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‎docs/src/man/models/contact.md‎

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## Contact
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## Contact Search
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| Parameter | Unit | Description |
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|---|---|---|
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|Contact Radius | $[m]$| Radius to get a list of potential contact pairs.|
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|Master| $[-]$| Block ID of the master nodes. |
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|Slave | $[-]$| Block ID of the slave nodes.|
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| Parameter | Unit | Description |
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| -------------- | ----- | ------------------------------------------------ |
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| Contact Radius | $[m]$ | Radius to get a list of potential contact pairs. |
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| Master | $[-]$ | Block ID of the master nodes. |
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| Slave | $[-]$ | Block ID of the slave nodes. |
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**Initialization**
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- Step 1 -
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Identification of all surface nodes of contact blocks.
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Identification of all surface nodes of contact blocks.
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- Step 2 -
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All surface nodes are stored in a list. This list exists at all cores with the current position of the surface nodes.
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All surface nodes are stored in a list. This list exists at all cores with the current position of the surface nodes.
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- Step 3 -
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Connect each surface node to a geometrical surface.
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Connect each surface node to a geometrical surface.
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The standard way for points in the parallelization is given here.
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$$\begin{bmatrix}1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \end{bmatrix}_{all}\rightarrow\begin{matrix} \begin{bmatrix} 4 \\ 6 \end{bmatrix}_{C1-global}\\
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$$
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\begin{bmatrix}1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \end{bmatrix}_{all}\rightarrow\begin{matrix} \begin{bmatrix} 4 \\ 6 \end{bmatrix}_{C1-global}\\
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\begin{bmatrix}1 \\ 2 \\ 7 \end{bmatrix}_{C2-global}\\
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\begin{bmatrix}3\\5\end{bmatrix}_{C3-global}\end{matrix}\leftarrow \rightarrow\begin{matrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix}_{C1-local}\\
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\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}_{C2-local}\\
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\begin{bmatrix}1\\2\end{bmatrix}_{C3-local}\end{matrix}$$
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\begin{bmatrix}1\\2\end{bmatrix}_{C3-local}\end{matrix}
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$$
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A new mapping is needed to the contact surface exchange list. The local numbering (right above) has to be mapped to the sublist and vice, versa.
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$$\begin{bmatrix}1 \\ 2 \\ 6 \\ 7 \end{bmatrix}_{surface}\leftarrow\rightarrow \begin{bmatrix}1 \\ 2 \\ 3 \\ 4 \end{bmatrix}_{surface}\rightarrow \begin{matrix} \begin{bmatrix} 6 \end{bmatrix}_{C1}\\
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$$
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\begin{bmatrix}1 \\ 2 \\ 6 \\ 7 \end{bmatrix}_{surface}\leftarrow\rightarrow \begin{bmatrix}1 \\ 2 \\ 3 \\ 4 \end{bmatrix}_{surface}\rightarrow \begin{matrix} \begin{bmatrix} 6 \end{bmatrix}_{C1}\\
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\begin{bmatrix}1 \\ 2 \\ 7 \end{bmatrix}_{C2}\\
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\begin{bmatrix}\,\,\,\,\end{bmatrix}_{C3}\end{matrix}\leftarrow \rightarrow\begin{matrix} \begin{bmatrix} 2 \end{bmatrix}_{C1-local}\\
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\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}_{C2-local}\\
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\begin{bmatrix}\,\,\,\,\end{bmatrix}_{C3-local}\end{matrix}$$
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\begin{bmatrix}\,\,\,\,\end{bmatrix}_{C3-local}\end{matrix}
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$$
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**List mapping**
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To fill the postion vector with the needed values mappings are needed.
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local point ids -> global point id -> reduced contact block point id
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The mapping has to be done in both directions, because the contact forces have to be applied to the local core point id.
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---
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**Computation**
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- Step 1 -
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Perform a nearest neighbor search with a user defined Contact Radius. The result is a list of potential contact pairs.
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Perform a nearest neighbor search with a user defined Contact Radius. The result is a list of potential contact pairs.
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- Step 2 -
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A sub list nearest neighbor search is performed.
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A sub list nearest neighbor search is performed.
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- Step 3 -
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All points found are than in contact.
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All points found are than in contact.
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The quasi code is given here
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end
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```
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The structure tries to minimize the search and the communication between cores.
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The structure tries to minimize the search and the communication between cores.
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**Synchronization**
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All cores (except core 1) send it's displacement values to core 1. Core 1 sends them back.
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!!! note "Efficiency"
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Smaller contact areas are more efficient in numerical analysis, because less synchronisation have to occur.
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Smaller contact areas are more efficient in numerical analysis, because less synchronisation have to occur.
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----
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---
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## Contact Models
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| Contact Model | Penalty Contact |
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|------------------------|:---------------:|
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| Contact Stiffness | ✔️|
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| Contact Model | Penalty Contact |
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| ----------------- | :-------------: |
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| Contact Stiffness | ✔️ |
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## Penalty Contact
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A bond-based contact formulation is computed. The contact stiffness $ c_{contact}$ is defined in YAML input file. Based on that utilzing the horizon $\delta$ the penalty stiffness shown in the table is computed. These parameter are comparable to the [bond-based formulation](@ref "Bond-based Peridynamics"). The algorithm was taken from [Peridigm](https://github.com/peridigm/peridigm/blob/master/src/contact/Peridigm_ShortRangeForceContactModel.cpp)
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| Dimension | Penalty Stiffness |
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|---|---|
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|plane strain:| $c_{Penalty} = \frac{48 c_{contact}}{\pi 5 \delta_{slave}^3}$ |
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|plane stress:| $c_{Penalty} = \frac{9 c_{contact}}{\pi \delta_{slave}^3}$ |
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|3D:| $c_{Penalty} = \frac{12 c_{contact}}{\pi \delta_{slave}^4}$ |
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A bond-based contact formulation is computed. The contact stiffness $ c\_{contact}$ is defined in YAML input file. Based on that utilzing the horizon $\delta$ the penalty stiffness shown in the table is computed. These parameter are comparable to the [bond-based formulation](@ref "Bond-based Peridynamics"). The algorithm was taken from [Peridigm](https://github.com/peridigm/peridigm/blob/master/src/contact/Peridigm_ShortRangeForceContactModel.cpp)
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| Dimension | Penalty Stiffness |
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| ------------- | ------------------------------------------------------------- |
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| plane strain: | $c_{Penalty} = \frac{48 c_{contact}}{\pi 5 \delta_{slave}^3}$ |
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| plane stress: | $c_{Penalty} = \frac{9 c_{contact}}{\pi \delta_{slave}^3}$ |
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| 3D: | $c_{Penalty} = \frac{12 c_{contact}}{\pi \delta_{slave}^4}$ |
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### Normal Contact
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The distance is currently computed as
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The distance is computed as
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$$d = |\underline{\mathbf{Y}}_{master}-\underline{\mathbf{Y}}_{slave}|$$
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Both can be adopted if needed. Utilizing the contact radius $r_{contact}$ the contact force densities are
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$$
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\mathbf{f}_{master} = c_{Penalty}\frac{(r_{contact}-d)}{\delta_{slave}}\mathbf{n}V_{slave}$$
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$$\mathbf{f}_{master}=c_{Penalty}\frac{(r_{contact}-d)}{\delta_{slave}}\mathbf{n}V_{slave}$$
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$$\mathbf{f}_{slave} = -c_{Penalty}\frac{(r_{contact}-d)}{\delta_{slave}}\mathbf{n}V_{master}$$
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Within PeriLab the functions compute_master_force_density and compute_slave_force_density are used to apply the force to the correct postions. This is needed, becaus if a parallel process is run, master and slave nodes not necessarily are on the same core.
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### Friction Contact
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| Contact Model | Penalty Contact |
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|------------------------|:---------------:|
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| Friction Coeffient | ✔️|
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| Contact Model | Penalty Contact |
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| ------------------ | :-------------: |
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| Friction Coeffient | ✔️ |
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The model is computed if the ""Friction Coeffient"" is greater than zero.
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For friction the normal forces
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$$\mathbf{f}_n=c_{Penalty}\frac{(r_{contact}-d)}{\delta_{slave}}\mathbf{n}$$
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are used. The tangential direction must be computed. Following the [Peridigm](https://github.com/peridigm/peridigm/blob/master/src/contact/Peridigm_ShortRangeForceContactModel.cpp) algorithm, the normal velocity of the master node $m$ and the slave node $s$ are
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$$\begin{matrix}
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v_{m} = \mathbf{v}_m\mathbf{n}\\
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v_{s} = \mathbf{v}_s\mathbf{n}
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\end{matrix}$$
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$$
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\begin{matrix}v_{m} = \mathbf{v}_m\mathbf{n}\\v_{s} = \mathbf{v}_s\mathbf{n}
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\end{matrix}
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$$
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To obtain the tangential part $tan$ this value hmust be substrated from the velocity.
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$$\begin{matrix}
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\mathbf{v}_{tan-m} = \mathbf{v}_m - v_{m}\mathbf{n}\\
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\mathbf{v}_{tan-s} = \mathbf{v}_s - v_{s}\mathbf{n}
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\end{matrix}$$
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$$\begin{matrix}\mathbf{v}_{tan-m} = \mathbf{v}_m - v_{m}\mathbf{n}\\\mathbf{v}_{tan-s} = \mathbf{v}_s - v_{s}\mathbf{n}\end{matrix}$$
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Tanking the average
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$$\mathbf{v}_{avg} = 0.5(\mathbf{v}_{tan-m}+\mathbf{v}_{tan-s})$$
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the relative velocity of each point can be determined.
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$$\begin{matrix}
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\mathbf{v}_{tan-m} = \mathbf{v}_{tan-m} - \mathbf{v}_{avg}\\
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\mathbf{v}_{tan-s} = \mathbf{v}_{tan-s} - \mathbf{v}_{avg}
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\end{matrix}$$
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$$\begin{matrix}\mathbf{v}_{tan-m}=\mathbf{v}_{tan-m}-\mathbf{v}_{avg}\\\mathbf{v}_{tan-s}=\mathbf{v}_{tan-s} - \mathbf{v}_{avg}\end{matrix}$$
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If the length of $\mathbf{v}_{tan-m}$ and $\mathbf{v}_{tan-s}$ is greater than zero, respectively, friction occurs. The friction force is computed as using the friction coefficient $\mu$.
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$$\begin{matrix}
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\mathbf{f}_{fric-m} = -\mu |\mathbf{f}_n|\frac{\mathbf{v}_{tan-m}}{|\mathbf{v}_{tan-m}|}\\
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\mathbf{f}_{fric-s} = -\mu |\mathbf{f}_n|\frac{\mathbf{v}_{tan-s}}{|\mathbf{v}_{tan-s}|}
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\end{matrix}$$
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$$\begin{matrix}\mathbf{f}_{fric-m} = -\mu |\mathbf{f}_n|\frac{\mathbf{v}_{tan-m}}{|\mathbf{v}_{tan-m}|}\\\mathbf{f}_{fric-s} = -\mu |\mathbf{f}_n|\frac{\mathbf{v}_{tan-s}}{|\mathbf{v}_{tan-s}|}\end{matrix}$$

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