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Nonlocal Constitutive Model Strategy

When the nonlocal constitutive model is considered, additional issues have to be taken into account. The response of the nonlocal material model in a given integration point depends on values at other integration points in its neighbourhood. Let us consider a case, when the integration point lies on the boundary or in the vicinity of the boundary between some partitions. In general, it would be possible to compute nonlocal quantity from its local as well as remote contributions in every integration point near the boundary. Nevertheless, the remote integration point values are influencing many local integration points (precisely their nonlocal values). As a consequence, this would lead to multiple requests (over a slow communication line) for the same remote integration point value from multiple integration points on the local partition. Due to the redundant requests for the same remote values and extremely fine grain communication pattern, it is necessary to reject this approach completely.

The proposed strategy can be considered as an enhanced version of the element- or node-cut approach, respectively. The cut runs again through element sides (node-cut) or elements (element-cut) dividing the whole mesh to several partitions. Generally, the local element response (or more precisely, the material model response in integration points of a local element lying near the boundary -- the size of averaging zone is a material property) can depend on some integration points lying on a remote partition. To get rid of this ``remote'' dependency, the so-called mirrored or remote-copy elements have been introduced. A remote-copy element is established for each element, which is on a remote partition, and values of any local integration point depend on it (they actually depend on element integration point values). After local quantities which are subjected to nonlocal averaging are computed at every integration point of local elements, their necessary exchange to the corresponding remote elements (and their integration points) values is done. After finishing the mutual exchange, the integration points of the remote-copy elements contain valid copies of the corresponding quantities and the nonlocal values at the integration points of the local elements can be easily computed, using values at the integration points of the remote-copy elements, instead of invoking expensive communication. No computation on the remote-copy elements is needed, because necessary element contributions and required output values are computed on those partitions which possess the counterparts of the remote-copy elements. It should be noted here, that the remote-copy element counterparts are local on the partition owning it. From this point of view, the remote-copy element can be viewed as a dummy element, which does not contribute to the characteristic equation on the local partition. It is only a mirror of another element, storing the copy of the local quantity (being subjected to averaging) of its master required by the local integration points depending on it. As a consequence, this approach leads to an effective communication, because the quantities to be averaged are sent or received to or from the remote integration points only once for a given time step.

If the local quantity, being averaged, can be computed from the current nodal values and element geometry (like a strain vector), then it needs not be transfered and can be directly computed from the nodal displacements. The transfer of nodal values to their remote-copy node mirrors (see element-cut) is probably more efficient than the transfer of local values to all element integration points. Moreover the values being averaged may be of vector or tensorial character. However, the remote-copy elements are still necessary to efficiently reduce the necessary communication. The transfer of local values at integration points is more general and can be used in more general situations with any nonlocal constitutive model.

Since each remote-copy element keeps typically an information about its source element 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. Each partition broadcasts its receive list to all other partitions, which, in turn, find out the corresponding local elements that will participate in the communication and insert them into their corresponding send map. This process is similar to the setup of the remote-copy node communication map in the element-cut approach.

In the node-cut strategy, enhanced by the use of remote-copy elements, generally two communication schemes have to be considered -- the first one for node-cut partitioning strategy (exchange of shared node data) and the second one for remote-copy element data exchange (see Table 3).

Mass contribution exchange for shared nodes;
while not finished loop
Assemble load vector ;
Exchange local values to be averaged for all integration points of remote-copy elements;
Compute local real nodal forces ;
Exchange real nodal force contributions for shared nodes;
Solve displacement increment from Eq.(1);
Compute acceleration & velocity fields (see Eqs (2) and (3));
Update displacement vector ;
Increment time ;

Table 3: Central difference node-cut stepping algorithm with nonlocal extension.

Similarly, in the element-cut strategy, two different communication schemes have to be considered. The first scheme for the exchange of remote-copy node data has to be supplemented by the second scheme for remote-copy element data exchange. This strategy, however, keeps its disadvantage in terms of duplicating the shared elements along the interface of adjacent partitions, thus increasing the computational load.

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Daniel Rypl