def visualize():
from skfem.visuals.matplotlib import plot, draw
ax = plot(basis[0], y, Nrefs=4, colorbar=True)
draw(basis[0], ax=ax)
plot(basis[1], y, ax=ax, Nrefs=4)
draw(basis[1], ax=ax)
return ax
def runTest(self):
m = self.mesh_type()
draw(m,
aspect=1.1,
facet_numbering=True,
node_numbering=True,
element_numbering=True)
def visualize():
from skfem.visuals.matplotlib import plot, draw
ax = plot(ib_dg,
vonmises1,
shading='gouraud',
colorbar='$\sigma_{\mathrm{mises}}$')
draw(mdefo, ax=ax)
plot(Ib_dg, vonmises2, ax=ax, Nrefs=3, shading='gouraud')
draw(Mdefo, ax=ax)
return ax
def triangulate(
corner_points,
max_refloops=20,
theta=0.8,
smooth_steps=50,
quality=0.9,
verbose=False,
**params,
):
from .criterion import avg_quality
from .initial import cdt
from .mark import adaptive_theta
from .refine import rgb
from .smooth import cpt
from .solve import laplace
m = process(
initial=cdt,
solve=laplace,
mark=adaptive_theta,
refine=rgb,
smooth=cpt,
criterion=avg_quality,
corner_points=corner_points,
max_refloops=max_refloops,
theta=theta,
smooth_steps=smooth_steps,
quality=quality,
verbose=verbose,
**params,
)
m.draw = types.MethodType(lambda self, **kwargs: draw(self, **kwargs), m)
m.show = types.MethodType(lambda self, **kwargs: show(self, **kwargs), m)
return m
def draw(self, *args, **kwargs):
"""Convenience wrapper for :func:`skfem.visuals.matplotlib.draw`."""
from skfem.visuals.matplotlib import draw, show
from skfem.assembly import CellBasis
ax = draw(CellBasis(self, self.elem()), *args, **kwargs)
ax.show = show
return ax
def visualize():
from skfem.visuals.matplotlib import plot, draw
M = MeshQuad(np.array(m.p + .5 * y[gb.nodal_dofs]), m.t)
ax = draw(M)
return plot(M,
sigma[sgb.nodal_dofs[0]],
ax=ax,
colorbar='$\sigma_{xx}$',
shading='gouraud')
def visualize():
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 = ib.zeros()
X[itr] = 1.0
plot(ib, X, Nrefs=5, shading='gouraud', ax=axi)
axi.set_aspect('equal')
i += 1
axi = axes[(1, 1)]
axi.set_axis_off()
draw(m, ax=axi)
axi.set_aspect('equal')
axi = axes[(2, 1)]
axi.set_axis_off()
X = ib.zeros()
X[np.array([56, 59, 64, 66])] = 1.0
plot(ib, X, Nrefs=5, shading='gouraud', ax=axi)
axi.set_aspect('equal')
axi = axes[(2, 2)]
axi.set_axis_off()
X = ib.zeros()
X[np.array([58, 61, 63, 65])] = 1.0
plot(ib, X, Nrefs=5, shading='gouraud', ax=axi)
axi.set_aspect('equal')
return axi
def process(initial=None,
solve=None,
mark=None,
refine=None,
smooth=None,
criterion=None,
max_refloops=6,
verbose=False,
**params):
if initial is not None:
if callable(initial):
mesh = initial(**params)
else:
mesh = initial
else:
raise Exception("The initial mesh not given.")
if verbose:
draw(mesh)
for itr in range(max_refloops):
estimators = solve(mesh, **params)
elements = mark(mesh, estimators, **params)
mesh = refine(mesh, elements, **params)
if verbose:
draw(mesh)
if smooth is not None:
mesh = smooth(mesh, **params)
if verbose:
draw(mesh)
if criterion is not None:
if criterion(mesh, **params):
break
if itr == max_refloops - 1:
warnings.warn("Criterion not satisfied in {} refinement loops.".format(
max_refloops))
return mesh
def visualize():
from skfem.visuals.matplotlib import plot, draw
ax = draw(M, boundaries_only=True)
return plot(M, X, shading="gouraud", ax=ax, colorbar=True)
dw1 = grad(w['u1'])
dw2 = grad(w['u2'])
return h * ((dw1[0] - dw2[0]) * n[0] + (dw1[1] - dw2[1]) * n[1])**2
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(.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 reversed(range(9)):
basis = Basis(m, e)
K = asm(laplace, basis)
M = asm(mass, basis)
#f = asm(load, basis)
I = m.interior_nodes()
Lh, eigen_uh = solve(*condense(K, M, I=I),
solver=solver_eigen_scipy_sym(sigma=20, k=1))
eigv = Lh[0]
print(eigv)
eigen_uh = eigen_uh.flatten()
rotf = asm(unit_load, ib)
psi = solve(*condense(stokes, rotf, D=ib.find_dofs()))
psi0, = ib.interpolator(psi)(np.zeros((2, 1)))
if __name__ == "__main__":
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import draw
from matplotlib.tri import Triangulation
print('psi0 = {} (cf. exact = 1/64 = {})'.format(psi0, 1 / 64))
M, Psi = ib.refinterp(psi, 3)
ax = draw(mesh)
ax.tricontour(Triangulation(*M.p, M.t.T), Psi)
name = splitext(argv[0])[0]
ax.get_figure().savefig(f'{name}_stream-lines.png')
refbasis = InteriorBasis(M, ElementTriP1())
velocity = np.vstack([
derivative(Psi, refbasis, refbasis, 1),
-derivative(Psi, refbasis, refbasis, 0)
])
ax = draw(mesh)
sparsity_factor = 2**3 # subsample the arrows
vector_factor = 2**3 # lengthen the arrows
x = M.p[:, ::sparsity_factor]
u = vector_factor * velocity[:, ::sparsity_factor]
ax.quiver(*x, *u, x[0])
from pathlib import Path
m = MeshTet.load(Path(__file__).parent / 'meshes' / 'beams.msh')
e1 = ElementTetP2()
e = ElementVectorH1(e1)
ib = Basis(m, e)
K = asm(linear_elasticity(*lame_parameters(200.0e9, 0.3)), ib)
rho = 8050.0
@BilinearForm
def mass(u, v, w):
from skfem.helpers import dot
return dot(rho * u, v)
M = asm(mass, ib)
L, x = solve(
*condense(K, M, D=ib.find_dofs()["fixed"]), solver=solver_eigen_scipy_sym()
)
if __name__ == "__main__":
from skfem.visuals.matplotlib import draw, show
sf = 2.0
draw(MeshTet(np.array(m.p + sf * x[ib.nodal_dofs, 0]), m.t))
show()
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)
X[np.array([58, 61, 63, 65])] = 1.0
plot(ib, X, Nrefs=5, shading='gouraud', ax=axi)
plt.axis('off')
plt.show()
M = asm(mass, gb)
D = gb.find_dofs({'left': m.facets_satisfying(lambda x: x[0] == 0.)})
y = gb.zeros()
I = gb.complement_dofs(D)
L, x = solve(*condense(K, M, I=I),
solver=solver_eigen_scipy_sym(k=6, sigma=0.0))
y = x[:, 4]
# calculate stress
sgb = gb.with_element(ElementVector(e))
C = linear_stress(lam, mu)
yi = gb.interpolate(y)
@LinearForm
def proj(v, _):
return ddot(C(sym_grad(yi)), v)
sigma = projection(proj, sgb, gb)
if __name__ == "__main__":
from skfem.visuals.matplotlib import plot, draw, show
M = MeshQuad(np.array(m.p + y[gb.nodal_dofs]), m.t)
ax = draw(M)
plot(M, sigma[sgb.nodal_dofs[0]], ax=ax, colorbar=True)
show()
def runTest(self):
m = self.mesh_type()
draw(m)
mapping = MappingIsoparametric(m, e1)
e = ElementVectorH1(e1)
gb = InteriorBasis(m, e, mapping, 2)
K = asm(linear_elasticity(1.0, 1.0), gb)
@BilinearForm
def mass(u, v, w):
return dot(u, v)
M = asm(mass, gb)
D = gb.get_dofs(lambda x: x[0] == 0.0).all()
y = np.zeros(gb.N)
I = gb.complement_dofs(D)
L, x = eigsh(K[I].T[I].T, k=6, M=M[I].T[I].T, which='SM')
y[I] = x[:, 4]
if __name__ == "__main__":
from skfem.visuals.matplotlib import draw, show
M = MeshQuad(np.array(m.p + y[gb.nodal_dofs]), m.t)
draw(M)
show()
def _repr_svg_(self) -> str:
from skfem.visuals.svg import draw
return draw(self, nrefs=2, boundaries_only=True).svg
def visualize():
from skfem.visuals.matplotlib import draw, plot
ax = draw(m)
return plot(m, u, ax=ax, shading='gouraud', colorbar=True)
elements that have nodal degrees-of-freedom. Support for higher-order elements
on mixed meshes is work-in-progress.
"""
from skfem import *
from skfem.models import laplace, unit_load
fname = 'docs/examples/meshes/mixedtriquad.msh'
out = ['cell_sets_dict'] # read boundary nodes from meshio
m = [
Mesh.load(fname, force_meshio_type='triangle'),
Mesh.load(fname, force_meshio_type='quad', out=out),
]
e = [ElementTriP1(), ElementQuad1()]
basis = list(map(Basis, m, e))
A = asm(laplace, basis)
f = asm(unit_load, basis)
y = solve(*condense(A, f, D=out[0]['boundary']['line']))
if __name__ == '__main__':
from os.path import splitext
from sys import argv
from skfem.visuals.matplotlib import plot, draw, savefig
ax = plot(basis[0], y, Nrefs=4)
draw(basis[0], ax=ax)
plot(basis[1], y, ax=ax, Nrefs=4)
draw(basis[1], ax=ax)
savefig(splitext(argv[0])[0] + '_solution.png')
