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Table 1 Elements of the Lie algebra \(\mathfrak g\) and the associated Lie group G

From: Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

Element \(\varvec{{\mathbf {Z}}}_{\varvec{I,r}} \varvec{\in {\mathfrak g}}\) Element \(\varvec{\exp \left( {{\mathbf {Z}}_{I,r}}\right) \in G}\) Action of \(\varvec{\exp \left( {{\mathbf {Z}}_{I,r}}\right) }\)
\({\mathbf {Z}}_{1,1} := \begin{pmatrix} 1 &{} - {\mathbf {x}}_0^1 \\ 0 &{} 0\end{pmatrix}\) \(\exp \left( {{\mathbf {Z}}_{1,1}}\right) = \begin{pmatrix} \mathrm{{e}} &{} ({1-\mathrm{{e}}}) {\mathbf {x}}_0^1 \\ 0 &{} 1\end{pmatrix}\) Dilatation centered on the position of node 1
\({\mathbf {Z}}_{1,2} := \begin{pmatrix} \mathrm{i} &{} -\mathrm{i}{\mathbf {x}}_0^1 \\ 0 &{} 0\end{pmatrix}\) \(\exp \left( {{\mathbf {Z}}_{1,2}}\right) = \begin{pmatrix} \mathrm{{e}}^{\mathrm{i}} &{} ({1-\mathrm{{e}}^{\mathrm{i}}}) {\mathbf {x}}_0^1 \\ 0 &{} 1\end{pmatrix}\) Rotation around the position of node 1
\({\mathbf {Z}}_{2,1} := \begin{pmatrix} 1 &{} -{\mathbf {x}}_0^2 \\ 0 &{} 0\end{pmatrix}\) \(\exp \left( {{\mathbf {Z}}_{2,1}}\right) = \begin{pmatrix} \mathrm{{e}} &{} ({1-\mathrm{{e}}}) {\mathbf {x}}_0^2 \\ 0 &{} 1\end{pmatrix}\) Dilatation centered on the position of node 2
\({\mathbf {Z}}_{2,2} := \begin{pmatrix} \mathrm{i} &{} -\mathrm{i}{\mathbf {x}}_0^2 \\ 0 &{} 0\end{pmatrix}\) \(\exp \left( {{\mathbf {Z}}_{2,2}}\right) = \begin{pmatrix} \mathrm{{e}}^{\mathrm{i}} &{} ({1-\mathrm{{e}}^{\mathrm{i}}}) {\mathbf {x}}_0^2 \\ 0 &{} 1\end{pmatrix}\) Rotation around the position of node 2