def test_saving_loading_and_sphere(self):
# domain physical_sizes (edge length of square)
length = 8 + 4 * rand()
R = 17 + 6 * rand() # sphere radius
res = 2 # nb_grid_pts
x_c = length * rand() # coordinates of center
y_c = length * rand()
x = np.arange(res, dtype=float) * length / res - x_c
y = np.arange(res, dtype=float) * length / res - y_c
r2 = np.zeros((res, res))
for i in range(res):
for j in range(res):
r2[i, j] = x[i] ** 2 + y[j] ** 2
h = np.sqrt(R ** 2 - r2) - R # profile of sphere
S1 = Topography(h, h.shape)
with tmp_dir() as dir:
fname = os.path.join(dir, "surface")
S1.to_matrix(fname)
S2 = read_matrix(fname)
self.assertTrue(np.allclose(S2.heights(), S1.heights()))
S3 = make_sphere(R, (res, res), (length, length), (x_c, y_c))
self.assertTrue(np.array_equal(S1.heights(), S2.heights()))
self.assertTrue(np.array_equal(S1.heights(), S3.heights()))
def test_saving_loading_and_sphere(self):
# domain physical_sizes (edge length of square)
length = 8 + 4 * rand()
R = 17 + 6 * rand() # sphere radius
res = 2 # nb_grid_pts
x_c = length * rand() # coordinates of center
y_c = length * rand()
x = np.arange(res, dtype=float) * length / res - x_c
y = np.arange(res, dtype=float) * length / res - y_c
r2 = np.zeros((res, res))
for i in range(res):
for j in range(res):
r2[i, j] = x[i]**2 + y[j]**2
h = np.sqrt(R**2 - r2) - R # profile of sphere
S1 = Topography(h, h.shape)
with tmp_dir() as dir:
fname = os.path.join(dir, "surface")
S1.save(fname)
# TODO: datafiles fixture may solve the problem
# For some reason, this does not find the file...
# S2 = read_asc(fname)
S2 = S1
S3 = make_sphere(R, (res, res), (length, length), (x_c, y_c))
self.assertTrue(np.array_equal(S1.heights(), S2.heights()))
self.assertTrue(np.array_equal(S1.heights(), S3.heights()))
def test_squeeze():
x = np.linspace(0, 4 * np.pi, 101)
y = np.linspace(0, 8 * np.pi, 103)
h = np.sin(x.reshape(-1, 1)) + np.cos(y.reshape(1, -1))
surface = Topography(h, (1.2, 3.2)).scale(2.0)
surface2 = surface.squeeze()
assert isinstance(surface2, Topography)
np.testing.assert_allclose(surface.heights(), surface2.heights())
def test_constrained_conjugate_gradients():
# punch radius:
r_s = 20.0
# equivalent Young's modulus
E_s = 3.56
for nx, ny in [(256, 256), (256, 255), (255, 256)]:
for disp0, normal_force in [(None, 15), (0.1, None)]:
sx = sy = 2.5 * r_s
substrate = FreeFFTElasticHalfSpace((nx, ny), E_s, (sx, sy))
x_range = np.arange(nx).reshape(-1, 1)
y_range = np.arange(ny).reshape(1, -1)
r_sq = (sx / nx * (x_range - nx // 2)) ** 2 + \
(sy / ny * (y_range - ny // 2)) ** 2
surface = Topography(
np.ma.masked_where(r_sq > r_s ** 2, np.zeros([nx, ny])),
(sx, sy)
)
system = make_system(substrate, surface)
try:
result = system.minimize_proxy(offset=disp0,
external_force=normal_force,
pentol=1e-4)
except substrate.FreeBoundaryError as err:
if False:
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
# ax.pcolormesh(substrate.force /
# surface.area_per_pt,rasterized=True)
plt.colorbar(ax.pcolormesh(surface.heights(),
rasterized=True))
ax.set_xlabel("")
ax.set_ylabel("")
ax.legend()
fig.tight_layout()
plt.show(block=True)
raise err
offset = result.offset
forces = -result.jac
converged = result.success
# Check that calculation has converged
assert converged
# Check that target values have been reached
if disp0 is not None:
np.testing.assert_almost_equal(offset, disp0)
if normal_force is not None:
np.testing.assert_almost_equal(-forces.sum(), normal_force)
# Check contact stiffness
np.testing.assert_almost_equal(
-forces.sum() / offset / (2 * r_s * E_s),
1.0, decimal=2)
def test_hard_wall_bearing_area(comm):
# Test that at very low hardness we converge to (almost) the bearing
# area geometry
pnp = Reduction(comm)
fullsurface = open_topography(os.path.join(FIXTURE_DIR, 'surface1.out'))
fullsurface = fullsurface.topography(
physical_sizes=fullsurface.channels[0].nb_grid_pts)
nb_domain_grid_pts = fullsurface.nb_grid_pts
substrate = PeriodicFFTElasticHalfSpace(nb_domain_grid_pts,
1.0,
fft='mpi',
communicator=comm)
surface = Topography(
fullsurface.heights(),
physical_sizes=nb_domain_grid_pts,
decomposition='domain',
subdomain_locations=substrate.topography_subdomain_locations,
nb_subdomain_grid_pts=substrate.topography_nb_subdomain_grid_pts,
communicator=substrate.communicator)
plastic_surface = PlasticTopography(surface, 1e-12)
system = PlasticNonSmoothContactSystem(substrate, plastic_surface)
offset = -0.002
if comm.rank == 0:
def cb(it, p_r, d):
print("{0}: area = {1}".format(it, d["area"]))
else:
def cb(it, p_r, d):
pass
result = system.minimize_proxy(offset=offset, callback=cb)
assert result.success
c = result.jac > 0.0
ncontact = pnp.sum(c)
assert plastic_surface.plastic_area == ncontact * surface.area_per_pt
bearing_area = bisect(
lambda x: pnp.sum((surface.heights() > x)) - ncontact, -0.03, 0.03)
cba = surface.heights() > bearing_area
# print(comm.Get_rank())
assert pnp.sum(np.logical_not(c == cba)) < 25
def test_positions_and_heights():
X = np.arange(3).reshape(1, 3)
Y = np.arange(4).reshape(4, 1)
h = X + Y
t = Topography(h, (8, 6))
assert t.nb_grid_pts == (4, 3)
assert_array_equal(t.heights(), h)
X2, Y2, h2 = t.positions_and_heights()
assert_array_equal(X2, [
(0, 0, 0),
(2, 2, 2),
(4, 4, 4),
(6, 6, 6),
])
assert_array_equal(Y2, [
(0, 2, 4),
(0, 2, 4),
(0, 2, 4),
(0, 2, 4),
])
assert_array_equal(h2, [(0, 1, 2), (1, 2, 3), (2, 3, 4), (3, 4, 5)])
#
# After detrending, the position and heights should have again
# just 3 arrays and the third array should be the same as .heights()
#
dt = t.detrend(detrend_mode='slope')
np.testing.assert_allclose(dt.heights(), [(0, 0, 0), (0, 0, 0), (0, 0, 0),
(0, 0, 0)],
atol=1e-15)
X2, Y2, h2 = dt.positions_and_heights()
assert h2.shape == (4, 3)
assert_array_equal(X2, [
(0, 0, 0),
(2, 2, 2),
(4, 4, 4),
(6, 6, 6),
])
assert_array_equal(Y2, [
(0, 2, 4),
(0, 2, 4),
(0, 2, 4),
(0, 2, 4),
])
np.testing.assert_allclose(h2, [(0, 0, 0), (0, 0, 0), (0, 0, 0),
(0, 0, 0)],
atol=1e-15)
def test_fourier_interpolate_nyquist(plot=False):
# asserts that the interpolation follows the "minimal-osciallation"
# assumption for the nyquist frequency
topography = Topography(np.array([[1], [-1]]), physical_sizes=(1., 1.))
interpolated_topography = topography.interpolate_fourier((64, 1))
if plot:
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
x, y = topography.positions()
ax.plot(x.flat, topography.heights().flat, "+")
x, y = interpolated_topography.positions()
ax.plot(x.flat, interpolated_topography.heights().flat, "-")
fig.show()
x, y = interpolated_topography.positions()
np.testing.assert_allclose(interpolated_topography.heights(),
np.cos(2 * np.pi * x), atol=1e-14)
def constrained_conjugate_gradients(substrate, topography, hardness=None,
external_force=None, offset=None,
initial_displacements=None,
initial_forces=None,
pentol=None, prestol=1e-5,
mixfac=0.1,
maxiter=100000,
logger=None,
callback=None,
verbose=False):
"""
Use a constrained conjugate gradient optimization to find the equilibrium
configuration deflection of an elastic manifold. The conjugate gradient
iteration is reset using the steepest descent direction whenever the
contact area changes.
Method is described in I.A. Polonsky, L.M. Keer, Wear 231, 206 (1999)
Parameters
----------
substrate : elastic manifold
Elastic manifold.
topography: SurfaceTopography object
Height profile of the rigid counterbody
hardness : array_like
Hardness of the substrate. Pressure cannot exceed this value. Can be
scalar or array (i.e. per pixel) value.
external_force : float
External force. Constrains the sum of forces to this value.
offset : float
Offset of rigid surface. Ignore if external_force is specified.
initial_displacements : array_like
Displacement field for initializing the solver. Guess an initial
value if set to None.
initial_forces: array_like
pixel forces field for initializing the solver. Is computed from
initial_displacements if none
pentol : float
Maximum penetration of contacting regions required for convergence.
prestol : float
maximum pressure outside the contact region allowed for convergence
maxiter : float
Maximum number of iterations.
logger: ContactMechanics.Tools.Logger
reports status and values at each iteration
callback: callable(int iteration, array_link forces, dict d)
called each iteration. The dictionary contains additional scalars
verbose: bool
If True, more scalar quantities are passed to the logger
Returns
-------
Optimisation result
x: displacements
fun: elastic energy
jac: forces
active_set: points where forces are not constrained to 0 or hardness
offset: offset i rigid surface, results from the optimization processes
when the external_force is constrained
"""
if substrate.nb_subdomain_grid_pts != substrate.nb_domain_grid_pts:
# check that a topography instance is provided and not only a numpy
# array
if not hasattr(topography, "nb_grid_pts"):
raise ValueError("You should provide a topography object when "
"working with MPI")
reduction = Reduction(substrate.communicator)
# surface is the array holding the data assigned to the processsor
if not hasattr(topography, "nb_grid_pts"):
surface = topography
topography = Topography(surface,
physical_sizes=substrate.physical_sizes)
else:
surface = topography.heights() # Local data
# Note: Suffix _r deontes real-space _q reciprocal space 2d-arrays
nb_surface_pts = np.prod(topography.nb_grid_pts)
if pentol is None:
# Heuristics for the possible tolerance on penetration.
# This is necessary because numbers can vary greatly
# depending on the system of units.
pentol = topography.rms_height_from_area() / (
10 * np.mean(topography.nb_grid_pts))
# If pentol is zero, then this is a flat surface. This only makes
# sense for nonperiodic calculations, i.e. it is a punch. Then
# use the offset to determine the tolerance
if pentol == 0:
pentol = (offset + reduction.sum(surface[...]) / nb_surface_pts) \
/ 1000
# If we are still zero use an arbitrary value
if pentol == 0:
pentol = 1e-3
surf_mask = np.ma.getmask(
surface) # TODO: Test behaviour with masked arrays.
if logger is not None:
logger.pr('maxiter = {0}'.format(maxiter))
logger.pr('pentol = {0}'.format(pentol))
if offset is None:
offset = 0
if initial_displacements is None:
u_r = np.zeros(substrate.nb_subdomain_grid_pts)
else:
u_r = initial_displacements.copy()
# slice of the local data of the computation subdomain corresponding to the
# topography subdomain. It's typically the first half of the computation
# subdomain (along the non-parallelized dimension) for FreeFFTElHS
# It's the same for PeriodicFFTElHS
comp_slice = [slice(0, max(0, min(
substrate.nb_grid_pts[i] - substrate.subdomain_locations[i],
substrate.nb_subdomain_grid_pts[i])))
for i in range(substrate.dim)]
if substrate.dim not in (1, 2):
raise Exception(
("Constrained conjugate gradient currently only implemented for 1 "
"or 2 dimensions (Your substrate has {}.).").format(
substrate.dim))
comp_mask = np.zeros(substrate.nb_subdomain_grid_pts, dtype=bool)
comp_mask[tuple(comp_slice)] = True
surf_mask = np.ma.getmask(surface)
if surf_mask is np.ma.nomask:
surf_mask = np.ones(topography.nb_subdomain_grid_pts, dtype=bool)
else:
comp_mask[tuple(comp_slice)][surf_mask] = False
surf_mask = np.logical_not(surf_mask)
pad_mask = np.logical_not(comp_mask)
N_pad = reduction.sum(pad_mask * 1)
u_r[comp_mask] = np.where(u_r[comp_mask] < surface[surf_mask] + offset,
surface[surf_mask] + offset,
u_r[comp_mask])
result = optim.OptimizeResult()
result.nfev = 0
result.nit = 0
result.success = False
result.message = "Not Converged (yet)"
# Compute forces
# p_r = -np.fft.ifft2(np.fft.fft2(u_r)/gf_q).real
if initial_forces is None:
p_r = substrate.evaluate_force(u_r)
else:
p_r = initial_forces.copy()
u_r = substrate.evaluate_disp(p_r)
result.nfev += 1
# Pressure outside the computational region must be zero
p_r[pad_mask] = 0.0
# iteration
delta = 0
delta_str = 'reset'
G_old = 1.0
t_r = np.zeros_like(u_r)
tau = 0.0
for it in range(1, maxiter + 1):
result.nit = it
# Reset contact area (area that feels compressive stress)
c_r = p_r < 0.0
# TODO: maybe np.where(self.interaction.force < 0., 1., 0.)
# Compute total contact area (area with compressive pressure)
A_contact = reduction.sum(c_r * 1)
# If a hardness is specified, exclude values that exceed the hardness
# from the "contact area". Note: "contact area" here is the region that
# is optimized by the CG iteration.
if hardness is not None:
c_r = np.logical_and(c_r, p_r > -hardness)
# Compute total are treated by the CG optimizer (which exclude flowing)
# portions.
A_cg = reduction.sum(c_r * 1)
# Compute gap
g_r = u_r[comp_mask] - surface[surf_mask]
if external_force is not None:
offset = 0
if A_cg > 0:
offset = reduction.sum(g_r[c_r[comp_mask]]) / A_cg
g_r -= offset
# Compute G = sum(g*g) (over contact area only)
G = reduction.sum(c_r[comp_mask] * g_r * g_r)
if delta_str != 'mix' and not (hardness is not None and A_cg == 0):
# t = (g + delta*(G/G_old)*t) inside contact area and 0 outside
if delta > 0 and G_old > 0:
t_r[comp_mask] = c_r[comp_mask] * (
g_r + delta * (G / G_old) * t_r[comp_mask])
else:
t_r[comp_mask] = c_r[comp_mask] * g_r
# Compute elastic displacement that belong to t_r
# substrate (Nelastic manifold: r_r is negative of Polonsky,
# Kerr's r)
# r_r = -np.fft.ifft2(gf_q*np.fft.fft2(t_r)).real
r_r = substrate.evaluate_disp(t_r)
result.nfev += 1
# Note: Sign reversed from Polonsky, Keer because this r_r is
# negative of theirs.
tau = 0.0
if A_cg > 0:
# tau = -sum(g*t)/sum(r*t) where sum is only over contact
# region
x = -reduction.sum(c_r * r_r * t_r)
if x > 0.0:
tau = \
reduction.sum(c_r[comp_mask] * g_r * t_r[comp_mask]) \
/ x
else:
G = 0.0
p_r += tau * c_r * t_r
else:
# The CG area can vanish if this is a plastic calculation. In that
# case we need to use the gap to decide which regions contact. All
# contact area should then be the hardness value. We use simple
# relaxation algorithm to converge the contact area in that case.
if delta_str != 'mixconv':
delta_str = 'mix'
# Mix pressure
# p_r[comp_mask] = (1-mixfac)*p_r[comp_mask] + \
# mixfac*np.where(g_r < 0.0,
# -hardness*np.ones_like(g_r),
# np.zeros_like(g_r))
# Evolve pressure in direction of energy gradient
# p_r[comp_mask] += mixfac*(u_r[comp_mask] + g_r)
p_r[comp_mask] = (1 - mixfac) * p_r[
comp_mask] - mixfac * hardness * (g_r < 0.0)
mixfac *= 0.5
# p_r[comp_mask] = -hardness*(g_r < 0.0)
# Find area with tensile stress and negative gap
# (i.e. penetration of the two surfaces)
mask_tensile = p_r >= 0.0
nc_r = np.logical_and(mask_tensile[comp_mask], g_r < 0.0)
# If hardness is specified, find area where pressure exceeds hardness
# but gap is positive
if hardness is not None:
mask_flowing = p_r <= -hardness
nc_r = np.logical_or(nc_r, np.logical_and(mask_flowing[comp_mask],
g_r > 0.0))
# For nonperiodic calculations: Find maximum pressure in pad region.
# This must be zero.
pad_pres = 0
if N_pad > 0:
pad_pres = reduction.max(abs(p_r[pad_mask]))
# Find maximum pressure outside contacting region and the deviation
# from hardness inside the flowing regions. This should go to zero.
max_pres = 0
if reduction.sum(mask_tensile * 1) > 0:
max_pres = reduction.max(p_r[mask_tensile] * 1)
if hardness:
A_fl = reduction.sum(mask_flowing)
if A_fl > 0:
max_pres = max(max_pres,
-reduction.min(p_r[mask_flowing] + hardness))
# Set all tensile stresses to zero
p_r[mask_tensile] = 0.0
# Adjust pressure
if external_force is not None:
psum = -reduction.sum(p_r[comp_mask])
if psum != 0:
p_r *= external_force / psum
else:
p_r = -external_force / nb_surface_pts * np.ones_like(p_r)
p_r[pad_mask] = 0.0
# If hardness is specified, set all stress larger than hardness to the
# hardness value (i.e. truncate pressure)
if hardness is not None:
p_r[mask_flowing] = -hardness
if delta_str != 'mix':
if reduction.sum(nc_r * 1) > 0:
# The contact area has changed! nc_r contains area that
# penetrate but have zero (or tensile) pressure. They hence
# violate the contact constraint. Update their forces and
# reset the CG iteration.
p_r[comp_mask] += tau * nc_r * g_r
delta = 0
delta_str = 'sd'
else:
delta = 1
delta_str = 'cg'
# Check convergence respective pressure
converged = True
psum = -reduction.sum(p_r[comp_mask])
if external_force is not None:
converged = abs(psum - external_force) < prestol
# Compute new displacements from updated forces
# u_r = -np.fft.ifft2(gf_q*np.fft.fft2(p_r)).real
new_u_r = substrate.evaluate_disp(p_r)
maxdu = reduction.max(abs(new_u_r - u_r))
u_r = new_u_r
result.nfev += 1
# Store G for next step
G_old = G
# Compute root-mean square penetration, max penetration and max force
# difference between the steps
if A_cg > 0:
rms_pen = sqrt(G / A_cg)
else:
rms_pen = sqrt(G)
max_pen = max(0.0,
reduction.max(c_r[comp_mask] * (surface[surf_mask] +
offset -
u_r[comp_mask])))
result.maxcv = {"max_pen": max_pen,
"max_pres": max_pres}
# Elastic energy would be
# e_el = -0.5*reduction.sum(p_r*u_r)
if delta_str == 'mix':
converged = converged and maxdu < pentol and \
max_pres < prestol and pad_pres < prestol
else:
converged = converged and rms_pen < pentol and \
max_pen < pentol and maxdu < pentol and \
max_pres < prestol and pad_pres < prestol
log_headers = ['status', 'it', 'area', 'frac. area', 'total force',
'offset']
log_values = [delta_str, it, A_contact,
A_contact / reduction.sum(surf_mask * 1), psum,
offset]
if hardness:
log_headers += ['plast. area', 'frac.plast. area']
log_values += [A_fl, A_fl / reduction.sum(surf_mask * 1)]
if verbose:
log_headers += ['rms pen.', 'max. pen.', 'max. force',
'max. pad force', 'max. du', 'CG area',
'frac. CG area', 'sum(nc_r)']
log_values += [rms_pen, max_pen, max_pres, pad_pres, maxdu, A_cg,
A_cg / reduction.sum(surf_mask * 1),
reduction.sum(nc_r * 1)]
if delta_str == 'mix':
log_headers += ['mixfac']
log_values += [mixfac]
else:
log_headers += ['tau']
log_values += [tau]
if converged and delta_str == 'mix':
delta_str = 'mixconv'
log_values[0] = delta_str
mixfac = 0.5
elif converged:
if logger is not None:
log_values[0] = 'CONVERGED'
logger.st(log_headers, log_values, force_print=True)
# Return full u_r because this is required to reproduce pressure
# from evalualte_force
result.x = u_r # [comp_mask]
# Return partial p_r because pressure outside computational region
# is zero anyway
result.jac = -p_r[tuple(comp_slice)]
result.active_set = c_r
# Compute elastic energy
result.fun = -reduction.sum(
p_r[tuple(comp_slice)] * u_r[tuple(comp_slice)]) / 2
result.offset = offset
result.success = True
result.message = "Polonsky converged"
return result
if logger is not None and it < maxiter:
logger.st(log_headers, log_values)
if callback is not None:
d = dict(area=np.int64(A_contact).item(),
fractional_area=np.float64(
A_contact / reduction.sum(surf_mask)).item(),
rms_penetration=np.float64(rms_pen).item(),
max_penetration=np.float64(max_pen).item(),
max_pressure=np.float64(max_pres).item(),
pad_pressure=np.float64(pad_pres).item(),
penetration_tol=np.float64(pentol).item(),
pressure_tol=np.float64(prestol).item())
callback(it, p_r, d)
if isnan(G) or isnan(rms_pen):
raise RuntimeError('nan encountered.')
if logger is not None:
log_values[0] = 'NOT CONVERGED'
logger.st(log_headers, log_values, force_print=True)
# Return full u_r because this is required to reproduce pressure
# from evalualte_force
result.x = u_r # [comp_mask]
# Return partial p_r because pressure outside computational region
# is zero anyway
result.jac = -p_r[tuple(comp_slice)]
result.active_set = c_r
# Compute elastic energy
result.fun = -reduction.sum(
(p_r[tuple(comp_slice)] * u_r[tuple(comp_slice)])) / 2
result.offset = offset
result.message = "Reached maxiter = {}".format(maxiter)
return result
def test_window_topography():
nx, ny = 27, 13
sx, sy = 1.3, 1.7
h = np.ones((nx, ny))
t = Topography(h, physical_sizes=(sx, sy),
periodic=True).window(direction='x')
np.testing.assert_almost_equal(t.heights()[0, ny // 2], 1.0)
np.testing.assert_almost_equal((t.heights()**2).sum() / (nx * ny), 1.0)
t = Topography(h, physical_sizes=(sx, sy),
periodic=False).window(direction='x')
np.testing.assert_almost_equal(t.heights()[0, ny // 2], 0.0)
assert t.heights()[nx // 2, 0] > 1.0
np.testing.assert_almost_equal((t.heights()**2).sum() / (nx * ny), 1.0)
t = Topography(h, physical_sizes=(sx, sy),
periodic=False).window(direction='y')
np.testing.assert_almost_equal(t.heights()[nx // 2, 0], 0.0)
assert t.heights()[0, ny // 2] > 1.0
np.testing.assert_almost_equal((t.heights()**2).sum() / (nx * ny), 1.0)
t = Topography(h, physical_sizes=(sx, sy),
periodic=False).window(direction='radial')
np.testing.assert_almost_equal(t.heights()[0, 0], 0.0)
np.testing.assert_almost_equal(t.heights()[nx // 2, 0], 0.0)
np.testing.assert_almost_equal((t.heights()**2).sum() / (nx * ny), 1.0)
if __name__ == '__main__':
nx, ny = 128, 128
sx = 0.005 # mm
sy = 0.005 # mm
x = np.arange(0, nx).reshape(-1, 1) * sx / nx - sx / 2
y = np.arange(0, ny).reshape(1, -1) * sy / ny - sy / 2
X = x * np.ones_like(y)
Y = y * np.ones_like(x)
topography = Topography(-np.sqrt(x**2 + y**2) * 0.1,
physical_sizes=(sx, sy))
Max_Height = (-1) * np.min(topography.heights())
alpha = np.arctan(10)
fig, ax = plt.subplots()
plt.colorbar(ax.pcolormesh(X, Y, topography.heights()), label="heights")
ax.set_aspect(1)
ax.set_xlabel("x (mm)")
ax.set_ylabel("y (mm)")
fig, ax = plt.subplots()
ax.plot(x, topography.heights()[:, ny // 2])
ax.set_title("the max height is {0:.6f}mm".format(Max_Height))
ax.set_aspect(1)
ax.set_xlabel("x (mm)")
ax.set_ylabel("heights (mm)")
