Algebraic Point Projection for Immersed Boundary Analysis on Low Degree NURBS Curves and Surfaces

Abstract

Point projection is an important geometric need when boundaries described by parametric curves and surfaces are immersed in domains. In problems where an immersed parametric boundary evolves with time as in solidification or fracture analysis, the projection from a point in the domain to the boundary is necessary to determine the interaction of the moving boundary with the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge, we present in this paper an algebraic procedure to project a point on to Non-uniform rational B-spline (NURBS) curves and surfaces. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bézier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain are then used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G 0 continuity—problems where the Newton–Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively.

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

“…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.…”