A penalty force acts only for negative gap; the local rank-one tangent follows the fixed contact direction in the initial-search route.
01 Model and scope
The penalty law activates only when the gap becomes negative; penetration is converted into a restoring force proportional to stiffness k. Increasing k reduces penetration but worsens conditioning, so the penalty value and mesh sensitivity must be reported rather than assuming exact contact.
Route assumptions and calculation checks
- Finite controlled penetration is an observable of the penalty law; zero penetration is not a universal acceptance condition.
- The initial-search regime fixes the selected face and normal during the nonlinear solve.
- Updated-search uses a geometry approximation; it is not a generally consistent finite-sliding tangent.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- Normal penalty stiffness.
- g(u)
- Signed normal gap.
- min(g,0)
- Zero in separation, negative during penetration.
- Declared contact-force direction for the selected pair.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Replace the hard constraint by a penalty potential
The penalty route approximates impenetrability with a normal spring active only after the gap becomes negative. It keeps the problem in displacement coordinates, but it trades exact closure for a finite penetration whose magnitude depends on stiffness and the structural compliance.
2 · Differentiate the fixed-direction force
For the frozen contact direction c, the contact residual is proportional to the negative part of the gap. In the closed regime its derivative with respect to displacement produces a rank-one tangent; in the open regime both the penalty force and its tangent vanish.
3 · Solve the piecewise nonlinear equilibrium
At each trial displacement, reevaluate the gap, contact force, and tangent, then assemble them with the structural terms. Newton iterations can move a pair between open and closed states, so convergence is assessed on the full equilibrium residual and the active-contact diagnostics.
4 · Choose stiffness and report its consequence
A larger k generally reduces penetration but worsens conditioning; an excessively small k gives a numerically easy solve with physically large overlap. Sweep k over a documented range and report both penetration and equilibrium error. The current initial-search route fixes the candidate face and normal.
Zero penetration is not the acceptance criterion for a penalty law; the declared tolerance and contact stiffness are part of the model.
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
EXPERIMENTAL_BOUNDED · static/quasi-static declared routes
Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.