<!-- GENERATED by tools/generators/itemDocsEmitter.py from definitions/ - do not edit -->
```{raw} latex
\clearpage
```

(sec-item-nodepointslope12)=
# NodePointSlope12

A 3D point/slope vector node for thin ANCF (absolute nodal coordinate formulation) plate elements, with 3 position and 2 $\times$ 3 slope coordinates, all {ref}`ODE2 <ODE2>`; the slope vectors are the derivatives of the position with respect to the two in-plane coordinates of the plate.

## Interface

- Provides: `Position`, `Orientation`, `PointSlope12`
- Node markers that can be attached: [](#sec-item-markernodeposition), [](#sec-item-markernoderigid), [](#sec-item-markernoderotationcoordinate)
- Objects that take this node: [](#sec-item-objectmasspoint), [](#sec-item-objectbeamgeometricallyexact), [](#sec-item-objectancfthinplate)

## Parameters

The parameters of the item; in a dictionary, its type is 'PointSlope12':

| name | type | size | default | description |
|---|---|---|---|---|
| **name** | String |  | '' | node's unique name |
| **referenceCoordinates** | Vector9D | 9 | [0.,0.,0.,1.,0.,0.,0.,1.,0.] | reference coordinates (x-pos,y-pos,z-pos; x-slopeX, y-slopeX, z-slopeX; x-slopeY, y-slopeY, z-slopeY) of node; global position of node without displacement |
| **initialCoordinates** | Vector9D | 9 | [0.,0.,0.,0.,0.,0.,0.,0.,0.] | initial displacement coordinates relative to reference coordinates |
| **initialVelocities** | Vector9D | 9 | [0.,0.,0.,0.,0.,0.,0.,0.,0.] | initial velocity coordinates |
| **visualization** | VNodePointSlope12 |  |  | parameters for visualization of item |

## Visualization parameters

The parameters of `VNodePointSlope12`, given as `visualization`:

| name | type | size | default | description |
|---|---|---|---|---|
| **show** | Bool |  | True | set true, if item is shown in visualization and false if it is not shown |
| **drawSize** | float |  | -1. | drawing size (diameter, dimensions of underlying cube, etc.) for item; size == -1.f means that default size is used |
| **color** | Float4 | 4 | [-1.,-1.,-1.,-1.] | Default RGBA color for nodes; 4th value is alpha-transparency; R=-1.f means, that default color is used |

## Drawing

Drawn as [all nodes](#sec-drawing-nodes); with `nodes.showNodalSlopes`, also its slope vectors.

Settings beyond those of all nodes: [`nodes.showNodalSlopes`](#sec-vsettingsnodes).

## Output variables

Available as `OutputVariableType` in sensors, `Get...Output()` and other functions:

```{tabularcolumns} \Y{0.25}\Y{0.25}\Y{0.5}
```

| output variable | symbol | description |
|---|---|---|
| Position | $\LU{0}{\pv}\cConfig = \LU{0}{[p_0,\, p_1,\, p_2]}\cConfig\tp$ | global 3D position vector of node (=displacement+reference position) |
| Displacement | $\LU{0}{\uv}\cConfig = \LU{0}{[q_0,\, q_1,\, q_2]}\cConfig\tp$ | global 3D displacement vector of node |
| Velocity | $\LU{0}{\vv}\cConfig = \LU{0}{[\dot q_0,\,\dot q_1,\,\dot q_2]}\cConfig\tp$ | global 3D velocity vector of node |
| Acceleration | $\LU{0}{\av}\cConfig = \LU{0}{[\ddot q_0,\,\ddot q_1,\,\ddot q_2]}\cConfig\tp$ | global 3D acceleration vector of node |
| CoordinatesTotal |  | displacement plus reference coordinates of node |
| Coordinates |  | coordinate vector of node (relative to reference configuration) |
| Coordinates_t |  | velocity coordinates vector of node |
| Coordinates_tt |  | acceleration coordinates vector of node |
| RotationMatrix | $[A_{00},\,A_{01},\,A_{02},\,A_{10},\,\ldots,\,A_{21},\,A_{22}]\cConfig\tp$ | vector with 9 components of the rotation matrix $\LU{0b}{\Rot}\cConfig$ in row-major format, in any configuration; the rotation matrix transforms local ($b$) to global (0) coordinates |
| Rotation | $[\varphi_0,\,\varphi_1,\,\varphi_2]\tp\cConfig$ | vector with 3 components of the Euler / Tait-Bryan angles in xyz-sequence |
| AngularVelocity | $\LU{0}{\tomega}\cConfig = \LU{0}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tp$ | global 3D angular velocity vector of node |
| AngularVelocityLocal | $\LU{b}{\tomega}\cConfig = \LU{b}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tp$ | local (body-fixed) 3D angular velocity vector of node |

(description-nodepointslope12)=
## Detailed description

### Coordinates

| index | symbol | kind | meaning | frame |
|---|---|---|---|---|
| 0, 1, 2 | $q_0,\,q_1,\,q_2$ | {ref}`ODE2 <ODE2>` | displacement of the node position $\rv$ | global |
| 3, 4, 5 | $q_3,\,q_4,\,q_5$ | {ref}`ODE2 <ODE2>` | change of the slope vector $\rv_x^\prime$ | global |
| 6, 7, 8 | $q_6,\,q_7,\,q_8$ | {ref}`ODE2 <ODE2>` | change of the slope vector $\rv_y^\prime$ | global |

### Configuration

$$
\rv = \rv\cRef + [q_0,\,q_1,\,q_2]\tp, \quad
\rv_x^\prime = \rv_{x,\mathrm{ref}}^\prime + [q_3,\,q_4,\,q_5]\tp, \quad
\rv_y^\prime = \rv_{y,\mathrm{ref}}^\prime + [q_6,\,q_7,\,q_8]\tp .
$$

### The slope vectors

The two slope vectors are the derivatives of the position of the mid-surface of a thin plate with
respect to its two in-plane coordinates, $\rv_x^\prime = \partial \rv / \partial x$ and
$\rv_y^\prime = \partial \rv / \partial y$. In a flat plate in the global $x$-$y$ plane they are
$[1,\;0,\;0]\tp$ and $[0,\;1,\;0]\tp$. They span the tangent plane of the mid-surface, and their
lengths and angle carry its in-plane strains. The default reference coordinates are those of a
flat plate in the global $x$-$y$ plane; a model gives the reference coordinates of every node.

### Frame and interpretation

All nine coordinates are global (absolute nodal coordinates). The node is used by
`ObjectANCFThinPlate`, whose element scales the slopes by `slopesScalingX` and `slopesScalingY`.

### Rotation

`MarkerNodeRigid` and the output variables of the rotation take an orthonormal frame of the slopes:
$\rv_x$ normalized is its $x$-axis, $\rv_y$ orthogonalized against it and normalized its $y$-axis, and the
$z$-axis - the normal of the surface - is their cross product. The angular velocity $\tomega$ and the rotation
Jacobian are not the derivatives of this frame but the least-squares fit
$\tomega = \left(-\tilde\rv_x\tilde\rv_x - \tilde\rv_y\tilde\rv_y\right)^{-1}\left(\tilde\rv_x\dot\rv_x + \tilde\rv_y\dot\rv_y\right)$,
symmetric in the two slopes; the two agree for slopes that keep their lengths and their right angle (#2219).
The derivative of $\Jm_{rot}\tp\vv$ with respect to the coordinates is not implemented: a connector whose
Jacobian needs it, e.g. a `RigidBodySpringDamper` at the node with an implicit solver, raises an error; a torque
and a joint do not need it.

### Action on the equations of motion

The nine coordinates lead to nine {ref}`ODE2 <ODE2>` equations, which the element provides; a force at the
node enters the first three.

(miniexample-nodepointslope12)=
## Mini example


```python
#a square plate clamped at one edge, from ANCF thin plate elements with position and slopes r_x, r_y
from exudyn.shells import ShellMesh
plate = ShellMesh(vertices=[[0,0,0],[1,0,0],[1,1,0],[0,1,0]], numberOfElementsX=2, numberOfElementsY=2,
                  youngsModulus=2e9, poissonsRatio=0, density=1000, thickness=0.01)
plate.CreateANCFThinPlateElements(mbs) #adds a NodePointSlope12 per mesh point
for node in plate.boundaryNodeNumbers['left']:
    mNode = mbs.AddMarker(MarkerNodeRigid(nodeNumber=node))
    mbs.CreateGenericJoint(itemNumbers=[oGround, mNode]) #clamped: position and orientation
for element in plate.elementNumbers:
    mbs.AddLoad(LoadMassProportional(markerNumber=mbs.AddMarker(MarkerBodyMass(bodyNumber=element)),
                                     loadVector=[0,0,-9.81]))

mbs.Assemble()
mbs.SolveStatic()

#a corner of the free edge; compare q*L^4/(8*D) = 0.0736 of a cantilever strip, D = E*h^3/12
corner = plate.vertexNodeNumbers[1]
exu.sys['testResult'] = mbs.GetNodeOutput(corner, exu.OutputVariableType.Displacement)[2]
```