Nopenetration

No-penetration is a constraint used in physics, engineering, and numerical simulation to prevent solid bodies or fluids from occupying the same physical space. It is a fundamental principle in contact mechanics and fluid–solid interaction, ensuring that entities that come into contact do not interpenetrate.

In the context of contact problems, no-penetration is often expressed through a gap function g(x,t) that measures the normal distance between surfaces. The gap must satisfy g(x,t) ≥ 0, with a corresponding normal contact force f_n that satisfies f_n ≥ 0 and f_n g = 0, a complementarity condition. When the gap is positive, bodies are separated and no contact force acts; when the gap is zero, contact forces prevent further intrusion.

In fluid dynamics, the no-penetration boundary condition states that the normal component of the fluid velocity at a solid boundary is zero (n · u = 0). This enforces impermeability, so fluid cannot flow through the boundary, while tangential motion may occur depending on other conditions such as slip or friction.

Numerical methods for enforcing no-penetration include the penalty method, which imposes a penalty for interpenetration; Lagrange multipliers, which enforce the constraint exactly; and augmented Lagrangian approaches, which combine these ideas for improved stability. In many problems, tangential interactions are modeled with friction laws (e.g., Coulomb friction) to describe realistic contact behavior.

Applications span mechanical joints and gears, brake systems, robotics, granular and particulate flows, and computer graphics simulations of rigid bodies. Limitations include potential numerical penetration due to discretization, the need for appropriate time stepping, and the complexity of accurately modeling friction and deformation at contacts.