def star_mesh():
figure()
# fake the corners:
pfix = [[0.25, 0.25], [-0.25, 0.25], [-0.25, -0.25], [0.25, -0.25]]
pts, tri = distmesh2d(star, huniform, 0.1, bbox, pfix)
plotmesh(pts, tri)
show()
def ex_disc_refine(h0):
print " meshing ..."
p1, t1 = distmesh2d(fd_disc, huniform, h0, bbox, [])
pts, mytri = fixmesh(p1, t1)
edges, tedges = edgelist(pts, mytri)
print " original mesh has %d nodes and %d edges" % (np.shape(pts)[0],
np.shape(edges)[0])
fig1 = plt.figure()
plt.subplot(1, 2, 1)
plotmesh(pts, mytri)
plt.title('original mesh')
rp, rt, e, ind = bdyrefine(pts, mytri, fd_disc, h0)
redges, tmp = edgelist(rp, rt)
print " refined mesh has %d nodes and %d edges" % (np.shape(rp)[0],
np.shape(redges)[0])
plt.subplot(1, 2, 2)
plotmesh(rp, rt)
plt.title('final refined mesh')
uh, inside = poisson(f_ex, fd_disc, h0, rp, rt, announce=True)
print " plotting ..."
fig4 = plt.figure()
ax = fig4.gca(projection='3d')
ax.plot_trisurf(rp[:, 0], rp[:, 1], uh, cmap=cm.jet, linewidth=0.2)
uexact = 1.0 - rp[:, 0]**2.0 - rp[:,
1]**2.0 # exact: u(x,y) = 1 - x^2 - y^2
err = max(abs(uh - uexact))
print "max error = %f" % err
def star_mesh():
figure()
# fake the corners:
pfix = [[0.25, 0.25], [-0.25, 0.25], [-0.25, -0.25], [0.25, -0.25]]
pts, tri = distmesh2d(star, huniform, 0.1, bbox, pfix)
plotmesh(pts, tri)
show()
def ex_ell(h0):
pfix = [[1, 1], [1, -1], [0, -1], [0, 0], [-1, 0], [-1, 1]]
plotmethod = 2
print " meshing ..."
fig1 = plt.figure()
p1, t1 = distmesh2d(fd_ell, huniform, h0, bbox, pfix)
pts, mytri = fixmesh(p1, t1)
plotmesh(pts, mytri)
uh, inside = poisson(f_ex, fd_ell, h0, pts, mytri, announce=True)
print " plotting ..."
fig2 = plt.figure()
if plotmethod == 1:
plotmesh(pts, tri, uh)
elif plotmethod == 2:
ax = fig2.gca(projection='3d')
ax.scatter(pts[:, 0], pts[:, 1], uh, c=uh, cmap=cm.jet)
else:
# seems unreliable
TRI = tri.Triangulation(pts[:, 0], pts[:, 1], triangles=mytri)
ax.plot_trisurf(pts[:, 0],
pts[:, 1],
uh,
TRI.triangles,
cmap=cm.jet,
linewidth=0.2)
def ell():
"""L-shaped domain from 'Finite Elements and Fast Iterative Solvers'
by Elman, Silvester, and Wathen."""
pfix = [[1,1], [1, -1], [0, -1], [0, 0], [-1, 0], [-1, 1]]
def d(pts):
return ddiff(drectangle(pts, -1, 1, -1, 1), drectangle(pts, -2, 0, -2, 0))
figure()
pts, tri = distmesh2d(d, huniform, 0.25, bbox, pfix)
plotmesh(pts, tri)
show()
def ex_disc(h0):
print " meshing ..."
p1, t1 = distmesh2d(fd_disc, huniform, h0, bbox, [])
pts, mytri = fixmesh(p1,t1)
fig1 = plt.figure()
plotmesh(pts, mytri)
uh, inside = poisson(f_ex,fd_disc,h0,pts,mytri,announce=True)
print " plotting ..."
fig2 = plt.figure()
ax = fig2.gca(projection='3d')
ax.plot_trisurf(pts[:,0], pts[:,1], uh, cmap=cm.jet, linewidth=0.2)
uexact = 1.0 - pts[:,0]**2.0 - pts[:,1]**2.0 # exact: u(x,y) = 1 - x^2 - y^2
err = max(abs(uh-uexact))
print "max error = %f" % err
def ell():
"""L-shaped domain from 'Finite Elements and Fast Iterative Solvers'
by Elman, Silvester, and Wathen."""
pfix = [[1, 1], [1, -1], [0, -1], [0, 0], [-1, 0], [-1, 1]]
def d(pts):
return ddiff(drectangle(pts, -1, 1, -1, 1),
drectangle(pts, -2, 0, -2, 0))
figure()
pts, tri = distmesh2d(d, huniform, 0.25, bbox, pfix)
plotmesh(pts, tri)
show()
def ex_disc(h0):
print " meshing ..."
p1, t1 = distmesh2d(fd_disc, huniform, h0, bbox, [])
pts, mytri = fixmesh(p1, t1)
fig1 = plt.figure()
plotmesh(pts, mytri)
uh, inside = poisson(f_ex, fd_disc, h0, pts, mytri, announce=True)
print " plotting ..."
fig2 = plt.figure()
ax = fig2.gca(projection='3d')
ax.plot_trisurf(pts[:, 0], pts[:, 1], uh, cmap=cm.jet, linewidth=0.2)
uexact = 1.0 - pts[:,
0]**2.0 - pts[:,
1]**2.0 # exact: u(x,y) = 1 - x^2 - y^2
err = max(abs(uh - uexact))
print "max error = %f" % err
def obscircle(h0,psifcn):
print " meshing and fixing mesh ..."
p1, t1 = distmesh2d(fd_disc, huniform, h0, bbox, [])
p1a, t1a = fixmesh(p1,t1)
print " ... original mesh has %d nodes" % np.shape(p1a)[0]
print " refining mesh once ..."
p2, t2, e, ind = bdyrefine(p1a,t1a,fd_disc,h0)
print " ... first refined mesh has %d nodes" % np.shape(p2)[0]
print " refining mesh again ..."
pts, mytri, e, ind = bdyrefine(p2,t2,fd_disc,h0)
print " ... second refined mesh has %d nodes" % np.shape(pts)[0]
print " solving ..."
tol = 1.0e-6
uh, ii, ierr = obstacle(psifcn, f_ex, tol, fd_disc, h0, pts, mytri, \
announce=True)
print " obstacle: %d iterations total to reach iteration tolerance %.2e" \
% (len(ierr), tol)
psi = np.maximum(psifcn(pts),-1.0*np.ones(np.shape(pts)[0]))
fig1 = plt.figure()
plotmesh(pts, mytri)
coin = (uh==psi) # coincidence set
plt.plot(pts[coin,0],pts[coin,1],'ro',markersize=8.0)
fig2 = plt.figure()
ax = fig2.gca(projection='3d')
#ax.plot_trisurf(pts[:,0], pts[:,1], psi, cmap=cm.Blues, linewidth=0.1)
ax.plot_trisurf(pts[:,0], pts[:,1], uh, cmap=cm.jet, linewidth=0.2)
ax.set_xlim3d(-1.0,1.0)
ax.set_ylim3d(-1.0,1.0)
ax.set_zlim3d(-0.25,1.25)
fig4 = plt.figure()
plt.semilogy(np.array(range(len(ierr)))+1.0,ierr,'o-')
plt.xlabel('j = iteration')
plt.ylabel('max diff successive iterations')
plt.grid(True)
if psifcn == psi_sphere:
print
print 'for sphere obstacle, exact solution is known:'
print ' (away from the obstacle: u(r) = b log(r) )'
apart = (uh > psi)
print ' max norm error = %.3e' % max(abs(uh[apart] - u_exact_sphere(pts[apart])))
def ex_ell(h0):
pfix = [[1,1], [1, -1], [0, -1], [0, 0], [-1, 0], [-1, 1]]
plotmethod = 2
print " meshing ..."
fig1 = plt.figure()
p1, t1 = distmesh2d(fd_ell, huniform, h0, bbox, pfix)
pts, mytri = fixmesh(p1,t1)
plotmesh(pts, mytri)
uh, inside = poisson(f_ex,fd_ell,h0,pts,mytri,announce=True)
print " plotting ..."
fig2 = plt.figure()
if plotmethod == 1:
plotmesh(pts, tri, uh)
elif plotmethod == 2:
ax = fig2.gca(projection='3d')
ax.scatter(pts[:,0], pts[:,1], uh, c=uh, cmap=cm.jet)
else:
# seems unreliable
TRI = tri.Triangulation(pts[:,0], pts[:,1], triangles=mytri)
ax.plot_trisurf(pts[:,0], pts[:,1], uh, TRI.triangles, cmap=cm.jet, linewidth=0.2)
def ex_disc_refine(h0):
print " meshing ..."
p1, t1 = distmesh2d(fd_disc, huniform, h0, bbox, [])
pts, mytri = fixmesh(p1,t1)
edges, tedges = edgelist(pts,mytri)
print " original mesh has %d nodes and %d edges" % (np.shape(pts)[0],np.shape(edges)[0])
fig1 = plt.figure()
plt.subplot(1,2,1)
plotmesh(pts, mytri)
plt.title('original mesh')
rp, rt, e, ind = bdyrefine(pts,mytri,fd_disc,h0)
redges, tmp = edgelist(rp,rt)
print " refined mesh has %d nodes and %d edges" % (np.shape(rp)[0],np.shape(redges)[0])
plt.subplot(1,2,2)
plotmesh(rp, rt)
plt.title('final refined mesh')
uh, inside = poisson(f_ex,fd_disc,h0,rp,rt,announce=True)
print " plotting ..."
fig4 = plt.figure()
ax = fig4.gca(projection='3d')
ax.plot_trisurf(rp[:,0], rp[:,1], uh, cmap=cm.jet, linewidth=0.2)
uexact = 1.0 - rp[:,0]**2.0 - rp[:,1]**2.0 # exact: u(x,y) = 1 - x^2 - y^2
err = max(abs(uh-uexact))
print "max error = %f" % err
def example_3_online():
figure()
pts, tri = distmesh2d(example3_online, example3_online_h, 0.02, bbox,
square)
plotmesh(pts, tri)
show()
def example_3b():
figure()
pts, tri = distmesh2d(example3, example3_h, 0.035, bbox, square)
plotmesh(pts, tri)
show()
def example_3a():
figure()
pts, tri = distmesh2d(example3, huniform, 0.15, bbox, square)
plotmesh(pts, tri, example3(pts))
show()
def example_2():
figure()
pts, tri = distmesh2d(example2, huniform, 0.1, bbox, [])
plotmesh(pts, tri)
show()
def example_1b():
figure()
pts, tri = distmesh2d(example1, huniform, 0.2, bbox, [])
plotmesh(pts, tri)
show()
def example_1b():
figure()
pts, tri = distmesh2d(example1, huniform, 0.2, bbox, [])
plotmesh(pts, tri)
show()
def circle_nonuniform():
figure()
# fake the corners:
pts, tri = distmesh2d(example1, circle_h, 0.1, bbox, [])
plotmesh(pts, tri)
show()
def example_2():
figure()
pts, tri = distmesh2d(example2, huniform, 0.1, bbox, [])
plotmesh(pts, tri)
show()
def annulus():
figure()
pts, tri = distmesh2d(example2, annulus_h, 0.04, bbox, square)
plotmesh(pts, tri)
show()
def example_3a():
figure()
pts, tri = distmesh2d(example3, huniform, 0.15, bbox, square)
plotmesh(pts, tri, example3(pts))
show()
def circle_nonuniform():
figure()
# fake the corners:
pts, tri = distmesh2d(example1, circle_h, 0.1, bbox, [])
plotmesh(pts, tri)
show()
def example_3b():
figure()
pts, tri = distmesh2d(example3, example3_h, 0.035, bbox, square)
plotmesh(pts, tri)
show()
def example_3_online():
figure()
pts, tri = distmesh2d(example3_online, example3_online_h, 0.02, bbox, square)
plotmesh(pts, tri)
show()
def annulus():
figure()
pts, tri = distmesh2d(example2, annulus_h, 0.04, bbox, square)
plotmesh(pts, tri)
show()
