Fast Quadtree/Octree adaptive meshing and re-meshing with linear mixed elements

Abstract

In this paper, we will focus on adaptive meshing and re-meshing. We present an original approach, based on Quadtree and Octree, to construct the initial mesh and refine it using mixed-elements. We propose a fast algorithm using a determined set of Patterns to handle transitions between fine and coarse regions, and to closely approximate surface boundaries. An optimized structure for storing edges permits to efficiently manage neighbor information, dramatically reducing the balancing time. Results, both in 2D and 3D, exhibit the multiple possibilities for local refinements. Comparisons in terms of accuracy, number of elements in the resulting mesh, and computation time, clearly position our method as a competitive alternative to generate a conform adapted mesh, which can be used with standard numerical methods.

Appendix A: Re-meshing file format

Whenever a mesh is generated, some additional data is stored. This file does not replace the mesh file, however it contains all the required information to re-build the balanced Quadtree or Octree state associated with that mesh. Combination of quadrants and octants is not allowed in a same file. Here is an example for the 3D case (.oct):

Header. It always has 3 integers representing the number of nodes, edges and octants. Next comes 3 doubles corresponding to the coordinates of each node. They are implicitly to an index, starting from 0.

List of edges. Each edge is defined by the indexes of its two nodes. The third integer represents the index of the mid-point of the edge. If 0, this edge was not split into 2 new sub-edges. For instance in the file example the second edge (1,2) was split by index 39. Therefore, sub-edges (1,39,0) and (2,39,0) should be present later in the list.

For each quadrant or octant cell, 2 lines will be used in the file. The first line contains: the number of nodes of this cell (4 or 8), the list of indexes defining this cell, and its refinement level. For instance the first octant in the file is composed of 8 nodes, where the first node index is 1315, the last one is 86, and its RL is 4. The second line corresponds to the number of input faces of $\mathcal {P}$ this octant intersects, followed by their indexes. An octant with 0 intersections completely remains inside the domain.

Geometric Transform (optional). Can be applied to all the nodes. This transform involves rotations and translations. These values will depend on the first generated mesh. If a Region of Interest (RoI) was provided (a second surface mesh), the output mesh will be aligned with this RoI.

Numbers of sub-elements that decompose each octant. If this integer value is a 0, it means that this octant was barely intersecting the surface at the refinement stage, and therefore it was later removed when projecting inside nodes that were close to the boundary. This information is still necessary to correctly reconstruct the mesh. Of course, octants without any node inside the domain are not stored. In the example, the first 3 octants were finally removed from the mesh; the fourth octant has 1 element, therefore if element index 0 must be refined, octant 0 will be refined. The fifth octant was also removed and the next one has element indexes 1 and 2. If any of these elements were to be refined, this octant will be refined. If both element indexes are asked to be refined, the octant will be refined only once.