def test_mesh_metric():
mesh = RectangleMesh(0, 0, 1, 1, 20, 20)
mesh = adapt(
interpolate(Constant(((10., 0.), (0., 10.))),
TensorFunctionSpace(mesh, 'CG', 1)))
#extract mesh metric
MpH = mesh_metric2(mesh)
# Plot element i
i = 20
t = linspace(0, 2 * pi, 101)
ind = MpH.function_space().dofmap().cell_dofs(i)
thecell = mesh.cells()[i]
centerxy = mesh.coordinates()[thecell, :].mean(0).repeat(3).reshape([2,
3]).T
cxy = mesh.coordinates()[thecell, :] - centerxy
pyplot(cxy[:, 0], cxy[:, 1], '-b')
H = MpH.vector().gather(ind).reshape(2, 2)
# H = array([[H[1],H[0]],[H[0],H[2]]])
#H = MpH.vector().gather(ind); H = array([[H[1],H[0]],[H[0],H[2]]])
#H = MpH.vector().array()[ind]; H = array([[H[1],H[0]],[H[0],H[2]]])
[v, w] = linalg.eig(H)
v /= pysqrt(3) #v = 1/pysqrt(v)/pysqrt(3)
elxy = array([pycos(t), pysin(t)]).T.dot(w).dot(diag(v)).dot(w.T)
hold('on')
pyplot(elxy[:, 0], elxy[:, 1], '-r')
hold('off')
axis('equal')
print('triangle area: %0.6f, ellipse axis product(*3*sqrt(3)/4): %0.6f' % (
pyabs(linalg.det(array([cxy[1, :] - cxy[0, :], cxy[2, :] - cxy[0, :]
]))) / 2, v[0] * v[1] * 3 * sqrt(3) / 4))
show()
def test_mesh_metric3D():
mesh = BoxMesh(0, 0, 0, 1, 1, 1, 20, 20, 20)
mesh = adapt(
interpolate(Constant(((100., 0., 0.), (0., 100., 0.), (0., 0., 100.))),
TensorFunctionSpace(mesh, 'CG', 1)))
#extract mesh metric
MpH = mesh_metric2(mesh)
for i in range(0, 40):
thecell = mesh.cells()[i]
ind = MpH.function_space().dofmap().cell_dofs(i)
coords = mesh.coordinates()[thecell, :]
r1 = coords[1, :] - coords[0, :]
r2 = coords[2, :] - coords[0, :]
r3 = coords[0:3, :].mean(0) - coords[3, :]
r12c = cross(r2, r1) / 6.
det1234 = pyabs((r12c * r3).sum())
H = MpH.vector().gather(ind).reshape(
3, 3) # H = array([[H[1],H[0]],[H[0],H[2]]])
[v, w] = linalg.eig(H)
print(
'tetrahedron volume: %0.6f, ellipsoid axis product(*sqrt(2)/12): %0.6f'
% (det1234, v.prod() * sqrt(2) / 12))
print v.prod() * sqrt(2) / 12 / (det1234)
def check_metric_ellipse(width=2e-2, eta = 0.02, Nadapt=6):
set_log_level(WARNING)
parameters["allow_extrapolation"] = True
### CONSTANTS
meshsz = 40
hd = Constant(width)
### SETUP MESH
mesh = RectangleMesh(Point(-0.5,-0.5),Point(0.5,0.5),1*meshsz,1*meshsz,"left/right")
### SETUP SOLUTION
angle = pi/8#rand()*pi/2
#testsol = 'tanh(x[0]/' + str(float(hd)) + ')' #tanh(x[0]/hd)
testsol = 'tanh((' + str(cos(angle)) + '*x[0]+'+ str(sin(angle)) + '*x[1])/' + str(float(hd)) + ')' #tanh(x[0]/hd)
ddtestsol = str(cos(angle)+sin(angle))+'*2*'+testsol+'*(1-pow('+testsol+',2))/'+str(float(hd)**2)
#testsol2 = 'tanh(x[1]/' + str(float(hd)) + ')' #tanh(x[0]/hd)
testsol2 = 'tanh((' + str(cos(angle)) + '*x[1]-'+ str(sin(angle)) + '*x[0])/' + str(float(hd)) + ')' #tanh(x[0]/hd)
ddtestsol2 = str(cos(angle)-sin(angle))+'*2*'+testsol2+'*(1-pow('+testsol2+',2))/'+str(float(hd)**2)
def boundary(x):
return x[0]-mesh.coordinates()[:,0].min() < DOLFIN_EPS or mesh.coordinates()[:,0].max()-x[0] < DOLFIN_EPS \
or mesh.coordinates()[:,1].min()+0.5 < DOLFIN_EPS or mesh.coordinates()[:,1].max()-x[1] < DOLFIN_EPS
# PERFORM ONE ADAPTATION ITERATION
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG" ,2); dis = TrialFunction(V); dus = TestFunction(V); u = Function(V); u2 = Function(V)
bc = DirichletBC(V, Expression(testsol), boundary)
bc2 = DirichletBC(V, Expression(testsol2), boundary)
R = interpolate(Expression(ddtestsol),V)
R2 = interpolate(Expression(ddtestsol2),V)
a = inner(grad(dis), grad(dus))*dx
L = R*dus*dx
L2 = R2*dus*dx
solve(a == L, u, bc)
solve(a == L2, u2, bc2)
H = metric_pnorm(u , eta, max_edge_length=1., max_edge_ratio=50); #Mp = project(H, TensorFunctionSpace(mesh, "CG", 1))
H2 = metric_pnorm(u2, eta, max_edge_length=1., max_edge_ratio=50); #Mp2 = project(H2, TensorFunctionSpace(mesh, "CG", 1))
H3 = metric_ellipse(H,H2); Mp3 = project(H3, TensorFunctionSpace(mesh, "CG", 1))
print("H11: %0.0f, H22: %0.0f, V: %0.0f,E: %0.0f" % (assemble(abs(H3[0,0])*dx),assemble(abs(H3[1,1])*dx),mesh.num_vertices(),mesh.num_cells()))
startTime = time()
if iii != 6:
# mesh2 = Mesh(adapt(Mp2))
mesh = Mesh(adapt(Mp3))
# mesh3 = adapt(Mp)
print("total time was %0.1fs" % (time()-startTime))
# PLOT MESH
figure(1); triplot(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells()); axis('equal'); axis('off'); box('off')
#figure(2); triplot(mesh2.coordinates()[:,0],mesh2.coordinates()[:,1],mesh2.cells()) #mesh = mesh2
#figure(3); triplot(mesh3.coordinates()[:,0],mesh3.coordinates()[:,1],mesh3.cells()) #mesh = mesh3
figure(4); testf = interpolate(u2,FunctionSpace(mesh,'CG',1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG" ,1))
zz = testf.vector().array()[vtx2dof]; zz[zz==1] -= 1e-16
tricontourf(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),zz,100)
show()
def minimal_example3D(width=2e-2, Nadapt=10, eta = 0.04):
### CONSTANTS
meshsz = 10
### SETUP MESH
mesh = BoxMesh(-0.5,-0.5,-0.5,0.5,0.5,0.5,meshsz,meshsz,meshsz)
### DERIVE FORCING TERM
angle = pi/8 #rand*pi/2
angle2 = pi/8 #rand*pi/2
sx = Symbol('sx'); sy = Symbol('sy'); sz = Symbol('sz'); width_ = Symbol('ww'); aa = Symbol('aa'); bb = Symbol('bb')
testsol = pytanh((sx*pycos(aa)*pysin(bb)+sy*pysin(aa)*pysin(bb)+sz*pycos(bb))/width_)
ddtestsol = str(diff(testsol,sx,sx)+diff(testsol,sy,sy)+diff(testsol,sz,sz)).replace('sx','x[0]').replace('sy','x[1]').replace('sz','x[2]')
#replace ** with pow
ddtestsol = ddtestsol.replace('tanh((x[0]*sin(aa)*cos(bb) + x[1]*cos(aa)*cos(bb) + x[2]*sin(bb))/ww)**2','pow(tanh((x[0]*sin(aa)*sin(bb) + x[1]*cos(aa)*sin(bb) + x[2]*cos(bb))/ww),2.)')
ddtestsol = ddtestsol.replace('cos(aa)**2','pow(cos(aa),2.)').replace('sin(aa)**2','pow(sin(aa),2.)').replace('ww**2','(ww*ww)').replace('cos(bb)**2','(cos(bb)*cos(bb))').replace('sin(bb)**2','(sin(bb)*sin(bb))')
#insert vaulues
ddtestsol = ddtestsol.replace('aa',str(angle)).replace('ww',str(width)).replace('bb',str(angle2))
testsol = str(testsol).replace('sx','x[0]').replace('sy','x[1]').replace('aa',str(angle)).replace('ww',str(width)).replace('bb',str(angle2)).replace('sz','x[2]')
ddtestsol = "-("+ddtestsol+")"
def boundary(x):
return x[0]-mesh.coordinates()[:,0].min() < DOLFIN_EPS or mesh.coordinates()[:,0].max()-x[0] < DOLFIN_EPS \
or mesh.coordinates()[:,1].min() < DOLFIN_EPS or mesh.coordinates()[:,1].max()-x[1] < DOLFIN_EPS \
or mesh.coordinates()[:,2].min() < DOLFIN_EPS or mesh.coordinates()[:,2].max()-x[2] < DOLFIN_EPS
# PERFORM TEN ADAPTATION ITERATIONS
fid = File("out.pvd")
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG" ,2); dis = TrialFunction(V); dus = TestFunction(V); u = Function(V)
a = inner(grad(dis), grad(dus))*dx
L = Expression(ddtestsol)*dus*dx
bc = DirichletBC(V, Expression(testsol), boundary)
solve(a == L, u, bc)
fid << u
startTime = time()
H = metric_pnorm(u, eta, max_edge_length=2., max_edge_ratio=50, CG1out=True)
#H = logproject(H)
if iii != Nadapt-1:
mesh = adapt(H)
L2error = errornorm(Expression(testsol), u, degree_rise=4, norm_type='L2')
log(INFO+1,"total (adapt+metric) time was %0.1fs, L2error=%0.0e, nodes: %0.0f" % (time()-startTime,L2error,mesh.num_vertices()))
plot(u,interactive=True)
plot(mesh,interactive=True)
def minimal_example3D(width=2e-2, Nadapt=10, eta = 0.04):
### CONSTANTS
meshsz = 10
### SETUP MESH
mesh = BoxMesh(Point(-0.5,-0.5,-0.5),Point(0.5,0.5,0.5),meshsz,meshsz,meshsz)
### DERIVE FORCING TERM
angle = pi/8 #rand*pi/2
angle2 = pi/8 #rand*pi/2
sx = Symbol('sx'); sy = Symbol('sy'); sz = Symbol('sz'); width_ = Symbol('ww'); aa = Symbol('aa'); bb = Symbol('bb')
testsol = pytanh((sx*pycos(aa)*pysin(bb)+sy*pysin(aa)*pysin(bb)+sz*pycos(bb))/width_)
ddtestsol = str(diff(testsol,sx,sx)+diff(testsol,sy,sy)+diff(testsol,sz,sz)).replace('sx','x[0]').replace('sy','x[1]').replace('sz','x[2]')
#replace ** with pow
ddtestsol = ddtestsol.replace('tanh((x[0]*sin(aa)*cos(bb) + x[1]*cos(aa)*cos(bb) + x[2]*sin(bb))/ww)**2','pow(tanh((x[0]*sin(aa)*sin(bb) + x[1]*cos(aa)*sin(bb) + x[2]*cos(bb))/ww),2.)')
ddtestsol = ddtestsol.replace('cos(aa)**2','pow(cos(aa),2.)').replace('sin(aa)**2','pow(sin(aa),2.)').replace('ww**2','(ww*ww)').replace('cos(bb)**2','(cos(bb)*cos(bb))').replace('sin(bb)**2','(sin(bb)*sin(bb))')
#insert vaulues
ddtestsol = ddtestsol.replace('aa',str(angle)).replace('ww',str(width)).replace('bb',str(angle2))
testsol = str(testsol).replace('sx','x[0]').replace('sy','x[1]').replace('aa',str(angle)).replace('ww',str(width)).replace('bb',str(angle2)).replace('sz','x[2]')
ddtestsol = "-("+ddtestsol+")"
def boundary(x):
return x[0]-mesh.coordinates()[:,0].min() < DOLFIN_EPS or mesh.coordinates()[:,0].max()-x[0] < DOLFIN_EPS \
or mesh.coordinates()[:,1].min() < DOLFIN_EPS or mesh.coordinates()[:,1].max()-x[1] < DOLFIN_EPS \
or mesh.coordinates()[:,2].min() < DOLFIN_EPS or mesh.coordinates()[:,2].max()-x[2] < DOLFIN_EPS
# PERFORM TEN ADAPTATION ITERATIONS
fid = File("out.pvd")
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG" ,2); dis = TrialFunction(V); dus = TestFunction(V); u = Function(V)
a = inner(grad(dis), grad(dus))*dx
L = Expression(ddtestsol)*dus*dx
bc = DirichletBC(V, Expression(testsol), boundary)
solve(a == L, u, bc)
fid << u
startTime = time()
H = metric_pnorm(u, eta, max_edge_length=2., max_edge_ratio=50, CG1out=True)
#H = logproject(H)
if iii != Nadapt-1:
mesh = adapt(H)
L2error = errornorm(Expression(testsol), u, degree_rise=4, norm_type='L2')
log(INFO+1,"total (adapt+metric) time was %0.1fs, L2error=%0.0e, nodes: %0.0f" % (time()-startTime,L2error,mesh.num_vertices()))
plot(u,interactive=True)
plot(mesh,interactive=True)
def test_mesh_metric3D():
mesh = BoxMesh(0,0,0,1,1,1,20,20,20)
mesh = adapt(interpolate(Constant(((100.,0.,0.),(0.,100.,0.),(0.,0.,100.))),TensorFunctionSpace(mesh,'CG',1)))
#extract mesh metric
MpH = mesh_metric2(mesh)
for i in range(0,40):
thecell = mesh.cells()[i]
ind = MpH.function_space().dofmap().cell_dofs(i)
coords = mesh.coordinates()[thecell,:]
r1 = coords[1,:]-coords[0,:]
r2 = coords[2,:]-coords[0,:]
r3 = coords[0:3,:].mean(0)-coords[3,:]
r12c = cross(r2,r1)/6.
det1234 = pyabs((r12c*r3).sum())
H = MpH.vector().gather(ind).reshape(3,3)# H = array([[H[1],H[0]],[H[0],H[2]]])
[v,w] = linalg.eig(H)
print('tetrahedron volume: %0.6f, ellipsoid axis product(*sqrt(2)/12): %0.6f' % (det1234,v.prod()*sqrt(2)/12))
print v.prod()*sqrt(2)/12/(det1234)
def test_mesh_metric():
mesh = RectangleMesh(0,0,1,1,20,20)
mesh = adapt(interpolate(Constant(((10.,0.),(0.,10.))),TensorFunctionSpace(mesh,'CG',1)))
#extract mesh metric
MpH = mesh_metric2(mesh)
# Plot element i
i = 20; t = linspace(0,2*pi,101)
ind = MpH.function_space().dofmap().cell_dofs(i)
thecell = mesh.cells()[i]
centerxy = mesh.coordinates()[thecell,:].mean(0).repeat(3).reshape([2,3]).T
cxy = mesh.coordinates()[thecell,:]-centerxy
pyplot(cxy[:,0],cxy[:,1],'-b')
H = MpH.vector().gather(ind).reshape(2,2);# H = array([[H[1],H[0]],[H[0],H[2]]])
#H = MpH.vector().gather(ind); H = array([[H[1],H[0]],[H[0],H[2]]])
#H = MpH.vector().array()[ind]; H = array([[H[1],H[0]],[H[0],H[2]]])
[v,w] = linalg.eig(H); v /= pysqrt(3) #v = 1/pysqrt(v)/pysqrt(3)
elxy = array([pycos(t),pysin(t)]).T.dot(w).dot(diag(v)).dot(w.T)
hold('on'); pyplot(elxy[:,0],elxy[:,1],'-r'); hold('off'); axis('equal')
print('triangle area: %0.6f, ellipse axis product(*3*sqrt(3)/4): %0.6f' % (pyabs(linalg.det(array([cxy[1,:]-cxy[0,:],cxy[2,:]-cxy[0,:]])))/2,v[0]*v[1]*3*sqrt(3)/4))
show()
def test_refine_metric():
# from mpi4py import MPI
import sys
# comm = MPI.COMM_WORLD
# mesh = Mesh("greenland.xml.gz")
mesh = UnitSquareMesh(100, 100)
V = FunctionSpace(mesh, "CG", 2)
f = interpolate(Expression("0.1*sin(50.*(2*x[0]-1)) + atan2(-0.1, (2.0*(2*x[0]-1) - sin(5.*(2*x[1]-1))))"), V)
eta = 0.01
#Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
#mesh = adapt(Mp)
if True:
level = 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh1 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh2 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh3 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh4 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh5 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh6 = adapt(Mp)
else:
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh1 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh2 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh3 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh4 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh5 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh6 = adapt(Mp)
# plot(Mp[0,0])
# from IPython import embed
# embed()
plot(mesh, title="initial mesh")
plot(new_mesh1, title="coarsen 1")
plot(new_mesh2, title="coarsen 2")
plot(new_mesh3, title="coarsen 3")
plot(new_mesh4, title="coarsen 4")
plot(new_mesh5, title="coarsen 5")
plot(new_mesh6, title="coarsen 6")
interactive()
def adv_convergence(width=2e-2,
delta=1e-2,
relp=1,
Nadapt=10,
use_adapt=True,
problem=3,
outname='',
use_reform=False,
CGorderL=[2, 3],
noplot=False,
Lx=3.):
### SETUP SOLUTION
sy = Symbol('sy')
width_ = Symbol('ww')
if problem == 3:
stepfunc = 0.5 + 165. / 104. / width_ * sy - 20. / 13. / width_**3 * sy**3 - 102. / 13. / width_**5 * sy**5 + 240. / 13. / width_**7 * sy**7
elif problem == 2:
stepfunc = 0.5 + 15. / 8. / width_ * sy - 5. / width_**3 * sy**3 + 6. / width_**5 * sy**5
elif problem == 1:
stepfunc = 0.5 + 1.5 / width_ * sy - 2 / width_**3 * sy**3
stepfunc = str(stepfunc).replace('sy',
'x[1]').replace('x[1]**2', '(x[1]*x[1])')
#REPLACE ** with pow
stepfunc = stepfunc.replace('x[1]**3', 'pow(x[1],3.)')
stepfunc = stepfunc.replace('x[1]**5', 'pow(x[1],5.)')
stepfunc = stepfunc.replace('x[1]**7', 'pow(x[1],7.)')
testsol = '(-ww/2 < x[1] && x[1] < ww/2 ? ' + stepfunc + ' : 0) + (ww/2<x[1] ? 1 : 0)'
testsol = testsol.replace('ww**2', '(ww*ww)').replace(
'ww**3',
'pow(ww,3.)').replace('ww**5',
'pow(ww,5.)').replace('ww**7', 'pow(ww,7.)')
testsol = testsol.replace('ww', str(width))
dp = Constant(1.)
fac = Constant(1 + 2. * delta)
delta = Constant(delta)
def left(x, on_boundary):
return x[0] + Lx / 2. < DOLFIN_EPS
def right(x, on_boundary):
return x[0] - Lx / 2. > -DOLFIN_EPS
def top_bottom(x, on_boundary):
return x[1] - 0.5 > -DOLFIN_EPS or x[1] + 0.5 < DOLFIN_EPS
class Inletbnd(SubDomain):
def inside(self, x, on_boundary):
return x[0] + Lx / 2. < DOLFIN_EPS
for CGorder in [2]: #CGorderL:
dofs = []
L2errors = []
for eta in 0.04 * pyexp2(-array(range(9)) * pylog(2) / 2):
### SETUP MESH
meshsz = int(
round(80 * 0.005 / (eta * (bool(use_adapt) == False) + 0.05 *
(bool(use_adapt) == True))))
if not bool(use_adapt) and meshsz > 80:
continue
mesh = RectangleMesh(-Lx / 2., -0.5, Lx / 2., 0.5, meshsz, meshsz,
"left/right")
# PERFORM TEN ADAPTATION ITERATIONS
for iii in range(Nadapt):
V = VectorFunctionSpace(mesh, "CG", CGorder)
Q = FunctionSpace(mesh, "CG", CGorder - 1)
W = V * Q
(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)
alpha = Expression(
"-0.25<x[0] && x[0]<0.25 && 0. < x[1] ? 1e4 : 0")
boundaries = FacetFunction("size_t", mesh)
#outletbnd = Outletbnd()
inletbnd = Inletbnd()
boundaries.set_all(0)
#outletbnd.mark(boundaries, 1)
inletbnd.mark(boundaries, 1)
ds = Measure("ds")[boundaries]
bc0 = DirichletBC(W.sub(0), Constant((0., 0.)), top_bottom)
bc1 = DirichletBC(W.sub(1), dp, left)
bc2 = DirichletBC(W.sub(1), Constant(0), right)
bc3 = DirichletBC(W.sub(0).sub(1), Constant(0.), left)
bc4 = DirichletBC(W.sub(0).sub(1), Constant(0.), right)
bcs = [bc0, bc1, bc2, bc3, bc4]
bndterm = dp * dot(v, Constant((-1., 0.))) * ds(1)
a = eta * inner(grad(u), grad(
v)) * dx - div(v) * p * dx + q * div(u) * dx + alpha * dot(
u, v) * dx #+bndterm
L = inner(Constant((0., 0.)), v) * dx - bndterm
U = Function(W)
solve(a == L, U, bcs)
u, ps = U.split()
#SOLVE CONCENTRATION
mm = mesh_metric2(mesh)
vdir = u / sqrt(inner(u, u) + DOLFIN_EPS)
if iii == 0 or use_reform == False:
Q2 = FunctionSpace(mesh, 'CG', 2)
c = Function(Q2)
q = TestFunction(Q2)
p = TrialFunction(Q2)
newq = (q + dot(vdir, dot(mm, vdir)) * inner(grad(q), vdir)
) #SUPG
if use_reform:
F = newq * (fac / (
(1 + exp(-c))**2) * exp(-c)) * inner(grad(c), u) * dx
J = derivative(F, c)
bc = DirichletBC(
Q2,
Expression("-log(" + str(float(fac)) + "/(" + testsol +
"+" + str(float(delta)) + ")-1)"), left)
# bc = DirichletBC(Q, -ln(fac/(Expression(testsol)+delta)-1), left)
problem = NonlinearVariationalProblem(F, c, bc, J)
solver = NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"][
"relaxation_parameter"] = relp
solver.solve()
else:
a2 = newq * inner(grad(p), u) * dx
bc = DirichletBC(Q2, Expression(testsol), left)
L2 = Constant(0.) * q * dx
solve(a2 == L2, c, bc)
if (not bool(use_adapt)) or iii == Nadapt - 1:
break
um = project(sqrt(inner(u, u)), FunctionSpace(mesh, 'CG', 2))
H = metric_pnorm(um,
eta,
max_edge_ratio=1 + 49 * (use_adapt != 2),
p=2)
H2 = metric_pnorm(c,
eta,
max_edge_ratio=1 + 49 * (use_adapt != 2),
p=2)
#H3 = metric_pnorm(ps , eta, max_edge_ratio=1+49*(use_adapt!=2), p=2)
H4 = metric_ellipse(H, H2)
#H5 = metric_ellipse(H3,H4,mesh)
mesh = adapt(H4)
if use_reform:
Q2 = FunctionSpace(mesh, 'CG', 2)
c = interpolate(c, Q2)
if use_reform:
c = project(fac / (1 + exp(-c)) - delta,
FunctionSpace(mesh, 'CG', 2))
L2error = bnderror(c, Expression(testsol), ds)
dofs.append(len(c.vector().array()) + len(U.vector().array()))
L2errors.append(L2error)
# fid = open("DOFS_L2errors_mesh_c_CG"+str(CGorder)+outname+".mpy",'w')
# pickle.dump([dofs[0],L2errors[0],c.vector().array().min(),c.vector().array().max()-1,mesh.cells(),mesh.coordinates(),c.vector().array()],fid)
# fid.close();
log(
INFO + 1,
"%1dX ADAPT<->SOLVE complete: DOF=%5d, error=%0.0e, min(c)=%0.0e,max(c)-1=%0.0e"
% (Nadapt, dofs[len(dofs) - 1], L2error,
c.vector().array().min(), c.vector().array().max() - 1))
# PLOT MESH + solution
figure()
testf = interpolate(c, FunctionSpace(mesh, 'CG', 1))
testfe = interpolate(Expression(testsol), FunctionSpace(mesh, 'CG', 1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG", 1))
zz = testf.vector().array()[vtx2dof]
zz[zz == 1] -= 1e-16
hh = tricontourf(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1],
mesh.cells(),
zz,
100,
cmap=get_cmap('binary'))
colorbar(hh)
hold('on')
triplot(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1],
mesh.cells(),
color='r',
linewidth=0.5)
hold('off')
axis('equal')
box('off')
# savefig(outname+'final_mesh_CG2.png',dpi=300) #; savefig('outname+final_mesh_CG2.eps',dpi=300)
#PLOT ERROR
figure()
xe = interpolate(Expression("x[0]"),
FunctionSpace(mesh, 'CG', 1)).vector().array()
ye = interpolate(Expression("x[1]"),
FunctionSpace(mesh, 'CG', 1)).vector().array()
I = xe - Lx / 2 > -DOLFIN_EPS
I2 = ye[I].argsort()
pyplot(ye[I][I2],
testf.vector().array()[I][I2] - testfe.vector().array()[I][I2],
'-b')
ylabel('error')
# PLOT L2error graph
figure()
pyloglog(dofs, L2errors, '-b.', linewidth=2, markersize=16)
xlabel('Degree of freedoms')
ylabel('L2 error')
# SAVE SOLUTION
dofs = array(dofs)
L2errors = array(L2errors)
fid = open("DOFS_L2errors_CG" + str(CGorder) + outname + ".mpy", 'w')
pickle.dump([dofs, L2errors], fid)
fid.close()
# #show()
# #LOAD SAVED SOLUTIONS
# fid = open("DOFS_L2errors_CG2"+outname+".mpy",'r')
# [dofs,L2errors] = pickle.load(fid)
# fid.close()
#
# PERFORM FITS ON LAST THREE POINTS
NfitP = 5
I = array(range(len(dofs) - NfitP, len(dofs)))
slope, ints = polyfit(pylog(dofs[I]), pylog(L2errors[I]), 1)
fid = open("DOFS_L2errors_CG2_fit" + outname + ".mpy", 'w')
pickle.dump([dofs, L2errors, slope, ints], fid)
fid.close()
#PLOT THEM TOGETHER
if CGorderL != [2]:
fid = open("DOFS_L2errors_CG3.mpy", 'r')
[dofs_old, L2errors_old] = pickle.load(fid)
fid.close()
slope2, ints2 = polyfit(pylog(dofs_old[I]), pylog(L2errors_old[I]), 1)
figure()
pyloglog(dofs,
L2errors,
'-b.',
dofs_old,
L2errors_old,
'--b.',
linewidth=2,
markersize=16)
hold('on')
pyloglog(dofs,
pyexp2(ints) * dofs**slope,
'-r',
dofs_old,
pyexp2(ints2) * dofs_old**slope2,
'--r',
linewidth=1)
hold('off')
xlabel('Degree of freedoms')
ylabel('L2 error')
legend([
'CG2', 'CG3',
"%0.2f*log(DOFs)" % slope,
"%0.2f*log(DOFs)" % slope2
]) #legend(['new data','old_data'])
# savefig('comparison.png',dpi=300) #savefig('comparison.eps');
if not noplot:
show()
def minimal_example3D(meshsz=20,
Nadapt=10,
dL=0.05,
eta=0.01,
returnmesh=False,
hax=False):
class inlet(SubDomain):
def inside(self, x, on_boundary):
return 0.5-DOLFIN_EPS < x[0] \
and -dL-DOLFIN_EPS < x[1] and x[1] < dL+DOLFIN_EPS \
and -dL-DOLFIN_EPS < x[2] and x[2] < dL+DOLFIN_EPS
def boundary(x, on_boundary): #not(boundary(x))
return on_boundary and ((0.5-DOLFIN_EPS >= x[0] \
or -dL-DOLFIN_EPS >= x[1] or x[1] >= dL+DOLFIN_EPS \
or -dL-DOLFIN_EPS >= x[2] or x[2] >= dL+DOLFIN_EPS))
def get_bnd_mesh(mesh):
coords = mesh.coordinates()
[bfaces, bfaces_IDs] = polyhedron_surfmesh(mesh)
bcoords = (coords[bfaces[:, 0], :] + coords[bfaces[:, 1], :] +
coords[bfaces[:, 2], :]) / 3.
I = (bcoords[:,0] > 0.5-DOLFIN_EPS) & (bcoords[:,1] < dL) & (bcoords[:,1] > -dL) \
& (bcoords[:,2] < dL) & (bcoords[:,2] > -dL)
I2 = (bcoords[:,0] > 0.5-DOLFIN_EPS) & (bcoords[:,1] < dL) & (bcoords[:,1] > -dL) \
& (bcoords[:,2] > dL)
I3 = (bcoords[:,0] > 0.5-DOLFIN_EPS) & (bcoords[:,1] < dL) & (bcoords[:,1] > -dL) \
& (bcoords[:,2] < -dL)
bfaces_IDs[I] = 97
if hax:
bfaces_IDs[I2] = 98
bfaces_IDs[I3] = 99 #problems
c1 = numpy.array([0.5, -dL, -dL]).repeat(coords.shape[0]).reshape(
[3, coords.shape[0]]).T
c2 = numpy.array([0.5, -dL, dL]).repeat(coords.shape[0]).reshape(
[3, coords.shape[0]]).T
c3 = numpy.array([0.5, dL, -dL]).repeat(coords.shape[0]).reshape(
[3, coords.shape[0]]).T
c4 = numpy.array([0.5, dL, dL]).repeat(coords.shape[0]).reshape(
[3, coords.shape[0]]).T
crnds = numpy.where((numpy.sqrt(((coords - c1)**2).sum(1)) < 1e3*DOLFIN_EPS) | \
(numpy.sqrt(((coords - c2)**2).sum(1)) < 1e3*DOLFIN_EPS) | \
(numpy.sqrt(((coords - c3)**2).sum(1)) < 1e3*DOLFIN_EPS) | \
(numpy.sqrt(((coords - c4)**2).sum(1)) < 1e3*DOLFIN_EPS))[0]
# return [bfaces,bfaces_IDs,crnds]
return [bfaces, bfaces_IDs]
### SETUP MESH
mesh = BoxMesh(-0.5, -0.5, -0.5, 0.5, 0.5, 0.5, meshsz, meshsz, meshsz)
fid = File("out.pvd")
# PERFORM TEN ADAPTATION ITERATIONS
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG", 2)
dis = TrialFunction(V)
dus = TestFunction(V)
u = Function(V)
a = inner(grad(dis), grad(dus)) * dx
boundaries = FacetFunction("size_t", mesh)
Inlet = inlet()
boundaries.set_all(0)
Inlet.mark(boundaries, 1)
ds = Measure("ds")[boundaries]
L = dus * ds(1) #+dus*dx
bc = DirichletBC(V, Constant(0.), boundary)
solve(a == L, u, bc)
fid << u
startTime = time()
H = metric_pnorm(u,
eta,
max_edge_length=2.,
max_edge_ratio=50,
CG1out=True)
#H = logproject(H)
if iii != Nadapt - 1:
[bfaces, bfaces_IDs] = get_bnd_mesh(mesh)
mesh = adapt(H, bfaces=bfaces, bfaces_IDs=bfaces_IDs)
log(
INFO + 1,
"total (adapt+metric) time was %0.1fs, nodes: %0.0f" %
(time() - startTime, mesh.num_vertices()))
plot(u, interactive=True)
plot(mesh, interactive=True)
def ellipse_convergence(asp=2,width=1e-2, Nadapt=10, eta_list=0.04*pyexp2(-array(range(15))*pylog(2)/2), \
use_adapt=True, problem=2, outname='', CGorderL = [2, 3], noplot=False, octaveimpl=False):
### SETUP SOLUTION
sx = Symbol('sx'); sy = Symbol('sy'); width_ = Symbol('ww'); asp_ = Symbol('a')
rrpy = pysqrt(sx*sx/asp_+sy*sy*asp_)
if problem == 2:
stepfunc = 0.5+165./104./width_*(rrpy-0.25)-20./13./width_**3*(rrpy-0.25)**3-102./13./width_**5*(rrpy-0.25)**5+240./13./width_**7*(rrpy-0.25)**7
elif problem == 1:
stepfunc = 0.5+15./8./width_*(rrpy-0.25)-5./width_**3*(rrpy-0.25)**3+6./width_**5*(rrpy-0.25)**5
else:
stepfunc = 0.5+1.5/width_*(rrpy-0.25)-2/width_**3*(rrpy-0.25)**3
ddstepfunc = str(diff(stepfunc,sx,sx)+diff(stepfunc,sy,sy)).replace('sx','x[0]').replace('sy','x[1]').replace('x[0]**2','(x[0]*x[0])').replace('x[1]**2','(x[1]*x[1])')
stepfunc = str(stepfunc).replace('sx','x[0]').replace('sy','x[1]').replace('x[0]**2','(x[0]*x[0])').replace('x[1]**2','(x[1]*x[1])')
#REPLACE ** with pow
stepfunc = stepfunc.replace('(a*(x[1]*x[1]) + (x[0]*x[0])/a)**(1/2)','sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a)')
ddstepfunc = ddstepfunc.replace('(a*(x[1]*x[1]) + (x[0]*x[0])/a)**(1/2)','sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a)')
ddstepfunc = ddstepfunc.replace('(a*(x[1]*x[1]) + (x[0]*x[0])/a)**(3/2)','pow(a*x[1]*x[1]+x[0]*x[0]/a,1.5)')
ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**2','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,2.)')
ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**3','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,3.)')
ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**4','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,4.)')
ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**5','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,5.)')
ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**6','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,6.)')
stepfunc = stepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**3','pow(sqrt(a*x[1]*x[1] + x[0]*x[0]/a) - 0.25,3.)')
stepfunc = stepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**5','pow(sqrt(a*x[1]*x[1] + x[0]*x[0]/a) - 0.25,5.)')
stepfunc = stepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**7','pow(sqrt(a*x[1]*x[1] + x[0]*x[0]/a) - 0.25,7.)')
testsol = '(0.25-ww/2<sqrt(x[0]*x[0]/a+x[1]*x[1]*a) && sqrt(x[0]*x[0]/a+x[1]*x[1]*a) < 0.25+ww/2 ? (' + stepfunc + ') : 0) + (0.25+ww/2<sqrt(x[0]*x[0]/a+x[1]*x[1]*a) ? 1 : 0)'
# testsol = '(0.25-ww/2<sqrt(x[0]*x[0]+x[1]*x[1]) && sqrt(x[0]*x[0]+x[1]*x[1]) < 0.25+ww/2 ? (' + stepfunc + ')) : (0.25<sqrt(x[0]*x[0]+x[1]*x[1]) ? 1 : 0)'
ddtestsol = '0.25-ww/2<sqrt(x[0]*x[0]/a+x[1]*x[1]*a) && sqrt(x[0]*x[0]/a+x[1]*x[1]*a) < 0.25+ww/2 ? (' + ddstepfunc + ') : 0'
# A = array([[0.5,0.5**3,0.5**5],[1,3*0.5**2,5*0.5**4],[0,6*0.5,20*0.5**3]]); b = array([0.5,0,0])
# from numpy.linalg import solve as pysolve #15/8,-5,6
# X = pysolve(A,b); from numpy import linspace; xx = linspace(-0.5,0.5,100)
# from pylab import plot as pyplot; pyplot(xx,X[0]*xx+X[1]*xx**3+X[2]*xx**5,'-b')
# rrpy = pysqrt(sx*sx+sy*sy)
#
# ddstepfunc = str(diff(stepfunc,sx,sx)+diff(stepfunc,sy,sy)).replace('sx','x[0]').replace('sy','x[1]').replace('x[0]**2','(x[0]*x[0])').replace('x[1]**2','(x[1]*x[1])')
# stepfunc = str(stepfunc).replace('sx','x[0]').replace('sy','x[1]').replace('x[0]**2','(x[0]*x[0])').replace('x[1]**2','(x[1]*x[1])')
# #REPLACE ** with pow
# ddstepfunc = ddstepfunc.replace('(a*(x[1]*x[1]) + (x[0]*x[0])/a)**(3/2)','pow(a*x[1]*x[1]+x[0]*x[0]/a,1.5)')
# ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**2','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,2.)')
# ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**3','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,3.)')
# ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**4','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,4.)')
# stepfunc = stepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**3','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,3.)')
# stepfunc = stepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**5','pow(sqrt(a*x[1]*x[1]+x[0]*x[0]/a) - 0.25,5.)')
# testsol = '(0.25-ww/2<sqrt(x[0]*x[0]+x[1]*x[1]) && sqrt(x[0]*x[0]+x[1]*x[1]) < 0.25+ww/2 ? (' + stepfunc + ') : 0) + (0.25+ww/2<sqrt(x[0]*x[0]+x[1]*x[1]) ? 1 : 0)'
## testsol = '(0.25-ww/2<sqrt(x[0]*x[0]+x[1]*x[1]) && sqrt(x[0]*x[0]+x[1]*x[1]) < 0.25+ww/2 ? (' + stepfunc + ')) : (0.25<sqrt(x[0]*x[0]+x[1]*x[1]) ? 1 : 0)'
# ddtestsol = '0.25-ww/2<sqrt(x[0]*x[0]+x[1]*x[1]) && sqrt(x[0]*x[0]+x[1]*x[1]) < 0.25+ww/2 ? (' + ddstepfunc + ') : 0'
# else: # problem == 0:
# rrpy = pysqrt(sx*sx+sy*sy)
# #'if(t<2*WeMax,0,if(t<4*WeMax,0.5+3/2/(2*WeMax)*(t-3*WeMax)-2/(2*WeMax)^3*(t-3*WeMax)^3,1))'; %0.5+3/2/dx*(x-xc)-2/dx^3*(x-xc)^3
# ddstepfunc = str(diff(stepfunc,sx,sx)+diff(stepfunc,sy,sy)).replace('sx','x[0]').replace('sy','x[1]').replace('x[0]**2','(x[0]*x[0])').replace('x[1]**2','(x[1]*x[1])')
# stepfunc = str(stepfunc).replace('sx','x[0]').replace('sy','x[1]').replace('x[0]**2','(x[0]*x[0])').replace('x[1]**2','(x[1]*x[1])')
# #REPLACE ** with pow
# ddstepfunc = ddstepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**2','pow(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25,2.)')
# ddstepfunc = ddstepfunc.replace('(a*(x[1]*x[1]) + (x[0]*x[0])/a)**(3/2)','pow(a*(x[1]*x[1]) + (x[0]*x[0])/a,1.5)')
# stepfunc = stepfunc.replace('(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25)**3','pow(sqrt(a*(x[1]*x[1]) + (x[0]*x[0])/a) - 0.25,3.)')
# testsol = '0.25-ww/2<sqrt(x[0]*x[0]+x[1]*x[1]) && sqrt(x[0]*x[0]+x[1]*x[1]) < 0.25+ww/2 ? (' + stepfunc + ') : (0.25<sqrt(x[0]*x[0]+x[1]*x[1]) ? 1 : 0)'
# ddtestsol = '0.25-ww/2<sqrt(x[0]*x[0]+x[1]*x[1]) && sqrt(x[0]*x[0]+x[1]*x[1]) < 0.25+ww/2 ? (' + ddstepfunc + ') : 0'
ddtestsol = ddtestsol.replace('a**2','(a*a)').replace('ww**2','(ww*ww)').replace('ww**3','pow(ww,3.)').replace('ww**4','pow(ww,4.)').replace('ww**5','pow(ww,5.)').replace('ww**6','pow(ww,6.)').replace('ww**7','pow(ww,7.)')
testsol = testsol.replace('ww**2','(ww*ww)').replace('ww**3','pow(ww,3.)').replace('ww**5','pow(ww,5.)').replace('ww**7','pow(ww,7.)')
ddtestsol = ddtestsol.replace('ww',str(width)).replace('a',str(asp))
testsol = testsol.replace('ww',str(width)).replace('a',str(asp))
ddtestsol = '-('+ddtestsol+')'
def boundary(x):
return x[0]+0.5 < DOLFIN_EPS or 0.5-x[0] < DOLFIN_EPS or x[1]+0.5 < DOLFIN_EPS or 0.5-x[1] < DOLFIN_EPS
for CGorder in CGorderL:
dofs = []
L2errors = []
#for eta in [0.16, 0.08, 0.04, 0.02, 0.01, 0.005, 0.0025] #, 0.0025/2, 0.0025/4, 0.0025/8]: #
for eta in eta_list:
### SETUP MESH
meshsz = int(round(80*0.005/(eta*(bool(use_adapt)==False)+0.05*(bool(use_adapt)==True))))
if (not bool(use_adapt)) and meshsz > 80:
continue
mesh = RectangleMesh(-0.0,-0.0,0.5*sqrt(asp),0.5/sqrt(asp),meshsz,meshsz,"left/right")
# PERFORM TEN ADAPTATION ITERATIONS
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG", CGorder); dis = TrialFunction(V); dus = TestFunction(V); u = Function(V)
#V2 = FunctionSpace(mesh, "CG", CGorder+2)
R = Expression(ddtestsol) #interpolate(Expression(ddtestsol),V2)
a = inner(grad(dis), grad(dus))*dx
L = R*dus*dx
bc = DirichletBC(V, Expression(testsol), boundary) #Constant(0.)
solve(a == L, u, bc)
if not bool(use_adapt):
break
H = metric_pnorm(u, eta, max_edge_ratio=1+49*(use_adapt!=2), p=2)
H = logproject(H)
if iii != Nadapt-1:
mesh = adapt(H, octaveimpl=octaveimpl, debugon=False)
L2error = errornorm(Expression(testsol), u, degree_rise=CGorder+2, norm_type='L2')
dofs.append(len(u.vector().array()))
L2errors.append(L2error)
log(INFO+1,"%1dX ADAPT<->SOLVE complete: DOF=%5d, error=%0.0e" % (Nadapt, dofs[len(dofs)-1], L2error))
# PLOT MESH + solution
figure()
testf = interpolate(u ,FunctionSpace(mesh,'CG',1))
testfe = interpolate(Expression(testsol),FunctionSpace(mesh,'CG',1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG" ,1))
zz = testf.vector().array()[vtx2dof]; zz[zz==1] -= 1e-16
hh=tricontourf(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),zz,100,cmap=get_cmap('binary'))
colorbar(hh)
axis('equal'); axis('off'); box('off'); xlim([0,0.5*sqrt(asp)]); ylim([0, 0.5/sqrt(asp)]); savefig('solution.png',dpi=300)
figure()
hold('on'); triplot(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),color='r',linewidth=0.5); hold('off')
axis('equal'); box('off')
axis('off'); xlim([0,0.5*sqrt(asp)]); ylim([0, 0.5/sqrt(asp)])
savefig(outname+'final_mesh_CG2.png',dpi=300) #; savefig('outname+final_mesh_CG2.eps',dpi=300)
#PLOT ERROR
figure()
zz = pyabs(testf.vector().array()-testfe.vector().array())[vtx2dof]; zz[zz==1] -= 1e-16
hh=tricontourf(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),zz,100,cmap=get_cmap('binary'))
colorbar(hh); axis('equal'); box('off'); title('error')
# PLOT L2error graph
figure()
pyloglog(dofs,L2errors,'-b.',linewidth=2,markersize=16); xlabel('Degree of freedoms'); ylabel('L2 error')
# SAVE SOLUTION
dofs = array(dofs); L2errors = array(L2errors)
fid = open("DOFS_L2errors_CG"+str(CGorder)+outname+".mpy",'w')
pickle.dump([dofs,L2errors],fid)
fid.close();
#LOAD SAVED SOLUTIONS
fid = open("DOFS_L2errors_CG2"+outname+".mpy",'r')
[dofs,L2errors] = pickle.load(fid)
fid.close()
# PERFORM FITS ON LAST THREE POINTS
NfitP = 9
I = array(range(len(dofs)-NfitP,len(dofs)))
slope,ints = polyfit(pylog(dofs[I]), pylog(L2errors[I]), 1)
if slope < -0.7:
fid = open("DOFS_L2errors_CG2_fit"+outname+".mpy",'w')
pickle.dump([dofs,L2errors,slope,ints],fid)
fid.close()
log(INFO+1,'succes')
else:
os.system('rm '+outname+'.lock')
log(INFO+1,'fail')
#PLOT THEM TOGETHER
if CGorderL != [2]:
fid = open("DOFS_L2errors_CG3.mpy",'r')
[dofs_old,L2errors_old] = pickle.load(fid)
fid.close()
slope2,ints2 = polyfit(pylog(dofs_old[I]), pylog(L2errors_old[I]), 1)
figure()
pyloglog(dofs,L2errors,'-b.',dofs_old,L2errors_old,'--b.',linewidth=2,markersize=16)
hold('on'); pyloglog(dofs,pyexp2(ints)*dofs**slope,'-r',dofs_old,pyexp2(ints2)*dofs_old**slope2,'--r',linewidth=1); hold('off')
xlabel('Degree of freedoms'); ylabel('L2 error')
legend(['CG2','CG3',"%0.2f*log(DOFs)" % slope, "%0.2f*log(DOFs)" % slope2]) #legend(['new data','old_data'])
# savefig('comparison.png',dpi=300) #savefig('comparison.eps');
if not noplot:
show()
def check_metric_ellipse(width=2e-2, eta=0.02, Nadapt=6):
set_log_level(WARNING)
parameters["allow_extrapolation"] = True
### CONSTANTS
meshsz = 40
hd = Constant(width)
### SETUP MESH
mesh = RectangleMesh(-0.5, -0.5, 0.5, 0.5, 1 * meshsz, 1 * meshsz,
"left/right")
### SETUP SOLUTION
angle = pi / 8 #rand()*pi/2
#testsol = 'tanh(x[0]/' + str(float(hd)) + ')' #tanh(x[0]/hd)
testsol = 'tanh((' + str(cos(angle)) + '*x[0]+' + str(
sin(angle)) + '*x[1])/' + str(float(hd)) + ')' #tanh(x[0]/hd)
ddtestsol = str(cos(angle) + sin(angle)
) + '*2*' + testsol + '*(1-pow(' + testsol + ',2))/' + str(
float(hd)**2)
#testsol2 = 'tanh(x[1]/' + str(float(hd)) + ')' #tanh(x[0]/hd)
testsol2 = 'tanh((' + str(cos(angle)) + '*x[1]-' + str(
sin(angle)) + '*x[0])/' + str(float(hd)) + ')' #tanh(x[0]/hd)
ddtestsol2 = str(cos(angle) - sin(
angle)) + '*2*' + testsol2 + '*(1-pow(' + testsol2 + ',2))/' + str(
float(hd)**2)
def boundary(x):
return x[0]-mesh.coordinates()[:,0].min() < DOLFIN_EPS or mesh.coordinates()[:,0].max()-x[0] < DOLFIN_EPS \
or mesh.coordinates()[:,1].min()+0.5 < DOLFIN_EPS or mesh.coordinates()[:,1].max()-x[1] < DOLFIN_EPS
# PERFORM ONE ADAPTATION ITERATION
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG", 2)
dis = TrialFunction(V)
dus = TestFunction(V)
u = Function(V)
u2 = Function(V)
bc = DirichletBC(V, Expression(testsol), boundary)
bc2 = DirichletBC(V, Expression(testsol2), boundary)
R = interpolate(Expression(ddtestsol), V)
R2 = interpolate(Expression(ddtestsol2), V)
a = inner(grad(dis), grad(dus)) * dx
L = R * dus * dx
L2 = R2 * dus * dx
solve(a == L, u, bc)
solve(a == L2, u2, bc2)
H = metric_pnorm(u, eta, max_edge_length=1., max_edge_ratio=50)
#Mp = project(H, TensorFunctionSpace(mesh, "CG", 1))
H2 = metric_pnorm(u2, eta, max_edge_length=1., max_edge_ratio=50)
#Mp2 = project(H2, TensorFunctionSpace(mesh, "CG", 1))
H3 = metric_ellipse(H, H2)
Mp3 = project(H3, TensorFunctionSpace(mesh, "CG", 1))
print("H11: %0.0f, H22: %0.0f, V: %0.0f,E: %0.0f" %
(assemble(abs(H3[0, 0]) * dx), assemble(
abs(H3[1, 1]) * dx), mesh.num_vertices(), mesh.num_cells()))
startTime = time()
if iii != 6:
# mesh2 = Mesh(adapt(Mp2))
mesh = Mesh(adapt(Mp3))
# mesh3 = adapt(Mp)
print("total time was %0.1fs" % (time() - startTime))
# PLOT MESH
figure(1)
triplot(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1], mesh.cells())
axis('equal')
axis('off')
box('off')
#figure(2); triplot(mesh2.coordinates()[:,0],mesh2.coordinates()[:,1],mesh2.cells()) #mesh = mesh2
#figure(3); triplot(mesh3.coordinates()[:,0],mesh3.coordinates()[:,1],mesh3.cells()) #mesh = mesh3
figure(4)
testf = interpolate(u2, FunctionSpace(mesh, 'CG', 1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG", 1))
zz = testf.vector().array()[vtx2dof]
zz[zz == 1] -= 1e-16
tricontourf(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1], mesh.cells(), zz, 100)
show()
def test_refine_metric():
# from mpi4py import MPI
import sys
# comm = MPI.COMM_WORLD
# mesh = Mesh("greenland.xml.gz")
mesh = UnitSquareMesh(100, 100)
V = FunctionSpace(mesh, "CG", 2)
f = interpolate(
Expression(
"0.1*sin(50.*(2*x[0]-1)) + atan2(-0.1, (2.0*(2*x[0]-1) - sin(5.*(2*x[1]-1))))"
), V)
eta = 0.01
#Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
#mesh = adapt(Mp)
if True:
level = 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh1 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh2 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh3 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh4 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh5 = adapt(Mp)
level *= 0.5
Mp = refine_metric(mesh_metric(mesh), level)
new_mesh6 = adapt(Mp)
else:
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh1 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh2 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh3 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh4 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh5 = adapt(Mp)
eta *= 2
Mp = metric_pnorm(f, mesh, eta, max_edge_ratio=5)
new_mesh6 = adapt(Mp)
# plot(Mp[0,0])
# from IPython import embed
# embed()
plot(mesh, title="initial mesh")
plot(new_mesh1, title="coarsen 1")
plot(new_mesh2, title="coarsen 2")
plot(new_mesh3, title="coarsen 3")
plot(new_mesh4, title="coarsen 4")
plot(new_mesh5, title="coarsen 5")
plot(new_mesh6, title="coarsen 6")
interactive()
def adv_convergence(width=2e-2, delta=1e-2, relp=1, Nadapt=10, use_adapt=True, problem=3, outname='', use_reform=False, CGorderL = [2, 3], noplot=False, Lx=3.):
### SETUP SOLUTION
sy = Symbol('sy'); width_ = Symbol('ww')
if problem == 3:
stepfunc = 0.5+165./104./width_*sy-20./13./width_**3*sy**3-102./13./width_**5*sy**5+240./13./width_**7*sy**7
elif problem == 2:
stepfunc = 0.5+15./8./width_*sy-5./width_**3*sy**3+6./width_**5*sy**5
elif problem == 1:
stepfunc = 0.5+1.5/width_*sy-2/width_**3*sy**3
stepfunc = str(stepfunc).replace('sy','x[1]').replace('x[1]**2','(x[1]*x[1])')
#REPLACE ** with pow
stepfunc = stepfunc.replace('x[1]**3','pow(x[1],3.)')
stepfunc = stepfunc.replace('x[1]**5','pow(x[1],5.)')
stepfunc = stepfunc.replace('x[1]**7','pow(x[1],7.)')
testsol = '(-ww/2 < x[1] && x[1] < ww/2 ? ' + stepfunc +' : 0) + (ww/2<x[1] ? 1 : 0)'
testsol = testsol.replace('ww**2','(ww*ww)').replace('ww**3','pow(ww,3.)').replace('ww**5','pow(ww,5.)').replace('ww**7','pow(ww,7.)')
testsol = testsol.replace('ww',str(width))
dp = Constant(1.)
fac = Constant(1+2.*delta)
delta = Constant(delta)
def left(x, on_boundary):
return x[0] + Lx/2. < DOLFIN_EPS
def right(x, on_boundary):
return x[0] - Lx/2. > -DOLFIN_EPS
def top_bottom(x, on_boundary):
return x[1] - 0.5 > -DOLFIN_EPS or x[1] + 0.5 < DOLFIN_EPS
class Inletbnd(SubDomain):
def inside(self, x, on_boundary):
return x[0] + Lx/2. < DOLFIN_EPS
for CGorder in [2]: #CGorderL:
dofs = []
L2errors = []
for eta in 0.04*pyexp2(-array(range(9))*pylog(2)/2):
### SETUP MESH
meshsz = int(round(80*0.005/(eta*(bool(use_adapt)==False)+0.05*(bool(use_adapt)==True))))
if not bool(use_adapt) and meshsz > 80:
continue
mesh = RectangleMesh(-Lx/2.,-0.5,Lx/2.,0.5,meshsz,meshsz,"left/right")
# PERFORM TEN ADAPTATION ITERATIONS
for iii in range(Nadapt):
V = VectorFunctionSpace(mesh, "CG", CGorder); Q = FunctionSpace(mesh, "CG", CGorder-1)
W = V*Q
(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)
alpha = Expression("-0.25<x[0] && x[0]<0.25 && 0. < x[1] ? 1e4 : 0")
boundaries = FacetFunction("size_t",mesh)
#outletbnd = Outletbnd()
inletbnd = Inletbnd()
boundaries.set_all(0)
#outletbnd.mark(boundaries, 1)
inletbnd.mark(boundaries, 1)
ds = Measure("ds")[boundaries]
bc0 = DirichletBC(W.sub(0), Constant((0., 0.)), top_bottom)
bc1 = DirichletBC(W.sub(1), dp , left)
bc2 = DirichletBC(W.sub(1), Constant(0), right)
bc3 = DirichletBC(W.sub(0).sub(1), Constant(0.), left)
bc4 = DirichletBC(W.sub(0).sub(1), Constant(0.), right)
bcs = [bc0,bc1,bc2,bc3,bc4]
bndterm = dp*dot(v,Constant((-1.,0.)))*ds(1)
a = eta*inner(grad(u), grad(v))*dx - div(v)*p*dx + q*div(u)*dx+alpha*dot(u,v)*dx#+bndterm
L = inner(Constant((0.,0.)), v)*dx -bndterm
U = Function(W)
solve(a == L, U, bcs)
u, ps = U.split()
#SOLVE CONCENTRATION
mm = mesh_metric2(mesh)
vdir = u/sqrt(inner(u,u)+DOLFIN_EPS)
if iii == 0 or use_reform == False:
Q2 = FunctionSpace(mesh,'CG',2); c = Function(Q2)
q = TestFunction(Q2); p = TrialFunction(Q2)
newq = (q+dot(vdir,dot(mm,vdir))*inner(grad(q),vdir)) #SUPG
if use_reform:
F = newq*(fac/((1+exp(-c))**2)*exp(-c))*inner(grad(c),u)*dx
J = derivative(F,c)
bc = DirichletBC(Q2, Expression("-log("+str(float(fac)) +"/("+testsol+"+"+str(float(delta))+")-1)"), left)
# bc = DirichletBC(Q, -ln(fac/(Expression(testsol)+delta)-1), left)
problem = NonlinearVariationalProblem(F,c,bc,J)
solver = NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"]["relaxation_parameter"] = relp
solver.solve()
else:
a2 = newq*inner(grad(p),u)*dx
bc = DirichletBC(Q2, Expression(testsol), left)
L2 = Constant(0.)*q*dx
solve(a2 == L2, c, bc)
if (not bool(use_adapt)) or iii == Nadapt-1:
break
um = project(sqrt(inner(u,u)),FunctionSpace(mesh,'CG',2))
H = metric_pnorm(um, eta, max_edge_ratio=1+49*(use_adapt!=2), p=2)
H2 = metric_pnorm(c, eta, max_edge_ratio=1+49*(use_adapt!=2), p=2)
#H3 = metric_pnorm(ps , eta, max_edge_ratio=1+49*(use_adapt!=2), p=2)
H4 = metric_ellipse(H,H2)
#H5 = metric_ellipse(H3,H4,mesh)
mesh = adapt(H4)
if use_reform:
Q2 = FunctionSpace(mesh,'CG',2)
c = interpolate(c,Q2)
if use_reform:
c = project(fac/(1+exp(-c))-delta,FunctionSpace(mesh,'CG',2))
L2error = bnderror(c,Expression(testsol),ds)
dofs.append(len(c.vector().array())+len(U.vector().array()))
L2errors.append(L2error)
# fid = open("DOFS_L2errors_mesh_c_CG"+str(CGorder)+outname+".mpy",'w')
# pickle.dump([dofs[0],L2errors[0],c.vector().array().min(),c.vector().array().max()-1,mesh.cells(),mesh.coordinates(),c.vector().array()],fid)
# fid.close();
log(INFO+1,"%1dX ADAPT<->SOLVE complete: DOF=%5d, error=%0.0e, min(c)=%0.0e,max(c)-1=%0.0e" % (Nadapt, dofs[len(dofs)-1], L2error,c.vector().array().min(),c.vector().array().max()-1))
# PLOT MESH + solution
figure()
testf = interpolate(c ,FunctionSpace(mesh,'CG',1))
testfe = interpolate(Expression(testsol),FunctionSpace(mesh,'CG',1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG" ,1))
zz = testf.vector().array()[vtx2dof]; zz[zz==1] -= 1e-16
hh=tricontourf(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),zz,100,cmap=get_cmap('binary'))
colorbar(hh)
hold('on'); triplot(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),color='r',linewidth=0.5); hold('off')
axis('equal'); box('off')
# savefig(outname+'final_mesh_CG2.png',dpi=300) #; savefig('outname+final_mesh_CG2.eps',dpi=300)
#PLOT ERROR
figure()
xe = interpolate(Expression("x[0]"),FunctionSpace(mesh,'CG',1)).vector().array()
ye = interpolate(Expression("x[1]"),FunctionSpace(mesh,'CG',1)).vector().array()
I = xe - Lx/2 > -DOLFIN_EPS; I2 = ye[I].argsort()
pyplot(ye[I][I2],testf.vector().array()[I][I2]-testfe.vector().array()[I][I2],'-b'); ylabel('error')
# PLOT L2error graph
figure()
pyloglog(dofs,L2errors,'-b.',linewidth=2,markersize=16); xlabel('Degree of freedoms'); ylabel('L2 error')
# SAVE SOLUTION
dofs = array(dofs); L2errors = array(L2errors)
fid = open("DOFS_L2errors_CG"+str(CGorder)+outname+".mpy",'w')
pickle.dump([dofs,L2errors],fid)
fid.close();
# #show()
# #LOAD SAVED SOLUTIONS
# fid = open("DOFS_L2errors_CG2"+outname+".mpy",'r')
# [dofs,L2errors] = pickle.load(fid)
# fid.close()
#
# PERFORM FITS ON LAST THREE POINTS
NfitP = 5
I = array(range(len(dofs)-NfitP,len(dofs)))
slope,ints = polyfit(pylog(dofs[I]), pylog(L2errors[I]), 1)
fid = open("DOFS_L2errors_CG2_fit"+outname+".mpy",'w')
pickle.dump([dofs,L2errors,slope,ints],fid)
fid.close()
#PLOT THEM TOGETHER
if CGorderL != [2]:
fid = open("DOFS_L2errors_CG3.mpy",'r')
[dofs_old,L2errors_old] = pickle.load(fid)
fid.close()
slope2,ints2 = polyfit(pylog(dofs_old[I]), pylog(L2errors_old[I]), 1)
figure()
pyloglog(dofs,L2errors,'-b.',dofs_old,L2errors_old,'--b.',linewidth=2,markersize=16)
hold('on'); pyloglog(dofs,pyexp2(ints)*dofs**slope,'-r',dofs_old,pyexp2(ints2)*dofs_old**slope2,'--r',linewidth=1); hold('off')
xlabel('Degree of freedoms'); ylabel('L2 error')
legend(['CG2','CG3',"%0.2f*log(DOFs)" % slope, "%0.2f*log(DOFs)" % slope2]) #legend(['new data','old_data'])
# savefig('comparison.png',dpi=300) #savefig('comparison.eps');
if not noplot:
show()
def minimal_example(width=2e-2, Nadapt=10, eta = 0.01):
### CONSTANTS
meshsz = 40
hd = Constant(width)
### SETUP MESH
mesh = RectangleMesh(Point(-0.5,-0.5),Point(0.5,0.5),1*meshsz,1*meshsz,"left/right")
### DERIVE FORCING TERM
angle = pi/8 #rand*pi/2
sx = Symbol('sx'); sy = Symbol('sy'); width_ = Symbol('ww'); aa = Symbol('aa')
testsol = pytanh((sx*pycos(aa)+sy*pysin(aa))/width_)
ddtestsol = str(diff(testsol,sx,sx)+diff(testsol,sy,sy)).replace('sx','x[0]').replace('sy','x[1]')
#replace ** with pow
ddtestsol = ddtestsol.replace('tanh((x[0]*sin(aa) + x[1]*cos(aa))/ww)**2','pow(tanh((x[0]*sin(aa) + x[1]*cos(aa))/ww),2.)')
ddtestsol = ddtestsol.replace('cos(aa)**2','pow(cos(aa),2.)').replace('sin(aa)**2','pow(sin(aa),2.)').replace('ww**2','(ww*ww)')
#insert vaulues
ddtestsol = ddtestsol.replace('aa',str(angle)).replace('ww',str(width))
testsol = str(testsol).replace('sx','x[0]').replace('sy','x[1]').replace('aa',str(angle)).replace('ww',str(width))
ddtestsol = "-("+ddtestsol+")"
def boundary(x):
return x[0]-mesh.coordinates()[:,0].min() < DOLFIN_EPS or mesh.coordinates()[:,0].max()-x[0] < DOLFIN_EPS \
or mesh.coordinates()[:,1].min()+0.5 < DOLFIN_EPS or mesh.coordinates()[:,1].max()-x[1] < DOLFIN_EPS
# PERFORM TEN ADAPTATION ITERATIONS
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG" ,2); dis = TrialFunction(V); dus = TestFunction(V); u = Function(V)
a = inner(grad(dis), grad(dus))*dx
L = Expression(ddtestsol)*dus*dx
bc = DirichletBC(V, Expression(testsol), boundary)
solve(a == L, u, bc)
startTime = time()
H = metric_pnorm(u, eta, max_edge_length=3., max_edge_ratio=None)
H = logproject(H)
if iii != Nadapt-1:
mesh = adapt(H)
L2error = errornorm(Expression(testsol), u, degree_rise=4, norm_type='L2')
log(INFO+1,"total (adapt+metric) time was %0.1fs, L2error=%0.0e, nodes: %0.0f" % (time()-startTime,L2error,mesh.num_vertices()))
# # PLOT MESH
# figure()
coords = mesh.coordinates().transpose()
# triplot(coords[0],coords[1],mesh.cells(),linewidth=0.1)
# #savefig('mesh.png',dpi=300) #savefig('mesh.eps');
figure() #solution
testf = interpolate(Expression(testsol),FunctionSpace(mesh,'CG',1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG" ,1))
zz = testf.vector().array()[vtx2dof]
hh=tricontourf(coords[0],coords[1],mesh.cells(),zz,100)
colorbar(hh)
# savefig('solution.png',dpi=300) #savefig('solution.eps');
figure() #analytical solution
testfe = interpolate(u,FunctionSpace(mesh,'CG',1))
zz = testfe.vector().array()[vtx2dof]
hh=tricontourf(coords[0],coords[1],mesh.cells(),zz,100)
colorbar(hh)
#savefig('analyt.png',dpi=300) #savefig('analyt.eps');
figure() #error
zz -= testf.vector().array()[vtx2dof]; zz[zz==1] -= 1e-16
hh=tricontourf(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),zz,100,cmap=get_cmap('binary'))
colorbar(hh)
hold('on'); triplot(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),color='r',linewidth=0.5); hold('off')
axis('equal'); box('off'); title('error')
show()
def maximal_example(eta_list=array([0.001]), Nadapt=5, timet=1.,
period=2 * pi):
### CONSTANTS
### SETUP SOLUTION
#testsol = '0.1*sin(50*x+2*pi*t/T)+atan(-0.1/(2*x - sin(5*y+2*pi*t/T)))';
sx = Symbol('sx')
sy = Symbol('sy')
sT = Symbol('sT')
st = Symbol('st')
spi = Symbol('spi')
testsol = 0.1 * pysin(50 * sx + 2 * spi * st / sT) + pyatan(
-0.1, 2 * sx - pysin(5 * sy + 2 * spi * st / sT))
ddtestsol = str(diff(testsol, sx, sx) + diff(testsol, sy, sy)).replace(
'sx', 'x[0]').replace('sy', 'x[1]').replace('spi', 'pi')
# replacing **P with pow(,P)
ddtestsol = ddtestsol.replace("(2*x[0] - sin(5*x[1] + 2*pi*st/sT))**2",
"pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.)")
ddtestsol = ddtestsol.replace("cos(5*x[1] + 2*pi*st/sT)**2",
"pow(cos(5*x[1] + 2*pi*st/sT),2.)")
ddtestsol = ddtestsol.replace(
"(pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.) + 0.01)**2",
"pow((pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.) + 0.01),2.)")
ddtestsol = ddtestsol.replace(
"(1 + 0.01/pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.))**2",
"pow(1 + 0.01/pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.),2.)")
ddtestsol = ddtestsol.replace("(2*x[0] - sin(5*x[1] + 2*pi*st/sT))**5",
"pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),5.)")
#insert values
ddtestsol = ddtestsol.replace('sT', str(period)).replace('st', str(timet))
testsol = str(testsol).replace('sx', 'x[0]').replace('sy', 'x[1]').replace(
'spi', 'pi').replace('sT', str(period)).replace('st', str(timet))
ddtestsol = "-(" + ddtestsol + ")"
error_list = []
dof_list = []
for eta in eta_list:
meshsz = 40
### SETUP MESH
# mesh = RectangleMesh(0.4,-0.1,0.6,0.3,1*meshsz,1*meshsz,"left/right") #shock
# mesh = RectangleMesh(-0.75,-0.3,-0.3,0.5,1*meshsz,1*meshsz,"left/right") #waves
mesh = RectangleMesh(-1.5, -0.25, 0.5, 0.75, 1 * meshsz, 1 * meshsz,
"left/right") #shock+waves
def boundary(x):
return near(x[0],mesh.coordinates()[:,0].min()) or near(x[0],mesh.coordinates()[:,0].max()) \
or near(x[1],mesh.coordinates()[:,1].min()) or near(x[1],mesh.coordinates()[:,1].max())
# PERFORM ONE ADAPTATION ITERATION
for iii in range(Nadapt):
startTime = time()
V = FunctionSpace(mesh, "CG", 2)
dis = TrialFunction(V)
dus = TestFunction(V)
u = Function(V)
# R = interpolate(Expression(ddtestsol),V)
a = inner(grad(dis), grad(dus)) * dx
L = Expression(ddtestsol) * dus * dx #
bc = DirichletBC(V, Expression(testsol), boundary)
solve(a == L, u, bc)
soltime = time() - startTime
startTime = time()
H = metric_pnorm(u, eta, max_edge_ratio=50, CG0H=3, p=4)
metricTime = time() - startTime
if iii != Nadapt - 1:
mesh = adapt(H)
TadaptTime = time() - startTime
L2error = errornorm(Expression(testsol),
u,
degree_rise=4,
norm_type='L2')
printstr = "%5.0f elements, %0.0e L2error, adapt took %0.0f %% of the total time, (%0.0f %% of which was the metric calculation)" \
% (mesh.num_cells(),L2error,TadaptTime/(TadaptTime+soltime)*100,metricTime/TadaptTime*100)
if len(eta_list) == 1:
print(printstr)
else:
error_list.append(L2error)
dof_list.append(len(u.vector().array()))
print(printstr)
if len(dof_list) > 1:
dof_list = array(dof_list)
error_list = array(error_list)
figure()
loglog(dof_list, error_list, '.b-', linewidth=2, markersize=16)
xlabel('Degree of freedoms')
ylabel('L2 error')
# # PLOT MESH
# figure()
coords = mesh.coordinates().transpose()
# triplot(coords[0],coords[1],mesh.cells(),linewidth=0.1)
# #savefig('mesh.png',dpi=300) #savefig('mesh.eps');
figure() #solution
testf = interpolate(Expression(testsol), FunctionSpace(mesh, 'CG', 1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG", 1))
zz = testf.vector().array()[vtx2dof]
hh = tricontourf(coords[0], coords[1], mesh.cells(), zz, 100)
colorbar(hh)
#savefig('solution.png',dpi=300) #savefig('solution.eps');
figure() #analytical solution
testfe = interpolate(u, FunctionSpace(mesh, 'CG', 1))
zz = testfe.vector().array()[vtx2dof]
hh = tricontourf(coords[0], coords[1], mesh.cells(), zz, 100)
colorbar(hh)
#savefig('analyt.png',dpi=300) #savefig('analyt.eps');
figure() #error
zz -= testf.vector().array()[vtx2dof]
zz[zz == 1] -= 1e-16
hh = tricontourf(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1],
mesh.cells(),
zz,
100,
cmap=get_cmap('binary'))
colorbar(hh)
hold('on')
triplot(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1],
mesh.cells(),
color='r',
linewidth=0.5)
hold('off')
axis('equal')
box('off')
title('error')
show()
def minimal_example(width=2e-2, Nadapt=10, eta=0.01):
### CONSTANTS
meshsz = 40
hd = Constant(width)
### SETUP MESH
mesh = RectangleMesh(Point(-0.5, -0.5), Point(0.5, 0.5), 1 * meshsz,
1 * meshsz, "left/right")
### DERIVE FORCING TERM
angle = pi / 8 #rand*pi/2
sx = Symbol('sx')
sy = Symbol('sy')
width_ = Symbol('ww')
aa = Symbol('aa')
testsol = pytanh((sx * pycos(aa) + sy * pysin(aa)) / width_)
ddtestsol = str(diff(testsol, sx, sx) + diff(testsol, sy, sy)).replace(
'sx', 'x[0]').replace('sy', 'x[1]')
#replace ** with pow
ddtestsol = ddtestsol.replace(
'tanh((x[0]*sin(aa) + x[1]*cos(aa))/ww)**2',
'pow(tanh((x[0]*sin(aa) + x[1]*cos(aa))/ww),2.)')
ddtestsol = ddtestsol.replace('cos(aa)**2', 'pow(cos(aa),2.)').replace(
'sin(aa)**2', 'pow(sin(aa),2.)').replace('ww**2', '(ww*ww)')
#insert vaulues
ddtestsol = ddtestsol.replace('aa', str(angle)).replace('ww', str(width))
testsol = str(testsol).replace('sx', 'x[0]').replace('sy', 'x[1]').replace(
'aa', str(angle)).replace('ww', str(width))
ddtestsol = "-(" + ddtestsol + ")"
def boundary(x):
return x[0]-mesh.coordinates()[:,0].min() < DOLFIN_EPS or mesh.coordinates()[:,0].max()-x[0] < DOLFIN_EPS \
or mesh.coordinates()[:,1].min()+0.5 < DOLFIN_EPS or mesh.coordinates()[:,1].max()-x[1] < DOLFIN_EPS
# PERFORM TEN ADAPTATION ITERATIONS
for iii in range(Nadapt):
V = FunctionSpace(mesh, "CG", 2)
dis = TrialFunction(V)
dus = TestFunction(V)
u = Function(V)
a = inner(grad(dis), grad(dus)) * dx
L = Expression(ddtestsol) * dus * dx
bc = DirichletBC(V, Expression(testsol), boundary)
solve(a == L, u, bc)
startTime = time()
H = metric_pnorm(u, eta, max_edge_length=3., max_edge_ratio=None)
H = logproject(H)
if iii != Nadapt - 1:
mesh = adapt(H)
L2error = errornorm(Expression(testsol),
u,
degree_rise=4,
norm_type='L2')
log(
INFO + 1,
"total (adapt+metric) time was %0.1fs, L2error=%0.0e, nodes: %0.0f"
% (time() - startTime, L2error, mesh.num_vertices()))
# # PLOT MESH
# figure()
coords = mesh.coordinates().transpose()
# triplot(coords[0],coords[1],mesh.cells(),linewidth=0.1)
# #savefig('mesh.png',dpi=300) #savefig('mesh.eps');
figure() #solution
testf = interpolate(Expression(testsol), FunctionSpace(mesh, 'CG', 1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG", 1))
zz = testf.vector().array()[vtx2dof]
hh = tricontourf(coords[0], coords[1], mesh.cells(), zz, 100)
colorbar(hh)
# savefig('solution.png',dpi=300) #savefig('solution.eps');
figure() #analytical solution
testfe = interpolate(u, FunctionSpace(mesh, 'CG', 1))
zz = testfe.vector().array()[vtx2dof]
hh = tricontourf(coords[0], coords[1], mesh.cells(), zz, 100)
colorbar(hh)
#savefig('analyt.png',dpi=300) #savefig('analyt.eps');
figure() #error
zz -= testf.vector().array()[vtx2dof]
zz[zz == 1] -= 1e-16
hh = tricontourf(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1],
mesh.cells(),
zz,
100,
cmap=get_cmap('binary'))
colorbar(hh)
hold('on')
triplot(mesh.coordinates()[:, 0],
mesh.coordinates()[:, 1],
mesh.cells(),
color='r',
linewidth=0.5)
hold('off')
axis('equal')
box('off')
title('error')
show()
def maximal_example(eta_list = array([0.001]), Nadapt=5, timet=1., period=2*pi):
### CONSTANTS
### SETUP SOLUTION
#testsol = '0.1*sin(50*x+2*pi*t/T)+atan(-0.1/(2*x - sin(5*y+2*pi*t/T)))';
sx = Symbol('sx'); sy = Symbol('sy'); sT = Symbol('sT'); st = Symbol('st'); spi = Symbol('spi')
testsol = 0.1*pysin(50*sx+2*spi*st/sT)+pyatan(-0.1,2*sx - pysin(5*sy+2*spi*st/sT))
ddtestsol = str(diff(testsol,sx,sx)+diff(testsol,sy,sy)).replace('sx','x[0]').replace('sy','x[1]').replace('spi','pi')
# replacing **P with pow(,P)
ddtestsol = ddtestsol.replace("(2*x[0] - sin(5*x[1] + 2*pi*st/sT))**2","pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.)")
ddtestsol = ddtestsol.replace("cos(5*x[1] + 2*pi*st/sT)**2","pow(cos(5*x[1] + 2*pi*st/sT),2.)")
ddtestsol = ddtestsol.replace("(pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.) + 0.01)**2","pow((pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.) + 0.01),2.)")
ddtestsol = ddtestsol.replace("(1 + 0.01/pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.))**2","pow(1 + 0.01/pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),2.),2.)")
ddtestsol = ddtestsol.replace("(2*x[0] - sin(5*x[1] + 2*pi*st/sT))**5","pow(2*x[0] - sin(5*x[1] + 2*pi*st/sT),5.)")
#insert values
ddtestsol = ddtestsol.replace('sT',str(period)).replace('st',str(timet))
testsol = str(testsol).replace('sx','x[0]').replace('sy','x[1]').replace('spi','pi').replace('sT',str(period)).replace('st',str(timet))
ddtestsol = "-("+ddtestsol+")"
error_list = []; dof_list = []
for eta in eta_list:
meshsz = 40
### SETUP MESH
# mesh = RectangleMesh(0.4,-0.1,0.6,0.3,1*meshsz,1*meshsz,"left/right") #shock
# mesh = RectangleMesh(-0.75,-0.3,-0.3,0.5,1*meshsz,1*meshsz,"left/right") #waves
mesh = RectangleMesh(-1.5,-0.25,0.5,0.75,1*meshsz,1*meshsz,"left/right") #shock+waves
def boundary(x):
return near(x[0],mesh.coordinates()[:,0].min()) or near(x[0],mesh.coordinates()[:,0].max()) \
or near(x[1],mesh.coordinates()[:,1].min()) or near(x[1],mesh.coordinates()[:,1].max())
# PERFORM ONE ADAPTATION ITERATION
for iii in range(Nadapt):
startTime = time()
V = FunctionSpace(mesh, "CG" ,2); dis = TrialFunction(V); dus = TestFunction(V); u = Function(V)
# R = interpolate(Expression(ddtestsol),V)
a = inner(grad(dis), grad(dus))*dx
L = Expression(ddtestsol)*dus*dx #
bc = DirichletBC(V, Expression(testsol), boundary)
solve(a == L, u, bc)
soltime = time()-startTime
startTime = time()
H = metric_pnorm(u, eta, max_edge_ratio=50, CG0H=3, p=4)
metricTime = time()-startTime
if iii != Nadapt-1:
mesh = adapt(H)
TadaptTime = time()-startTime
L2error = errornorm(Expression(testsol), u, degree_rise=4, norm_type='L2')
printstr = "%5.0f elements, %0.0e L2error, adapt took %0.0f %% of the total time, (%0.0f %% of which was the metric calculation)" \
% (mesh.num_cells(),L2error,TadaptTime/(TadaptTime+soltime)*100,metricTime/TadaptTime*100)
if len(eta_list) == 1:
print(printstr)
else:
error_list.append(L2error); dof_list.append(len(u.vector().array()))
print(printstr)
if len(dof_list) > 1:
dof_list = array(dof_list); error_list = array(error_list)
figure()
loglog(dof_list,error_list,'.b-',linewidth=2,markersize=16); xlabel('Degree of freedoms'); ylabel('L2 error')
# # PLOT MESH
# figure()
coords = mesh.coordinates().transpose()
# triplot(coords[0],coords[1],mesh.cells(),linewidth=0.1)
# #savefig('mesh.png',dpi=300) #savefig('mesh.eps');
figure() #solution
testf = interpolate(Expression(testsol),FunctionSpace(mesh,'CG',1))
vtx2dof = vertex_to_dof_map(FunctionSpace(mesh, "CG" ,1))
zz = testf.vector().array()[vtx2dof]
hh=tricontourf(coords[0],coords[1],mesh.cells(),zz,100)
colorbar(hh)
#savefig('solution.png',dpi=300) #savefig('solution.eps');
figure() #analytical solution
testfe = interpolate(u,FunctionSpace(mesh,'CG',1))
zz = testfe.vector().array()[vtx2dof]
hh=tricontourf(coords[0],coords[1],mesh.cells(),zz,100)
colorbar(hh)
#savefig('analyt.png',dpi=300) #savefig('analyt.eps');
figure() #error
zz -= testf.vector().array()[vtx2dof]; zz[zz==1] -= 1e-16
hh=tricontourf(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),zz,100,cmap=get_cmap('binary'))
colorbar(hh)
hold('on'); triplot(mesh.coordinates()[:,0],mesh.coordinates()[:,1],mesh.cells(),color='r',linewidth=0.5); hold('off')
axis('equal'); box('off'); title('error')
show()
