3D alpha shape test

Example of isolating tetrahedron elements within the boundary of a cube ant its inscribed sphere. The tetrahedra element edge length is around 0.5 in general. It is indicated that the higher resolution of boundary recognition is provided by alpha=0.5, close to the element edge length.

Three types of elements with alpha=0.5

alpha=0.5

alpha=1

alpha=2

2D Alpha shape test

An example of  implementing alpha shape method in boundary recognition, using Mathematica.

Random points

alpha=3

alpha=2

alpha=1

alpha=0.6

alpha=0.4

alpha=0.3

Boundary recognition

Based on the idea of alpha shape, one can devise a boundary recognition scheme by eliminating the elements whose   circumsphere contains no nodes which are within the boundary. This means, first one must have those nodes identified, then those nodes and the coefficient (defines the extended version of circumsphere) are used for boundary recognition. Here is a example of identifying the boundary of those nodes lying inside a cube but outside its inscribed sphere.

The first four plots show original meshed cube with inside view, the next six plots are when elements inside the sphere are eliminated. The rest is what we have when we change , i.e. the coefficient determining whether one tetrahedron should be eliminated because its extended circumsphere, whose radius depends on , contains nodes within the boundary. Here is reduces, resulting in more elements erased from the picture.

Alpha shape

Alpha shape is a generalized way determining the geometric objects by a set of points, of which convex hull is a special case. It can be applied to FEM for boundary recognition. Essentially this method identifies the boundary of a set of points by eliminating those triangles (generated by triangulation of the points) which can be put into empty circles whose diagram are defined by threshold alpha. It is intuitive for a more simple criteria: eliminating the triangles whose circumcircles contains no points in the defined set. Below is a simple demo by Mathematica which tries to identify the boundary of a subset of points within circle radius 5 from a set points randomly distributed. The red arc isolates the point set   inside, the first step is to eliminate the triangles whose circumcirlces contains no points in , the next step is to eliminate the triangles contain nodes other than those in . More delicate results can be reached by tuning alpha circle instead of using simple circumcirlces.

TetGen+Triangle=Meshpy

TetGen is a well-tested mesh generator for 3D tetrahedron meshing. It can generate conforming Delaunay tetrahedra and apply volume control. Meshpy combine tetgen with another well known 2D triangulation code Triangle into one package, which is very useful.

A cube with Quality mesh and additional points added generated by TetGen