Cavity operators

Remesher implementation: the cavity operator

Given a mesh entity $e$ (a vertex or an edge in the present case), the cavity $C(e)$ is the set of mesh elements that contain the entity $e$.

A cavity $C(e)$ can be filled from a vertex $v$ as the set of elements created from $\mathbf{v}$ and the faces of the boundary of the cavity $\partial C(e)$ (oriented outwards). $$\mathcal{F}(\mathbf{v},C(e))=\{K=(\mathbf{v},{\mathbf{g}}_1,\cdots ,{\mathbf{g}}_d)|g=({\mathbf{g}}_1,\cdots ,{\mathbf{g}}_d)\in \partial C(e),\mathbf{v}\notin g\}$$

Edge collapse using cavity operators

The collapse of edge $e_{i,j}=({\mathbf{x}}_i,{\mathbf{x}}_j)$ can be performed as follow

• check the tags
  • if $dim(e_{i,j})>dim({\mathbf{x}}_i)$ and $dim(e_{i,j})>dim({\mathbf{x}}_j)$, do not collapse
  • if $dim(e_{i,j})>dim({\mathbf{x}}_i)$ and $dim(e_{i,j})=dim({\mathbf{x}}_j)$, swap indices $i$ and $j$
• build the vertex cavity $C({\mathbf{x}}_i)$
• fill the vertex cavity from ${\mathbf{x}}_j$: $F=\mathcal{F}({\mathbf{x}}_j,C({\mathbf{x}}_i))$
• if $F$ passes all the criteria for swaps
  • remove all the elements from $C({\mathbf{x}}_i)$ from the mesh
  • insert all the elements from $F$ to the mesh

What to do with the metric ${\mathcal{M}}_i$?

Edge split using cavity operators

The collapse of edge $e_{i,j}$ can be performed as follow

• build the edge cavity $C(e_{i,j})$
• create the midpoint $\mathbf{x}=({\mathbf{x}}_i+{\mathbf{x}}_j)/2$, project it onto the geometry if needed, and compute the metric $\mathcal{M}=\exp ((\log ({\mathcal{M}}_i)+\log ({\mathcal{M}}_j))/2)$
• fill the vertex cavity from $\mathbf{x}$: $F=\mathcal{F}(\mathbf{x},C({\mathbf{e}}_{i,j}))$
• if $F$ passes all the criteria for swaps
  • remove all the elements from $C({\mathbf{e}}_{i,j})$ from the mesh
  • insert verted $\mathbf{x}$ (with metric $\mathcal{M}$) to the mesh
  • insert all the elements from $F$ to the mesh

Edge swap using cavity operators

The swap of edge $e_{i,j}$ can be performed as follow

• build the edge cavity $C(e_{i,j})$
• compute $q_{min}={\min }_{K\in C(e_{i,j})}q(K)$
• if $q_{min}$ is too low, loop over all the vertices $\mathbf{x}$ on the boundary of C (that are neither ${\mathbf{x}}_i$ nor ${\mathbf{x}}_j$)
  • fill the vertex cavity from $\mathbf{x}$: $F=\mathcal{F}(\mathbf{x},C({\mathbf{e}}_{i,j}))$
  • if $F$ passes all the criteria, it is a valid candidate
• if valid candidates are available, let $F_0$ be the best one (the one with for which the minimum element quality is maximum)
  • remove all the elements from $C({\mathbf{e}}_{i,j})$ from the mesh
  • insert all the elements from $F_0$ to the mesh

Smoothing using cavity operators

The smoothnig of vertex ${\mathbf{x}}_i$ can be performed as follow

• build the vertex cavity $C({\mathbf{x}}_i)$
• compute $q_{min}={\min }_{K\in C({\mathbf{x}}_i)}q(K)$
• compute the subset of the vertices in $C({\mathbf{x}}_i)$ that are tagged in the same entity as ${\mathbf{x}}_i$ or one of its children, and average the positions of these vertices to obtain the new position $\mathbf{x}$
• if ${\mathbf{x}}_i$ is on a boundary, project $\mathbf{x}$ on the geometry
• fill the vertex cavity from $\mathbf{x}$: $F=\mathcal{F}(\mathbf{x},C({\mathbf{x}}_i))$
• if ${\min }_{K\in F}q(K)<q_{min}$:
  • locate the element of $C({\mathbf{x}}_i)$ that contains $\mathbf{x}$ (or is the closest), and interpolate the metric in $\mathbf{x}$ using the barycentric coordinates
  • update the location and metric of vertex $i$