Obtuse angle in triangulation

Alexxxx · May 21, 2023, 2:53pm

Hi!

I’m solving 2D exterior problem. And in my triangulation there are some triangles with angles>pi/2. How can I generate the triangulation only with acute angles. Below is my code on C++.

Thank you.
Ng_Meshing_Parameters mp;
mp.maxh = 0.5;
mp.fineness = 1;
mp.second_order = 0;
Ng_Geometry_2D * geom;
Ng_Init();
geom = Ng_LoadGeometry_2D(filename.c_str());
Ng_GenerateMesh_2D (geom, &m_tria_mesh.m_pNg_mesh, &mp);
Ng_SaveMesh(m_tria_mesh.m_pNg_mesh, volFileName);

joachim · May 21, 2023, 3:24pm

Hi Alex,

there is no guarantee that interior angles are acute. For people working with finite elements it is usually not important.

If you need it: You can hack into the mesh optimizer, and penalize non-acute triangles. Starting point is CalcTriangleBadness (and its gradient below):

github.com

NGSolve/netgen/blob/c8992733979107b5831b4881ac06a4e15a21f19a/libsrc/meshing/smoothing2.cpp#L13


      
          #include "meshing.hpp"
          #include <opti.hpp>
          
          
namespace netgen
          {
          
          
  static const double c_trig = 0.14433756;      // sqrt(3.0) / 12
            static const double c_trig4 = 0.57735026;     // sqrt(3.0) / 3
          
          

          
  inline double CalcTriangleBadness (double x2, double x3, double y3, 
          				     double metricweight, double h)
            {
              // badness = sqrt(3.0) / 12 * (\sum l_i^2) / area - 1 
              // p1 = (0, 0), p2 = (x2, 0), p3 = (x3, y3);
          
          
    double cir_2 = (x2*x2 + x3*x3 + y3*y3 - x2*x3);
              double area = x2 * y3;
              
              if (area <= 1e-24 * cir_2)
                return 1e10;

Joachim

Alexxxx · May 21, 2023, 5:51pm

thanks a lot. I will try penalize non-acute triangles.

I’m a beginner in finite elements. The term Laplace(U) transferes to integration of normal derivative of U by the triangle curve. I use Dirichlet Cell as a poligon of circlepoints(of the inscribed circle) of triangles around. If triangulation consists of non-acute triangles, the poligon is not convex and i dont know how what to do with it. May be some other algorithms for calculating Laplace(U) on triangels.

Thank you