Anisotropic Mesh Generation in High-Dimensional Spaces
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Solution Overview
Problem
Current methods for generating anisotropic meshes in high-dimensional spaces are computationally intensive due to the complexity of Delaunay triangulation, which increases significantly with dimension, making computations prohibitive in spaces like 6D or 10D.
Innovation Solution
An iterative method is introduced that reduces computation volume by selecting additional points based on distance comparisons, allowing for the determination of updated intersection results between Voronoi cells and the element, and enabling parallel processing of Voronoi cells, using a safety radius theorem to identify non-contributing bisecting hyperplanes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If embedding in larger dimensions is used to generate anisotropic meshes, then mesh quality and anisotropy control are improved, but computation volume increases prohibitively
Solution Approach 1:
The patent extracts only the necessary Voronoi cells for mesh generation by identifying and removing points that do not contribute to the Voronoi diagram within the domain of interest. This is achieved by computing distances from domain boundaries and filtering out points beyond a certain threshold, thereby reducing the number of points that require full Delaunay triangulation processing while maintaining mesh quality.
Solution Approach 2:
The patent segments the computation space by dividing the set of points into multiple subsets based on their spatial distribution and contribution to the Voronoi diagram. By processing these subsets separately and selectively, the computation volume is reduced while still generating the complete anisotropic mesh structure needed for high-quality results.
2Reliability
If all points are processed to compute Voronoi cells, then complete and accurate mesh coverage is achieved, but computation time increases significantly
Solution Approach 1:
The patent performs preliminary actions by pre-computing and storing distance information from each point to the domain boundaries before generating the Voronoi diagram. This preliminary computation enables rapid identification and exclusion of non-contributing points during the mesh generation process, ensuring complete coverage is maintained while significantly reducing the time required for full point processing.
3Manufacturing precision
If iterative processing with multiple iterations is performed, then mesh accuracy and convergence are improved, but computation load increases
Solution Approach 1:
The patent applies local quality by performing iterative refinement only in regions where it is necessary to maintain mesh accuracy. By identifying areas requiring refinement and applying iterative processing locally rather than globally, the patent achieves the required mesh accuracy while minimizing the overall computation load and improving productivity.
Data Source
AI summary
Method for processing data defining an element in a space E or:a/ we define a set of n points in E;b/ determined is the intersection between said element and a cell Voroni determined from the point that associates him x, implementing an iterative process that has no current iteration k, it selects an additional point x of the game distinct from x points and the intersection between calculi is half of the space bounded by the hyperplane E mediator of the segment that contains x, and resulting intersection calculi in no previous iteration, the selection of X is function of a comparison between x and x and twice the maximum distance between a bridge and a result of intersection when iterative processing.