def get_circle_boundary(radius, centre, bd_eta_max):
# centre is an object of point data type
I = bd_eta_max.n
r = np.empty(I+1, dtype=object)
# Find point on circles such that line from circle
# to eta_max boundary is normal to circle
# theta = arctan((y - yc) / (x - xc))
points = bd_eta_max.Points
for xi in range(I+1):
point = points[xi]
xp, yp = point.x, point.y
dist = np.sqrt((xp - centre.x)**2 + (yp - centre.y)**2)
costheta = (xp - centre.x) / dist
sintheta = (yp - centre.y) / dist
x = centre.x + radius*costheta
y = centre.y + radius*sintheta
r[xi] = Point(x, y)
# x = np.empty(I+1)
# y = np.empty(I+1)
# for i in range(I+1):
# x[i] = r[i].x
# y[i] = r[i].y
# plt.plot(x, y, 'k.')
# plt.show()
bd = Boundary(I, BdType.eta_min)
bd.populate_boundary_ndar_points(r)
return bd
def get_xi_boundaries(start, end, I):
# For the flow past a cylinder, the xi boundaries are the same line
# The points are stretched closer to the start
delta = 4.5 # trial and error?
stretching_f = lambda xi: 1 + (np.tanh(delta*(xi - 1)) / np.tanh(delta))
r = BoundaryGrids.fit_line(start, end, I, stretching_f)
bd2 = Boundary(I, BdType.xi_min)
bd2.populate_boundary_ndar_points(r)
bd3 = Boundary(I, BdType.xi_max)
bd3.populate_boundary_ndar_points(r)
return (bd2, bd3)
def get_trap_grid(I, J):
# Grid as per ASM2010 AIAA code verification paper
# eta_min
start_point = Point(0., 0.)
end_point = Point(0.3, 0.0)
r = BoundaryGrids.fit_line(start_point, end_point, I)
bd1 = Boundary(I, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
# eta_max
start_point = Point(0., 0.1)
end_point = Point(0.3, 0.005)
r = BoundaryGrids.fit_line(start_point, end_point, I)
bd2 = Boundary(I, BdType.eta_max)
bd2.populate_boundary_ndar_points(r)
# xi_min
start_point = Point(0., 0.)
end_point = Point(0., 0.1)
r = BoundaryGrids.fit_line(start_point, end_point, J)
bd3 = Boundary(J, BdType.xi_min)
bd3.populate_boundary_ndar_points(r)
# xi_max
start_point = Point(0.3, 0.)
end_point = Point(0.3, 0.005)
r = BoundaryGrids.fit_line(start_point, end_point, J)
bd4 = Boundary(J, BdType.xi_max)
bd4.populate_boundary_ndar_points(r)
Grid_obj = StructuredGridGeneration.get_univariate_grid(bd1, bd2, bd3, bd4)
Grid_obj.save_to_vtk('trap.vtk')
Grid_obj.save_to_txt('trap.txt')
def get_eta_max_boundary(p1, p2, p3, p4, I):
# The eta_max boundary is a rectangle with 4 points.
# Specify from top right point anti-clockwise
# We also need to return the indices of the array where
# p = pi, i = 1 to 4
# The actual starting point is midpoint of p1 and p4
start_point = 0.5 * (p1 + p4)
# Totally 5 segments
# L1: start to p1 with I / 4 points
r1 = BoundaryGrids.fit_line(start_point, p1, int(I/4))
# L2: p1 to p2 with 3*I/16 points
r2 = BoundaryGrids.fit_line(p1, p2, int(3*I/16))
# L3: p2 to p3 with I/8 points
r3 = BoundaryGrids.fit_line(p2, p3, int(I/8))
# L4: p3 to p4 with 3*I/16 points
r4 = BoundaryGrids.fit_line(p3, p4, int(3*I/16))
# L5: p4 to start with I/4 points
r5 = BoundaryGrids.fit_line(p4, start_point, int(I/4))
# Combine them (numpy.append)
# Skip p1 common
r = np.append(r1[:-1], r2)
# Skip p2 common
r = np.append(r[:-1], r3)
# Skip p3 common
r = np.append(r[:-1], r4)
# Skip p4 common
r = np.append(r[:-1], r5)
# I is number of number of grid points - 1
I = len(r) - 1
bd = Boundary(I, BdType.eta_max)
bd.populate_boundary_ndar_points(r)
for i in range(len(r)):
if r[i] == p1:
i1 = i
elif r[i] == p2:
i2 = i
elif r[i] == p3:
i3 = i
elif r[i] == p4:
i4 = i
i_list = [i1, i2, i3, i4]
return (bd, i_list)
def get_oblique_shock_grid(delta, I):
# Grid as per ASM2010 AIAA code verification paper
# Ramp at I/2
J = int(I / 2)
x_min = -1
x_corner = 0
y = (x_corner - x_min)
x_end_dist = (x_corner - x_min)
# eta_min
start_point = Point(x_min, 0)
corner_point = Point(x_corner, 0)
end_point = Point(x_end_dist * np.cos(delta), x_end_dist * np.sin(delta))
r1 = BoundaryGrids.fit_line(start_point, corner_point, int(I / 2))
r2 = BoundaryGrids.fit_line(corner_point, end_point, int(I / 2))
r = np.append(r1[:-1], r2)
bd1 = Boundary(I, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
I = len(bd1.Points) - 1
# eta_max
start_point = Point(x_min, y)
corner_point = Point(x_corner, y)
# end_point = Point(x_end_dist*np.cos(delta), x_end_dist*np.sin(delta) + y)
end_point = Point(x_end_dist * np.cos(delta), y)
# r1 = BoundaryGrids.fit_line(start_point, corner_point, int(I/2))
# r2 = BoundaryGrids.fit_line(corner_point, end_point, int(I/2))
# r = np.append(r1[:-1], r2)
r = BoundaryGrids.fit_line(start_point, end_point, I)
bd2 = Boundary(I, BdType.eta_max)
bd2.populate_boundary_ndar_points(r)
# xi_min
start_point = Point(x_min, 0)
end_point = Point(x_min, y)
r = BoundaryGrids.fit_line(start_point, end_point, J)
bd3 = Boundary(J, BdType.xi_min)
bd3.populate_boundary_ndar_points(r)
# xi_max
start_point = Point(x_end_dist * np.cos(delta), x_end_dist * np.sin(delta))
end_point = Point(x_end_dist * np.cos(delta), y)
r = BoundaryGrids.fit_line(start_point, end_point, J)
bd4 = Boundary(J, BdType.xi_max)
bd4.populate_boundary_ndar_points(r)
Grid_obj = StructuredGridGeneration.get_univariate_grid(bd1, bd2, bd3, bd4)
Grid_obj.save_to_vtk('oblique_shock.vtk')
Grid_obj.save_to_txt('oblique_shock.txt')
def get_shock_BL_grid(delta, I):
# Grid as per ASM2010 AIAA code verification paper
# Ramp at I/2
J = int(I / 2)
x_min = 0
x_corner = 0.1
x_mid = 0.2 * x_corner
y = 0.0368
x_end_dist = x_corner - x_min
# eta_max
start_point = Point(x_min, y)
turning_point = Point(x_mid, (x_mid - x_min) * np.tan(delta) + y)
end_point = Point(x_corner, -(x_corner-x_mid)*np.tan(delta) + \
turning_point.y)
r1 = BoundaryGrids.fit_line(start_point, turning_point, int(I * 0.2))
r2 = BoundaryGrids.fit_line(turning_point, end_point, int(I * 0.8))
r = np.append(r1[:-1], r2)
bd2 = Boundary(I, BdType.eta_max)
bd2.populate_boundary_ndar_points(r)
I = len(bd2.Points) - 1
# eta_min
start_point = Point(x_min, 0.)
end_point = Point(x_corner, .0)
r = BoundaryGrids.fit_line(start_point, end_point, I)
bd1 = Boundary(I, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
dt = 3.0
# xi_min
start_point = Point(x_min, 0)
end_point = Point(x_min, y)
stretching_f = lambda xi: 1 + (np.tanh(dt * (xi - 1)) / np.tanh(dt))
r = BoundaryGrids.fit_line(start_point, end_point, J, stretching_f)
bd3 = Boundary(J, BdType.xi_min)
bd3.populate_boundary_ndar_points(r)
# xi_max
start_point = Point(x_corner, 0.)
end_point = Point(x_corner, -(x_corner-x_mid)*np.tan(delta) + \
turning_point.y)
r = BoundaryGrids.fit_line(start_point, end_point, J, stretching_f)
bd4 = Boundary(J, BdType.xi_max)
bd4.populate_boundary_ndar_points(r)
Grid_obj = StructuredGridGeneration.get_univariate_grid(
bd1, bd2, bd3, bd4, stretching_f)
Grid_obj.save_to_vtk('shock_BL.vtk')
Grid_obj.save_to_txt('shock_BL.txt')
def get_airfoil_dat_gridgen():
xt, yt = zip(*np.loadtxt('top'))
xb, yb = zip(*np.loadtxt('bottom'))
CPt = []
CPb = []
for i in range(len(xt)):
CPt.append(Point(xt[i], yt[i]))
for i in range(len(xb)):
CPb.append(Point(xb[i], yb[i]))
I = 200
# xi_spacing_f = lambda xi, I: 0.5 * I * (1 - np.cos(xi * np.pi / I))
xi_spacing_f = lambda xi, I: xi
s_CP_f = lambda j, n, I: j * I / n
s_CP_inverse_f = lambda s, n, I: s * n / I
rt = fit_cubic_spline(CPt, I, xi_spacing_f, s_CP_f, s_CP_inverse_f)
rb = fit_cubic_spline(CPb, I, xi_spacing_f, s_CP_f, s_CP_inverse_f)
bd1 = Boundary(I, BdType.eta_min)
bd1.populate_boundary_ndar_points(rt)
bd2 = Boundary(I, BdType.eta_min)
bd2.populate_boundary_ndar_points(rb)
xtnew = []
ytnew = []
xbnew = []
ybnew = []
for i in range(len(rt)):
xtnew.append(rt[i].x)
ytnew.append(rt[i].y)
for i in range(len(rb)):
xbnew.append(rb[i].x)
ybnew.append(rb[i].y)
np.savetxt('top-new', zip(xtnew, ytnew))
np.savetxt('bottom-new', zip(xbnew, ybnew))
plt.plot(xtnew, ytnew, '*-')
plt.plot(xbnew, ybnew, '.-')
plt.axis('equal')
plt.show()
def get_exp_plane_grid(I, J):
delta = -10 * np.pi / 180
x_min = -1
x_corner = 0
y_max = (x_corner - x_min) * 1.5
x_end_dist = (x_corner - x_min) * 2
x_max = x_end_dist * np.cos(delta)
y_min = x_end_dist * np.sin(delta)
x_min = 0
x_max = 1
y_min = 0
y_max = 0.2
start_point = Point(x_min, y_min)
end_point = Point(x_max, y_min)
r = BoundaryGrids.fit_line(start_point, end_point, int(I))
bd1 = Boundary(len(r) - 1, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
start_point = Point(x_min, y_max)
end_point = Point(x_max, y_max)
r = BoundaryGrids.fit_line(start_point, end_point, int(I))
bd2 = Boundary(len(r) - 1, BdType.eta_max)
bd2.populate_boundary_ndar_points(r)
start_point = Point(x_min, y_min)
end_point = Point(x_min, y_max)
r = BoundaryGrids.fit_line(start_point, end_point, int(J))
bd3 = Boundary(len(r) - 1, BdType.xi_min)
bd3.populate_boundary_ndar_points(r)
start_point = Point(x_max, y_min)
end_point = Point(x_max, y_max)
r = BoundaryGrids.fit_line(start_point, end_point, int(J))
bd4 = Boundary(len(r) - 1, BdType.xi_max)
bd4.populate_boundary_ndar_points(r)
Grid_obj = StructuredGridGeneration.get_univariate_grid(bd1, bd2, \
bd3, bd4)
# Grid_obj.save_to_vtk('ExpFan-IB.vtk', 'ExpFan-IB')
# Grid_obj.save_to_txt('ExpFan-IB.txt')
# Grid_obj.save_to_vtk('RampChannel-IB.vtk', 'RampChannel-IB')
Grid_obj.save_to_txt('Rchannel-IB-2.txt')
def get_naca_grid_gridgen():
# Use points from naca equation as control points so that the trailing
# edge is sharp
x, y = get_IB_naca_4_dig('0012', chord_len=1.0, num_points=66)
CP = []
for i in range(len(x)):
CP.append(Point(x[i], y[i]))
################ Generating around airfoil
I = 100
# xi_spacing_f = lambda xi, I: 0.5 * I * (1 - np.cos(xi * np.pi / I))
xi_spacing_f = lambda xi, I: xi
s_CP_f = lambda j, n, I: j * I / n
s_CP_inverse_f = lambda s, n, I: s * n / I
r = fit_cubic_spline(CP, I, xi_spacing_f, s_CP_f, s_CP_inverse_f)
bd1 = Boundary(I, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
x = []
y = []
for i in range(len(r)):
x.append(r[i].x)
y.append(r[i].y)
return (x, y)
def get_ramp_grid(I, J):
# Grid as per ASM2010 AIAA code verification paper
# Ramp at I/3
x_bl = 0.
y_bl = 0.
x_rs = 0.25
y_rs = 0.0
x_re = 0.5
y_re = 0.067
x_br = 1.0
y_br = 0.067
x_tl = 0.0
y_tl = 0.2
x_tr = 1.0
y_tr = 0.2
# eta_min
start_point = Point(x_bl, y_bl)
end_point = Point(x_rs, y_rs)
r1 = BoundaryGrids.fit_line(start_point, end_point, int(I / 4))
start_point = Point(x_rs, y_rs)
end_point = Point(x_re, y_re)
r2 = BoundaryGrids.fit_line(start_point, end_point, int(I / 4))
start_point = Point(x_re, y_re)
end_point = Point(x_br, y_br)
r3 = BoundaryGrids.fit_line(start_point, end_point, int(I / 2))
r = np.append(r1[:-1], r2)
r = np.append(r[:-1], r3)
bd1 = Boundary(I, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
I = len(bd1.Points) - 1
# eta_max
start_point = Point(x_tl, y_tl)
end_point = Point(x_tr, y_tr)
r = BoundaryGrids.fit_line(start_point, end_point, I)
bd2 = Boundary(I, BdType.eta_max)
bd2.populate_boundary_ndar_points(r)
# xi_min
start_point = Point(x_bl, y_bl)
end_point = Point(x_tl, y_tl)
r = BoundaryGrids.fit_line(start_point, end_point, J)
bd3 = Boundary(J, BdType.xi_min)
bd3.populate_boundary_ndar_points(r)
# xi_max
start_point = Point(x_br, y_br)
end_point = Point(x_tr, y_tr)
r = BoundaryGrids.fit_line(start_point, end_point, J)
bd4 = Boundary(J, BdType.xi_max)
bd4.populate_boundary_ndar_points(r)
print len(bd1.Points), len(bd2.Points), len(bd3.Points), len(bd4.Points)
Grid_obj = StructuredGridGeneration.get_univariate_grid(bd1, bd2, bd3, bd4)
Grid_obj.save_to_vtk('ramp_channel_uniform.vtk')
Grid_obj.save_to_txt('ramp_channel_uniform.txt')
def get_plane_IB(delta, I, J):
global plt
x_min = -1.0
x_max = 5.0
y_min = -5.0
y_max = 5.0
airfoil_start = x_min + 43.5 / 100 * (x_max - x_min)
airfoil_end = x_min + 57.5 / 100 * (x_max - x_min)
airfoil_y_s = y_min + 48 / 100 * (y_max - y_min)
airfoil_y_e = y_min + 52 / 100 * (y_max - y_min)
dx = 0.01
dy = 0.01
# x_min = -6.0 * 0.01
# x_max = 21.0 * 0.01
# y_min = -5.0 * 0.01
# y_max = 5.0 * 0.01
# airfoil_start = x_min + 15.38/100*(x_max - x_min)
# airfoil_end = x_min + 26.92/100*(x_max - x_min)
# airfoil_start = -1 * 0.01
# airfoil_end = 1 * 0.01
# airfoil_y_s = y_min + 40./100*(y_max - y_min)
# airfoil_y_e = y_min + 60./100*(y_max - y_min)
# airfoil_y_s = -2.5 * 0.01
# airfoil_y_e = 2.5 * 0.01
# dx = 0.02 * 0.01
# dy = 0.02 * 0.01
length1 = abs(airfoil_start - y_min)
length2 = abs(x_max - airfoil_end)
delta1 = BoundaryGrids.find_hyp_tan_stretching_factor(
dx, int(I / 4), length1)
delta2 = BoundaryGrids.find_hyp_tan_stretching_factor(
dx, int(3 * I / 4), length2)
stretching_f1 = lambda xi: 1 + (np.tanh(delta1 *
(xi - 1)) / np.tanh(delta1))
stretching_f2 = lambda xi: 1 + (np.tanh(delta2 *
(xi - 1)) / np.tanh(delta2))
end_point = Point(x_min, y_min)
start_point = Point(airfoil_start, y_min)
r1_t = BoundaryGrids.fit_line(start_point, end_point, int(I / 4),
stretching_f1)
r1 = np.empty(len(r1_t), dtype=object)
for i in range(len(r1)):
r1[i] = r1_t[len(r1) - 1 - i]
num = int((airfoil_end - airfoil_start) / dx) + 1
start_point = Point(airfoil_start, y_min)
end_point = Point(airfoil_end, y_min)
r2 = BoundaryGrids.fit_line(start_point, end_point, num)
start_point = Point(airfoil_end, y_min)
end_point = Point(x_max, y_min)
r3 = BoundaryGrids.fit_line(start_point, end_point, int(3 * I / 4),
stretching_f2)
r = np.append(r1[:-1], r2)
r = np.append(r[:-1], r3)
bd1 = Boundary(len(r) - 1, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
f = plt.figure()
plt = bd1.plot_boundary(plt, 'eta_min')
end_point = Point(x_min, y_max)
start_point = Point(airfoil_start, y_max)
r1_t = BoundaryGrids.fit_line(start_point, end_point, int(I / 4),
stretching_f1)
r1 = np.empty(len(r1_t), dtype=object)
for i in range(len(r1)):
r1[i] = r1_t[len(r1) - 1 - i]
num = int((airfoil_end - airfoil_start) / dx) + 1
start_point = Point(airfoil_start, y_max)
end_point = Point(airfoil_end, y_max)
r2 = BoundaryGrids.fit_line(start_point, end_point, num)
start_point = Point(airfoil_end, y_max)
end_point = Point(x_max, y_max)
r3 = BoundaryGrids.fit_line(start_point, end_point, int(3 * I / 4),
stretching_f2)
r = np.append(r1[:-1], r2)
r = np.append(r[:-1], r3)
bd2 = Boundary(len(r) - 1, BdType.eta_max)
bd2.populate_boundary_ndar_points(r)
plt = bd2.plot_boundary(plt, 'eta_max')
I = len(r) - 1
# start_point = Point(x_min, y_min)
# end_point = Point(x_min, y_max)
# A = 3
# L = 1
# yc = 0.5
# stretching_f = lambda xi: (L*xi) + A*(yc - L*xi)*(1 - xi)*xi
# r = BoundaryGrids.fit_line(start_point, end_point, int(J), stretching_f)
# bd3 = Boundary(len(r)-1, BdType.xi_min)
# bd3.populate_boundary_ndar_points(r)
# plt = bd3.plot_boundary(plt, 'xi_min')
# start_point = Point(x_max, y_min)
# end_point = Point(x_max, y_max)
# A = 3
# L = 1
# yc = 0.5
# stretching_f = lambda xi: (L*xi) + A*(yc - L*xi)*(1 - xi)*xi
# r = BoundaryGrids.fit_line(start_point, end_point, int(J), stretching_f)
# bd4 = Boundary(len(r)-1, BdType.xi_max)
# bd4.populate_boundary_ndar_points(r)
# plt = bd4.plot_boundary(plt, 'xi_max')
# plt.show()
num = int((airfoil_y_e - airfoil_y_s) / dy) + 1
K = int(J / 2) + int(J / 2) + num
Grid_obj = TwoDGrid(I + 1, K + 1)
Grid_obj.set_boundary(bd1)
Grid_obj.set_boundary(bd2)
Grid = Grid_obj.Grid
length1 = abs(airfoil_y_s - y_min)
length2 = abs(y_max - airfoil_y_e)
delta1 = BoundaryGrids.find_hyp_tan_stretching_factor(
dy, int(J / 2), length1)
delta2 = BoundaryGrids.find_hyp_tan_stretching_factor(
dy, int(J / 2), length2)
stretching_f1 = lambda xi: 1 + (np.tanh(delta1 *
(xi - 1)) / np.tanh(delta1))
stretching_f2 = lambda xi: 1 + (np.tanh(delta2 *
(xi - 1)) / np.tanh(delta2))
for xi in range(0, I + 1):
P_eta_min = Grid[xi, 0]
P_eta_max = Grid[xi, K]
end_point = P_eta_min
start_point = Point(P_eta_min.x, airfoil_y_s)
r1_t = BoundaryGrids.fit_line(start_point, end_point, int(J / 2),
stretching_f1)
r1 = np.empty(len(r1_t), dtype=object)
for i in range(len(r1)):
r1[i] = r1_t[len(r1) - 1 - i]
# num = int((airfoil_y_e - airfoil_y_s)/ 0.005) + 1
start_point = Point(P_eta_min.x, airfoil_y_s)
end_point = Point(P_eta_max.x, airfoil_y_e)
r2 = BoundaryGrids.fit_line(start_point, end_point, num)
start_point = Point(P_eta_max.x, airfoil_y_e)
end_point = P_eta_max
r3 = BoundaryGrids.fit_line(start_point, end_point, int(J / 2),
stretching_f2)
r = np.append(r1[:-1], r2)
r = np.append(r[:-1], r3)
# K = int(J/2) + int(J/2) + num
# print len(r1), len(r2), len(r3), len(r), J, K
Grid[xi, :] = r
# Grid_obj = StructuredGridGeneration.get_univariate_grid(bd1, bd2, \
# bd3, bd4, stretching_f)
Grid_obj.save_to_vtk('Circle-IB.vtk', 'Circle-IB')
Grid_obj.save_to_txt('Circle-IB.txt')
def get_Boundaries_Problem1():
# Grid points go from xi = 0 to I. Hence I+1 grid points
# n + 1 = number of control points
global plt
f = plt.figure()
################ Extract grid
filename = 'n0012.dat'
CP = extract_control_points(filename)
CP_x = []
CP_y = []
for p in CP:
CP_x.append(p.x)
CP_y.append(p.y)
################ Generating around airfoil
I = 100
xi_spacing_f = lambda xi, I: 0.5 * I * (1 - np.cos(xi * np.pi / I))
s_CP_f = lambda j, n, I: j * I / n
s_CP_inverse_f = lambda s, n, I: s * n / I
r = fit_cubic_spline(CP, I, xi_spacing_f, s_CP_f, s_CP_inverse_f)
bd1 = Boundary(I, BdType.eta_min)
bd1.populate_boundary_ndar_points(r)
#plt = bd1.plot_boundary(plt, '1 - Control points')
#plt.plot(CP_x, CP_y, marker='.', markevery=3, label = '1 - Control points')
chord = np.sqrt((CP_x[-1] - CP_x[0])**2 + (CP_y[-1] - CP_y[0])**2)
################ Generating the lines
# Clustering near the start
I = 100
start = {'x': 1, 'y': bd1.Points[-1].y}
end = {'x': 21, 'y': 0}
length = np.sqrt((end['x'] - start['x'])**2 + (end['y'] - start['y'])**2)
ds1 = 0.001 * chord
delta = find_hyp_tan_stretching_factor(ds1, I, length)
stretching_f = lambda xi: 1 + (np.tanh(delta * (xi - 1)) / np.tanh(delta))
r = fit_line(start, end, I, stretching_f)
bd2 = Boundary(I, BdType.xi_max)
bd2.populate_boundary_ndar_points(r)
#plt = bd2.plot_boundary(plt, 'Boundary 2')
# Clustering near the end
I = 100
start = {'x': 0, 'y': 0}
end = {'x': -20, 'y': 0}
length = np.sqrt((end['x'] - start['x'])**2 + (end['y'] - start['y'])**2)
ds1 = 0.001 * chord
delta = find_hyp_tan_stretching_factor(ds1, I, length)
stretching_f = lambda xi: 1 + (np.tanh(delta * (xi - 1)) / np.tanh(delta))
r = fit_line(start, end, I, stretching_f)
bd3 = Boundary(I, BdType.xi_min)
bd3.populate_boundary_ndar_points(r)
#plt = bd3.plot_boundary(plt, 'Boundary 3')
################ Plotting the semicircle
I = 100
Radius = 20.5
t1 = np.pi
t2 = 0
centre = {'x': 0.5, 'y': 0}
r = fit_semicircle(Radius, t1, t2, centre, I)
bd4 = Boundary(I, BdType.eta_max)
bd4.populate_boundary_ndar_points(r)
#plt = bd4.plot_boundary(plt, 'Boundary 4')
#plt.legend(loc='best')
#plt.show()
return (bd1, bd2, bd3, bd4)