U-splines: splines over unstructured meshes. (English) Zbl 1507.65059
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 quadrilateral cells in the same mesh) and T-junctions are also supported, although the continuity of interfaces with triangle and tetrahedral cells is limited in the present work. The U-spline algorithm introduces a new technique for using local null space solutions to construct basis functions for the global spline null space problem. 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.
MSC:
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 65D07 | Numerical computation using splines |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
Keywords:
unstructured splines; unstructured mesh; computer-aided geometric design (CAGD); isogeometric analysis (IGA); computer-aided engineering (CAE); finite element analysis (FEA)References:
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