  Level set-based immersed boundary techniques operate on nonconforming meshes
while providing a crisp definition of interface and external boundaries. In
such techniques, an isocontour of a level set field interpolated from nodal
level set values defines a problem's geometry. If the interface is explicitly
tracked, the intersected elements are typically divided into sub-elements to
which a phase needs to be assigned. Due to loss of information in the
discretization of the level set field, certain geometrical configurations allow
for ambiguous phase assignment of sub-elements, and thus ambiguous definition
of the interface. The study presented here focuses on analyzing these
topological ambiguities in embedded geometries constructed from discretized
level set fields on hexahedral meshes. The analysis is performed on
three-dimensional problems where several intersection configurations can
significantly affect the problem's topology. This is in contrast to
two-dimensional problems where ambiguous topological features exist only in one
intersection configuration and identifying and resolving them is
straightforward. A set of rules that resolve these ambiguities for two-phase
problems is proposed, and algorithms for their implementations are provided.
The influence of these rules on the evolution of the geometry in the
optimization process is investigated with linear elastic topology optimization
problems. These problems are solved by an explicit level set topology
optimization framework that uses the extended finite element method to predict
physical responses. This study shows that the choice of a rule to resolve
topological features can result in drastically different final geometries.
However, for the problems studied in this paper, the performances of the
optimized design do not differ.
