def visualize():
from skfem.visuals.matplotlib import plot
return plot(ibasis2, u2, Nrefs=4, colorbar=True, shading='gouraud')
w=[basis['psi'].interpolate(velocity[i::2]) for i in range(2)])
psi = solve(*condense(A, vorticity, D=basis['psi'].find_dofs()['floor'].all()))
if __name__ == '__main__':
from functools import partial
from os.path import splitext
from sys import argv
from matplotlib.tri import Triangulation
from skfem.visuals.matplotlib import plot, savefig
name = splitext(argv[0])[0]
plot(mesh, pressure)
savefig(f'{name}-pressure.png', bbox_inches='tight', pad_inches=0)
mesh.save(f'{name}-velocity.vtk',
{'velocity': velocity[basis['u'].nodal_dofs].T})
fig, ax = subplots()
ax.plot(
*mesh.p[:,
mesh.facets[:,
np.concatenate(list(mesh.boundaries.values()))]],
color='k')
n_streamlines = 11
plot = partial(ax.tricontour,
Triangulation(*mesh.p, mesh.t.T),
from matplotlib.tri import Triangulation
from skfem.visuals.matplotlib import plot, draw, savefig
name = splitext(argv[0])[0]
mesh.save(f'{name}_velocity.vtk',
{'velocity': velocity[basis['u'].nodal_dofs].T})
print(basis['psi'].interpolator(psi)(np.zeros((2, 1)))[0],
'(cf. exact 1/64)')
print(
basis['p'].interpolator(pressure)(np.array([[-0.5, 0.5], [0.5, 0.5]])),
'(cf. exact -/+ 1/8)')
ax = draw(mesh)
plot(basis['p'], pressure, ax=ax)
savefig(f'{name}_pressure.png')
ax = draw(mesh)
velocity1 = velocity[basis['u'].nodal_dofs]
ax.quiver(*mesh.p, *velocity1, mesh.p[0, :]) # colour by buoyancy
savefig(f'{name}_velocity.png')
ax = draw(mesh)
ax.tricontour(Triangulation(*mesh.p, mesh.t.T),
psi[basis['psi'].nodal_dofs.flatten()])
savefig(f'{name}_stream-function.png')
@BilinearForm
def hdg(u, ut, v, vt, w):
from skfem.helpers import grad, dot
# outwards normal
n = w.n * (-1.)**w.idx[0]
return dot(n, grad(u)) * (vt - v) + dot(n, grad(v)) * (ut - u)\
+ 1e1 / w.h * (ut - u) * (vt - v)
@LinearForm
def load(v, vt, w):
return 1. * v
A = asm(laplace, ibasis) + asm(hdg, tbasis)
f = asm(load, ibasis)
y = solve(*condense(A, f, D=ibasis.get_dofs()))
(u1, ibasis1), (u2, ibasis2) = ibasis.split(y)
if __name__ == '__main__':
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import plot, savefig
plot(ibasis1, u1, Nrefs=3, colorbar=True, shading='gouraud')
savefig(splitext(argv[0])[0] + '_p1dg.png')
plot(ibasis2, u2, Nrefs=4, colorbar=True, shading='gouraud')
savefig(splitext(argv[0])[0] + '_skeleton.png')
n = w.n
x = w.x
return -dot(grad(v), n) * (x[0] == 1.0)
A = asm(laplace, ib)
B = asm(facetbilinf, fb)
b = asm(facetlinf, fb)
D = ib.find_dofs({'plate': m.facets_satisfying(lambda x: (x[1] == 0.0))})
I = ib.complement_dofs(D)
import scipy.sparse
b = scipy.sparse.csr_matrix(b)
K = scipy.sparse.bmat([[A + B, b.T], [b, None]], 'csr')
import numpy as np
f = np.concatenate((ib.zeros(), -1.0 * np.ones(1)))
I = np.append(I, K.shape[0] - 1)
x = solve(*condense(K, f, I=I))
if __name__ == "__main__":
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import plot, savefig
plot(m, x[:-1], colorbar=True)
savefig(splitext(argv[0])[0] + '_solution.png')
def visualize():
from skfem.visuals.matplotlib import plot
return plot(m, x, shading='gouraud', colorbar=True)
@BilinearForm
def advection(u, v, w):
from skfem.helpers import grad
_, y = w.x
velocity_0 = 6 * y * (height - y) # parabolic plane Poiseuille
return v * velocity_0 * grad(u)[0]
dofs = basis.find_dofs({
'inlet': mesh.facets_satisfying(lambda x: x[0] == 0.),
'floor': mesh.facets_satisfying(lambda x: x[1] == 0.)
})
interior = basis.complement_dofs(dofs)
A = asm(laplace, basis) + peclet * asm(advection, basis)
t = np.zeros(basis.N)
t[dofs['floor'].all()] = 1.
t = solve(*condense(A, np.zeros_like(t), t, I=interior))
basis0 = basis.with_element(ElementQuad0())
t0 = solve(asm(mass, basis0), asm(mass, basis, basis0) @ t)
if __name__ == '__main__':
from pathlib import Path
from skfem.visuals.matplotlib import plot, savefig
plot(mesh, t0)
savefig(Path(__file__).with_suffix('.png'),
bbox_inches='tight',
pad_inches=0)
print(f'v={v} c')
if __name__ == '__main__':
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import plot, savefig
import matplotlib.pyplot as plt
from skfem.utils import derivative
B_x = derivative(A, global_basis, global_basis, 1)
B_y = -derivative(A, global_basis, global_basis, 0)
E_x = -derivative(U, global_basis, global_basis, 0)
E_y = -derivative(U, global_basis, global_basis, 1)
fig = plt.figure(figsize=(11.52, 5.12))
ax1 = plt.subplot(1, 2, 1)
plot(global_basis, np.sqrt(B_x**2 + B_y**2), ax=ax1, colorbar=True)
ax1.set_title('Magnetic flux density (Tesla)')
ax1.set_aspect('equal')
ax1.set_yticks([])
ax2 = plt.subplot(1, 2, 2)
plot(global_basis, np.sqrt(E_x**2 + E_y**2), ax=ax2, colorbar=True)
ax2.set_title('Electric field strength (V/m)')
ax2.set_aspect('equal')
ax2.set_yticks([])
savefig(splitext(argv[0])[0] + '_solution.png')
psi = solve(*condense(A, vorticity, D=basis['psi'].get_dofs('floor')))
if __name__ == '__main__':
from functools import partial
from os.path import splitext
from sys import argv
from matplotlib.tri import Triangulation
from skfem.visuals.matplotlib import plot, savefig
name = splitext(argv[0])[0]
plot(mesh, pressure)
savefig(f'{name}-pressure.png', bbox_inches='tight', pad_inches=0)
mesh.save(f'{name}-velocity.vtk',
{'velocity': velocity[basis['u'].nodal_dofs].T})
fig, ax = subplots()
ax.plot(*mesh.p[:, mesh.facets[:, np.concatenate(
list(mesh.boundaries.values()))]],
color='k')
n_streamlines = 11
contour = partial(ax.tricontour,
Triangulation(*mesh.p, mesh.t.T),
psi[basis['psi'].nodal_dofs.flatten()],
linewidths=1.)
def visualize():
from skfem.visuals.matplotlib import draw, plot
ax = draw(m)
return plot(ib, x, ax=ax, shading='gouraud', colorbar=True, nrefs=2)
def runTest(self):
m = self.mesh_type()
plot(m, m.p[0])
def visualize():
from skfem.visuals.matplotlib import plot, show
return plot(basis, u, shading='gouraud', colorbar=True)
facet_basis = FacetBasis(mesh, element, facets=mesh.boundaries['perimeter'])
H = heat_transfer_coefficient * asm(convection, facet_basis)
core_basis = InteriorBasis(mesh, basis.elem, elements=mesh.subdomains['core'])
f = joule_heating * asm(unit_load, core_basis)
temperature = solve(L + H, f)
if __name__ == '__main__':
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import draw, plot, savefig
T0 = {
'skfem':
basis.interpolator(temperature)(np.zeros((2, 1)))[0],
'exact':
(joule_heating * radii[0]**2 / 4 / thermal_conductivity['core'] *
(2 * thermal_conductivity['core'] / radii[1] /
heat_transfer_coefficient +
(2 * thermal_conductivity['core'] / thermal_conductivity['annulus'] *
np.log(radii[1] / radii[0])) + 1))
}
print('Central temperature:', T0)
ax = draw(mesh)
plot(basis, temperature, ax=ax, colorbar=True)
savefig(splitext(argv[0])[0] + '_solution.png')
ib = InteriorBasis(m, e, intorder=5)
if __name__ == "__main__":
import matplotlib.pyplot as plt
from skfem.visuals.matplotlib import plot, draw
f, axes = plt.subplots(3, 3)
ixs = [(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 0)]
i = 0
for itr in ib.nodal_dofs[:, 4]:
axi = axes[ixs[i]]
axi.set_axis_off()
X = np.zeros(ib.N)
X[itr] = 1.0
plot(ib, X, Nrefs=5, shading='gouraud', ax=axi)
i += 1
axi = axes[(1, 1)]
axi.set_axis_off()
draw(m, ax=axi)
axi = axes[(2, 1)]
axi.set_axis_off()
X = np.zeros(ib.N)
X[np.array([56, 59, 64, 66])] = 1.0
plot(ib, X, Nrefs=5, shading='gouraud', ax=axi)
axi = axes[(2, 2)]
axi.set_axis_off()
X = np.zeros(ib.N)
b = asm(load, V)
@BilinearForm
def diffusion_form(u, v, w):
return sum(v.grad * (q(w["w"]) * u.grad))
def diffusion_matrix(u):
return asm(diffusion_form, V, w=V.interpolate(u))
dirichlet = apply(u_exact, mesh.p) # P1 nodal interpolation
plot(V, dirichlet).get_figure().savefig(str(output_dir.joinpath("exact.png")))
def residual(u):
r = b - diffusion_matrix(u) @ u
r[boundary] = 0.0
return r
u = np.zeros(V.N)
u[boundary] = dirichlet[boundary]
result = root(residual, u, method="krylov")
if result.success:
u = result.x
print("Success. Residual =", np.linalg.norm(residual(u), np.inf))
"""One-dimensional Poisson."""
import numpy as np
from skfem import *
from skfem.models.poisson import laplace, unit_load
m = MeshLine(np.linspace(0, 1, 10))
e = ElementLineP1()
basis = InteriorBasis(m, e)
A = asm(laplace, basis)
b = asm(unit_load, basis)
x = solve(*condense(A, b, D=basis.find_dofs()))
if __name__ == "__main__":
from skfem.visuals.matplotlib import plot, show
plot(m, x)
show()
from matplotlib.animation import FuncAnimation
import matplotlib.pyplot as plt
from skfem.visuals.matplotlib import plot
parser = ArgumentParser(description='heat equation in a rectangle')
parser.add_argument(
'-g',
'--gif',
action='store_true',
help='write animated GIF',
)
args = parser.parse_args()
ax = plot(mesh, u_init[basis.nodal_dofs.flatten()], shading='gouraud')
title = ax.set_title('t = 0.00')
field = ax.get_children()[0] # vertex-based temperature-colour
fig = ax.get_figure()
fig.colorbar(field)
def update(event):
t, u = event
u0 = {'skfem': (probe @ u)[0], 'exact': (probe @ exact(t))[0]}
print('{:4.2f}, {:5.3f}, {:+7.4f}'.format(t, u0['skfem'],
u0['skfem'] - u0['exact']))
title.set_text(f'$t$ = {t:.2f}')
field.set_array(u[basis.nodal_dofs.flatten()])
solver=solver_iter_krylov(minres),
M=build_pc_diag(asm(mass, basis['p'])))
velocity = flow(pressure)
basis['psi'] = basis['u'].with_element(ElementQuad2())
psi = np.zeros(A.shape[0])
vorticity = asm(rot, basis['psi'], w=basis['u'].interpolate(velocity))
psi = solve(*condense(asm(laplace, basis['psi']), vorticity, D=basis['psi'].find_dofs()))
psi0 = basis['psi'].interpolator(psi)(np.zeros((2, 1)))[0]
if __name__ == '__main__':
from os.path import splitext
from sys import argv
from matplotlib.tri import Triangulation
from skfem.visuals.matplotlib import plot, draw
print(psi0)
name = splitext(argv[0])[0]
plot(mesh, pressure, colorbar=True).get_figure().savefig(
f'{name}_pressure.png')
ax = draw(mesh)
ax.tricontour(Triangulation(*mesh.p, mesh.to_meshtri().t.T),
psi[basis['psi'].nodal_dofs.flatten()])
ax.get_figure().savefig(f'{name}_stream-lines.png')
from skfem import *
from skfem.models.poisson import laplace, unit_load
m = MeshTri().refined(4)
e = ElementTriP1()
basis = InteriorBasis(m, e)
A = asm(laplace, basis)
b = asm(unit_load, basis)
x = solve(*condense(A, b, I=m.interior_nodes()))
if __name__ == "__main__":
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import plot, savefig
plot(m, x, shading='gouraud', colorbar=True)
savefig(splitext(argv[0])[0] + '_solution.png')
M = asm(mass, basis)
u_exact = 2 * np.arctan2(*basis.doflocs[::-1]) / np.pi
u_error = u - u_exact
error_L2 = np.sqrt(u_error @ M @ u_error)
conductance = {'skfem': u @ A @ u,
'exact': 2 * np.log(2) / np.pi}
@Functional
def port_flux(w):
from skfem.helpers import dot, grad
return dot(w.n, grad(w['u']))
current = {}
for port, boundary in mesh.boundaries.items():
fbasis = FacetBasis(mesh, elements, facets=boundary)
current[port] = asm(port_flux, fbasis, u=fbasis.interpolate(u))
if __name__ == '__main__':
from skfem.visuals.matplotlib import plot, show
print('L2 error:', error_L2)
print('conductance:', conductance)
print('Current in through ports:', current)
plot(basis, u)
show()
import numpy as np
from .ex17 import mesh, basis, radii, joule_heating, thermal_conductivity
annulus = np.unique(basis.element_dofs[:, mesh.subdomains['annulus']])
temperature = basis.zeros()
core = basis.complement_dofs(annulus)
core_basis = Basis(mesh, basis.elem, elements=mesh.subdomains['core'])
L = asm(laplace, core_basis)
f = asm(unit_load, core_basis)
temperature = solve(*condense(thermal_conductivity['core'] * L,
joule_heating * f,
D=annulus))
T0 = {
"skfem": (basis.probes(np.zeros((2, 1))) @ temperature)[0],
"exact": joule_heating * radii[0] ** 2 / 4 / thermal_conductivity["core"],
}
if __name__ == '__main__':
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import draw, plot
ax = draw(mesh)
plot(mesh, temperature[basis.nodal_dofs.flatten()],
ax=ax, colorbar=True)
ax.get_figure().savefig(splitext(argv[0])[0] + '_solution.png')
eta_E = edge_jump.elemental(fbasis[0], **w)
tmp = np.zeros(m.facets.shape[1])
np.add.at(tmp, fbasis[0].find, eta_E)
eta_E = np.sum(0.5 * tmp[m.t2f], axis=0)
return eta_K + eta_E
if __name__ == "__main__":
from skfem.visuals.matplotlib import draw, plot, show
draw(m)
for itr in range(9): # 9 adaptive refinements
if itr > 0:
m = m.refined(adaptive_theta(eval_estimator(m, u)))
basis = InteriorBasis(m, e)
K = asm(laplace, basis)
f = asm(load, basis)
I = m.interior_nodes()
u = solve(*condense(K, f, I=I))
if __name__ == "__main__":
draw(m)
plot(m, u, shading='gouraud')
show()
from matplotlib.animation import FuncAnimation
import matplotlib.pyplot as plt
from skfem.visuals.matplotlib import plot
parser = ArgumentParser(description='heat equation in a rectangle')
parser.add_argument(
'-g',
'--gif',
action='store_true',
help='write animated GIF',
)
args = parser.parse_args()
ax = plot(mesh, u_init, shading='gouraud')
title = ax.set_title('t = 0.00')
field = ax.get_children()[0] # vertex-based temperature-colour
fig = ax.get_figure()
fig.colorbar(field)
def update(event):
t, u = event
u0 = {
'skfem': basis.interpolator(u)(np.zeros((2, 1)))[0],
'exact':
np.exp(-diffusivity * np.pi**2 * t / 4 * sum(halfwidth**-2))
}
print('{:4.2f}, {:5.3f}, {:+7.4f}'.format(t, u0['skfem'],
u0['skfem'] - u0['exact']))
