Improved Dixon Resultant for Generating Signed Algebraic Level Sets and Algebraic Boolean Operations on Closed Parametric Surfaces

Vaitheeswaran, Pavan Kumar; Subbarayan, Ganesh

doi:10.1016/j.cad.2021.103004

Cited by 4 publications

(1 citation statement)

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“…Such iterations are generally non-robust, as Newton-Raphson iterations yield non-unique foot (nearest) points on the parametric surface near regions of large curvature. In order to circumvent these challenges, we have recently proposed techniques founded on algebraic geometry to estimate unsigned and signed distance measures from parametric boundaries, termed algebraic level sets [34,35], which have been further developed for point projection [41] and phase merging [42]. Algebraic level sets are briefly reviewed below prior to describing mesh refinement algorithms based on algebraic level sets.…”

Section: Brief Review Of Algebraic Level Setsmentioning

confidence: 99%

Efficient Local Refinement near Parametric Boundaries Using kd-Tree Data Structure and Algebraic Level Sets

2022

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In analysis of problems with parametric spline boundaries that are immersed or inserted into an underlying domain, the discretization on the underlying domain usually does not conform to the inserted boundaries. While the fixed underlying discretization is of great convenience as the immersed boundaries evolve, the field approximations near the inserted boundaries require refinement in the underlying domain, as do the quadrature cells. In this paper, a kd-tree data structure together with a sign-based and/or distance-based refinement strategy is proposed for local refinement near the inserted boundaries as well as for adaptive quadrature near the boundaries. The developed algorithms construct and utilize implicit forms of parametric Non-Uniform Rational B-Spline (NURBS) surfaces to algebraically (and non-iteratively) estimate distance as well as sign relative to the inserted boundary. The kd-tree local refinement is demonstrated to produce fewer sub-cells for the same accuracy of solution as compared to the classical quad/oct tree-based subdivision. Consistent with the kd-tree data structure, we describe a new a priori refinement algorithm based on the signed and unsigned distance from the inserted boundary. We first demonstrate the local refinement strategy coupled with the the kd-tree data structure by constructing Truncated Hierarchical B-spline (THB-spline) “meshes”. We next demonstrate the accuracy and efficiency of the developed local refinement strategy through adaptive quadrature near NURBS boundaries inserted within volumetric three-dimensional NURBS discretizations.

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