In this dissertation we present a novel method for deformable interface tracking
in 2D and 3D|deformable simplicial complexes (DSC). Deformable interfaces
are used in several applications, such as
fluid simulation, image analysis, reconstruction
or structural optimization.
In the DSC method, the interface (curve in 2D; surface in 3D) is represented
explicitly as a piecewise linear curve or surface. However, the domain is also
subject to discretization: triangulation in 2D; tetrahedralization in 3D. This
way, the interface can be alternatively represented as a set of edges/triangles
separating triangles/tetrahedra marked as outside from those marked as inside.
Such an approach allows for robust topological adaptivity. Among other advantages
of the deformable simplicial complexes there are: space adaptivity,
ability to handle and preserve sharp features, possibility for topology control.
We demonstrate those strengths in several applications.
In particular, a novel, DSC-based fluid dynamics solver has been developed
during the PhD project. A special feature of this solver is that due to the fact
that DSC maintains an explicit interface representation, surface tension is more
easily dealt with.
One particular advantage of DSC is the fact that as an alternative to topology
adaptivity, topology control is also possible. This is exploited in the construction
of cut loci on tori where a front expands from a single point on a torus and stops
when it self-intersects.