A noniterative method for robustly computing the intersections between a line and a curve or surface

@article{Xiao2019ANM,
  title={A noniterative method for robustly computing the intersections between a line and a curve or surface},
  author={Xiao Xiao and Laurent Bus{\'e} and Fehmi Cirak},
  journal={International Journal for Numerical Methods in Engineering},
  year={2019},
  volume={120},
  pages={382 - 390}
}
The need to compute the intersections between a line and a high‐order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a noniterative method for computing intersections by solving a matrix singular value decomposition and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one‐shot using only standard linear algebra techniques… 
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4 Citations

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AlgebRe gèOmétrie Modelisation et AlgoriTHmes

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POEMA aims to train scientists at the interplay of algebra, geometry and computer science for polynomial optimization problems and to foster scientific and technological advances, stimulating interdisciplinary and intersectoriality knowledge exchange between algebraists, geometers, computer scientists and industrial actors facing real-life optimization problems.

24 References

Implicit matrix representations of rational Bézier curves and surfaces

Implicit matrix representations of rational Bézier curves and surfaces

Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures

Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement

Matrix-based implicit representations of rational algebraic curves and applications

Implicitization using moving curves and surfaces

This paper presents a radically new approach to the century old problem of computing the implicit equation of a parametric surface using a determinant which is one fourth the size of the conventional expression based on Dixon's resultant.

Shape Interrogation for Computer Aided Design and Manufacturing

This book provides the mathematical fundamentals as well as algorithms for various shape interrogation methods including nonlinear polynomial solvers, intersection problems, differential geometry of intersection curves, distance functions, curve and surface interrogation, umbilics and lines of curvature, geodesics, and offset curves and surfaces.

The generation of arbitrary order curved meshes for 3D finite element analysis

A procedure for generating curved meshes, suitable for high-order finite element analysis, is described. The strategy adopted is based upon curving a generated initial mesh with planar edges and…

Computational Contact Mechanics

The mathematical structure of the contact formulation for finite element methods is derived on the basis of a continuum description of contact, and several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed.

Curvilinear mesh generation using a variational framework