Cartesian Mesh Generation Using Exact Arithmetic Tie-Breaking
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Solution Overview
Problem
Current CAD-based Cartesian mesh generation techniques require massively parallel computing resources to achieve high accuracy and robustness for large numbers of cells, especially when modeling electromagnetic fields and complex geometries, and are limited by memory efficiency and ray-facet intersection tests.
Innovation Solution
A highly efficient data storage ray tracing method combined with a highly accurate ray-facet intersection test and a novel exact arithmetic tie-breaking algorithm for rays intersecting facet edges and vertices, allowing for the generation of over a trillion cells on a single processor workstation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If massively parallel computing resources are used for CAD-based Cartesian mesh generation, then manufacturing precision and reliability are improved, but device complexity and cost increase
Solution Approach 1:
The patent extracts and optimizes the critical computational components (ray-facet intersection tests and tie-breaking algorithms) from the overall mesh generation system. By focusing computational effort on these specific algorithms rather than using brute-force parallel processing across the entire system, high precision is achieved with reduced computational complexity.
Solution Approach 2:
The patent changes the computational parameters by using exact arithmetic for tie-breaking decisions and optimized intersection test parameters. This allows the system to achieve high mesh generation accuracy through parameter optimization rather than through increased parallel computing power.
2Measurement precision
If the number of cells is increased to achieve high resolution, then measurement precision is improved, but memory usage increases
Solution Approach 1:
The patent performs preliminary action by pre-calculating and storing only the essential ray-facet intersection data and surface position information before mesh generation. This preliminary preparation allows the system to handle trillion-cell meshes by having all necessary geometric data ready in an optimized format, reducing the memory burden during the actual mesh generation process.
Solution Approach 2:
The patent applies local quality by storing different types of data with different levels of detail and precision where needed. Critical intersection points and surface positions are stored with high precision, while less critical data uses reduced precision storage. This localized optimization of data quality allows high resolution meshes with efficient memory usage.
3Manufacturing precision
If exact arithmetic tie-breaking algorithms are implemented for rays intersecting facet edges and vertices, then manufacturing precision is improved, but computational time increases
Solution Approach 1:
The patent applies partial action by using exact arithmetic tie-breaking only for the critical cases of ray-facet intersections at edges and vertices, rather than applying it universally to all intersections. This selective application of exact arithmetic provides the necessary precision for accurate mesh generation while avoiding the excessive computational time that would result from applying exact arithmetic to every intersection test.
Data Source
AI summary
A highly accurate and robust cubic cell mesh generator “Cubegen” capable of a trillion plus cell meshes on a single processor 4-Gigabyte main memory workstation has been developed. The cells are generated in Yee format for the Finite Difference Time Domain method. Three key techniques were employed to achieve this capability: a highly efficient data storage ray tracing method, a highly accurate ray-facet intersection test, and a novel exact arithmetic tie-breaking algorithm for rays intersecting facet edges and vertices.