# How to calculate efficiently mesh edges midpoints?

I have a 2D mesh of triangles used in Finite Element method to discretize the domain. I want to calculate the midpoints of all the edges because I want to use $\mathbb{P}^2$ elements. I am using MatLab. The data that I have is:

• Matrix containing nodes coordinates, $x$ and $y$. The size is $N\times 2$, where $N$ is the number of nodes.

• Matrix containing each triangle nodes. The size is $NT\times 3$, where $NT$ is the number of triangles in the mesh and 3 columns because each triangle has three nodes. For example, a row of this matrix could have $[10, 4 ,22]$ so the triangle is formed by these nodes and in order to know their coordinates I just take the rows $10, 4$ and $22$ from the node coordinate matrix (the previous one).

• Array with boundary nodes.

I know in advance that the number of nodes (including midpoints) is going to be $2N\times N_b -3$ where $N_b$ is the number of nodes in the boundary of the mesh, so I can preallocate the matrix. The new node matrix will contain the previous nodes coordinates and the new ones and the element matrix will now have 6 columns because one node will be added per edge.

What my code is doing is to run across all triangles and calculate their edges midpoints and if it is not yet in the coordinate node matrix include it (here I am losing many time checking if it is yet inside or not). The main problem is that as interior edges are common to two triangles many times the midpoint of an edge is already calculated so I am wasting time calculating the midpoint and after it checking that is already in the matrix.

Could you give me any advice about how to calculate more efficiently the midpoints of edges and include them in the matrix. Note: I do not have a matrix with the edges of the mesh.

Thanks!

• As a possible alternate solution. Use gmsh to create your mesh and then just write a small script to import the mesh into matlab. Its not too hard as I have done this myself. This will allow alot more flexibility i think in terms of different kinds of elements and order of elements. Jun 22 '15 at 16:03
• In this case I am using Distmesh but it only returns edges at the boundary. The main advantage of Distmesh is that I can create the mesh automatically from my code. However, Gmsh does not allow it easily. But thanks for the suggestion. Jun 22 '15 at 18:05

• The 'index' of the cell is the number of the row of your $N_T\times 3$ matrix in which you store the indices of the three vertices of the $N_T$ cells. Jun 24 '15 at 11:43
On way to do this is to iterate over the matrix containing the triangle nodes. Each row contains three node indices, and thus three two-element vectors of potential edges. Insert these vectors into your hash table, using the sorted version of the vector as a key. (An example: If my edges are $[1, 3]$, $[3, 2]$, $[2, 1]$, my hash table should look like {[1, 2]: [[2, 1]], [1, 3]: [[1, 3]], [2, 3]:[[3, 2]]}, where I'm using Python dictionary notation to represent the hash table because Python dictionaries are implemented as hash tables in the standard library.) Then reading off the keys of the hash table will give you unique edges, and you can use this information to insert midpoints.