Two and three-quarter dimensional meshing facilitators

1996, Engineering with Computers

Abstract

This paper presents generated enhancements for robust 'two and three-quarter dimensional meshing', including:

Figures (17)

Fig. 1. Mesh generation using automated interval assignment aad submapping tools. more powerful meshing aigorithms continues, but no one technique has appeared which can fill the needs of all meshing tasks. Therefore, various mesh gener- ation research efforts continue in an attempt to automate these meshing processes (Fig. 1). The transformation of a two.dimensional (2D) mesh into a three dimensional (3D) volume and shell elements has long been a useful tool in the arsenal of algorithms available for 3D meshing [1]. A 2D mesh simply extruded in some general third dimension produces what is termed a two and one half dimensional (2.5D) volume mesh. However, any number of these 2.5D pieces can be combined to form extremely complex meshes as shown in Fig. 2. As can be seen in this figure, these pieces can be extruded in logically orthogonal directions with respect to each other. We have termed this general meshing method of combining orthog- onally independent 2.5D pieces as being two and three quarter dimensional (2.75D).

the proper tools, including those described in this paper, the propagation can be largely eliminated. Figure 4 shows how these tools would be effective at reducing the decomposition of this object to only one cut. Implicit for this type of decomposition to work effectively would be the use of the following techniques which are presented in this paper. In their simplest form, 2.5D tools depend only on a 2D mesh as input. Thus they have often been developed independent of any formal geometric description of the 3D geometry. This disconnection from the actual geometry presents problems in generalizing the algorithms to general sweeps, in accurately combining nodes shared between merged pieces, and in applying and verifying boundary conditions. It also prevents the effective use of general 3D CAD data when available. Two and Three-Quarter Dimensional Meshing Facilitators

Fig. 3. Simple geometry showing the effect of propagating decompositions.

Fig. 4. Non-propagating decomposition allowed with the proper 2.75D meshing tools.

Fig. 5. Surface vertex type examples. This paper discusses these tools and their interaction. First, the idea of ‘surface vertex types’, ‘i-j-k’ traversals, and ‘hard and soft interval settings’ will be defined. These definitions are needed for proper discussion of both the automated interval assignment process and submapping techniques. Finally, the n-faceted volume mapping, submapping and general sweeping tools are discussed.

The ‘surface vertex type’ classifications control the ‘i-j-k’ traversals of the automated interval assignment and submapping algorithms (defined in Section 2.2). It also controls which vertices will be selected as ‘logical corners’ for the mapping algorithm.

Table 1. Traversal tables assignments. The node at the first vertex is assigned a coordinate of (0, 0) in ‘i-j*’ space, and nodes along the first curve (assumed to be in the ‘+i’ direction) are incremented in ‘i’ (e.g. (1,0), (2,0), etc). As vertices are reached which change the ‘i-j’ direction, as described above, the appropriate coordinate is incremented or decremented. Extending the previous example, if an end vertex is reached and the node-at that vertex was assigned (9, 0) as its (i, }) coordinate, the next node would increment in the +j direction (i.e. its ‘i~j’ coordinate would be (9, 1)) (see Table 1). Surface vertex classifications and ‘i-j’ traversals provide a method to classify surface boundary curves and to assign all meshed nodes on. the surface boundary an ‘i-j’ coordinate in the local ‘i-j’ co- ordinate system.

Fig. 6. Local i-j traversal classifications and resulting ‘unfoided’ surface geomeiry.

Fig. 7. Initial interval settings and final interval assignment.

Fig. 8. Example of submapping geometry and resulting mesh. eee ag Bones Fane ee See ae Se Kee eS RS ot ee To find the ‘cut node’ in ‘i-j’ space each node in the nodal loop is given an ‘i-j’ coordinate according to the ‘i-j’ space traversals for submapping as defined

Fig. 9, Corner and cut node.

The equations to calculate the interpolated nodes depend on the ‘i-j’ direction of the cut and the sign of the direction. For the case where the cut is in the ‘+i’ direction (see Fig. 10), the nodes for the interpolated edge are found by the following equation:

is a usable tool that automates 3D meshing for objects that have n-number of faces. Where surface mapping requires a logical quadrilateral, volume mapping requires a logical hexahedral. Volume submapping is another step to further improve 3D meshing. Volume submapping will automatically use pseudo decompo- sition, like submapping (see Section 4.2), to break volumes down into logical hexahedrons. Two and Three-Quarter Dimensional Meshing Facilitators