The sign convention distinguishes separation, closure, and penetration. Contact pressure is compressive and nonnegative.
01 Model and scope
The signed normal gap encodes the unilateral condition: separated surfaces carry no compressive pressure, while closed surfaces may carry pressure but cannot interpenetrate. The three Kuhn–Tucker conditions express those alternatives without choosing a penalty stiffness.
Route assumptions and calculation checks
- Barycentric weights bi distribute the master-side force across the selected triangle.
- The audit can expose gap, pressure, active/open status, and complementarity residual.
- Selected master face and normal belong to the search route; this is not general surface-to-surface contact.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- Initial signed normal gap.
- n, bᵢ
- Master normal and barycentric projection weights.
- g(u)
- Current normal separation.
- Nonnegative compressive contact pressure in the documented sign convention.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Define the pair and sign convention
Choose the candidate slave point, the selected master face, its outward normal n, and the barycentric weights on that face. The signed gap is positive when the surfaces are separated, zero at touch, and negative after penetration. These choices define the contact search route as well as the equations.
The selected face and normal are frozen in the documented initial-search route; this is not a general surface-to-surface sliding model.
2 · State the unilateral law
Normal contact can push the bodies apart but cannot pull them together. Thus the gap cannot be negative in the exact constraint model, pressure cannot be tensile, and pressure is nonzero only when the gap closes. Together these three conditions are the Signorini complementarity conditions.
A penalty method permits a controlled negative gap. Do not apply the exact zero-penetration condition to a penalty result.
3 · Transfer a point reaction to the face
The contact force at the slave point is balanced by an equal-and-opposite reaction on the master face. The barycentric shape values distribute that reaction to the three master nodes while preserving its resultant. This distribution is assembled with the bulk-element residual.
A_c denotes the discrete contact measure used by the route. The contact work and sign of the assembled residual follow the implementation convention.
4 · Check complementarity and equilibrium
Inspect the computed gap, pressure, active/open flag, and complementarity residual together. A plausible force alone is insufficient: it can hide a tensile multiplier, a badly penetrated pair, or an incorrectly selected face. Also check global force balance and sensitivity to the contact mesh/search parameters.
04 Algorithm and implementation
Read the linked versioned source for the actual assembly and solver behavior. The derivation above explains the mathematics and is not a verbatim source excerpt.
05 Verification and limitations
SHARED KINEMATICS · not a maturity label
The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.