continuum mechanics, this leads to a unilateral constraint or a variational inequality formulation. In dynamics, the equations of motion are augmented by these contact constraints.

Enforcement methods vary. Penalty methods impose a repulsive force that increases as penetration grows, which is simple but can allow small penetrations unless stiffness is very high. Lagrange multiplier methods enforce nonpenetration exactly by introducing contact forces as unknowns subject to complementarity. Augmented Lagrangian and mortar methods provide improved stability for complex contact. Friction adds further considerations, with models such as Coulomb friction that govern tangential forces and potential slipping at the contact interface.

Applications of nonpenetration concepts include computer graphics and animation, robotics and haptics, physics-based simulation, CAD and virtual prototyping, and the study of granular materials and soils. It also underpins mathematical formulations of obstacle problems and variational inequalities.