Tetrahedral Meshing for Self-Intersecting Boundaries
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Solution Overview
Problem
Existing tetrahedral meshing methods fail when dealing with self-intersecting boundaries in geometric models, leading to incomplete mesh generation and potential quality issues, as they require watertight and intersection-free input meshes, which are not always achievable due to modeling errors or numerical imprecision.
Innovation Solution
An improved meshing method detects intersecting faces in the initial tetrahedral mesh, creates cavities around the intersections, generates independent meshes within these cavities, and stitches them back together with the initial mesh, ensuring intersection-free cavities while respecting the input boundary faces, allowing for successful mesh refinement and quality improvement without modifying the input surface mesh.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional tetrahedral meshing methods are used on surface meshes with self-intersecting boundaries, then the mesh generation process fails or produces incomplete meshes, but manual repair or intersection removal can cause quality issues or violate user input
Solution Approach 1:
The method segments the problematic intersecting region from the rest of the mesh by creating cavities around self-intersecting faces. This allows the intersecting portion to be handled separately through sculpting and remeshing operations, while the non-intersecting portions retain their original high-quality mesh structure. The segmentation isolates the complexity to manageable regions.
Solution Approach 2:
The method extracts the self-intersecting faces and their associated intersecting regions from the original mesh. By identifying unrecovered faces after initial meshing and creating cavities around them, the problematic elements are taken out and handled through a specialized sculpting and remeshing process, preventing them from compromising the entire mesh.
2Productivity
If surface mesh repair is performed to remove intersections before meshing, then mesh generation can complete, but the original boundary faces may be violated or mesh quality deteriorates
Solution Approach 1:
The method performs preliminary meshing on the entire surface mesh including intersecting regions, then identifies unrecovered faces that correspond to intersections. This preliminary action allows the algorithm to proceed without immediate repair, and only the necessary regions are subsequently sculpted and remeshed, preserving original boundary faces where possible.
Solution Approach 2:
The method applies local remeshing only to specific cavities containing self-intersecting faces, rather than repairing the entire surface mesh. This localized approach maintains high mesh quality in non-problematic regions while addressing intersections only where they occur, preserving the accuracy of boundary faces outside the sculpted regions.
3Device complexity
If existing Delaunay algorithm is used directly on intersecting boundaries, then the process is simple, but the mesh generation fails to complete
Solution Approach 1:
The method maintains the simplicity of the Delaunay algorithm for the majority of the mesh while segmenting out problematic intersecting regions. The complex sculpting and cavity creation operations are applied only to localized regions containing self-intersections, allowing the bulk of the meshing to proceed using the simple and efficient Delaunay approach.
Solution Approach 2:
The method introduces intermediary structures (cavities and sculpted regions) between the input surface mesh and the final tetrahedral mesh. These intermediary elements mediate the handling of self-intersecting faces, allowing the Delaunay algorithm to operate on a modified mesh that respects boundary faces while avoiding direct conflicts from intersections.
Data Source
AI summary
Existing tetrahedral meshing which require watertight and intersection free triangular mesh as input can be enhanced to tolerate and respect self-intersecting boundaries. After an initial tetrahedral mesh constructed with all input nodes, intersecting faces can be recovered using the proposed method. Starting from one of the intersecting faces, a set of intersecting faces around it can be detected, and a surrounding region can be sculpted out to form a void. Two intersection-free and topologically separate cavities can then be defined within the void and meshed independently. After stitching the meshes from the two cavities back to the original initial mesh, the existing tetrahedral meshing can continue with the regular meshing steps such as refinement etc. to finish tetrahedral meshing. In one special case when we have two points at the same location, in order to insert the second of the two points, a method can offset the second point inwardly into the mesh region a small little bit and then revert back the second point to its original location after meshing is done. The user input is not altered or mesh quality would not be affected because the point movement is small.