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Element-cut Concept

In this approach, the cut runs through elements. Those elements, which are divided by the cut, are duplicated on each neighbouring partition and are marked as duplicated elements. The elements belonging solely to a particular partition will be referred to as local ones. The geometry of a duplicated element is described using node(s) assigned to the same partition and by node(s) owned by a different partition, which local copies are called remote-copy nodes (see Fig. 2). The remote-copy nodes on a particular partition are assigned with local DOFs. In order to guarantee the correctness of the solution, a modification of the single processor algorithm in terms of exchange of remote-copy node internal forces values has to be made. The nodal internal forces are computed only for local partition nodes. The nodal internal forces for remote-copy nodes are received from the partitions owning the local counterparts of remote-copy node. Thus, each partition has to set all remote-copy node internal forces according to the values received from remote partitions, where corresponding remote node counterparts are local nodes. Similarly, each partition has to send its local node values to other partitions, where corresponding remote copies exist. The correct masses for remote-copy nodes must be established by an operation analogous to internal force contribution exchange before the time-stepping algorithm starts. The solution algorithm, extended by mass and internal forces exchange for remote-copy nodes, is summarized in Table 2.

Receive masses for remote-copy nodes;
while not finished loop
Assemble load vector ;
Compute local real nodal forces ;
Exchange real nodal force values for remote-copy nodes;
Solve displacement increment from Eq.(1);
Compute acceleration & velocity fields (see Eqs (2) and (3));
Update displacement vector ;
Increment time ;

Table 2: Central difference element-cut stepping algorithm.

The process of mutual exchange of internal nodal force values must be repeated for each time step to guarantee the correctness of the solution. In order to efficiently handle this exchange, each partition assembles its send and receive communication maps for all partitions. The send map contains the local node numbers, for which the exchange, in terms of sending the local nodal values to a particular remote partition, is required. The receive map contains the remote-copy node numbers, for which the exchange, in terms of receiving the nodal values from a particular remote partition, is required. The exchange process consists in packing the local nodal values according to the send maps and sending them to the corresponding partitions, followed by receiving remote partitions values and unpacking their contributions according to the receive maps. This process requires the corresponding send maps on the local partition and receive maps on the remote partitions to be identical. This can be ensured, for example, by sorting the send and receive maps according to unique global node numbers.

Since each remote-copy node typically keeps information about its local counterpart on the remote partition, the receive maps can be assembled locally on each partition. However, the send maps, must be established by the mutual partition communication, in which each partition broadcasts its receive list to all partitions. Based on broadcasted remote partition receive lists, each partition selects corresponding local nodes which will participate in communication by inserting them into its communication send maps.

Next: Nonlocal Constitutive Model Strategy Up: Local Constitutive Model Strategy Previous: Node-cut Concept

Daniel Rypl