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 = 0.13255958653113628
Newton iteration 1energy = -0.008785678644368577<A u 1 , A u 1 >_{-1}^0.5 = 1.1107598523198008e-05
Newton iteration 2energy = -0.008785678706057953<A u 2 , A u 2 >_{-1}^0.5 = 2.8075898252642246e-13
Newton iteration 3energy = -0.008785678706057952<A u 3 , A u 3 >_{-1}^0.5 = 3.741902157776e-17
energy = -0.008785678706057952
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 = 0.13255958653113625
Newton iteration 1
Energy: -0.00878567864436858
err = 1.1107598523178109e-05
Newton iteration 2
Energy: -0.008785678706057952
err = 2.807455664350966e-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 = 8.749999999999998<A u 0 , A u 0 >_{-1}^0.5 = 0.0011109609185296946
Newton iteration 1energy = 8.75005276617483<A u 1 , A u 1 >_{-1}^0.5 = 0.009148322973133765
Newton iteration 2energy = 8.751477850270941<A u 2 , A u 2 >_{-1}^0.5 = 0.04772739026732187
Newton iteration 3energy = 8.758125165185952<A u 3 , A u 3 >_{-1}^0.5 = 0.11337401648217023
Newton iteration 4energy = 8.753021775938693<A u 4 , A u 4 >_{-1}^0.5 = 0.06546139967597644
Newton iteration 5energy = 17.103342030948674<A u 5 , A u 5 >_{-1}^0.5 = 2.7931046809405693
Newton iteration 6energy = 9.681624244626434<A u 6 , A u 6 >_{-1}^0.5 = 7.121338031267679
Newton iteration 7energy = -411.66269569145055<A u 7 , A u 7 >_{-1}^0.5 = 28.867702165200257
Newton iteration 8energy = 3.663918575946365<A u 8 , A u 8 >_{-1}^0.5 = 3.427818843815449
Newton iteration 9energy = -18.995359467921844<A u 9 , A u 9 >_{-1}^0.5 = 6.697070258969443
Newton iteration 10energy = -74.50022980913081<A u 10 , A u 10 >_{-1}^0.5 = 11.955595040587365
Newton iteration 11energy = -18.598626544437593<A u 11 , A u 11 >_{-1}^0.5 = 7.017303947035795
Newton iteration 12energy = 32.02731805741041<A u 12 , A u 12 >_{-1}^0.5 = 8.02903487065813
Newton iteration 13energy = 5.478032449287156<A u 13 , A u 13 >_{-1}^0.5 = 2.3997455854448324
Newton iteration 14energy = -159.8138035930931<A u 14 , A u 14 >_{-1}^0.5 = 17.9716084482156
Newton iteration 15energy = -18.513252215288958<A u 15 , A u 15 >_{-1}^0.5 = 6.632060780213734
Newton iteration 16energy = -15.75237975642085<A u 16 , A u 16 >_{-1}^0.5 = 5.4720311076794905
Newton iteration 17energy = 11.700383454561234<A u 17 , A u 17 >_{-1}^0.5 = 1.9998306026547423
Newton iteration 18energy = -149.55175018310547<A u 18 , A u 18 >_{-1}^0.5 = 17.939989756084167
Newton iteration 19energy = -122.21556091308594<A u 19 , A u 19 >_{-1}^0.5 = 16.464950840685667
Newton iteration 20energy = 20.068984985351562<A u 20 , A u 20 >_{-1}^0.5 = 756.492828790009
Newton iteration 21energy = -112312975360.0<A u 21 , A u 21 >_{-1}^0.5 = 473916.53343712824
Newton iteration 22energy = 4194304.0<A u 22 , A u 22 >_{-1}^0.5 = 851.2840572008482
Newton iteration 23energy = 6291456.0<A u 23 , A u 23 >_{-1}^0.5 = 1372.9056640273288
Newton iteration 24energy = 2097152.0<A u 24 , A u 24 >_{-1}^0.5 = 2046.5928896695846
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 = 0.4145019207496359<A u 0 , A u 0 >_{-1}^0.5 = 0.2809270463814763
Newton iteration 1energy = 0.37689785802772924<A u 1 , A u 1 >_{-1}^0.5 = 0.01917437047193973
Newton iteration 2energy = 0.3767134210404818<A u 2 , A u 2 >_{-1}^0.5 = 9.628937134602255e-05
Newton iteration 3energy = 0.37671341640458017<A u 3 , A u 3 >_{-1}^0.5 = 2.5054562992371867e-09
Newton iteration 4energy = 0.37671341640458006<A u 4 , A u 4 >_{-1}^0.5 = 3.003282889895448e-16
t = 0.1
Newton iteration 0energy = 0.3438812865238592<A u 0 , A u 0 >_{-1}^0.5 = 0.23603934740782906
Newton iteration 1energy = 0.31710944329874874<A u 1 , A u 1 >_{-1}^0.5 = 0.013442999343193882
Newton iteration 2energy = 0.3170188793463516<A u 2 , A u 2 >_{-1}^0.5 = 4.660092984275238e-05
Newton iteration 3energy = 0.31701887826051917<A u 3 , A u 3 >_{-1}^0.5 = 5.76804827520762e-10
Newton iteration 4energy = 0.3170188782605193<A u 4 , A u 4 >_{-1}^0.5 = 3.331991791505268e-16
t = 0.2
Newton iteration 0energy = 0.2954943565138458<A u 0 , A u 0 >_{-1}^0.5 = 0.18345925931027077
Newton iteration 1energy = 0.27916403033890114<A u 1 , A u 1 >_{-1}^0.5 = 0.008014651163108892
Newton iteration 2energy = 0.2791318699362322<A u 2 , A u 2 >_{-1}^0.5 = 1.6250158077627237e-05
Newton iteration 3energy = 0.279131869804198<A u 3 , A u 3 >_{-1}^0.5 = 6.857115501841529e-11
Newton iteration 4energy = 0.27913186980419785<A u 4 , A u 4 >_{-1}^0.5 = 3.416831081582394e-16
t = 0.30000000000000004
Newton iteration 0energy = 0.26692789507920534<A u 0 , A u 0 >_{-1}^0.5 = 0.13349658930568847
Newton iteration 1energy = 0.2582050355110707<A u 1 , A u 1 >_{-1}^0.5 = 0.0041885131065817005
Newton iteration 2energy = 0.2581962576290239<A u 2 , A u 2 >_{-1}^0.5 = 4.366756818971012e-06
Newton iteration 3energy = 0.25819625761948967<A u 3 , A u 3 >_{-1}^0.5 = 4.8506189386333065e-12
Newton iteration 4energy = 0.25819625761948944<A u 4 , A u 4 >_{-1}^0.5 = 3.588377772533667e-16
t = 0.4
Newton iteration 0energy = 0.2520431142368918<A u 0 , A u 0 >_{-1}^0.5 = 0.09236175483868561
Newton iteration 1energy = 0.24783887867786308<A u 1 , A u 1 >_{-1}^0.5 = 0.0019843467589302
Newton iteration 2energy = 0.2478369092241838<A u 2 , A u 2 >_{-1}^0.5 = 9.685217011892036e-07
Newton iteration 3energy = 0.2478369092237148<A u 3 , A u 3 >_{-1}^0.5 = 2.3476531523095486e-13
Newton iteration 4energy = 0.24783690922371485<A u 4 , A u 4 >_{-1}^0.5 = 3.7248806965135973e-16
t = 0.5
Newton iteration 0energy = 0.24499322447001137<A u 0 , A u 0 >_{-1}^0.5 = 0.061646841904519516
Newton iteration 1energy = 0.24311100787346115<A u 1 , A u 1 >_{-1}^0.5 = 0.0008779623506489409
Newton iteration 2energy = 0.2431106224096563<A u 2 , A u 2 >_{-1}^0.5 = 1.8822078760924155e-07
Newton iteration 3energy = 0.24311062240963854<A u 3 , A u 3 >_{-1}^0.5 = 8.81645854287601e-15
t = 0.6
Newton iteration 0energy = 0.2418735481479401<A u 0 , A u 0 >_{-1}^0.5 = 0.04016635875651366
Newton iteration 1energy = 0.24107181028212027<A u 1 , A u 1 >_{-1}^0.5 = 0.0003715367480243826
Newton iteration 2energy = 0.24107174125819192<A u 2 , A u 2 >_{-1}^0.5 = 3.3668941389797124e-08
Newton iteration 3energy = 0.2410717412581913<A u 3 , A u 3 >_{-1}^0.5 = 4.617523763320809e-16
t = 0.7
Newton iteration 0energy = 0.24055438873764584<A u 0 , A u 0 >_{-1}^0.5 = 0.025782033859246597
Newton iteration 1energy = 0.2402233339155409<A u 1 , A u 1 >_{-1}^0.5 = 0.00015331214259508866
Newton iteration 2energy = 0.2402233221629405<A u 2 , A u 2 >_{-1}^0.5 = 5.795968682048955e-09
Newton iteration 3energy = 0.24022332216294046<A u 3 , A u 3 >_{-1}^0.5 = 4.3395457490668377e-16
t = 0.7999999999999999
Newton iteration 0energy = 0.2400118618581594<A u 0 , A u 0 >_{-1}^0.5 = 0.016425332383409562
Newton iteration 1energy = 0.23987730469401888<A u 1 , A u 1 >_{-1}^0.5 = 6.283809116192321e-05
Newton iteration 2energy = 0.23987730271968516<A u 2 , A u 2 >_{-1}^0.5 = 1.0154930066600787e-09
Newton iteration 3energy = 0.2398773027196851<A u 3 , A u 3 >_{-1}^0.5 = 4.805333945709333e-16
t = 0.8999999999999999
Newton iteration 0energy = 0.23979158193635391<A u 0 , A u 0 >_{-1}^0.5 = 0.010464445156609924
Newton iteration 1energy = 0.23973691889059928<A u 1 , A u 1 >_{-1}^0.5 = 2.620620881241072e-05
Newton iteration 2energy = 0.23973691854721502<A u 2 , A u 2 >_{-1}^0.5 = 1.999460969185541e-10
Newton iteration 3energy = 0.23973691854721504<A u 3 , A u 3 >_{-1}^0.5 = 4.457911895517942e-16
t = 0.9999999999999999
Newton iteration 0energy = 0.2397018609405314<A u 0 , A u 0 >_{-1}^0.5 = 0.00673541582696503
Newton iteration 1energy = 0.23967920289205236<A u 1 , A u 1 >_{-1}^0.5 = 1.1556763084947155e-05
Newton iteration 2energy = 0.23967920282527283<A u 2 , A u 2 >_{-1}^0.5 = 5.0994587785631396e-11
Newton iteration 3energy = 0.2396792028252728<A u 3 , A u 3 >_{-1}^0.5 = 4.714110044332649e-16
t = 1.0999999999999999
Newton iteration 0energy = 0.23966435830447477<A u 0 , A u 0 >_{-1}^0.5 = 0.0044525423614036955
Newton iteration 1energy = 0.2396544535095991<A u 1 , A u 1 >_{-1}^0.5 = 5.694142646676037e-06
Newton iteration 2energy = 0.23965445349338746<A u 2 , A u 2 >_{-1}^0.5 = 1.7725040441281417e-11
Newton iteration 3energy = 0.2396544534933875<A u 3 , A u 3 >_{-1}^0.5 = 4.567722193531726e-16
t = 1.2
Newton iteration 0energy = 0.23964767142737942<A u 0 , A u 0 >_{-1}^0.5 = 0.0030979081418175307
Newton iteration 1energy = 0.23964287580252736<A u 1 , A u 1 >_{-1}^0.5 = 3.284053675981641e-06
Newton iteration 2energy = 0.2396428757971349<A u 2 , A u 2 >_{-1}^0.5 = 7.767210031814492e-12
Newton iteration 3energy = 0.23964287579713484<A u 3 , A u 3 >_{-1}^0.5 = 4.662852056770271e-16
t = 1.3
Newton iteration 0energy = 0.23963934988363764<A u 0 , A u 0 >_{-1}^0.5 = 0.002326255780763496
Newton iteration 1energy = 0.2396366454535445<A u 1 , A u 1 >_{-1}^0.5 = 2.2181767038475607e-06
Newton iteration 2energy = 0.23963664545108435<A u 2 , A u 2 >_{-1}^0.5 = 3.983602113181903e-12
Newton iteration 3energy = 0.2396366454510843<A u 3 , A u 3 >_{-1}^0.5 = 4.444524444123482e-16
t = 1.4000000000000001
Newton iteration 0energy = 0.23963447926266948<A u 0 , A u 0 >_{-1}^0.5 = 0.0019006816819441919
Newton iteration 1energy = 0.23963267365152316<A u 1 , A u 1 >_{-1}^0.5 = 1.6879462608466211e-06
Newton iteration 2energy = 0.2396326736500986<A u 2 , A u 2 >_{-1}^0.5 = 2.298272966166801e-12
Newton iteration 3energy = 0.23963267365009866<A u 3 , A u 3 >_{-1}^0.5 = 4.578265035668924e-16
t = 1.5000000000000002
Newton iteration 0energy = 0.23963111549003374<A u 0 , A u 0 >_{-1}^0.5 = 0.0016629280504840886
Newton iteration 1energy = 0.23962973322336584<A u 1 , A u 1 >_{-1}^0.5 = 1.3855651852362916e-06
Newton iteration 2energy = 0.23962973322240586<A u 2 , A u 2 >_{-1}^0.5 = 1.458620838855635e-12
Newton iteration 3energy = 0.239629733222406<A u 3 , A u 3 >_{-1}^0.5 = 4.518386193786961e-16
t = 1.6000000000000003
Newton iteration 0energy = 0.23962848049094237<A u 0 , A u 0 >_{-1}^0.5 = 0.0015187632040498532
Newton iteration 1energy = 0.23962732741809964<A u 1 , A u 1 >_{-1}^0.5 = 1.1887244240849515e-06
Newton iteration 2energy = 0.23962732741739326<A u 2 , A u 2 >_{-1}^0.5 = 1.0010923706915846e-12
Newton iteration 3energy = 0.2396273274173932<A u 3 , A u 3 >_{-1}^0.5 = 4.2566588833671387e-16
t = 1.7000000000000004
Newton iteration 0energy = 0.23962625398674228<A u 0 , A u 0 >_{-1}^0.5 = 0.0014193843873493643
Newton iteration 1energy = 0.23962524682330952<A u 1 , A u 1 >_{-1}^0.5 = 1.0456329093933632e-06
Newton iteration 2energy = 0.23962524682276293<A u 2 , A u 2 >_{-1}^0.5 = 7.300993583012621e-13
Newton iteration 3energy = 0.2396252468227629<A u 3 , A u 3 >_{-1}^0.5 = 4.534110875500262e-16
t = 1.8000000000000005
Newton iteration 0energy = 0.23962429644294805<A u 0 , A u 0 >_{-1}^0.5 = 0.0013419591604952315
Newton iteration 1energy = 0.23962339612718114<A u 1 , A u 1 >_{-1}^0.5 = 9.330532164753765e-07
Newton iteration 2energy = 0.23962339612674582<A u 2 , A u 2 >_{-1}^0.5 = 5.563574228766939e-13
Newton iteration 3energy = 0.23962339612674577<A u 3 , A u 3 >_{-1}^0.5 = 4.0237775990779104e-16
t = 1.9000000000000006
Newton iteration 0energy = 0.23962254072403272<A u 0 , A u 0 >_{-1}^0.5 = 0.0012763226584932732
Newton iteration 1energy = 0.23962172630379305<A u 1 , A u 1 >_{-1}^0.5 = 8.399033658821723e-07
Newton iteration 2energy = 0.23962172630344042<A u 2 , A u 2 >_{-1}^0.5 = 4.3694470025439145e-13
Newton iteration 3energy = 0.23962172630344036<A u 3 , A u 3 >_{-1}^0.5 = 4.3805270472932427e-16
t = 2.0000000000000004
Newton iteration 0energy = 0.2396209495967649<A u 0 , A u 0 >_{-1}^0.5 = 0.001217971596189921
Newton iteration 1energy = 0.2396202079282432<A u 1 , A u 1 >_{-1}^0.5 = 7.604668960370086e-07
Newton iteration 2energy = 0.23962020792795408<A u 2 , A u 2 >_{-1}^0.5 = 3.503108574361001e-13
Newton iteration 3energy = 0.2396202079279541<A u 3 , A u 3 >_{-1}^0.5 = 4.2155579625369558e-16
t = 2.1000000000000005
Newton iteration 0energy = 0.2396194989264497<A u 0 , A u 0 >_{-1}^0.5 = 0.00116483587636887
Newton iteration 1energy = 0.23961882055016812<A u 1 , A u 1 >_{-1}^0.5 = 6.914942047786559e-07
Newton iteration 2energy = 0.23961882054992897<A u 2 , A u 2 >_{-1}^0.5 = 2.8492948852230904e-13
Newton iteration 3energy = 0.23961882054992895<A u 3 , A u 3 >_{-1}^0.5 = 4.041623035830586e-16
t = 2.2000000000000006
Newton iteration 0energy = 0.23961817091741788<A u 0 , A u 0 >_{-1}^0.5 = 0.0011158806899792928
Newton iteration 1energy = 0.23961754835794213<A u 1 , A u 1 >_{-1}^0.5 = 6.309368258451742e-07
Newton iteration 2energy = 0.2396175483577431<A u 2 , A u 2 >_{-1}^0.5 = 2.342551544364737e-13
Newton iteration 3energy = 0.23961754835774307<A u 3 , A u 3 >_{-1}^0.5 = 3.7659915046707365e-16
t = 2.3000000000000007
Newton iteration 0energy = 0.23961695130021213<A u 0 , A u 0 >_{-1}^0.5 = 0.001070516865471795
Newton iteration 1energy = 0.23961637832545668<A u 1 , A u 1 >_{-1}^0.5 = 5.773776869002892e-07
Newton iteration 2energy = 0.23961637832529<A u 2 , A u 2 >_{-1}^0.5 = 1.941499345816214e-13
Newton iteration 3energy = 0.23961637832529<A u 3 , A u 3 >_{-1}^0.5 = 4.188228630245016e-16
t = 2.400000000000001
Newton iteration 0energy = 0.23961582808585102<A u 0 , A u 0 >_{-1}^0.5 = 0.0010283540627855198
Newton iteration 1energy = 0.23961529935305356<A u 1 , A u 1 >_{-1}^0.5 = 5.297627567252096e-07
Newton iteration 2energy = 0.23961529935291329<A u 2 , A u 2 >_{-1}^0.5 = 1.6202966695387348e-13
Newton iteration 3energy = 0.23961529935291334<A u 3 , A u 3 >_{-1}^0.5 = 3.999392740516295e-16
t = 2.500000000000001
Newton iteration 0energy = 0.23961479095503746<A u 0 , A u 0 >_{-1}^0.5 = 0.0009890970729079993
Newton iteration 1energy = 0.23961430181779786<A u 1 , A u 1 >_{-1}^0.5 = 4.872666035427371e-07
Newton iteration 2energy = 0.23961430181767918<A u 2 , A u 2 >_{-1}^0.5 = 1.3597696579476573e-13
Newton iteration 3energy = 0.23961430181767918<A u 3 , A u 3 >_{-1}^0.5 = 3.8936866568800425e-16
t = 2.600000000000001
Newton iteration 0energy = 0.2396138309145451<A u 0 , A u 0 >_{-1}^0.5 = 0.0009525015914657539
Newton iteration 1energy = 0.23961337730105753<A u 1 , A u 1 >_{-1}^0.5 = 4.4921981867275515e-07
Newton iteration 2energy = 0.2396133773009566<A u 2 , A u 2 >_{-1}^0.5 = 1.1472689377689044e-13
Newton iteration 3energy = 0.2396133773009566<A u 3 , A u 3 >_{-1}^0.5 = 4.0613019318331927e-16
t = 2.700000000000001
Newton iteration 0energy = 0.23961294007376774<A u 0 , A u 0 >_{-1}^0.5 = 0.0009183547757921597
Newton iteration 1energy = 0.23961251839967962<A u 1 , A u 1 >_{-1}^0.5 = 4.150665658147821e-07
Newton iteration 2energy = 0.2396125183995934<A u 2 , A u 2 >_{-1}^0.5 = 9.723092503762368e-14
t = 2.800000000000001
Newton iteration 0energy = 0.2396121114812304<A u 0 , A u 0 >_{-1}^0.5 = 0.0008864662194052564
Newton iteration 1energy = 0.23961171858168256<A u 1 , A u 1 >_{-1}^0.5 = 3.8433776208680404e-07
Newton iteration 2energy = 0.23961171858160873<A u 2 , A u 2 >_{-1}^0.5 = 8.284192585188487e-14
t = 2.9000000000000012
Newton iteration 0energy = 0.2396113389953494<A u 0 , A u 0 >_{-1}^0.5 = 0.0008566633749442762
Newton iteration 1energy = 0.2396109720692427<A u 1 , A u 1 >_{-1}^0.5 = 3.56632954760139e-07
Newton iteration 2energy = 0.2396109720691791<A u 2 , A u 2 >_{-1}^0.5 = 7.08144198455852e-14
t = 3.0000000000000013
Newton iteration 0energy = 0.23961061717765086<A u 0 , A u 0 >_{-1}^0.5 = 0.0008287889263646279
Newton iteration 1energy = 0.2396102737406834<A u 1 , A u 1 >_{-1}^0.5 = 3.3160737847372484e-07
Newton iteration 2energy = 0.23961027374062843<A u 2 , A u 2 >_{-1}^0.5 = 6.083862558389641e-14
t = 3.1000000000000014
Newton iteration 0energy = 0.23960994120243972<A u 0 , A u 0 >_{-1}^0.5 = 0.0008026990506789688
Newton iteration 1energy = 0.2396096190469689<A u 1 , A u 1 >_{-1}^0.5 = 3.089623021042399e-07
Newton iteration 2energy = 0.23960961904692127<A u 2 , A u 2 >_{-1}^0.5 = 5.243570434427366e-14
t = 3.2000000000000015
Newton iteration 0energy = 0.2396093067794342<A u 0 , A u 0 >_{-1}^0.5 = 0.0007782621114843857
Newton iteration 1energy = 0.23960900393990864<A u 1 , A u 1 >_{-1}^0.5 = 2.884375804322967e-07
Newton iteration 2energy = 0.23960900393986703<A u 2 , A u 2 >_{-1}^0.5 = 4.5379186834519644e-14
t = 3.3000000000000016
Newton iteration 0energy = 0.23960871008705956<A u 0 , A u 0 >_{-1}^0.5 = 0.0007553575814402258
Newton iteration 1energy = 0.23960842481012126<A u 1 , A u 1 >_{-1}^0.5 = 2.698057545765207e-07
Newton iteration 2energy = 0.23960842481008499<A u 2 , A u 2 >_{-1}^0.5 = 3.942665697811389e-14
t = 3.4000000000000017
Newton iteration 0energy = 0.239608147714714<A u 0 , A u 0 >_{-1}^0.5 = 0.0007338751008235229
Newton iteration 1energy = 0.2396078784332731<A u 1 , A u 1 >_{-1}^0.5 = 2.528672818893669e-07
Newton iteration 2energy = 0.23960787843324122<A u 2 , A u 2 >_{-1}^0.5 = 3.4403333422175943e-14
t = 3.5000000000000018
Newton iteration 0energy = 0.23960761661268348<A u 0 , A u 0 >_{-1}^0.5 = 0.0007137136279143401
Newton iteration 1energy = 0.2396073619233964<A u 1 , A u 1 >_{-1}^0.5 = 2.374466157006292e-07
Newton iteration 2energy = 0.23960736192336823<A u 2 , A u 2 >_{-1}^0.5 = 3.0137895183934865e-14
t = 3.600000000000002
Newton iteration 0energy = 0.23960711404861687<A u 0 , A u 0 >_{-1}^0.5 = 0.0006947806591328947
Newton iteration 1energy = 0.23960687269229808<A u 1 , A u 1 >_{-1}^0.5 = 2.2338894227617265e-07
Newton iteration 2energy = 0.23960687269227304<A u 2 , A u 2 >_{-1}^0.5 = 2.6475425404173353e-14
t = 3.700000000000002
Newton iteration 0energy = 0.23960663756966025<A u 0 , A u 0 >_{-1}^0.5 = 0.0006769915073033545
Newton iteration 1energy = 0.23960640841422393<A u 1 , A u 1 >_{-1}^0.5 = 2.1055743624529568e-07
Newton iteration 2energy = 0.2396064084142018<A u 2 , A u 2 >_{-1}^0.5 = 2.3313036877749828e-14
t = 3.800000000000002
Newton iteration 0energy = 0.2396061849694749<A u 0 , A u 0 >_{-1}^0.5 = 0.000660268631502265
Newton iteration 1energy = 0.239605966995068<A u 1 , A u 1 >_{-1}^0.5 = 1.9883093309711906e-07
Newton iteration 2energy = 0.23960596699504824<A u 2 , A u 2 >_{-1}^0.5 = 2.065902890547247e-14
t = 3.900000000000002
Newton iteration 0energy = 0.23960575425948316<A u 0 , A u 0 >_{-1}^0.5 = 0.0006445410144723969
Newton iteration 1energy = 0.2396055465455128<A u 1 , A u 1 >_{-1}^0.5 = 1.8810194233256467e-07
Newton iteration 2energy = 0.23960554654549515<A u 2 , A u 2 >_{-1}^0.5 = 1.8347648711839195e-14
t = 4.000000000000002
Newton iteration 0energy = 0.23960534364377498<A u 0 , A u 0 >_{-1}^0.5 = 0.000629743584834913
Newton iteration 1energy = 0.2396051453575804<A u 1 , A u 1 >_{-1}^0.5 = 1.7827494164831928e-07
Newton iteration 2energy = 0.23960514535756444<A u 2 , A u 2 >_{-1}^0.5 = 1.636304282394239e-14
t = 4.100000000000001
Newton iteration 0energy = 0.239604951497189<A u 0 , A u 0 >_{-1}^0.5 = 0.0006158166819322605
Newton iteration 1energy = 0.23960476188413732<A u 1 , A u 1 >_{-1}^0.5 = 1.692649062807602e-07
Newton iteration 2energy = 0.23960476188412316<A u 2 , A u 2 >_{-1}^0.5 = 1.4662402859105416e-14
t = 4.200000000000001
Newton iteration 0energy = 0.23960457634614438<A u 0 , A u 0 >_{-1}^0.5 = 0.0006027055613937356
Newton iteration 1energy = 0.23960439472096592<A u 1 , A u 1 >_{-1}^0.5 = 1.6099603650454517e-07
Newton iteration 2energy = 0.23960439472095302<A u 2 , A u 2 >_{-1}^0.5 = 1.3206294622538514e-14
t = 4.300000000000001
Newton iteration 0energy = 0.23960421685186575<A u 0 , A u 0 >_{-1}^0.5 = 0.0005903599395996152
Newton iteration 1energy = 0.23960404259105963<A u 1 , A u 1 >_{-1}^0.5 = 1.5340065262841853e-07
Newton iteration 2energy = 0.23960404259104776<A u 2 , A u 2 >_{-1}^0.5 = 1.1852161151991052e-14
t = 4.4
Newton iteration 0energy = 0.2396038717956791<A u 0 , A u 0 >_{-1}^0.5 = 0.0005787335752183923
Newton iteration 1energy = 0.239603704330855<A u 1 , A u 1 >_{-1}^0.5 = 1.46418233678702e-07
Newton iteration 2energy = 0.23960370433084424<A u 2 , A u 2 >_{-1}^0.5 = 1.0748221722086695e-14
t = 4.5
Newton iteration 0energy = 0.23960354006611378<A u 0 , A u 0 >_{-1}^0.5 = 0.0005677838859552262
Newton iteration 1energy = 0.2396033788781438<A u 1 , A u 1 >_{-1}^0.5 = 1.3999457887557826e-07
Newton iteration 2energy = 0.2396033788781339<A u 2 , A u 2 >_{-1}^0.5 = 9.755204030558368e-15
t = 4.6
Newton iteration 0energy = 0.2396032206475711<A u 0 , A u 0 >_{-1}^0.5 = 0.0005574715986071908
Newton iteration 1energy = 0.23960306526144695<A u 1 , A u 1 >_{-1}^0.5 = 1.3408107468246175e-07
Newton iteration 2energy = 0.23960306526143796<A u 2 , A u 2 >_{-1}^0.5 = 8.937102320550931e-15
t = 4.699999999999999
Newton iteration 0energy = 0.23960291261035707<A u 0 , A u 0 >_{-1}^0.5 = 0.0005477604304859817
Newton iteration 1energy = 0.2396027625906625<A u 1 , A u 1 >_{-1}^0.5 = 1.2863405340373862e-07
Newton iteration 2energy = 0.2396027625906542<A u 2 , A u 2 >_{-1}^0.5 = 8.133917206458277e-15
t = 4.799999999999999
Newton iteration 0energy = 0.23960261510190292<A u 0 , A u 0 >_{-1}^0.5 = 0.0005386168002506194
Newton iteration 1energy = 0.2396024700488202<A u 1 , A u 1 >_{-1}^0.5 = 1.2361423086153643e-07
Newton iteration 2energy = 0.23960247004881263<A u 2 , A u 2 >_{-1}^0.5 = 7.475563372220586e-15
t = 4.899999999999999
Newton iteration 0energy = 0.23960232733902043<A u 0 , A u 0 >_{-1}^0.5 = 0.0005300095661919587
Newton iteration 1energy = 0.2396021868848042<A u 1 , A u 1 >_{-1}^0.5 = 1.1898621269510796e-07
Newton iteration 2energy = 0.23960218688479706<A u 2 , A u 2 >_{-1}^0.5 = 6.923928011706675e-15
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 = 0.3939295546929309
Newton iteration 1
err = 3.930938818539888e-15
The minimization problem is solved by the solution of the PDE:
\[\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} 1 \cdot v \quad \forall ~ v \in V^{ho,disc}\]
under the constraint
\[u(v) = u^{lo}(v) \quad \text{ for all vertices } v \in V(T) \text{ for all } T.\]