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

Abstract

Complex physical problems with emerging topologies such as phase nucleation and growth during solidification or electromigration, as well as design problems such as topology optimization are commonly modeled using phase field or level set methods. However, the implicit representations of the evolving interfaces in these methods implicitize geometrical parameters such as normals and curvatures that dictate phase evolution, and recover the exact interface geometry only in the limit of mesh refinement. Recently, an explicit interface method termed Enriched Isogeometric Analysis (EIGA) was developed to model physical problems with evolving phases. This method uses signed algebraic level sets generated from the Dixon resultant to capture the influence of the interface on its neighborhood. The signed algebraic level sets allow modeling topological changes through algebraic Boolean operations using R-functions. However, a challenge in generating algebraic level sets from parametric surfaces is that the Dixon resultant fails to implicitize or lacks a sign for common parametric surfaces such as spheres or cylinders. The lack of sign prevents implicit Boolean operations using R-functions to capture topological changes. In this paper, a maximal-rank submatrix approach is used to recover implicitizations of parametric surfaces with trivially singular Dixon resultants. Furthermore, a multivariate polynomial square root method is developed to recover sign from unsigned resultants. The generated signed algebraic level sets are then demonstrated to solve three-dimensional heat conduction in a solid with multiple, arbitrarily-shaped voids. Algebraic Boolean operations using R-functions on these signed level sets are used to naturally capture coalescence of voids without needing surface–surface intersection calculation.