4.3.5. Splines, adding holes and locally controlling the mesh resolution
Actually, the previous mesh is not a fish - it does not have an eye! One could excuse this by arguing that it is a flatfish, viewed from the bottom, so that the migrated eye is not visible. However, then the mouth is wrong…
So, we must add an eye and we will introduce spline boundaries and how to control the local mesh size. To that end, only slight modifications are necessary. In the define_geometry(), we now add the eye points and boundary lines:
eye_size=0.125*S
eye_center_x,eye_center_y=-S*0.25,S*0.3
eye_resolution=self.default_resolution*0.2 # Fine mesh near the eye
p_eye_center=self.point(eye_center_x,eye_center_y,size=eye_resolution)
p_eye_north=self.point(eye_center_x,eye_center_y+eye_size,size=eye_resolution)
p_eye_west=self.point(eye_center_x-eye_size,eye_center_y,size=eye_resolution)
p_eye_east=self.point(eye_center_x+eye_size,eye_center_y,size=eye_resolution)
p_eye_south=self.point(eye_center_x,eye_center_y-0.25*eye_size,size=eye_resolution)
self.circle_arc(p_eye_west,p_eye_north,center=p_eye_center,name="eye")
self.circle_arc(p_eye_north,p_eye_east,center=p_eye_center,name="eye")
self.spline([p_eye_west,p_eye_south,p_eye_east],name="eye")
With the keyword argument size in the point() calls, the local mesh size (resolution) can be set. If it is not passed, it will default to default_resolution of the GmshTemplate. The upper part of the eye are just circular segments. We have split it into two segments, since the total angle of the circular arc is 180\(\:\mathrm{^\circ}\). With a single py:meth:~pyoomph.meshes.gmsh.GmshTemplate.circle_arc from p_eye_west to p_eye_east, it would be ambiguous, whether the arc should cover the north or the south direction. For that reason, any py:meth:~pyoomph.meshes.gmsh.GmshTemplate.circle_arc with an opening angle \(\geq180\:\mathrm{^\circ}\) must be slit into multiple segments. The bottom part of the eye is a spline(), which takes a list of points it has to pass.
We have to tell gmsh that the eye should be removed from the fish. It is actually a hole in the mesh, so we have to pass the keyword holes to the plane_surface():
self.plane_surface("mouth","fin","curved",name=self.domain_name,holes=[["eye"]])
holes must take a list holes, which again are each lists of boundary lines or boundary line names.
Fig. 4.16 Using a spline and local mesh size control, we add a hole to the fish mesh to create its eye.
In the problem class, a NeumannBC is added to the "eye" boundary
eqs += NeumannBC(u=1*meter) @ "eye"
so that the result looks as depicted in Fig. 4.16. Since our PoissonEquation has a coefficient in meter**2, the NeumannBC boundary condition has to be in units of meter.