2-D Insertion

In the 2-D part of the present study, we use three-node constant-strain triangular volumetric elements. The cohesive elements thus have four node with the nodes ordered in counter-clockwise fashion as seen in Figure 2.8. Cohesive elements are inserted in place of existing or proposed edges by duplicating the edge and possibly its nodes to form the new four-node system. In order to insure proper insertion, more information is needed. In particular, each edge must know the two nodes connected to it as well as the volumetric elements it borders. Another list is also needed, which contains, for each node, the list of all edges and volumetric elements connected to it. This additional information is created once at the beginning of the simulation, and is updated as cohesive elements are inserted adaptively. Consistency is the key to a successful cohesive element insertion. We always define the top volumetric element of an edge to be the element pointed to by the normal vector of the edge. The normal vector is itself consistently calculated as pointing to the right of the segment from the first to the second node (Figure 2.8).

Figure 2.8: 2-D Cohesive element representation.

Inconsistent numbering and calculations can result in insertion of invalid cohesive elements which may be ``criss-crossed'' as demonstrated in Figure 2.9. Criss-crossed cohesive elements are a result of incorrect initiation of top/bottom elements and left/right nodes.

Figure 2.9: 2-D cohesive element insertion: (a) proposed edge for cohesive insertion, (b) inserted cohesive element, (c) ``criss-crossed'' cohesive element.

Once all the edge and node information has been defined, cohesive element insertion can be initiated. A selected edge can only be made cohesive if it is not already cohesive, or this edge is an internal edge that has not been flagged by the user. External boundary edges are restricted from cohesive insertion because cohesive elements required to be sandwiched between two volumetric elements.

The valid proposed edge is now ready for cohesive insertion. Standard cohesive insertion always requires the duplication of the proposed edge. The nodes of the proposed edge are only duplicated if these nodes are already connected to an existing cohesive or duplicated edge. These nodes can be easily recognized as they will be flagged as cohesive nodes. Nodes flagged as ``normal'' (i.e., non-cohesive) have no cohesive or duplicated edges in their lists and as a result they will not be duplicated. Nodal duplication schematically splits the mesh at the node. The original edges and volumetric elements connected to this node are also split between the new node halves. This is similar to the 1-D case where the cohesive node system is composed of two halves, each of which is connected to its own elements causing the redistribution of nodal masses. In 2-D, nodal masses are obtained from the volumetric elements to which the node is connected. When a node is duplicated, the mass of the original and duplicated halves must be recalculated based only on the volumetric elements connected to each half. The sum of these masses should equal the original mass present prior to any insertion around the selected node. In addition, conservation of momentum must also be maintained for each node by duplicating the nodal displacements, velocities and accelerations, as well as any other pertinent flags and markers.

Figure 2.10: Connectivity update of nodes and elements.

Only five different cases can be encountered during cohesive element insertion. These cases are presented graphically in Figure 2.11. The first case is for an existing cohesive element, which, for obvious reason, cannot have any more cohesive elements inserted in its place. The second case involves a proposed edge that has never been made cohesive and is not connected to any other cohesive edges. Insertion for this case involves only the duplication of the proposed edge. The resulting cohesive element is considered dormant because it shares the left and right node sets. Cases $3$ and $4$ are mirror images of each other. They each involve a normal proposed edge that is connected to one other cohesive edge or element. Element insertion for these cases requires the duplication of the connecting node as well as the duplication of the edge. The final case is concerned with an edge that is bordered by two other cohesive edges. This insertion requires the duplication of both the left and right nodes.

Figure 2.11: Common 2-D insertion cases.

In order to clarify the various insertion cases, we present in Figure 2.12 an illustrative example of insertion for a simple 2-D mesh. The first proposed cohesive edge is a non-cohesive edge having non-cohesive nodes. This situation corresponds to Case #2 in Figure 2.11, for which only the edge is duplicated and the two nodes are flagged cohesive. Although a cohesive element is added to the general list of cohesive elements, this element has no impact on the structural solution since no displacement jump is possible. Next, we insert the neighbor edge to the right of the first element. This edge is also non-cohesive but it contains one cohesive node resulting from the first insertion. Depending on the orientation of the normal vector of this edge, this situation corresponds to Cases #3 or #4 in Figure 2.11. Finally, the third cohesive element insertion is similar in concept to the second one, except that the edge is not oriented horizontally.

Figure 2.12: Illustrative example of three cohesive element insertions using Cases #2 and #3 in Figure 2.11.

In order to illustrate insertion Case #5, we consider the illustrative example shown in Figure 2.13. After the insertion of the first two cohesive elements, the middle edge is still a ``normal'' edge although both of its nodes are flagged as cohesive. Insertion of a cohesive element along this edge thus cause the duplication of both the left and right nodes.

Figure 2.13: Illustration of insertion Case #5 in Figure 2.11.