Semi-implicit representation of sharp features with level sets

Abstract

The present contribution enriches the nowadays "classical" level set implicit representation of geometries with topological information in order to correctly represent sharp features. For this, sharp features are classified according to their positions within elements of the level set support. Based on this additional information, sub-elements and interface-mesh used in a finite element context for integration and application of boundary conditions are modified to match exactly to the sharp features. In order to analyze evolving geometries, Boolean operations on these semi-implicit representations are derived so that the minimal additional information to represent correctly the new geometry is stored. This approach has been successfully applied to complex two-dimensional geometries. It computes in a robust way numerous Boolean operations and guarantees the precision and the convergence rate of the numerical simulations.

Quantifying discretization errors for soft tissue simulation in computer assisted surgery: A preliminary study

“…To circumvent or simplify the mesh generation issue, implicit/immersed boundary approaches have been proposed, where the mesh does not conform to the geometry, which is treated by implicit functions such as level sets and enriched finite element methods. This idea was proposed in [81] and later generalized in [84,85], [58,27], [75,66] and [109,126,46], in combination with (goal-oriented) error estimates. Although promising, applications of such approaches to patient-specific geometries remain in their infancy, see a review of related methods in [19].…”

Combination of topology optimization and Lie derivative-based shape optimization for electro-mechanical design

“…Furthermore, the topological complexity of classical CAD based shape optimization is reduced since geometric entities, such as holes, can be merged or removed without degenerating the model, but unfortunately cannot be created, except if the method is combined with a topological derivative (see for instance [15,16,17,18,19] or [20] for more theoretical aspects). However, the use of a level set representation of the geometry makes it difficult to take into account geometries with sharp angles (see for instance [21] or [22]). It limits hence the range of systems that can be optimized.…”