I used the following code to test the resulting good mesh (the final 
picture above, which I used to think was good, but in fact not), the result 
is :"number of boundary_id = 0 faces: 24". So in the triangulation, there 
still exists 24 faces whose boundary_id is 0. 
I think when refine the triangulation in dealii, the newly created interior 
faces will be associated with numbers::internal_face_boundary_id.  What 
causes this?   


int i = 0;
   typename Triangulation<2>::active_cell_iterator 
            cell1 = triangulation.begin_active(),
            endc1 = triangulation.end();
   for(; cell1!=endc1; cell1++)
      for(unsigned int face = 0; face<GeometryInfo<2>::faces_per_cell; 
face++){
         if(cell1->face(face)->boundary_id() == 0)
            i ++;
      }
   std::cout<<"number of boundary_id = 0 faces: "<<i<<std::endl;


在 2018年4月30日星期一 UTC+8下午4:54:56,zeng写道:
>
> this phenomenon also happens if the line "if(cell->face(face)->boundary_id() 
> == 0)" is replaced with "if(cell->at_boundary() == true)". 
>

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