Rigid motions in a free-body stiffness nullspace

Let $D\subset {\mathbb{R}}^d$, $d\in \{2,3\}$, be a connected polygonal or polyhedral domain with a finite conforming affine simplicial mesh ${\mathcal{T}}_h$ and no essential boundary constraints. Define the continuous P1 vector space $S_h=\{v\in C^0(\bar{D};{\mathbb{R}}^d):v{\mathrm{\mid }}_K\text{ is affine for every }K\in {\mathcal{T}}_h\}$ and its global nodal basis $({\varphi }_i{)}_{i=1}^N$. For constants $\mu >0$ and $\lambda \ge 0$, set $\epsilon (v)=(\mathrm{\nabla }v+\mathrm{\nabla }{v}^T)\mathrm{/}2$, $k(v,w)=2\mu {\int }_D\epsilon (v):\epsilon (w)+\lambda {\int }_D\mathrm{div}v\mathrm{div}w$, and $K_{ij}=k({\varphi }_j,{\varphi }_i)$. For every infinitesimal rigid motion $r(x)=a+\mathrm{\Omega }x$, with $a\in {\mathbb{R}}^d$ and ${\mathrm{\Omega }}^T=-\mathrm{\Omega }$, its nodal coefficient vector lies in $\ker K$. Thus the range of any complete nullspace basis supplied to Algebraic gauge for a rigid-body stiffness nullspace must contain these translation and rotation vectors. This example does not claim they exhaust the nullspace on every mesh.