A problem about the function near()

General
1

Hello, everyone;
I am trying to write code for Circular cylinder. Thear are two ways to set boundary conditions.
This is my code:

from fenics import *
import numpy as np
import matplotlib.pyplot as plt
from mshr import *

#create mesh
domain1 = Rectangle(Point(0.0, 0.0), Point(10.0, 1.0))
domain2 = Circle(Point(0.5, 0.5), 0.25)
mesh = generate_mesh(domain1 - domain2, 100)
x = SpatialCoordinate(mesh)
#plot(mesh)

#define boundary
class Right(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and near(x[0], 10.0)

class Left(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and near(x[0], 0.0)

class Bottom(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and near(x[1], 0.0)
    
class Top(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and near(x[1], 1.0)

class Internal_Boundary(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary and near((x[0]-0.5)**2 + (x[1]-0.5)**2-0.25**2, 0)
        #return on_boundary and x[0]>0.2 and x[0]<0.8 and x[1]>0.2 and x[1] <0.8
bottom = Bottom()
top = Top()
left = Left()
right = Right()
internal_boundary = Internal_Boundary()
#boundary marks
bmf = MeshFunction("size_t", mesh, mesh.topology().dim()-1, 0)
bmf.set_all(0)
bottom.mark(bmf, 1)
right.mark(bmf, 2)
top.mark(bmf, 3)
left.mark(bmf, 4)
internal_boundary.mark(bmf, 5)
ds = Measure("ds", domain= mesh, subdomain_data= bmf)

print(assemble(1*ds(5)))

In this code,
I want to use

return on_boundary and near((x[0]-0.5)**2 + (x[1]-0.5)**2-0.25**2, 0)

or

return on_boundary and x[0]>0.2 and x[0]<0.8 and x[1]>0.2 and x[1] <0.8

to define the internal cricle boundary, but why the first seem to be true (Boundary Integral is pi/2) and the second way seem to be false(Boundary Integral is 0.).
Thanks!

2

This is because your mesh is very coarse, and thus the accuracy of the circle is reduced. Increasing the tolerance in near solves the problem:

from fenics import *
import numpy as np
import matplotlib.pyplot as plt
from mshr import *

#create mesh
domain1 = Rectangle(Point(0.0, 0.0), Point(10.0, 1.0))
domain2 = Circle(Point(0.5, 0.5), 0.25)
mesh = generate_mesh(domain1 - domain2, 100)

class Internal_Boundary(SubDomain):
    def inside(self, x, on_boundary):
        return near((x[0]-0.5)**2 + (x[1]-0.5)**2, 0.25**2, 1e-2) and on_boundary

internal_boundary = Internal_Boundary()
#boundary marks
bmf = MeshFunction("size_t", mesh, mesh.topology().dim()-1)

internal_boundary.mark(bmf, 5)
ds = Measure("ds", domain= mesh, subdomain_data= bmf)

print(assemble(1*ds(5)))
File("mesh.pvd")<< bmf