NC mesh with hanging element?

[dongzhangim]

hello, all
i'm new to mfem. I wanna use MFEM for the simulation of elastic deformation of a nonconforming mesh with hanging element. Hanging element means when a quad or hex element is refined, some subdivided elements are deleted leaving the remaining ones(one) hanging. That's my understanding.

First, I tested two elements with one refined (see Figure A). The results (generated by ex21) are shown in Figure B.

The mesh files are as follows.

MFEM NC mesh v1.0

dimension
2

elements
6
0 1 3 0 0 1 4 3
0 2 3 3 2 3 4 5
0 2 3 0 1 8 10 7
0 2 3 0 8 2 9 10
0 2 3 0 7 10 6 4
0 2 3 0 10 9 5 6

boundary
9
3 1 0 1
3 1 1 8
3 1 2 8
3 1 3 4
3 1 4 6
3 1 5 6
1 1 0 3
2 1 2 8
2 1 5 9

vertex_parents
5
6 4 5
7 1 4
8 1 2
9 2 5
10 6 8

coordinates
6
2
0 0
1 0
2 0
0 1
1 1
2 1

mfem_mesh_end

Second, I try to delete the element with id 5 by removing its id from the element of id 1 as depicted in Figure C, however, it does not work with errors:

Verification failed: (geom == Geometry::SEGMENT || geom == Geometry::TRIANGLE || geom == Geometry::SQUARE || geom == Geometry::CUBE || geom == Geometry::PRISM || geom == Geometry::PYRAMID || geom == Geometry::TETRAHEDRON) is false:
--> Element type 0 is not supported by NCMesh.
... in function: void mfem::CheckSupportedGeom(Geometry::Type)
... in file: mesh/ncmesh.cpp:128

The NC mesh files are as follows:

MFEM NC mesh v1.0
dimension
2

elements
5
0 1 3 0 0 1 4 3
0 2 3 3 2 3 4
0 2 3 0 1 8 10 7
0 2 3 0 8 2 9 10
0 2 3 0 7 10 6 4

boundary
9
3 1 0 1
3 1 1 8
3 1 2 8
3 1 3 4
3 1 4 6
3 1 10 6
1 1 0 3
2 1 2 8
2 1 10 9

vertex_parents
5
6 4 5
7 1 4
8 1 2
9 2 5
10 6 8

coordinates
6
2
0 0
1 0
2 0
0 1
1 1
2 1

mfem_mesh_end

So, how can we tackle nc mesh with hanging elements?

[v-dobrev]

One problem that I see is: vertex id 5 is no longer needed, so you can remove it from coordinates and also from vertex_parents. Also, some of the vertices that you previously described in the vertex_parents section have to be added as regular vertices due to the removal of vertex with id 5.

[dongzhangim]

Dear Veselin,
Thanks for your help. After deleting the hanging node with an ID of 5 and making two nodes with coordinates of (1.5, 1) and (2, 0.5) regular, it works as shown below, although the refinement hierarchy cannot be stored.

MFEM NC mesh v1.0
dimension
2

elements
4
0 1 3 0 0 1 4 3
0 2 3 0 1 8 9 7
0 2 3 0 8 2 6 9
0 2 3 0 7 9 5 4

boundary
9
3 1 0 1
3 1 1 8
3 1 2 8
3 1 3 4
3 1 4 5
3 1 5 9
3 1 6 9
1 1 0 3
2 1 2 6

vertex_parents
3
7 1 4
8 1 2
9 5 8

coordinates
7
2
0 0
1 0
2 0
0 1
1 1
1.5 1
2 0.5

mfem_mesh_end

Also, we can make the single node with the coordinate of (1 0.5) nonregular, it works as well.

MFEM NC mesh v1.0
dimension
2

elements
4
0 1 3 0 0 1 5 4
0 2 3 0 1 2 7 9
0 2 3 0 2 3 8 7
0 2 3 0 9 7 6 5

boundary
9
3 1 0 1
3 1 1 2
3 1 2 3
3 1 4 5
3 1 5 6
3 1 6 7
3 1 7 8
1 1 0 4
2 1 3 8

vertex_parents
1
9 1 5

coordinates
9
2
0 0
1 0
1.5 0
2 0
0 1
1 1
1.5 1
1.5 0.5
2 0.5

mfem_mesh_end

[dongzhangim]

is it possible to add a hanging point onto another hanging point? as follows. // m is a hanging node of p1 and p2 // q is a hanging node of p1 and m

I've tried the following codes, it gives error when running NCMesh::NCMesh(const Mesh *mesh)

#include "mfem.hpp"
#include <fstream>

using namespace mfem;

int main()
{
   Mesh mesh(2, 0, 0);

   int p0 = mesh.AddVertex(0, 0);
   int p1 = mesh.AddVertex(1, 0);
   int p2 = mesh.AddVertex(1, 1);
   int p3 = mesh.AddVertex(0, 1);

   int m  = mesh.AddVertex(1, 0.5);
   int q = mesh.AddVertex(1, 0.25);

   int r0 = mesh.AddVertex(1.25, 0);
   int r1 = mesh.AddVertex(1.25, 0.25);


   // the left is a 1*1 element
   mesh.AddQuad(p0, p1, p2, p3);

   //  the right is a 1/4*1/4 element
   mesh.AddQuad(p1, r0, r1, q);

   // m is a hanging node of p1 and p2 
   mesh.AddVertexParents(m,  p1, p2); // 1/2
   // q is a hanging node of p1 and m 
   mesh.AddVertexParents(q, p1, m);  // 1/4

   mesh.FinalizeMesh(0);              // 0：no refinement
   mesh.Save("quarter.mesh");

   return 0;
}

[dylan-copeland]

For nonuniform spacing in an NCMesh, see miniapps/meshing/ref321.cpp introduced by PR 4371, e.g. Refinement(elem, type, 0.25) for 1/4 spacing. The easiest way to generate an NCMesh like this is probably to start with a coarse conforming mesh and add refinements of elements as in ref321.cpp.

[v-dobrev]

@dylan-copeland, what if the whole mesh is just the two elements from the illustration? Do we have a method AddVertexParents where we can specify the 1/4 spacing directly, modifying the sample code above?

[dylan-copeland]

We have protected methods in NCMesh to set spacing, but the public interface only supports refinement as in the miniapp ref321. Element refinement is the intended way to produce an NCMesh.

[dongzhangim]

what we want is to generate a NCmesh from an octree or b+tree voxels. Therefore, the AddVertexParents function which can specify the 1/4 spacing directly will be more interesting. Otherwise, i have to conduct MFEM refinement during the bool operation for the octree or b+ tree. Also during the refinement, some voxel may be deleted. I'm not sure whether the some child element can be deleted during the refinement.

[dylan-copeland]

It may be possible to do this with refinement and derefinement, although not optimal. If new capabilities would be more convenient, you can make a PR with the necessary code changes.