A Constraint-Based System to Ensure the Preservation of Sharp Geometric Features in Hexahedral Meshes (original) (raw)

Abstract

Generating a full hexahedral mesh for any 3D geometric domain is still a challenging problem. Among the different attempts, the octree-based methods are the most efficient from an engineering point of view. But the main drawback of such methods is the lack of control near the boundary. In this work, we propose an a posteriori technique based on the notion of the fundamental mesh in order to improve the mesh quality near the boundary. This approach is based on the resolution of a constraint problem defined on the topology of the CAD model that we have to discretize.

Figures (14)

Fig. 1. In a, two primal sheets (respectively in red and blue) intersect each other along a chord (in green) that is shown alone in b

Fig. 1. In a, two primal sheets (respectively in red and blue) intersect each other along a chord (in green) that is shown alone in b

Fig. 2. Two examples of fundamental chords This definition ensures that if you consider a geometric curve cg delimiting two geometric surfaces, then, on both surfaces, quadrilaterals adjacent to curve cg belong to a single primal chord (see Fig. 2). This definition specifies nothing about the hexahedra having an edge associated to curve cg but no faces on the geometric boundary. Indeed, any number of chords can be fundamental to the same geometric curve. In order to define the notion of fundamental sheets, we now introduce the definition of geometric surfaces capture.

Fig. 2. Two examples of fundamental chords This definition ensures that if you consider a geometric curve cg delimiting two geometric surfaces, then, on both surfaces, quadrilaterals adjacent to curve cg belong to a single primal chord (see Fig. 2). This definition specifies nothing about the hexahedra having an edge associated to curve cg but no faces on the geometric boundary. Indeed, any number of chords can be fundamental to the same geometric curve. In order to define the notion of fundamental sheets, we now introduce the definition of geometric surfaces capture.

Fig. 3. The same geometric 3D domain where a curved surface is captured by a single fundamental sheet in a and three non-fundamental sheets in b The first condition ensures that the geometric surface is captured by a single primal sheet. For instance, let us consider Fig. 3-a, all the hexahedra belonging to the green primal sheet are along a single curved surface. On the contrary, in Fig. 3-b, this surface is partially captured by three primal sheets. The second condition guarantees that the corresponding dual sheet is locally a 2-manifold.

Fig. 3. The same geometric 3D domain where a curved surface is captured by a single fundamental sheet in a and three non-fundamental sheets in b The first condition ensures that the geometric surface is captured by a single primal sheet. For instance, let us consider Fig. 3-a, all the hexahedra belonging to the green primal sheet are along a single curved surface. On the contrary, in Fig. 3-b, this surface is partially captured by three primal sheets. The second condition guarantees that the corresponding dual sheet is locally a 2-manifold.

Fig. 4. A hexahedral mesh (in a) and one level 1 fundamental sheet (in b), two level 2 fundamental sheets (in c) and four level 3 fundamental sheets (in d)

Fig. 4. A hexahedral mesh (in a) and one level 1 fundamental sheet (in b), two level 2 fundamental sheets (in c) and four level 3 fundamental sheets (in d)

Fig. 5. Dual representation of the fundamental sheets given in Fig. 4

Fig. 5. Dual representation of the fundamental sheets given in Fig. 4

Fig. 6. The four types of curves we handle The “locality” term is due to the fact that locally to a curve, it is impos- sible to distinguish a level 1 from a level 2 fundamental sheet and a level 3 from a level 2 fundamental sheet. Moreover a propagation of constraints along geometric curves and surfaces can alter this classification.

Fig. 6. The four types of curves we handle The “locality” term is due to the fact that locally to a curve, it is impos- sible to distinguish a level 1 from a level 2 fundamental sheet and a level 3 from a level 2 fundamental sheet. Moreover a propagation of constraints along geometric curves and surfaces can alter this classification.

Fig. 7. Propagation of geometric vertices’ constraints along geometric curves

Fig. 7. Propagation of geometric vertices’ constraints along geometric curves

Fig. 8. The labeling of vertex V is increased (left) or decreased by 2 (right)

Fig. 8. The labeling of vertex V is increased (left) or decreased by 2 (right)

Table 1. Validity rules for 3, 4 and 5-valent geometric vertices

Table 1. Validity rules for 3, 4 and 5-valent geometric vertices

Fig. 9. A portion of the research tree

Fig. 9. A portion of the research tree

Fig. 10. Fundamental sheets to insert

Fig. 10. Fundamental sheets to insert

Fig. 11. A diamond-shaped domain discretized with a THex mesh then improved by inserting fundamental sheets the final mesh are given in Fig. 11-a and Fig. 11-b. In Fig. 11-c and 11-d, two points of view are used to highlight the pairs of level 3 fundamental sheets that go across the volume. To drive the sheet insertion inside the volume, we use an advancing front algorithm that depends on the underlying hexahedral mesh. The result can thus differ depending on the mesh but the success of sharp feature detection does not rely on the base mesh. The second example is called the hook model. Snapshots from four view angles are given in Fig. 12. We can see that the patterns of THex meshes are no longer along curves where the insertion of fundamental sheets allows us to get surface chords on both sides of each geometric curve. In Fig. 13 are given the three sets of fundamental sheets.

Fig. 11. A diamond-shaped domain discretized with a THex mesh then improved by inserting fundamental sheets the final mesh are given in Fig. 11-a and Fig. 11-b. In Fig. 11-c and 11-d, two points of view are used to highlight the pairs of level 3 fundamental sheets that go across the volume. To drive the sheet insertion inside the volume, we use an advancing front algorithm that depends on the underlying hexahedral mesh. The result can thus differ depending on the mesh but the success of sharp feature detection does not rely on the base mesh. The second example is called the hook model. Snapshots from four view angles are given in Fig. 12. We can see that the patterns of THex meshes are no longer along curves where the insertion of fundamental sheets allows us to get surface chords on both sides of each geometric curve. In Fig. 13 are given the three sets of fundamental sheets.

Fig. 12. The resulting mesh for the hook model

Fig. 12. The resulting mesh for the hook model

Fig. 13. Level 1, level 2 and level 3 fundamental sheets for the hook model respec- tively in a, b and c

Fig. 13. Level 1, level 2 and level 3 fundamental sheets for the hook model respec- tively in a, b and c

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