Cell Cut-based Method

The cell cut-based surface extraction method resulted from the development of the cell attribute-based method for crack visualization. The cell attribute-based method does not define crack geometry, therefore, its functionality was extended in the cell cut-based method. Moreover, the cell attribute-based method generates global decomposition for whole solution domain, while the cell cut-based method makes local decomposition and use the effective augmentation strategy.

Figure 2.5 illustrates the application of the cell cut-based surface extraction method in 2D. Lines between centers of particles are connections of the computational lattice. Black lines represent the unbroken lattice connections. Thin red lines represent the broken lattice connections, while red tubes visualize cracks. A simple cell cut-based method generates surfaces according to the information

2. VISUALIZATION METHODS AND DISTRIBUTED SOFTWARE FOR DISCRETE… 29

about the way in which cracks cut cells of the local decomposition assembled from the lattice connections. In 2D, the method simply connects middle points of the broken connections.

Fig. 2.5. Illustration of cell cut-based method

The simplified schema of the cell cut-based surface extraction method is presented in Figure 2.6. During the preparation of data structures, the lists of the connected neighbours were assembled for all lattice nodes. The following module generated a local space decomposition from the centres of the particles and the one-dimensional lattice connections between the particles employed in the lattice- based DEM computations.

A schema of the local decompositions method is presented in Figure 2.7. Local decompositions were generated only in the fractured regions, which were identified by marking the neighbourhood of the broken connections. In fact, these nodes are the centres of the particles. Another loop runs through the marked nodes to generate cells in the fractured regions. The developed method includes a very important condition, which checks if the current node has been already processed in the process of visualizing the preceding time steps. It saves computational resources by employing the already available cells and augmenting the local decomposition according only to the connections, which were broken during the last time step.

Several types of higher dimensionality cells might be considered to generate a suitable space decomposition based on the 1D connections of the processed lattice. Therefore, a loop running through the considered cell types should be also required. In our case, the generated 3D topology consists of the pyramids with triangular or quadrilateral bases, which are treated as different cell types and generated separately. To speed up computations, the node sets (triplets or quadruplets) were generated for identifying all possible triangular or quadrilateral

30 2. VISUALIZATION METHODS AND DISTRIBUTED SOFTWARE FOR DISCRETE …

bases. In the 2D case, the triangles were considered and the pairs of nodes were generated.

Fig. 2.6. A simplified schema of the cell cut-based method

New cells were assembled from the lattice connections in a loop, running through the generated node sets, which were connected to the node, considered in the main loop through the marked nodes of the lattice. A complex condition checks the presence of all connections in the assembled cell. The connections between the nodes of a set, forming the base of the pyramid, as well as the connections between the base of the pyramid and the considered node, were checked. In the case of a pyramid with a quadrilateral base, this condition also checks the absence of the diagonals of the quadrilateral base. Another condition checks if the current cell has not been generated yet. If all conditions are satisfied,

2. VISUALIZATION METHODS AND DISTRIBUTED SOFTWARE FOR DISCRETE… 31

32 2. VISUALIZATION METHODS AND DISTRIBUTED SOFTWARE FOR DISCRETE …

the cell is assembled from the relevant connections. The assembled cell should be associated with the lattice connections that compose it in the case of using the geometric cell centre-based visualization method. However, the surface extraction based on the cell cut neither needs this association, nor executes. The following module fills the data structures and augments the cell list, appending a new cell to the list of the previously generated cells. It is worth noting that some cells of the list had already been generated by visualizing the preceding time steps and do not need to be generated at this stage. Finally, the loops through the generated node sets, the considered cell types and the marked nodes were completed. The resulting local decompositions cover the fractured regions and can be effectively used for extracting the crack surfaces.

The main loop (Figure 2.6) runs through all cells of the resulting decomposition. The method determined the graphics primitives required to represent the part of the crack surface passing through the cell according to the number and order of the broken connections. Initially, all possible topological states of the surface cutting the cell were explored and stored in a case table during the preparation of data structures. In the loop running through the cells, the index for looking up the relevant topological state of the processed cell in the case table was obtained. Another loop runs through all connections of the processed cell for obtaining the midpoints of the broken connections. The included condition checked if the processed connection was broken and determined if the coordinates of the midpoint had to be calculated. Finally, the graphics primitives were created according to the index of the state table. The vertices of the generated primitives were placed at the calculated midpoints of the broken connections. The main loop running through all cells of the local decomposition was completed, when the crack surfaces were described by graphics primitives in all cut cells.