Updated search and finite sliding

Physical formulationRESEARCH · approximate geometric tangent

Recomputing the closest face, projection, or normal as trial displacement changes is a separate geometric regime.

01 Model and scope

Recomputing the master face, projection and normal as the body moves changes the contact geometry as well as the gap. A tangent that differentiates only the gap omits those geometric derivatives; the documented updated-search route is therefore not a general finite-sliding formulation.

Route assumptions and calculation checks

The rank-one term alone does not prove all geometric derivatives.
The source classifies this regime as approximate unless the full tangent is independently established.
General surface-to-surface sliding, self-contact, impact, and contact dynamics remain outside the demonstrated claim.

Useful foundations and prerequisites

Equilibrium and virtual work Units, signs and error measures

02 Essential equations and symbols

c = c(u); fcontact(u) = k min(g(u),0)c(u)

Definitions and conventions

c(u): Trial-state-dependent contact direction.
g(u): Current normal gap.
$\partial{}g/\partial{}u$: Gap derivative.
$\partial{}c/\partial{}u$: Contact-direction derivative omitted by a gap-only tangent.

01Updated geometry

02Closest point + normal

03Differentiate all terms

04Approximation boundary

Updated search and finite sliding · method mapUpdated geometry changes projection, gap, and normal; the exact derivative must include all three effects.

3 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Update the contact geometry at every trial

In an updated-search formulation, the closest master location, projection coordinates, and contact normal can all depend on the trial displacement. The force therefore changes through both the signed gap and the evolving contact direction; a frozen-normal tangent misses part of this dependency.

\mathbf c=\mathbf c(\mathbf u),\qquad g=g(\mathbf u)

2 · Differentiate every geometric dependency

The product rule creates a gap-derivative term and a direction-derivative term. If the closest-point projection also changes, its derivative enters through c(u) and g(u). These terms can change the convergence rate and consistency of the Newton tangent, especially when the active face switches.

\mathbf K_c= k\,\mathbf c\otimes\nabla g + k\min(g,0)\,\frac{\partial\mathbf c}{\partial\mathbf u}

3 · Treat the current route as approximate

The audited updated-search path uses a geometric approximation and does not demonstrate a fully consistent derivative for all projection, normal, and face-switch terms. Keep this classification visible; do not infer general finite-sliding contact from the presence of a changing search point.

\text{full sliding tangent}=\text{gap term}+\text{normal term}+\text{projection term}+\text{active-set terms}

General surface-to-surface sliding, self-contact, impact, and contact dynamics remain outside this demonstrated claim.

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

RESEARCH · approximate geometric tangent

Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.

Readiness audit

06 Sources and related sheets

Readiness audit Contact limitations

Continue the reasoning

Gap and complementarity Linear-static active set with multipliers Nonlinear frictionless penalty Regularized Coulomb friction

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