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Suppose I have a mesh consisting of a set $M$ of conformal elements that fill the region $R=[0,1]\times[0,1]$. Suppose that I also have a 2D shape $S\subset R$ whose boundary $\partial S$ is piecewise linear, but which is not necessarily convex nor necessarily simply connected. Without making any assumptions on the particular shape of the elements in the mesh, how can I efficiently determine the smallest set of elements whose union contains all of the shape $S$?

Though my question is posed in 2D, I'm interested in an approach that would extend to 3D as well.

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  • $\begingroup$ Can't you, in a first step, find all of the cells that are intersected by the piecewise linear boundary $\partial S$? If your cells are simplices, then this intersection should be relatively straightforward to find by just looping over the line segments that make up $\partial S$. Once you have the cells that are intersected, you will need to find all cells that are either on the inside or outside of a set of connected intersected cells. $\endgroup$ – Wolfgang Bangerth Jan 17 '17 at 1:03
  • $\begingroup$ What makes you think that the smallest set is the best set? $\endgroup$ – Philip Roe Nov 13 '17 at 17:40
  • $\begingroup$ What is the meaning of "conformal elements"? $\endgroup$ – Futurologist Jun 12 '18 at 17:14
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In case you do not want to take a simple convex hull (results in overestimations and not composed only of elements), here is an algorithm for, maybe, solving a specialized version of your problem:

  1. Find all the simply connected components (using e.g. union-find algorithm).
  2. For each component, find the alpha shape.

Naturally extends to 3D. Haven't proved if it's the smallest though.

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Let's first also assume you have some way to validate that a point is within the shape $S$. Given you have that implemented, one could use the vertices of some given element to check if it is within the shape $S$.

The next step would be to find an element that does lie within $S$, which could be done by finding an element that contains/shares one of the vertices used to create the piecewise shape $S$. This could be efficiently done using something like a KD Tree, Spatial Hashtable, or similar data structure.

Once you have found a single element that is within the shape $S$, you can then use a Breadth/Depth First Search based on neighboring elements and their vertices to find the elements within $S$.

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    $\begingroup$ I don't think this will work because the fraction of $S$ that may lie within a cell may be very small and, in particular, not contain any vertex or other special point. I suspect that querying individual points will not work, but that you need to start with queries that test intersection of a cell and $S$. Because the boundary of $S$ is composed of line segments, that should be feasible. $\endgroup$ – Wolfgang Bangerth Jan 17 '17 at 16:36
  • $\begingroup$ @WolfgangBangerth That is certainly an edge case that is feasible. Provided $S$ is sufficiently large, I am sure the above approach could capture majority of the cells, but there could be cells near the boundary of $S$ that fail due to the condition you describe. Perhaps one could use the intersection test you describe for the elements along the boundary of $S$ and then use a Breadth/Depth First traversal for the interior based on an neighboring elements' centroid or vertex? $\endgroup$ – spektr Jan 17 '17 at 20:09

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