Contact

Contact Search

Initialization

• Step 1 -

Identification of all surface nodes of contact blocks.

• Step 2 -

All surface nodes are stored in a list. This list exists at all cores with the current position of the surface nodes.

• Step 3 -

Connect each surface node to a geometrical surface.

The standard way for points in the parallelization is given here.

\[\begin{bmatrix}1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \end{bmatrix}_{all}\rightarrow\begin{matrix} \begin{bmatrix} 4 \\ 6 \end{bmatrix}_{C1-global}\\ \begin{bmatrix}1 \\ 2 \\ 7 \end{bmatrix}_{C2-global}\\ \begin{bmatrix}3\\5\end{bmatrix}_{C3-global}\end{matrix}\leftarrow \rightarrow\begin{matrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix}_{C1-local}\\ \begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}_{C2-local}\\ \begin{bmatrix}1\\2\end{bmatrix}_{C3-local}\end{matrix}\]

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.

\[\begin{bmatrix}1 \\ 2 \\ 6 \\ 7 \end{bmatrix}_{surface}\leftarrow\rightarrow \begin{bmatrix}1 \\ 2 \\ 3 \\ 4 \end{bmatrix}_{surface}\rightarrow \begin{matrix} \begin{bmatrix} 6 \end{bmatrix}_{C1}\\ \begin{bmatrix}1 \\ 2 \\ 7 \end{bmatrix}_{C2}\\ \begin{bmatrix}\,\,\,\,\end{bmatrix}_{C3}\end{matrix}\leftarrow \rightarrow\begin{matrix} \begin{bmatrix} 2 \end{bmatrix}_{C1-local}\\ \begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}_{C2-local}\\ \begin{bmatrix}\,\,\,\,\end{bmatrix}_{C3-local}\end{matrix}\]

List mapping To fill the postion vector with the needed values mappings are needed.

local point ids -> global point id -> reduced contact block point id The mapping has to be done in both directions, because the contact forces have to be applied to the local core point id.

Computation

• Step 1 -

Perform a nearest neighbor search with a user defined Contact Radius. The result is a list of potential contact pairs.

• Step 2 -

A sub list nearest neighbor search is performed.

• Step 3 -

All points found are than in contact.

The quasi code is given here

# synchronize all contact nodes
synchronize_contact_points(datamanager, "Deformed Coordinates", "NP1",
                                               all_positions)

## loop over all contact pairs
for (cm, block_contact_params) in pairs(contact_params)
    contact_search(...)
end

function contact_search(...)
    if n < search_frequency
        potential_contact_list = global_search(...) # finds all points within the search radius
        local_search(...)
        return
    else
        synchronize_contact_points(datamanager, "Deformed Coordinates", "NP1",
                                               all_positions[potential_contact_list])
        local_search(...)   # finds all points within the contact radius
        return
    end
end

The structure tries to minimize the search and the communication between cores.

Synchronization All cores (except core 1) send it's displacement values to core 1. Core 1 sends them back.

Contact Models

Penalty Contact

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. The algorithm was taken from [Peridigm](https://github.com/peridigm/peridigm/blob/master/src/contact/PeridigmShortRangeForceContactModel.cpp)

Normal Contact

The distance is currently computed as

\[d = |\underline{\mathbf{Y}}_{master}-\underline{\mathbf{Y}}_{slave}|\]

The normal is

\[\mathbf{n} = \frac{\underline{\mathbf{Y}}_{master}-\underline{\mathbf{Y}}_{slave}}{d}\]

Both can be adopted if needed. Utilizing the contact radius $r_{contact}$ the contact force densities are

\[ \mathbf{f}_{master} = c_{Penalty}\frac{(r_{contact}-d)}{\delta_{slave}}\mathbf{n}V_{slave}\]

\[\mathbf{f}_{slave} = -c_{Penalty}\frac{(r_{contact}-d)}{\delta_{slave}}\mathbf{n}V_{master}\]

Within PeriLab the functions computemasterforcedensity and computeslaveforcedensity 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.

Friction Contact

The model is computed if the ""Friction Coeffient"" is greater than zero. For friction the normal forces $\mathbf{f}_n=c_{Penalty}\frac{(r_{contact}-d)}{\delta_{slave}}\mathbf{n}$ are used. The tangential direction must be computed. Following the Peridigm algorithm, the normal velocity of the master node $m$ and the slave node $s$ are

\[\begin{matrix} v_{m} = \mathbf{v}_m\mathbf{n}\\ v_{s} = \mathbf{v}_s\mathbf{n} \end{matrix}\]

To obtain the tangential part $tan$ this value hmust be substrated from the velocity. $\begin{matrix} \mathbf{v}_{tan-m} = \mathbf{v}_m - v_{m}\mathbf{n}\\ \mathbf{v}_{tan-s} = \mathbf{v}_s - v_{s}\mathbf{n} \end{matrix}$

Tanking the average $\mathbf{v}_{avg} = 0.5(\mathbf{v}_{tan-m}+\mathbf{v}_{tan-s})$ the relative velocity of each point can be determined. $\begin{matrix} \mathbf{v}_{tan-m} = \mathbf{v}_{tan-m} - \mathbf{v}_{avg}\\ \mathbf{v}_{tan-s} = \mathbf{v}_{tan-s} - \mathbf{v}_{avg} \end{matrix}$ 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 $\mu$.

\[\begin{matrix} \mathbf{f}_{fric-m} = -\mu |\mathbf{f}_n|\frac{\mathbf{v}_{tan-m}}{|\mathbf{v}_{tan-m}|}\\ \mathbf{f}_{fric-s} = -\mu |\mathbf{f}_n|\frac{\mathbf{v}_{tan-s}}{|\mathbf{v}_{tan-s}|} \end{matrix}\]