def distmesh2d(fd, fh, h0, bbox, pfix=None, fig='gcf'):
"""
distmesh2d: 2-D Mesh Generator using Distance Functions.
Usage
-----
>>> p, t = distmesh2d(fd, fh, h0, bbox, pfix)
Parameters
----------
fd: Distance function d(x,y)
fh: Scaled edge length function h(x,y)
h0: Initial edge length
bbox: Bounding box, (xmin, ymin, xmax, ymax)
pfix: Fixed node positions, shape (nfix, 2)
fig: Figure to use for plotting, or None to disable plotting.
Returns
-------
p: Node positions (Nx2)
t: Triangle indices (NTx3)
Example: (Uniform Mesh on Unit Circle)
>>> fd = lambda p: sqrt((p**2).sum(1))-1.0
>>> p, t = distmesh2d(fd, huniform, 2, (-1,-1,1,1))
Example: (Rectangle with circular hole, refined at circle boundary)
>>> fd = lambda p: ddiff(drectangle(p,-1,1,-1,1), dcircle(p,0,0,0.5))
>>> fh = lambda p: 0.05+0.3*dcircle(p,0,0,0.5)
>>> p, t = distmesh2d(fd, fh, 0.05, (-1,-1,1,1),
[(-1,-1), (-1,1), (1,-1), (1,1)])
Example: (Polygon)
>>> pv=[(-0.4, -0.5), (0.4, -0.2), (0.4, -0.7), (1.5, -0.4), (0.9, 0.1),
(1.6, 0.8), (0.5, 0.5), (0.2, 1.0), (0.1, 0.4), (-0.7, 0.7),
(-0.4, -0.5)]
>>> fd = lambda p: dpoly(p, pv)
>>> p, t = distmesh2d(fd, huniform, 0.1, (-1,-1, 2,1), pv)
Example: (Ellipse)
>>> fd = lambda p: p[:,0]**2/2**2 + p[:,1]**2/1**2 - 1
>>> p, t = dm.distmesh2d(fd, dm.huniform, 0.2, (-2,-1, 2,1))
Example: (Square, with size function point and line sources)
>>> fd = lambda p: dm.drectangle(p,0,1,0,1)
>>> fh = lambda p: np.minimum(np.minimum(
0.01+0.3*abs(dm.dcircle(p,0,0,0)),
0.025+0.3*abs(dm.dpoly(p,[(0.3,0.7),(0.7,0.5)]))), 0.15)
>>> p, t = dm.distmesh2d(fd, fh, 0.01, (0,0,1,1), [(0,0),(1,0),(0,1),(1,1)])
Example: (NACA0012 airfoil)
>>> hlead=0.01; htrail=0.04; hmax=2; circx=2; circr=4
>>> a=.12/.2*np.array([0.2969,-0.1260,-0.3516,0.2843,-0.1036])
>>> a0=a[0]; a1=np.hstack((a[5:0:-1], 0.0))
>>> fd = lambda p: dm.ddiff(
dm.dcircle(p,circx,0,circr),
(abs(p[:,1])-np.polyval(a1, p[:,0]))**2-a0**2*p[:,0])
>>> fh = lambda p: np.minimum(np.minimum(
hlead+0.3*dm.dcircle(p,0,0,0),
htrail+0.3*dm.dcircle(p,1,0,0)), hmax)
>>> fixx = 1.0-htrail*np.cumsum(1.3**np.arange(5))
>>> fixy = a0*np.sqrt(fixx)+np.polyval(a1, fixx)
>>> fix = np.vstack((
np.array([(circx-circr,0),(circx+circr,0),
(circx,-circr),(circx,circr),
(0,0),(1,0)]),
np.vstack((fixx, fixy)).T,
np.vstack((fixx, -fixy)).T))
>>> box = (circx-circr,-circr, circx+circr,circr)
>>> h0 = min(hlead, htrail, hmax)
>>> p, t = dm.distmesh2d(fd, fh, h0, box, fix)
"""
if fig == 'gcf':
import matplotlib.pyplot as plt
fig = plt.gcf()
dptol=.001; ttol=.1; Fscale=1.2; deltat=.2; geps=.001*h0;
deps=np.sqrt(np.finfo(np.double).eps)*h0;
densityctrlfreq=30;
# Extract bounding box
xmin, ymin, xmax, ymax = bbox
if pfix is not None:
pfix = np.array(pfix, dtype='d')
# 0. Prepare a figure.
if fig is not None:
from distmesh.plotting import SimplexCollection
fig.clf()
ax = fig.gca()
c = SimplexCollection()
ax.add_collection(c)
ax.set_xlim(xmin, xmax)
ax.set_ylim(ymin, ymax)
ax.set_aspect('equal')
ax.set_axis_off()
fig.canvas.draw()
# 1. Create initial distribution in bounding box (equilateral triangles)
x, y = np.mgrid[xmin:(xmax+h0):h0,
ymin:(ymax+h0*np.sqrt(3)/2):h0*np.sqrt(3)/2]
x[:, 1::2] += h0/2 # Shift even rows
p = np.vstack((x.flat, y.flat)).T # List of node coordinates
# 2. Remove points outside the region, apply the rejection method
p = p[fd(p)<geps] # Keep only d<0 points
r0 = 1/fh(p)**2 # Probability to keep point
p = p[np.random.random(p.shape[0])<r0/r0.max()] # Rejection method
if pfix is not None:
p = ml.setdiff_rows(p, pfix) # Remove duplicated nodes
pfix = ml.unique_rows(pfix); nfix = pfix.shape[0]
p = np.vstack((pfix, p)) # Prepend fix points
else:
nfix = 0
N = p.shape[0] # Number of points N
count = 0
pold = float('inf') # For first iteration
while True:
count += 1
# 3. Retriangulation by the Delaunay algorithm
dist = lambda p1, p2: np.sqrt(((p1-p2)**2).sum(1))
if (dist(p, pold)/h0).max() > ttol: # Any large movement?
pold = p.copy() # Save current positions
t = spspatial.Delaunay(p).vertices # List of triangles
pmid = p[t].sum(1)/3 # Compute centroids
t = t[fd(pmid) < -geps] # Keep interior triangles
# 4. Describe each bar by a unique pair of nodes
bars = np.vstack((t[:, [0,1]],
t[:, [1,2]],
t[:, [2,0]])) # Interior bars duplicated
bars.sort(axis=1)
bars = ml.unique_rows(bars) # Bars as node pairs
# 5. Graphical output of the current mesh
if fig is not None:
c.set_simplices((p, t))
fig.canvas.draw()
# 6. Move mesh points based on bar lengths L and forces F
barvec = p[bars[:,0]] - p[bars[:,1]] # List of bar vectors
L = np.sqrt((barvec**2).sum(1)) # L = Bar lengths
hbars = fh(p[bars].sum(1)/2)
L0 = (hbars*Fscale
*np.sqrt((L**2).sum()/(hbars**2).sum())) # L0 = Desired lengths
# Density control - remove points that are too close
if (count % densityctrlfreq) == 0 and (L0 > 2*L).any():
ixdel = np.setdiff1d(bars[L0 > 2*L].reshape(-1), np.arange(nfix))
p = p[np.setdiff1d(np.arange(N), ixdel)]
N = p.shape[0]; pold = float('inf')
continue
F = L0-L; F[F<0] = 0 # Bar forces (scalars)
Fvec = F[:,None]/L[:,None].dot([[1,1]])*barvec # Bar forces (x,y components)
Ftot = ml.dense(bars[:,[0,0,1,1]],
np.repeat([[0,1,0,1]], len(F), axis=0),
np.hstack((Fvec, -Fvec)),
shape=(N, 2))
Ftot[:nfix] = 0 # Force = 0 at fixed points
p += deltat*Ftot # Update node positions
# 7. Bring outside points back to the boundary
d = fd(p); ix = d>0 # Find points outside (d>0)
if ix.any():
dgradx = (fd(p[ix]+[deps,0])-d[ix])/deps # Numerical
dgrady = (fd(p[ix]+[0,deps])-d[ix])/deps # gradient
dgrad2 = dgradx**2 + dgrady**2
p[ix] -= (d[ix]*np.vstack((dgradx, dgrady))/dgrad2).T # Project
# 8. Termination criterion: All interior nodes move less than dptol (scaled)
if (np.sqrt((deltat*Ftot[d<-geps]**2).sum(1))/h0).max() < dptol:
break
# Clean up and plot final mesh
p, t = dmutils.fixmesh(p, t)
if fig is not None:
c.set_simplices((p, t))
fig.canvas.draw()
return p, t
def distmeshnd(fd, fh, h0, bbox, pfix=None, fig='gcf'):
"""
distmeshnd: N-D Mesh Generator using Distance Functions.
Usage
-----
>>> p, t = distmesh2d(fd, fh, h0, bbox, pfix)
Parameters
----------
fd: Distance function d(x,y)
fh: Scaled edge length function h(x,y)
h0: Initial edge length
bbox: Bounding box, (xmin, ymin, zmin, ..., xmax, ymax, zmax, ...)
pfix: Fixed node positions, shape (nfix, dim)
fig: Figure to use for plotting, or None to disable plotting.
Returns
-------
p: Node positions (np, dim)
t: Triangle indices (nt, dim+1)
Example: (Unit ball)
>>> dim = 3
>>> fd = lambda p: sqrt((p**2).sum(1))-1.0
>>> bbox = np.vstack((-np.ones(dim), np.ones(dim)))
>>> p, t = distmeshnd(fd, huniform, 2, bbox)
"""
if fig == 'gcf':
import matplotlib.pyplot as plt
fig = plt.gcf()
bbox = np.array(bbox).reshape(2, -1)
dim = bbox.shape[1]
ptol=.001; ttol=.1; L0mult=1+.4/2**(dim-1); deltat=.1; geps=1e-1*h0;
deps=np.sqrt(np.finfo(np.double).eps)*h0;
if pfix is not None:
pfix = np.array(pfix, dtype='d')
nfix = len(pfix)
else:
pfix = np.empty((0, dim))
nfix = 0
# 0. Prepare a figure.
if fig is not None:
if dim == 2:
from distmesh.plotting import SimplexCollection
fig.clf()
ax = fig.gca()
c = SimplexCollection()
ax.add_collection(c)
ax.set_xlim(bbox[:,0])
ax.set_ylim(bbox[:,1])
ax.set_aspect('equal')
ax.set_axis_off()
fig.canvas.draw()
elif dim == 3:
import mpl_toolkits.mplot3d
from distmesh.plotting import axes_simpplot3d
fig.clf()
ax = fig.gca(projection='3d')
ax.set_xlim(bbox[:,0])
ax.set_ylim(bbox[:,1])
ax.set_zlim(bbox[:,2])
fig.canvas.draw()
else:
print("Plotting only supported in dimensions 2 and 3.")
fig = None
# 1. Create initial distribution in bounding box (equilateral triangles)
p = np.mgrid[tuple(slice(min, max+h0, h0) for min, max in bbox.T)]
p = p.reshape(dim, -1).T
# 2. Remove points outside the region, apply the rejection method
p = p[fd(p)<geps] # Keep only d<0 points
r0 = fh(p)
p = np.vstack((pfix,
p[np.random.rand(p.shape[0]) < r0.min()**dim / r0**dim]))
N = p.shape[0]
count = 0
pold = float('inf') # For first iteration
while True:
# 3. Retriangulation by the Delaunay algorithm
dist = lambda p1, p2: np.sqrt(((p1-p2)**2).sum(1))
if (dist(p, pold)/h0).max() > ttol: # Any large movement?
pold = p.copy() # Save current positions
t = spspatial.Delaunay(p).vertices # List of triangles
pmid = p[t].sum(1)/(dim+1) # Compute centroids
t = t[fd(pmid) < -geps] # Keep interior triangles
# 4. Describe each bar by a unique pair of nodes
bars = np.vstack((t[:,pair] for pair in
itertools.combinations(range(dim+1), 2)))
bars.sort(axis=1)
bars = ml.unique_rows(bars) # Bars as node pairs
# 5. Graphical output of the current mesh
if fig is not None:
if dim == 2:
c.set_simplices((p,t))
fig.canvas.draw()
elif dim == 3:
if count % 5 == 0:
ax.cla()
axes_simpplot3d(ax, p, t, p[:,1] > 0)
ax.set_title('Retriangulation #%d' % count)
fig.canvas.draw()
else:
print('Retriangulation #%d' % count)
count += 1
# 6. Move mesh points based on bar lengths L and forces F
barvec = p[bars[:,0]] - p[bars[:,1]] # List of bar vectors
L = np.sqrt((barvec**2).sum(1)) # L = Bar lengths
hbars = fh(p[bars].sum(1)/2)
L0 = (hbars*L0mult*((L**dim).sum()/(hbars**dim).sum())**(1.0/dim))
F = L0-L; F[F<0] = 0 # Bar forces (scalars)
Fvec = F[:,None]/L[:,None].dot(np.ones((1,dim)))*barvec # Bar forces (x,y components)
Ftot = ml.dense(bars[:,[0]*dim+[1]*dim],
np.repeat([list(range(dim))*2], len(F), axis=0),
np.hstack((Fvec, -Fvec)),
shape=(N, dim))
Ftot[:nfix] = 0 # Force = 0 at fixed points
p += deltat*Ftot # Update node positions
# 7. Bring outside points back to the boundary
d = fd(p); ix = d>0 # Find points outside (d>0)
if ix.any():
def deps_vec(i): a = [0]*dim; a[i] = deps; return a
dgrads = [(fd(p[ix]+deps_vec(i))-d[ix])/deps for i in range(dim)]
dgrad2 = sum(dgrad**2 for dgrad in dgrads)
p[ix] -= (d[ix]*np.vstack(dgrads)/dgrad2).T # Project
# 8. Termination criterion: All interior nodes move less than dptol (scaled)
maxdp = deltat*np.sqrt((Ftot[d<-geps]**2).sum(1)).max()
if maxdp < ptol*h0:
break
return p, t
def distmesh2d(fd,
fh,
h0,
bbox,
pfix=None,
fig='gcf',
dptol=.001,
ttol=.1,
Fscale=1.2,
deltat=.2,
geps_multiplier=.001,
densityctrlfreq=30):
"""
distmesh2d: 2-D Mesh Generator using Distance Functions.
Usage
-----
>>> p, t = distmesh2d(fd, fh, h0, bbox, pfix)
Parameters
----------
fd: Distance function d(x,y)
fh: Scaled edge length function h(x,y)
h0: Initial edge length
bbox: Bounding box, (xmin, ymin, xmax, ymax)
pfix: Fixed node positions, shape (nfix, 2)
fig: Figure to use for plotting, or None to disable plotting.
Returns
-------
p: Node positions (Nx2)
t: Triangle indices (NTx3)
Example: (Uniform Mesh on Unit Circle)
>>> fd = lambda p: sqrt((p**2).sum(1))-1.0
>>> p, t = distmesh2d(fd, huniform, 2, (-1,-1,1,1))
Example: (Rectangle with circular hole, refined at circle boundary)
>>> fd = lambda p: ddiff(drectangle(p,-1,1,-1,1), dcircle(p,0,0,0.5))
>>> fh = lambda p: 0.05+0.3*dcircle(p,0,0,0.5)
>>> p, t = distmesh2d(fd, fh, 0.05, (-1,-1,1,1),
[(-1,-1), (-1,1), (1,-1), (1,1)])
Example: (Polygon)
>>> pv=[(-0.4, -0.5), (0.4, -0.2), (0.4, -0.7), (1.5, -0.4), (0.9, 0.1),
(1.6, 0.8), (0.5, 0.5), (0.2, 1.0), (0.1, 0.4), (-0.7, 0.7),
(-0.4, -0.5)]
>>> fd = lambda p: dpoly(p, pv)
>>> p, t = distmesh2d(fd, huniform, 0.1, (-1,-1, 2,1), pv)
Example: (Ellipse)
>>> fd = lambda p: p[:,0]**2/2**2 + p[:,1]**2/1**2 - 1
>>> p, t = dm.distmesh2d(fd, dm.huniform, 0.2, (-2,-1, 2,1))
Example: (Square, with size function point and line sources)
>>> fd = lambda p: dm.drectangle(p,0,1,0,1)
>>> fh = lambda p: np.minimum(np.minimum(
0.01+0.3*abs(dm.dcircle(p,0,0,0)),
0.025+0.3*abs(dm.dpoly(p,[(0.3,0.7),(0.7,0.5)]))), 0.15)
>>> p, t = dm.distmesh2d(fd, fh, 0.01, (0,0,1,1), [(0,0),(1,0),(0,1),(1,1)])
Example: (NACA0012 airfoil)
>>> hlead=0.01; htrail=0.04; hmax=2; circx=2; circr=4
>>> a=.12/.2*np.array([0.2969,-0.1260,-0.3516,0.2843,-0.1036])
>>> a0=a[0]; a1=np.hstack((a[5:0:-1], 0.0))
>>> fd = lambda p: dm.ddiff(
dm.dcircle(p,circx,0,circr),
(abs(p[:,1])-np.polyval(a1, p[:,0]))**2-a0**2*p[:,0])
>>> fh = lambda p: np.minimum(np.minimum(
hlead+0.3*dm.dcircle(p,0,0,0),
htrail+0.3*dm.dcircle(p,1,0,0)), hmax)
>>> fixx = 1.0-htrail*np.cumsum(1.3**np.arange(5))
>>> fixy = a0*np.sqrt(fixx)+np.polyval(a1, fixx)
>>> fix = np.vstack((
np.array([(circx-circr,0),(circx+circr,0),
(circx,-circr),(circx,circr),
(0,0),(1,0)]),
np.vstack((fixx, fixy)).T,
np.vstack((fixx, -fixy)).T))
>>> box = (circx-circr,-circr, circx+circr,circr)
>>> h0 = min(hlead, htrail, hmax)
>>> p, t = dm.distmesh2d(fd, fh, h0, box, fix)
"""
if fig == 'gcf':
import matplotlib.pyplot as plt
fig = plt.gcf()
geps = geps_multiplier * h0
deps = np.sqrt(np.finfo(np.double).eps) * h0
# Extract bounding box
xmin, ymin, xmax, ymax = bbox
if pfix is not None:
pfix = np.array(pfix, dtype='d')
# 0. Prepare a figure.
if fig is not None:
from distmesh.plotting import SimplexCollection
fig.clf()
ax = fig.gca()
c = SimplexCollection()
ax.add_collection(c)
ax.set_xlim(xmin, xmax)
ax.set_ylim(ymin, ymax)
ax.set_aspect('equal')
ax.set_axis_off()
fig.canvas.draw()
# 1. Create initial distribution in bounding box (equilateral triangles)
x, y = np.mgrid[xmin:(xmax + h0):h0,
ymin:(ymax + h0 * np.sqrt(3) / 2):h0 * np.sqrt(3) / 2]
x[:, 1::2] += h0 / 2 # Shift even rows
p = np.vstack((x.flat, y.flat)).T # List of node coordinates
# 2. Remove points outside the region, apply the rejection method
p = p[fd(p) < geps] # Keep only d<0 points
r0 = 1 / fh(p)**2 # Probability to keep point
p = p[np.random.random(p.shape[0]) < r0 / r0.max()] # Rejection method
if pfix is not None:
p = ml.setdiff_rows(p, pfix) # Remove duplicated nodes
pfix = ml.unique_rows(pfix)
nfix = pfix.shape[0]
p = np.vstack((pfix, p)) # Prepend fix points
else:
nfix = 0
N = p.shape[0] # Number of points N
count = 0
pold = float('inf') # For first iteration
while True:
count += 1
# 3. Retriangulation by the Delaunay algorithm
dist = lambda p1, p2: np.sqrt(((p1 - p2)**2).sum(1))
if (dist(p, pold) / h0).max() > ttol: # Any large movement?
pold = p.copy() # Save current positions
t = spspatial.Delaunay(p).vertices # List of triangles
pmid = p[t].sum(1) / 3 # Compute centroids
t = t[fd(pmid) < -geps] # Keep interior triangles
# 4. Describe each bar by a unique pair of nodes
bars = np.vstack((t[:, [0, 1]], t[:, [1, 2]],
t[:, [2, 0]])) # Interior bars duplicated
bars.sort(axis=1)
bars = ml.unique_rows(bars) # Bars as node pairs
# 5. Graphical output of the current mesh
if fig is not None:
c.set_simplices((p, t))
fig.canvas.draw()
# 6. Move mesh points based on bar lengths L and forces F
barvec = p[bars[:, 0]] - p[bars[:, 1]] # List of bar vectors
L = np.sqrt((barvec**2).sum(1)) # L = Bar lengths
hbars = fh(p[bars].sum(1) / 2)
L0 = (hbars * Fscale * np.sqrt(
(L**2).sum() / (hbars**2).sum())) # L0 = Desired lengths
# Density control - remove points that are too close
if (count % densityctrlfreq) == 0 and (L0 > 2 * L).any():
ixdel = np.setdiff1d(bars[L0 > 2 * L].reshape(-1), np.arange(nfix))
p = p[np.setdiff1d(np.arange(N), ixdel)]
N = p.shape[0]
pold = float('inf')
continue
F = L0 - L
F[F < 0] = 0 # Bar forces (scalars)
Fvec = F[:, None] / L[:, None].dot(
[[1, 1]]) * barvec # Bar forces (x,y components)
Ftot = ml.dense(bars[:, [0, 0, 1, 1]],
np.repeat([[0, 1, 0, 1]], len(F), axis=0),
np.hstack((Fvec, -Fvec)),
shape=(N, 2))
Ftot[:nfix] = 0 # Force = 0 at fixed points
p += deltat * Ftot # Update node positions
# 7. Bring outside points back to the boundary
d = fd(p)
ix = d > 0 # Find points outside (d>0)
if ix.any():
dgradx = (fd(p[ix] + [deps, 0]) - d[ix]) / deps # Numerical
dgrady = (fd(p[ix] + [0, deps]) - d[ix]) / deps # gradient
dgrad2 = dgradx**2 + dgrady**2
p[ix] -= (d[ix] * np.vstack(
(dgradx, dgrady)) / dgrad2).T # Project
# 8. Termination criterion: All interior nodes move less than dptol (scaled)
if (np.sqrt((deltat * Ftot[d < -geps]**2).sum(1)) / h0).max() < dptol:
break
# Clean up and plot final mesh
p, t = dmutils.fixmesh(p, t)
if fig is not None:
c.set_simplices((p, t))
fig.canvas.draw()
return p, t