As described earlier, a key component of the adaptive CVFE scheme relies on the ability to insert dynamically cohesive elements in a finite element mesh. This section provides some details on three issues associated with the dynamic element insertion process. The first one is the criterion used to insert the cohesive elements: in this work, we select the regions for insertion based either on a bounding box approach which grows as elements are inserted, or a stress-based approach where cohesive elements are placed in regions where stress levels have reached a predetermined threshold. The second issue is that of database management: the insertion algorithm requires a more detailed knowledge of the current mesh, including connectivity information for the nodes, edges and elements, as well as other flags not necessary with standard pre-simulation cohesive element insertions. Finally, we address the ``mechanics'' of the insertion process, i.e., how to insert cohesive elements with minimal disturbance of the solution. Various approaches including selective damping or pre-stretching are considered.