A noniterative method for robustly computing the intersections between a line and a curve or surface
Abstract
The need to compute the intersections between a line and a highorder 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 (SVD) and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in oneshot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used NewtonRaphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree $q$ of the shape functions and is $2q \times 3q$ for curves and $6 q^2 \times 8 q^2$ for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used highorder finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitroproject/.
 Publication:

International Journal for Numerical Methods in Engineering
 Pub Date:
 October 2019
 DOI:
 10.1002/nme.6136
 arXiv:
 arXiv:1902.01814
 Bibcode:
 2019IJNME.120..382X
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Computational Geometry;
 Mathematics  Algebraic Geometry
 EPrint:
 doi:10.1002/nme.6136