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.

通常使用相场或能级组方法对具有新兴拓扑的复杂物理问题（例如，凝固或电迁移过程中的相成核和生长）以及设计问题（例如拓扑优化）进行建模。但是，这些方法中不断演变的界面的隐式表示隐含了指示相演化的几何参数（例如法线和曲率），并且仅在网格细化的范围内恢复了精确的界面几何形状。最近，开发了一种被称为“等距几何分析”（EIGA）的显式接口方法，以对具有演化阶段的物理问题进行建模。此方法使用从Dixon结果生成的带符号代数能级集来捕获界面对其邻域的影响。带符号的代数级集允许使用R函数通过代数布尔运算对拓扑变化进行建模。但是，从参数曲面生成代数能级集的一个挑战是Dixon结果无法隐含或缺少常见参数曲面（例如球体或圆柱体）的符号。缺少符号可防止使用R函数捕获拓扑变化的隐式布尔运算。在本文中，采用最大秩子矩阵方法来恢复具有奇异Dixon结果的参数面的隐式化。此外，开发了一种多元多项式平方根方法来从无符号结果中恢复符号。然后证明了生成的带符号代数能级集可以解决具有多个任意形状的空隙的固体中的三维热传导。