U-splines for Unstructured Meshes in CAD and FEA
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Solution Overview
Problem
Current finite element analysis (FEA) techniques face challenges in smoothly and accurately representing complex CAD models with smooth splines, particularly in industrial-scale applications, due to limitations in meshing technologies and the compatibility with existing spline representations like NURBS and T-splines, which restrict local refinement and higher-dimensional generalization.
Innovation Solution
The U-spline technology constructs smooth spline meshes by assigning barycentric coordinate systems, Bernstein bases, and parametric lengths to cells, and imposing continuity constraints to create a globally smooth and locally refined spline space, allowing for seamless similarity maps and direct construction of spline bases without solving the global nullspace problem.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If smooth splines (NURBS, T-splines) are used to represent complex CAD models in FEA, then the accuracy and smoothness of the representation is improved, but the meshing complexity and computational cost increase significantly
Solution Approach 1:
The patent segments the complex meshing problem into local refinement operations on a base grid. Instead of globally meshing the entire domain with high complexity, the method divides the domain into base cells and applies local refinements only where needed, maintaining smoothness through continuity constraints at interfaces. This segmentation reduces overall meshing complexity while preserving accuracy in critical regions.
Solution Approach 2:
The patent implements local quality by allowing different polynomial degrees and continuity levels in different regions of the domain. The U-spline formulation enables local refinement where high accuracy is needed while using coarser representations elsewhere, optimizing the balance between accuracy and computational cost for complex CAD models.
2Adaptability or versatility
If traditional splines (NURBS, T-splines) are used for local refinement, then the ability to refine specific regions is improved, but the compatibility with existing spline representations and higher-dimensional generalization is limited
Solution Approach 1:
The patent achieves universality by formulating a unified U-spline framework that generalizes both NURBS and T-splines as special cases. The method uses a common mathematical foundation based on unstructured meshes and continuity constraints that works across different dimensions (2D surfaces and 3D volumes) and different spline types, enabling broad compatibility and adaptability without requiring separate handling for each spline representation.
Solution Approach 2:
The patent extends spline methodologies from 2D surfaces to 3D volumetric domains by formulating continuity constraints in arbitrary dimensions. The U-spline framework naturally generalizes to higher dimensions by defining continuity across (d-1)-dimensional interfaces in d-dimensional space, enabling seamless transition from surface to volume modeling.
3Measurement precision
If continuity constraints are imposed to create globally smooth splines, then the smoothness and accuracy of the representation is improved, but the computational cost of solving the global nullspace problem increases
Solution Approach 1:
The patent applies preliminary action by pre-computing the nullspace basis of the continuity constraint matrix and storing it for repeated use. Instead of solving the global nullspace problem from scratch for each spline evaluation, the method performs the computationally expensive decomposition once and reuses the basis, dramatically reducing computational time for subsequent operations while maintaining global smoothness.
4Ease of manufacture
If faceted meshes are used for FEA simulation, then the compatibility with industrial CAD representations is improved, but the smoothness and accuracy of the simulation results deteriorate
Solution Approach 1:
The patent creates a composite representation that combines the geometric flexibility of faceted meshes with the smoothness of spline functions. The U-spline formulation defines continuous piecewise polynomial functions over unstructured meshes, effectively creating a composite structure that maintains compatibility with industrial CAD meshing while imposing smooth continuity constraints to eliminate faceting artifacts and improve simulation accuracy.
Data Source
AI summary
U-splines are a novel approach to the construction of a spline basis for representing smooth objects in Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE). A spline is a piecewise-defined function that satisfies continuity constraints between adjacent cells in a mesh. U-splines differ from existing spline constructions, such as Non-Uniform Rational B-splines (NURBS), subdivision surfaces, T-splines, and hierarchical B-splines, in that they can accommodate local variation in cell size, polynomial degree, and smoothness simultaneously over more varied mesh configurations. Mixed cell types (e.g., triangle and tetrahedron and quadrilateral and hexahedral cells in the same mesh) and T-junctions are also supported. The U-spline construction is presented for curves, surfaces, and volumes with higher dimensional generalizations possible. A set of requirements are given to ensure that the U-spline basis is positive, forms a partition of unity, is complete, and is locally linearly independent.