local¶
Module: core
Category: mat3dvaluator
Type string: "local"
Parameters¶
| Name | Description | Default | Units |
|---|---|---|---|
local |
local | [] |
Description¶
The local mat3d valuator uses the local element node numbering to construct an orthonormal \(3 \times 3\) matrix.
First the element in which a point \(\mathbf{r}\) is located is determined. From this the nodal coordinates of the element are defined, \(\mathbf{p}_n\), with \(n\) ranging from 1 to the number of nodes of the element.
Let \(l\) be the local parameter. Then, define,
\[
\mathbf{a}=\mathbf{p}_{l_2}-\mathbf{p}_{l_1},\quad\mathbf{d}=\mathbf{p}_{l_3}-\mathbf{p}_{l_1}
\]
and the corresponding unit vectors \(\hat{\mathbf{a}}=\mathbf{a}/\left||\mathbf{a} \right||\) and \(\hat{\mathbf{d}}=\mathbf{d}/\left||\mathbf{d} \right||\).
Then we construct three perpendicular vectors,
\[\begin{align}
& {{\mathbf{e}}_{1}}=\hat{\mathbf{a}} \\
& {{\mathbf{e}}_{3}}=\hat{\mathbf{a}}\times \hat{\mathbf{d}} \\
& {{\mathbf{e}}_{2}}={{\mathbf{e}}_{3}}\times {{\mathbf{e}}_{1}} \\
\end{align}\]
From this, we define the orthonormal matrix,
\[
\mathbf{Q}=\left[ \begin{matrix} {\mathbf{e}_{1}} & {\mathbf{e}_{2}} & {\mathbf{e}_{3}} \end{matrix} \right]
\]