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An affine shear field passes the frictionless contact patch test

Statement

On the unit square with $Γ_{D} = {0} \times [1 / 4, 3 / 4]$ , $Γ_{C} = {1} \times [1 / 4, 3 / 4]$ , and $n = (1, 0)$ , use any matching boundary-conforming affine triangular mesh that resolves both segments as unions of boundary edges, and take obstacle $ψ = 0$ . Let $u (x, y) = (0, x)$ and define the load $F (v) = a (u, v)$ on $V$ . Then $u$ is the unique continuous solution and, on every such mesh, the same field is the unique nodal P1 solution. Its contact gap is identically zero.

Facts & Assumptions

Given: The elasticity and Signorini conventions of Small-strain elasticity with a frictionless contact gap and The vector Signorini contact set and variational inequality, with the square and boundary segments above.

[A1]

Countable Choice is used by the imported continuous and discrete Stampacchia theorems. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[A2]

The Axiom of Choice is inherited by the trace and partial-boundary Korn interfaces in the continuous well-posedness theorem. (The Axiom of Choice, Continuous frictionless elasticity contact is well posed)

[F1]

$V$ consists of $H^{1}$ vector fields with zero Dirichlet trace, and $a$ is the isotropic symmetric-strain elasticity form. (Small-strain elasticity with a frictionless contact gap)

[F2]

The feasible set is $K = {v \in V : T_{C} v \cdot n \geq ψ}$ and the nodal scheme defines finite-dimensional $V_{h}$ and a nonempty closed convex set $K_{h} \subseteq K$ . (The vector Signorini contact set and variational inequality, Conforming nodal P1 Signorini contact discretization, Broken and conforming polynomial spaces on a matching mesh, Every finite-dimensional normed space is Banach, The convex-obstacle nodal set is conforming)

[F3]

The elasticity form is bounded by $M$ and coercive by $α > 0$ on $V$ . (The isotropic contact elasticity form is well defined and bounded, Korn coercivity with a partial Dirichlet segment)

[F4]

The continuous VI has a unique solution. (Continuous frictionless elasticity contact is well posed)

[F5]

On a Hilbert space, a nonempty closed convex set and bounded coercive form yield a unique VI solution under Countable Choice. (Stampacchia's variational inequality)

Proof

1.1F1F2givenalgebra

The affine field $u$ lies in $H^{1} (Ω; R^{2})$ and vanishes on the left side, so $u \in V$ . On the right side $u \cdot n = 0 = ψ$ , hence $u \in K$ and has zero gap. Every affine function restricts to a continuous piecewise-affine function on the given matching mesh, so $u \in V_{h}$ ; its contact nodal inequalities hold with equality, so $u \in K_{h}$ .

2.1A1A2F2F3F4F5step 1.1algebra∎

The gradient is $\nabla u = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array})$ , so $div u = 0$ and $ε (u) = (\begin{array}{cc} 0 & 1 / 2 \\ 1 / 2 & 0 \end{array})$ . By [F3], $F (v) = a (u, v)$ is bounded. For every $v \in K$ , $a (u, v - u) - F (v - u) = 0$ , so $u$ solves the continuous VI; uniqueness follows from [F4]. For every $v_{h} \in K_{h} \subseteq K$ , the same identity holds, so $u$ solves the discrete VI. The inherited $H^{1}$ inner product makes finite-dimensional $V_{h}$ a Hilbert space by [F2]; the restricted form is bounded and coercive by [F3], and [F5] gives uniqueness.

Depends on

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part XI Chapter 53: Stokes equations: Basic ideas (standard reference, not scraped)
Walter Zulehner, Numerical Methods for Continuum Mechanics 1, complete 80-page lecture notes (standard reference, not scraped)