This page was generated from unit-3.8-nonlmin/nonlmin.ipynb.
3.8 Nonlinear minimization problems¶
We consider problems of the form
\[\text{find } u \in V \text{ s.t. } E(u) \leq E(v) \quad \forall~ v \in V.\]
[1]:
from netgen import gui
from ngsolve import *
Scalar minimization problems¶
As a first example we take \(V = H^1_0\) and
\[E(u) = \int_{\Omega} \vert \nabla u \vert^2 + u^4 - fu ~dx.\]
The minimization is equivalent to solving the nonlinear PDE:
\[- \Delta u + 4 u^3 = f \text{ in } \Omega\]
We solve the PDE with a Newton iteration.
[2]:
from netgen.geom2d import unit_square
mesh = Mesh (unit_square.GenerateMesh(maxh=0.2))
V = H1(mesh, order=4, dirichlet=[1,2,3,4])
u = V.TrialFunction()
To solve the problem we use the Variation integrator. Based on the symbolic description of the energy functional, it is able to
evaluate the energy functional (
Energy)
\[E(u) \qquad (E:V \to \mathbb{R})\]
compute the Gateau derivative for a given \(u\) (
Apply):
\[A(u)(v) = E'(u)(v) \qquad (A(u): V \to \mathbb{R})\]
compute the second derivative (
AssembleLinearization)
\[(\delta A)(w)(u,v) \qquad (\delta A(w): V\times V \to \mathbb{R})\]
[3]:
a = BilinearForm (V, symmetric=True)
a += Variation ( (grad(u)*grad(u) + u**4-u) * dx)
Equivalent to:
a += (2 * grad(u) * grad(v) + 4*u*u*u*v - 1 * v)*dx
(which has the same form as the problems in the nonlinear example)
We recall the Newton iteration (cf. unit-3.7 ) we make the loop:
Given an initial guess \(u^0\)
loop over \(i=0,..\) until convergence:
Compute linearization: $A u^i + \delta `A(u^i) :nbsphinx-math:Delta `u^{i} = 0 $:
\(f^i = A u^i\)
\(B^i = \delta A(u^i)\)
Solve \(B^i \Delta u^i = -f^i\)
Update \(u^{i+1} = u^i + \Delta u^{i}\)
Evaluate stopping criteria
Evaluate \(E(u^{i+1})\)
As a stopping criteria we take \(\langle A u^i,\Delta u^i \rangle = \langle A u^i, A u^i \rangle_{(B^i)^{-1}}< \varepsilon\).
[4]:
def SolveNonlinearMinProblem(a,gfu,tol=1e-13,maxits=25):
res = gfu.vec.CreateVector()
du = gfu.vec.CreateVector()
for it in range(maxits):
print ("Newton iteration {:3}".format(it),end="")
print ("energy = {:16}".format(a.Energy(gfu.vec)),end="")
#solve linearized problem:
a.Apply (gfu.vec, res)
a.AssembleLinearization (gfu.vec)
inv = a.mat.Inverse(V.FreeDofs())
du.data = inv * res
#update iteration
gfu.vec.data -= du
#stopping criteria
stopcritval = sqrt(abs(InnerProduct(du,res)))
print ("<A u",it,", A u",it,">_{-1}^0.5 = ", stopcritval)
if stopcritval < tol:
break
Redraw(blocking=True)
[5]:
gfu = GridFunction (V)
gfu.vec[:] = 0
Draw(gfu,mesh,"u")
SolveNonlinearMinProblem(a,gfu)
print ("energy = ", a.Energy(gfu.vec))
Newton iteration 0energy = 0.0<A u 0 , A u 0 >_{-1}^0.5 = 0.13255956597624674
Newton iteration 1energy = -0.008785675919637857<A u 1 , A u 1 >_{-1}^0.5 = 1.1107598218154633e-05
Newton iteration 2energy = -0.008785675981327226<A u 2 , A u 2 >_{-1}^0.5 = 2.8074411839327077e-13
Newton iteration 3energy = -0.008785675981327228<A u 3 , A u 3 >_{-1}^0.5 = 3.203464773686272e-17
energy = -0.008785675981327228
Again, a Newton for minimization is shipped with NGSolve:
[6]:
from ngsolve.solvers import *
gfu.vec[:] = 0
NewtonMinimization(a,gfu)
Redraw()
Newton iteration 0
energy = 0.0
err = 0.13255956597624674
Newton iteration 1
energy = -0.008785675919637857
err = 1.1107598218139178e-05
Newton iteration 2
energy = -0.008785675981327228
err = 2.807422534806441e-13
Nonlinear elasticity¶
We consider a beam which is fixed on one side and is subject to gravity only. We assume a Neo-Hookean hyperelastic material. The model is a nonlinear minimization problem with
\[E(v) := \int_{\Omega} \frac{\mu}{2} ( \operatorname{tr}(F^T F-I)+\frac{2 \mu}{\lambda} \operatorname{det}(F^T F)^{\frac{\lambda}{2\mu}} - 1) - \gamma ~ (f,v) ~~ dx\]
where \(\mu\) and \(\lambda\) are the Lamé parameters and \(F = I + D v\) where \(v: \Omega \to \mathbb{R}^2\) is the sought for displacement.
[7]:
import netgen.geom2d as geom2d
from ngsolve import *
geo = geom2d.SplineGeometry()
pnums = [ geo.AddPoint (x,y,maxh=0.01) for x,y in [(0,0), (1,0), (1,0.1), (0,0.1)] ]
for p1,p2,bc in [(0,1,"bot"), (1,2,"right"), (2,3,"top"), (3,0,"left")]:
geo.Append(["line", pnums[p1], pnums[p2]], bc=bc)
mesh = Mesh(geo.GenerateMesh(maxh=0.05))
[8]:
# E module and poisson number:
E, nu = 210, 0.2
# Lamé constants:
mu = E / 2 / (1+nu)
lam = E * nu / ((1+nu)*(1-2*nu))
V = H1(mesh, order=2, dirichlet="left", dim=mesh.dim)
u = V.TrialFunction()
#gravity:
force = CoefficientFunction( (0,-1) )
[9]:
def Pow(a, b):
return exp (log(a)*b)
def NeoHook (C):
return 0.5 * mu * (Trace(C-I) + 2*mu/lam * Pow(Det(C), -lam/2/mu) - 1)
I = Id(mesh.dim)
F = I + Grad(u)
C = F.trans * F.trans
factor = Parameter(1.0)
a = BilinearForm(V, symmetric=True)
a += Variation( NeoHook (C).Compile() * dx
-factor * (InnerProduct(force,u) ).Compile() * dx)
We want to solve the minimization problem for \(\gamma = 5\). Due to the high nonlinearity in the problem, the Newton iteration will not convergence with any initial guess. We approach the case \(\gamma = 5\) by solving problems with \(\gamma = i/10\) for \(i=1,..,50\) and taking the solution of the previous problem as an initial guess.
[10]:
gfu = GridFunction(V)
gfu.vec[:] = 0
Draw (gfu, mesh, "u")
SetVisualization (deformation=True)
res = gfu.vec.CreateVector()
du = gfu.vec.CreateVector()
for loadstep in range(50):
print ("loadstep", loadstep)
factor.Set ((loadstep+1)/10)
SolveNonlinearMinProblem(a,gfu)
Redraw()
loadstep 0
Newton iteration 0energy = 8.75<A u 0 , A u 0 >_{-1}^0.5 = 0.0010077888576925991
Newton iteration 1energy = 9.354191959620758<A u 1 , A u 1 >_{-1}^0.5 = 0.6976209150349882
Newton iteration 2energy = 13.284089564731062<A u 2 , A u 2 >_{-1}^0.5 = 5.138275393070173
Newton iteration 3energy = -1683.444975483786<A u 3 , A u 3 >_{-1}^0.5 = 57.99398254445939
Newton iteration 4energy = -533.9836497682816<A u 4 , A u 4 >_{-1}^0.5 = 32.59228501286862
Newton iteration 5energy = -18.29025550505594<A u 5 , A u 5 >_{-1}^0.5 = 5.420132272736724
Newton iteration 6energy = -120.37628146392308<A u 6 , A u 6 >_{-1}^0.5 = 15.337704384267557
Newton iteration 7energy = 4.700450661744071<A u 7 , A u 7 >_{-1}^0.5 = 3.890136254392822
Newton iteration 8energy = -8.702773841650924<A u 8 , A u 8 >_{-1}^0.5 = 4.501094895708501
Newton iteration 9energy = 53.857920095324516<A u 9 , A u 9 >_{-1}^0.5 = 10.530409335207905
Newton iteration 10energy = -1.297882080078125<A u 10 , A u 10 >_{-1}^0.5 = 4.230571774827796
Newton iteration 11energy = -2.6806640625<A u 11 , A u 11 >_{-1}^0.5 = 4.228737878607998
Newton iteration 12energy = 226.7734375<A u 12 , A u 12 >_{-1}^0.5 = 21.498940696277163
Newton iteration 13energy = -7.875<A u 13 , A u 13 >_{-1}^0.5 = 4.375970957074128
Newton iteration 14energy = 96.0<A u 14 , A u 14 >_{-1}^0.5 = 21.880818650966397
Newton iteration 15energy = -2024.0<A u 15 , A u 15 >_{-1}^0.5 = 73.19234500661636
Newton iteration 16energy = -5632.0<A u 16 , A u 16 >_{-1}^0.5 = 251.21103142689546
Newton iteration 17energy = 0.0<A u 17 , A u 17 >_{-1}^0.5 = 2605.6338329036394
Newton iteration 18energy = -25165824.0<A u 18 , A u 18 >_{-1}^0.5 = 5326.44264973658
Newton iteration 19energy = -60817408.0<A u 19 , A u 19 >_{-1}^0.5 = 11524.323912056532
Newton iteration 20energy = 201326592.0<A u 20 , A u 20 >_{-1}^0.5 = 12753.48851433997
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 1
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 2
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 3
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 4
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 5
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 6
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 7
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 8
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 9
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 10
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 11
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 12
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 13
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 14
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 15
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 16
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 17
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 18
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 19
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 20
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 21
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 22
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 23
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 24
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 25
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 26
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 27
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 28
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 29
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 30
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 31
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 32
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 33
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 34
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 35
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 36
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 37
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 38
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 39
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 40
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 41
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 42
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 43
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 44
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 45
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 46
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 47
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 48
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
loadstep 49
Newton iteration 0energy = nan<A u 0 , A u 0 >_{-1}^0.5 = nan
Newton iteration 1energy = nan<A u 1 , A u 1 >_{-1}^0.5 = nan
Newton iteration 2energy = nan<A u 2 , A u 2 >_{-1}^0.5 = nan
Newton iteration 3energy = nan<A u 3 , A u 3 >_{-1}^0.5 = nan
Newton iteration 4energy = nan<A u 4 , A u 4 >_{-1}^0.5 = nan
Newton iteration 5energy = nan<A u 5 , A u 5 >_{-1}^0.5 = nan
Newton iteration 6energy = nan<A u 6 , A u 6 >_{-1}^0.5 = nan
Newton iteration 7energy = nan<A u 7 , A u 7 >_{-1}^0.5 = nan
Newton iteration 8energy = nan<A u 8 , A u 8 >_{-1}^0.5 = nan
Newton iteration 9energy = nan<A u 9 , A u 9 >_{-1}^0.5 = nan
Newton iteration 10energy = nan<A u 10 , A u 10 >_{-1}^0.5 = nan
Newton iteration 11energy = nan<A u 11 , A u 11 >_{-1}^0.5 = nan
Newton iteration 12energy = nan<A u 12 , A u 12 >_{-1}^0.5 = nan
Newton iteration 13energy = nan<A u 13 , A u 13 >_{-1}^0.5 = nan
Newton iteration 14energy = nan<A u 14 , A u 14 >_{-1}^0.5 = nan
Newton iteration 15energy = nan<A u 15 , A u 15 >_{-1}^0.5 = nan
Newton iteration 16energy = nan<A u 16 , A u 16 >_{-1}^0.5 = nan
Newton iteration 17energy = nan<A u 17 , A u 17 >_{-1}^0.5 = nan
Newton iteration 18energy = nan<A u 18 , A u 18 >_{-1}^0.5 = nan
Newton iteration 19energy = nan<A u 19 , A u 19 >_{-1}^0.5 = nan
Newton iteration 20energy = nan<A u 20 , A u 20 >_{-1}^0.5 = nan
Newton iteration 21energy = nan<A u 21 , A u 21 >_{-1}^0.5 = nan
Newton iteration 22energy = nan<A u 22 , A u 22 >_{-1}^0.5 = nan
Newton iteration 23energy = nan<A u 23 , A u 23 >_{-1}^0.5 = nan
Newton iteration 24energy = nan<A u 24 , A u 24 >_{-1}^0.5 = nan
Allen-Cahn equation¶
The Allen-Cahn equations describe the process of phase separation and is the (\(L^2\)) gradient-flow equation to the energy
\[E(v) = \int_{\Omega} \varepsilon \vert \nabla v \vert^2~+~v^2(1-v^2) ~ dx\]
i.e. the solution to the Allen-Cahn equation solves
\[\partial_t u = \frac{\delta E}{\delta u}\]
The quantity \(u\) is an indicator for a phase where \(-1\) refers to one phase and \(1\) to another phase.
The equation has two driving forces:
\(u\) is pulled into one of the two minima (\(-1\) and \(1\)) of the nonlinear term \(u^2(1-u^2)\) (separation of the phases)
the diffusion term scaled with \(\varepsilon\) enforces a smooth transition between the two phases. \(\varepsilon\) determines the size of the transition layer
We use the "SymbolicEnergy" feature to formulate the energy minimization problem and combine it with an implicit Euler discretization:
\[M u^{n+1} - M u^n = \Delta t \underbrace{\frac{\delta E}{\delta u}}_{=:A(u)} (u^{n+1})\]
which we can interpreted as a nonlinear minimization problem again with the energy
\[E^{IE}(v) = \int_{\Omega} \frac{\varepsilon}{2} \vert \nabla v \vert^2~+~v^2(1-v^2) + \frac{1}{2\Delta t} \vert v - u^n \vert^2 ~ dx\]
To solve the nonlinear equation at every time step we again rely on Newton’s method.
[11]:
from ngsolve import *
from netgen.geom2d import *
periodic = SplineGeometry()
pnts = [ (0,0), (1,0), (1,1), (0,1) ]
pnums = [periodic.AppendPoint(*p) for p in pnts]
lright = periodic.Append ( ["line", pnums[0], pnums[1]],bc="periodic")
btop = periodic.Append ( ["line", pnums[1], pnums[2]], bc="periodic")
periodic.Append ( ["line", pnums[3], pnums[2]], leftdomain=0, rightdomain=1, copy=lright, bc="periodic")
periodic.Append ( ["line", pnums[0], pnums[3]], leftdomain=0, rightdomain=1, copy=btop, bc="periodic")
mesh = Mesh (periodic.GenerateMesh(maxh=0.2))
V = Periodic(H1(mesh, order=4, dirichlet=[]))
u = V.TrialFunction()
eps = 4e-3
dt = 1e-1
gfu = GridFunction(V)
gfuold = GridFunction(V)
a = BilinearForm (V, symmetric=False)
a += Variation( (eps/2*grad(u)*grad(u) + ((1-u*u)*(1-u*u))
+ 0.5/dt*(u-gfuold)*(u-gfuold)) * dx)
[12]:
from math import pi
gfu = GridFunction(V)
gfu.Set(sin(2*pi*x))
#gfu.Set(sin(1e8*x)) #<- essentially a random function
Draw(gfu,mesh,"u")
SetVisualization (deformation=True)
t = 0
[13]:
for timestep in range(50):
gfuold.vec.data = gfu.vec
SolveNonlinearMinProblem(a,gfu)
Redraw()
t += dt
print("t = ", t)
Newton iteration 0energy = 0.4145028222324427<A u 0 , A u 0 >_{-1}^0.5 = 0.2809231437108451
Newton iteration 1energy = 0.3769004492603473<A u 1 , A u 1 >_{-1}^0.5 = 0.019176258851511047
Newton iteration 2energy = 0.3767159759702836<A u 2 , A u 2 >_{-1}^0.5 = 9.630978095557888e-05
Newton iteration 3energy = 0.3767159713324164<A u 3 , A u 3 >_{-1}^0.5 = 2.5088242526781963e-09
Newton iteration 4energy = 0.3767159713324165<A u 4 , A u 4 >_{-1}^0.5 = 2.9432235471649787e-16
t = 0.1
Newton iteration 0energy = 0.3438886461859239<A u 0 , A u 0 >_{-1}^0.5 = 0.2359984908217805
Newton iteration 1energy = 0.31712741612159595<A u 1 , A u 1 >_{-1}^0.5 = 0.013451967104651883
Newton iteration 2energy = 0.3170367312150467<A u 2 , A u 2 >_{-1}^0.5 = 4.664634007251611e-05
Newton iteration 3energy = 0.3170367301270973<A u 3 , A u 3 >_{-1}^0.5 = 5.773857887750828e-10
Newton iteration 4energy = 0.3170367301270973<A u 4 , A u 4 >_{-1}^0.5 = 3.08565630314406e-16
t = 0.2
Newton iteration 0energy = 0.2955289399084663<A u 0 , A u 0 >_{-1}^0.5 = 0.18333789232253322
Newton iteration 1energy = 0.2792210737880485<A u 1 , A u 1 >_{-1}^0.5 = 0.008020422172486526
Newton iteration 2energy = 0.2791888669894087<A u 2 , A u 2 >_{-1}^0.5 = 1.6284517755710286e-05
Newton iteration 3energy = 0.27918886685681554<A u 3 , A u 3 >_{-1}^0.5 = 6.887919196209894e-11
Newton iteration 4energy = 0.2791888668568154<A u 4 , A u 4 >_{-1}^0.5 = 3.8195616342709453e-16
t = 0.30000000000000004
Newton iteration 0energy = 0.26701067131949136<A u 0 , A u 0 >_{-1}^0.5 = 0.1332965629931126
Newton iteration 1energy = 0.25831419019650714<A u 1 , A u 1 >_{-1}^0.5 = 0.004187366441885053
Newton iteration 2energy = 0.25830541710911625<A u 2 , A u 2 >_{-1}^0.5 = 4.373249880739843e-06
Newton iteration 3energy = 0.2583054170995535<A u 3 , A u 3 >_{-1}^0.5 = 4.876079452815707e-12
Newton iteration 4energy = 0.2583054170995535<A u 4 , A u 4 >_{-1}^0.5 = 3.3911551407263503e-16
t = 0.4
Newton iteration 0energy = 0.25217703918435974<A u 0 , A u 0 >_{-1}^0.5 = 0.09212611622514374
Newton iteration 1energy = 0.24799424785134225<A u 1 , A u 1 >_{-1}^0.5 = 0.0019800924735281803
Newton iteration 2energy = 0.24799228683269622<A u 2 , A u 2 >_{-1}^0.5 = 9.67462013132836e-07
Newton iteration 3energy = 0.24799228683222818<A u 3 , A u 3 >_{-1}^0.5 = 2.3518965740291886e-13
Newton iteration 4energy = 0.24799228683222818<A u 4 , A u 4 >_{-1}^0.5 = 3.7443809554783123e-16
t = 0.5
Newton iteration 0energy = 0.2451661962247548<A u 0 , A u 0 >_{-1}^0.5 = 0.06142315219753689
Newton iteration 1energy = 0.2432975937684904<A u 1 , A u 1 >_{-1}^0.5 = 0.0008740614555038883
Newton iteration 2energy = 0.24329721172242827<A u 2 , A u 2 >_{-1}^0.5 = 1.8733189359437555e-07
Newton iteration 3energy = 0.24329721172241084<A u 3 , A u 3 >_{-1}^0.5 = 8.753432890250982e-15
t = 0.6
Newton iteration 0energy = 0.24207025658945602<A u 0 , A u 0 >_{-1}^0.5 = 0.03998628290418452
Newton iteration 1energy = 0.24127567813807244<A u 1 , A u 1 >_{-1}^0.5 = 0.0003690821207900638
Newton iteration 2energy = 0.24127561002317857<A u 2 , A u 2 >_{-1}^0.5 = 3.339132356832766e-08
Newton iteration 3energy = 0.241275610023178<A u 3 , A u 3 >_{-1}^0.5 = 5.046020325102631e-16
t = 0.7
Newton iteration 0energy = 0.240763108616933<A u 0 , A u 0 >_{-1}^0.5 = 0.025661409740772207
Newton iteration 1energy = 0.24043513963300997<A u 1 , A u 1 >_{-1}^0.5 = 0.00015217410938014536
Newton iteration 2energy = 0.2404351280542411<A u 2 , A u 2 >_{-1}^0.5 = 5.74933457643747e-09
Newton iteration 3energy = 0.24043512805424122<A u 3 , A u 3 >_{-1}^0.5 = 3.6246171248700585e-16
t = 0.7999999999999999
Newton iteration 0energy = 0.24022546483918178<A u 0 , A u 0 >_{-1}^0.5 = 0.0163721279344063
Newton iteration 1energy = 0.2400917774171359<A u 1 , A u 1 >_{-1}^0.5 = 6.263686267925896e-05
Newton iteration 2energy = 0.2400917754554265<A u 2 , A u 2 >_{-1}^0.5 = 1.030685997804339e-09
Newton iteration 3energy = 0.24009177545542662<A u 3 , A u 3 >_{-1}^0.5 = 4.2241803868323616e-16
t = 0.8999999999999999
Newton iteration 0energy = 0.24000627440773312<A u 0 , A u 0 >_{-1}^0.5 = 0.010486464502981605
Newton iteration 1energy = 0.23995138191747395<A u 1 , A u 1 >_{-1}^0.5 = 2.6656671604408124e-05
Newton iteration 2energy = 0.23995138156218296<A u 2 , A u 2 >_{-1}^0.5 = 2.2716190158479914e-10
Newton iteration 3energy = 0.239951381562183<A u 3 , A u 3 >_{-1}^0.5 = 4.2576384690540574e-16
t = 0.9999999999999999
Newton iteration 0energy = 0.2399157882973673<A u 0 , A u 0 >_{-1}^0.5 = 0.006844755296894013
Newton iteration 1energy = 0.23989238972747978<A u 1 , A u 1 >_{-1}^0.5 = 1.2489092164822844e-05
Newton iteration 2energy = 0.23989238964949075<A u 2 , A u 2 >_{-1}^0.5 = 7.427829017843829e-11
Newton iteration 3energy = 0.23989238964949086<A u 3 , A u 3 >_{-1}^0.5 = 4.670596197216282e-16
t = 1.0999999999999999
Newton iteration 0energy = 0.23987666835532992<A u 0 , A u 0 >_{-1}^0.5 = 0.004664493817844102
Newton iteration 1energy = 0.23986579907505193<A u 1 , A u 1 >_{-1}^0.5 = 6.9604933639867725e-06
Newton iteration 2energy = 0.23986579905082767<A u 2 , A u 2 >_{-1}^0.5 = 3.38272859986765e-11
Newton iteration 3energy = 0.23986579905082775<A u 3 , A u 3 >_{-1}^0.5 = 4.415766972746844e-16
t = 1.2
Newton iteration 0energy = 0.2398579963528002<A u 0 , A u 0 >_{-1}^0.5 = 0.003421663848452132
Newton iteration 1energy = 0.23985214669664828<A u 1 , A u 1 >_{-1}^0.5 = 4.710134579855553e-06
Newton iteration 2energy = 0.2398521466855556<A u 2 , A u 2 >_{-1}^0.5 = 1.859709543405875e-11
Newton iteration 3energy = 0.23985214668555557<A u 3 , A u 3 >_{-1}^0.5 = 4.435203152048984e-16
t = 1.3
Newton iteration 0energy = 0.2398475475690217<A u 0 , A u 0 >_{-1}^0.5 = 0.002752600824904295
Newton iteration 1energy = 0.23984376149489278<A u 1 , A u 1 >_{-1}^0.5 = 3.6652764335416655e-06
Newton iteration 2energy = 0.23984376148817577<A u 2 , A u 2 >_{-1}^0.5 = 1.153143661066987e-11
Newton iteration 3energy = 0.2398437614881757<A u 3 , A u 3 >_{-1}^0.5 = 4.751156991531967e-16
t = 1.4000000000000001
Newton iteration 0energy = 0.23984051163305103<A u 0 , A u 0 >_{-1}^0.5 = 0.0024031293480801267
Newton iteration 1energy = 0.23983762559026806<A u 1 , A u 1 >_{-1}^0.5 = 3.0872203347781752e-06
Newton iteration 2energy = 0.23983762558550256<A u 2 , A u 2 >_{-1}^0.5 = 7.81224707708281e-12
Newton iteration 3energy = 0.23983762558550256<A u 3 , A u 3 >_{-1}^0.5 = 4.554551406752829e-16
t = 1.5000000000000002
Newton iteration 0energy = 0.23983499320021812<A u 0 , A u 0 >_{-1}^0.5 = 0.002212931109753908
Newton iteration 1energy = 0.2398325456786902<A u 1 , A u 1 >_{-1}^0.5 = 2.71324568440958e-06
Newton iteration 2energy = 0.23983254567500942<A u 2 , A u 2 >_{-1}^0.5 = 5.665173141427404e-12
Newton iteration 3energy = 0.23983254567500942<A u 3 , A u 3 >_{-1}^0.5 = 4.726602517996779e-16
t = 1.6000000000000003
Newton iteration 0energy = 0.2398302381096158<A u 0 , A u 0 >_{-1}^0.5 = 0.00209600786221311
Newton iteration 1energy = 0.23982804221769077<A u 1 , A u 1 >_{-1}^0.5 = 2.4404707967652944e-06
Newton iteration 2energy = 0.23982804221471282<A u 2 , A u 2 >_{-1}^0.5 = 4.324024549280457e-12
Newton iteration 3energy = 0.23982804221471277<A u 3 , A u 3 >_{-1}^0.5 = 4.4003037232717784e-16
t = 1.7000000000000004
Newton iteration 0energy = 0.2398259387400405<A u 0 , A u 0 >_{-1}^0.5 = 0.0020119537358292837
Newton iteration 1energy = 0.2398239153146432<A u 1 , A u 1 >_{-1}^0.5 = 2.2239964856029444e-06
Newton iteration 2energy = 0.23982391531217007<A u 2 , A u 2 >_{-1}^0.5 = 3.4244584591071713e-12
Newton iteration 3energy = 0.23982391531217015<A u 3 , A u 3 >_{-1}^0.5 = 4.735097125096318e-16
t = 1.8000000000000005
Newton iteration 0energy = 0.23982196287589924<A u 0 , A u 0 >_{-1}^0.5 = 0.0019431518886679814
Newton iteration 1energy = 0.2398200753888226<A u 1 , A u 1 >_{-1}^0.5 = 2.0425057977835395e-06
Newton iteration 2energy = 0.23982007538673666<A u 2 , A u 2 >_{-1}^0.5 = 2.782691845653111e-12
Newton iteration 3energy = 0.2398200753867367<A u 3 , A u 3 >_{-1}^0.5 = 4.3181437412949555e-16
t = 1.9000000000000006
Newton iteration 0energy = 0.2398182479358837<A u 0 , A u 0 >_{-1}^0.5 = 0.0018821371909694491
Newton iteration 1energy = 0.23981647706316297<A u 1 , A u 1 >_{-1}^0.5 = 1.8850840284019326e-06
Newton iteration 2energy = 0.23981647706138623<A u 2 , A u 2 >_{-1}^0.5 = 2.301654568685749e-12
Newton iteration 3energy = 0.23981647706138623<A u 3 , A u 3 >_{-1}^0.5 = 4.065593619486878e-16
t = 2.0000000000000004
Newton iteration 0energy = 0.2398147596177571<A u 0 , A u 0 >_{-1}^0.5 = 0.0018257325375343942
Newton iteration 1energy = 0.23981309325352684<A u 1 , A u 1 >_{-1}^0.5 = 1.7456684941452875e-06
Newton iteration 2energy = 0.23981309325200312<A u 2 , A u 2 >_{-1}^0.5 = 1.927554569081422e-12
Newton iteration 3energy = 0.2398130932520031<A u 3 , A u 3 >_{-1}^0.5 = 4.332672216391466e-16
t = 2.1000000000000005
Newton iteration 0energy = 0.2398114756305933<A u 0 , A u 0 >_{-1}^0.5 = 0.00177253811819305
Newton iteration 1energy = 0.23980990492349186<A u 1 , A u 1 >_{-1}^0.5 = 1.6205829233983481e-06
Newton iteration 2energy = 0.23980990492217882<A u 2 , A u 2 >_{-1}^0.5 = 1.6286739972797851e-12
Newton iteration 3energy = 0.23980990492217874<A u 3 , A u 3 >_{-1}^0.5 = 4.418668025273975e-16
t = 2.2000000000000006
Newton iteration 0energy = 0.2398083792159379<A u 0 , A u 0 >_{-1}^0.5 = 0.0017218913657836535
Newton iteration 1energy = 0.23980689696317328<A u 1 , A u 1 >_{-1}^0.5 = 1.5073985458880097e-06
Newton iteration 2energy = 0.23980689696203722<A u 2 , A u 2 >_{-1}^0.5 = 1.3852078240876382e-12
Newton iteration 3energy = 0.23980689696203722<A u 3 , A u 3 >_{-1}^0.5 = 3.9938194820822333e-16
t = 2.3000000000000007
Newton iteration 0energy = 0.2398054565059059<A u 0 , A u 0 >_{-1}^0.5 = 0.0016734406119765272
Newton iteration 1energy = 0.23980405647728192<A u 1 , A u 1 >_{-1}^0.5 = 1.4043855040843581e-06
Newton iteration 2energy = 0.23980405647629582<A u 2 , A u 2 >_{-1}^0.5 = 1.1841647727200964e-12
Newton iteration 3energy = 0.2398040564762958<A u 3 , A u 3 >_{-1}^0.5 = 4.247566867110549e-16
t = 2.400000000000001
Newton iteration 0energy = 0.2398026953982045<A u 0 , A u 0 >_{-1}^0.5 = 0.0016269699860902154
Newton iteration 1energy = 0.2398013720319713<A u 1 , A u 1 >_{-1}^0.5 = 1.310234983066994e-06
Newton iteration 2energy = 0.23980137203111307<A u 2 , A u 2 >_{-1}^0.5 = 1.016525806629807e-12
Newton iteration 3energy = 0.239801372031113<A u 3 , A u 3 >_{-1}^0.5 = 4.639260787377315e-16
t = 2.500000000000001
Newton iteration 0energy = 0.2398000850401235<A u 0 , A u 0 >_{-1}^0.5 = 0.0015823266755118544
Newton iteration 1energy = 0.23979883329107027<A u 1 , A u 1 >_{-1}^0.5 = 1.2239101236459694e-06
Newton iteration 2energy = 0.23979883329032123<A u 2 , A u 2 >_{-1}^0.5 = 8.756180125687046e-13
Newton iteration 3energy = 0.23979883329032126<A u 3 , A u 3 >_{-1}^0.5 = 4.2664004919119956e-16
t = 2.600000000000001
Newton iteration 0energy = 0.239797615563624<A u 0 , A u 0 >_{-1}^0.5 = 0.001539390031738783
Newton iteration 1energy = 0.23979643081609273<A u 1 , A u 1 >_{-1}^0.5 = 1.1445610184530363e-06
Newton iteration 2energy = 0.2397964308154376<A u 2 , A u 2 >_{-1}^0.5 = 7.565200023581615e-13
Newton iteration 3energy = 0.23979643081543767<A u 3 , A u 3 >_{-1}^0.5 = 4.432499036739255e-16
t = 2.700000000000001
Newton iteration 0energy = 0.2397952779287728<A u 0 , A u 0 >_{-1}^0.5 = 0.0014980579568581748
Newton iteration 1energy = 0.2397941559392485<A u 1 , A u 1 >_{-1}^0.5 = 1.0714732744944532e-06
Newton iteration 2energy = 0.23979415593867454<A u 2 , A u 2 >_{-1}^0.5 = 6.553641968942448e-13
Newton iteration 3energy = 0.23979415593867454<A u 3 , A u 3 >_{-1}^0.5 = 4.1874593965469807e-16
t = 2.800000000000001
Newton iteration 0energy = 0.23979306381744558<A u 0 , A u 0 >_{-1}^0.5 = 0.0014582405541298975
Newton iteration 1energy = 0.23979200067199585<A u 1 , A u 1 >_{-1}^0.5 = 1.0040351240407334e-06
Newton iteration 2energy = 0.23979200067149176<A u 2 , A u 2 >_{-1}^0.5 = 5.691109095262611e-13
Newton iteration 3energy = 0.23979200067149187<A u 3 , A u 3 >_{-1}^0.5 = 4.155201030438557e-16
t = 2.9000000000000012
Newton iteration 0energy = 0.2397909655529058<A u 0 , A u 0 >_{-1}^0.5 = 0.0014198569182498524
Newton iteration 1energy = 0.23978995763305<A u 1 , A u 1 >_{-1}^0.5 = 9.41715334679448e-07
Newton iteration 2energy = 0.23978995763260663<A u 2 , A u 2 >_{-1}^0.5 = 4.953430609681837e-13
Newton iteration 3energy = 0.23978995763260666<A u 3 , A u 3 >_{-1}^0.5 = 4.062288222243957e-16
t = 3.0000000000000013
Newton iteration 0energy = 0.2397889760344917<A u 0 , A u 0 >_{-1}^0.5 = 0.0013828333391534655
Newton iteration 1energy = 0.2397880199885049<A u 1 , A u 1 >_{-1}^0.5 = 8.840477145492321e-07
Newton iteration 2energy = 0.2397880199881142<A u 2 , A u 2 >_{-1}^0.5 = 4.320760055005469e-13
Newton iteration 3energy = 0.23978801998811416<A u 3 , A u 3 >_{-1}^0.5 = 4.274265081484502e-16
t = 3.1000000000000014
Newton iteration 0energy = 0.23978708868229578<A u 0 , A u 0 >_{-1}^0.5 = 0.0013471021789486233
Newton iteration 1energy = 0.2397861814004151<A u 1 , A u 1 >_{-1}^0.5 = 8.30619822227915e-07
Newton iteration 2energy = 0.23978618140007008<A u 2 , A u 2 >_{-1}^0.5 = 3.776448366640841e-13
Newton iteration 3energy = 0.23978618140007008<A u 3 , A u 3 >_{-1}^0.5 = 4.2135651035695886e-16
t = 3.2000000000000015
Newton iteration 0energy = 0.2397852973891652<A u 0 , A u 0 >_{-1}^0.5 = 0.0013126010930648345
Newton iteration 1energy = 0.2397844359818262<A u 1 , A u 1 >_{-1}^0.5 = 7.810644622035656e-07
Newton iteration 2energy = 0.23978443598152122<A u 2 , A u 2 >_{-1}^0.5 = 3.307692287274495e-13
Newton iteration 3energy = 0.23978443598152122<A u 3 , A u 3 >_{-1}^0.5 = 4.032754271097354e-16
t = 3.3000000000000016
Newton iteration 0energy = 0.23978359647845926<A u 0 , A u 0 >_{-1}^0.5 = 0.00127927244252396
Newton iteration 1energy = 0.23978277825700822<A u 1 , A u 1 >_{-1}^0.5 = 7.350530890752462e-07
Newton iteration 2energy = 0.239782778256738<A u 2 , A u 2 >_{-1}^0.5 = 2.9022431212867816e-13
Newton iteration 3energy = 0.239782778256738<A u 3 , A u 3 >_{-1}^0.5 = 4.700343128758156e-16
t = 3.4000000000000017
Newton iteration 0energy = 0.23978198066655104<A u 0 , A u 0 >_{-1}^0.5 = 0.0012470628218648177
Newton iteration 1energy = 0.23978120312604315<A u 1 , A u 1 >_{-1}^0.5 = 6.922905624613361e-07
Newton iteration 2energy = 0.2397812031258036<A u 2 , A u 2 >_{-1}^0.5 = 2.5513449494556696e-13
Newton iteration 3energy = 0.23978120312580356<A u 3 , A u 3 >_{-1}^0.5 = 4.1233551729270967e-16
t = 3.5000000000000018
Newton iteration 0energy = 0.23978044502934848<A u 0 , A u 0 >_{-1}^0.5 = 0.0012159226630448416
Newton iteration 1energy = 0.23977970583314206<A u 1 , A u 1 >_{-1}^0.5 = 6.525108847400067e-07
Newton iteration 2energy = 0.23977970583292926<A u 2 , A u 2 >_{-1}^0.5 = 2.2468937918330332e-13
Newton iteration 3energy = 0.2397797058329293<A u 3 , A u 3 >_{-1}^0.5 = 3.962806355020745e-16
t = 3.600000000000002
Newton iteration 0energy = 0.23977898497228517<A u 0 , A u 0 >_{-1}^0.5 = 0.0011858058930362665
Newton iteration 1energy = 0.23977828193819856<A u 1 , A u 1 >_{-1}^0.5 = 6.154736727079722e-07
Newton iteration 2energy = 0.23977828193800924<A u 2 , A u 2 >_{-1}^0.5 = 1.9821307859912326e-13
Newton iteration 3energy = 0.23977828193800924<A u 3 , A u 3 >_{-1}^0.5 = 4.269934002657767e-16
t = 3.700000000000002
Newton iteration 0energy = 0.2397775962033396<A u 0 , A u 0 >_{-1}^0.5 = 0.001156669631793492
Newton iteration 1energy = 0.23977692729118208<A u 1 , A u 1 >_{-1}^0.5 = 5.809611925994214e-07
Newton iteration 2energy = 0.23977692729101333<A u 2 , A u 2 >_{-1}^0.5 = 1.75183457391464e-13
Newton iteration 3energy = 0.23977692729101338<A u 3 , A u 3 >_{-1}^0.5 = 4.3723109513723124e-16
t = 3.800000000000002
Newton iteration 0energy = 0.23977627470871887<A u 0 , A u 0 >_{-1}^0.5 = 0.0011284739221579224
Newton iteration 1energy = 0.23977563800903526<A u 1 , A u 1 >_{-1}^0.5 = 5.487758354709267e-07
Newton iteration 2energy = 0.2397756380088847<A u 2 , A u 2 >_{-1}^0.5 = 1.5508731897276287e-13
Newton iteration 3energy = 0.23977563800888474<A u 3 , A u 3 >_{-1}^0.5 = 4.564160512717691e-16
t = 3.900000000000002
Newton iteration 0energy = 0.2397750167308999<A u 0 , A u 0 >_{-1}^0.5 = 0.001101181486078326
Newton iteration 1energy = 0.23977441045479161<A u 1 , A u 1 >_{-1}^0.5 = 5.187379450384153e-07
Newton iteration 2energy = 0.239774410454657<A u 2 , A u 2 >_{-1}^0.5 = 1.3755493842188126e-13
Newton iteration 3energy = 0.23977441045465703<A u 3 , A u 3 >_{-1}^0.5 = 4.095649200609425e-16
t = 4.000000000000002
Newton iteration 0energy = 0.2397738187487627<A u 0 , A u 0 >_{-1}^0.5 = 0.001074757503214583
Newton iteration 1energy = 0.23977324121866778<A u 1 , A u 1 >_{-1}^0.5 = 4.906839324571569e-07
Newton iteration 2energy = 0.23977324121854737<A u 2 , A u 2 >_{-1}^0.5 = 1.2223585670298852e-13
Newton iteration 3energy = 0.23977324121854743<A u 3 , A u 3 >_{-1}^0.5 = 4.3171662182331956e-16
t = 4.100000000000001
Newton iteration 0energy = 0.23977267745958777<A u 0 , A u 0 >_{-1}^0.5 = 0.0010491694090460192
Newton iteration 1energy = 0.23977212710091575<A u 1 , A u 1 >_{-1}^0.5 = 4.644646276698405e-07
Newton iteration 2energy = 0.23977212710080792<A u 2 , A u 2 >_{-1}^0.5 = 1.0872608116116177e-13
Newton iteration 3energy = 0.23977212710080795<A u 3 , A u 3 >_{-1}^0.5 = 4.1875454396182396e-16
t = 4.200000000000001
Newton iteration 0energy = 0.23977158976271706<A u 0 , A u 0 >_{-1}^0.5 = 0.001024386710280996
Newton iteration 1energy = 0.23977106509624785<A u 1 , A u 1 >_{-1}^0.5 = 4.3994382997375445e-07
Newton iteration 2energy = 0.23977106509615115<A u 2 , A u 2 >_{-1}^0.5 = 9.693884643545344e-14
t = 4.300000000000001
Newton iteration 0energy = 0.23977055274470216<A u 0 , A u 0 >_{-1}^0.5 = 0.0010003808158115914
Newton iteration 1energy = 0.2397700523796682<A u 1 , A u 1 >_{-1}^0.5 = 4.169970270670945e-07
Newton iteration 2energy = 0.23977005237958127<A u 2 , A u 2 >_{-1}^0.5 = 8.652038019647517e-14
t = 4.4
Newton iteration 0energy = 0.239769563665782<A u 0 , A u 0 >_{-1}^0.5 = 0.0009771248817590904
Newton iteration 1energy = 0.23976908629356442<A u 1 , A u 1 >_{-1}^0.5 = 3.955102583501198e-07
Newton iteration 2energy = 0.2397690862934863<A u 2 , A u 2 >_{-1}^0.5 = 7.735034192969801e-14
t = 4.5
Newton iteration 0energy = 0.23976861994755688<A u 0 , A u 0 >_{-1}^0.5 = 0.0009545936693686939
Newton iteration 1energy = 0.23976816433593098<A u 1 , A u 1 >_{-1}^0.5 = 3.7537910334617193e-07
Newton iteration 2energy = 0.23976816433586048<A u 2 , A u 2 >_{-1}^0.5 = 6.931270608433821e-14
t = 4.6
Newton iteration 0energy = 0.23976771916173206<A u 0 , A u 0 >_{-1}^0.5 = 0.0009327634146648947
Newton iteration 1energy = 0.2397672841496067<A u 1 , A u 1 >_{-1}^0.5 = 3.5650777846654474e-07
Newton iteration 2energy = 0.23976728414954324<A u 2 , A u 2 >_{-1}^0.5 = 6.220969450540064e-14
t = 4.699999999999999
Newton iteration 0energy = 0.23976685901982447<A u 0 , A u 0 >_{-1}^0.5 = 0.0009116117088952585
Newton iteration 1energy = 0.23976644351242657<A u 1 , A u 1 >_{-1}^0.5 = 3.388083290629074e-07
Newton iteration 2energy = 0.23976644351236912<A u 2 , A u 2 >_{-1}^0.5 = 5.586161326439167e-14
t = 4.799999999999999
Newton iteration 0energy = 0.2397660373637349<A u 0 , A u 0 >_{-1}^0.5 = 0.0008911173888813033
Newton iteration 1energy = 0.23976564032819303<A u 1 , A u 1 >_{-1}^0.5 = 3.2219990501297664e-07
Newton iteration 2energy = 0.2397656403281411<A u 2 , A u 2 >_{-1}^0.5 = 5.02831172942925e-14
t = 4.899999999999999
Newton iteration 0energy = 0.23976525215710134<A u 0 , A u 0 >_{-1}^0.5 = 0.000871260436470312
Newton iteration 1energy = 0.23976487261839036<A u 1 , A u 1 >_{-1}^0.5 = 3.0660811023793785e-07
Newton iteration 2energy = 0.23976487261834337<A u 2 , A u 2 >_{-1}^0.5 = 4.532196011625237e-14
t = 4.999999999999998
Minimal energy extension (postscript in unit-2.1.3 )¶
\[u \in V^{ho,disc}, \quad u^{lo,cont} \in V^{lo,cont}, \quad \lambda \in V^{lo,disc},\]
[14]:
from netgen.geom2d import unit_square
mesh = Mesh(unit_square.GenerateMesh(maxh=0.1))
fes_ho = Discontinuous(H1(mesh, order=10))
fes_lo = H1(mesh, order=1, dirichlet=".*")
fes_lam = Discontinuous(H1(mesh, order=1))
fes = FESpace([fes_ho, fes_lo, fes_lam])
uho, ulo, lam = fes.TrialFunction()
\[\int_{\Omega} \frac12 \Vert \nabla u \Vert^2 - u + \sum_T \sum_{V \in V(T)} ((u-u^{lo})\cdot \lambda)|_{V} \longrightarrow \operatorname{min}!\]
[15]:
a = BilinearForm(fes)
a += Variation(0.5 * grad(uho)*grad(uho)*dx
- 1*uho*dx
+ (uho-ulo)*lam*dx(element_vb=BBND))
gfu = GridFunction(fes)
solvers.Newton(a=a, u=gfu)
Draw(gfu.components[0])
Newton iteration 0
err = 0.3929793997717297
Newton iteration 1
err = 3.956677249565376e-15
The minimization problem is solved by the solution of the PDE:
\[ \begin{align}\begin{aligned} \int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} 1 \cdot v \quad \forall ~ v \in V^{ho,disc}\\under the constraint\end{aligned}\end{align} \]
\[u(v) = u^{lo}(v) \quad \text{ for all vertices } v \in V(T) \text{ for all } T.\]