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 = 0.1325595659762467
Newton iteration 1energy = -0.008785675919637855<A u 1 , A u 1 >_{-1}^0.5 = 1.110759821811827e-05
Newton iteration 2energy = -0.008785675981327228<A u 2 , A u 2 >_{-1}^0.5 = 2.8075378767027274e-13
Newton iteration 3energy = -0.008785675981327228<A u 3 , A u 3 >_{-1}^0.5 = 3.7881417672072186e-17
energy = -0.008785675981327228
Again, a Newton for minimization is shipped with NGSolve:
[6]:
from ngsolve.solvers import *
gfu.vec[:] = 0
NewtonMinimization(a,gfu)
Redraw()
Newton iteration 0
energy = 0.0
err = 0.1325595659762467
Newton iteration 1
energy = -0.008785675919637857
err = 1.1107598218108251e-05
Newton iteration 2
energy = -0.00878567598132723
err = 2.8075134013106847e-13
Nonlinear elasticity¶
We consider a beam which is fixed on one side and is subject to gravity only. We assume a Neo-Hookean hyperelastic material. The model is a nonlinear minimization problem with
\[E(v) := \int_{\Omega} \frac{\mu}{2} ( \operatorname{tr}(F^T F-I)+\frac{2 \mu}{\lambda} \operatorname{det}(F^T F)^{\frac{\lambda}{2\mu}} - 1) - \gamma ~ (f,v) ~~ dx\]
where \(\mu\) and \(\lambda\) are the Lamé parameters and \(F = I + D v\) where \(v: \Omega \to \mathbb{R}^2\) is the sought for displacement.
[7]:
import netgen.geom2d as geom2d
from ngsolve import *
geo = geom2d.SplineGeometry()
pnums = [ geo.AddPoint (x,y,maxh=0.01) for x,y in [(0,0), (1,0), (1,0.1), (0,0.1)] ]
for p1,p2,bc in [(0,1,"bot"), (1,2,"right"), (2,3,"top"), (3,0,"left")]:
geo.Append(["line", pnums[p1], pnums[p2]], bc=bc)
mesh = Mesh(geo.GenerateMesh(maxh=0.05))
[8]:
# E module and poisson number:
E, nu = 210, 0.2
# Lamé constants:
mu = E / 2 / (1+nu)
lam = E * nu / ((1+nu)*(1-2*nu))
V = H1(mesh, order=2, dirichlet="left", dim=mesh.dim)
u = V.TrialFunction()
#gravity:
force = CoefficientFunction( (0,-1) )
[9]:
def Pow(a, b):
return exp (log(a)*b)
def NeoHook (C):
return 0.5 * mu * (Trace(C-I) + 2*mu/lam * Pow(Det(C), -lam/2/mu) - 1)
I = Id(mesh.dim)
F = I + Grad(u)
C = F.trans * F.trans
factor = Parameter(1.0)
a = BilinearForm(V, symmetric=True)
a += Variation( NeoHook (C).Compile() * dx
-factor * (InnerProduct(force,u) ).Compile() * dx)
We want to solve the minimization problem for \(\gamma = 5\). Due to the high nonlinearity in the problem, the Newton iteration will not convergence with any initial guess. We approach the case \(\gamma = 5\) by solving problems with \(\gamma = i/10\) for \(i=1,..,50\) and taking the solution of the previous problem as an initial guess.
[10]:
gfu = GridFunction(V)
gfu.vec[:] = 0
Draw (gfu, mesh, "u")
SetVisualization (deformation=True)
res = gfu.vec.CreateVector()
du = gfu.vec.CreateVector()
for loadstep in range(50):
print ("loadstep", loadstep)
factor.Set ((loadstep+1)/10)
SolveNonlinearMinProblem(a,gfu)
Redraw()
loadstep 0
Newton iteration 0energy = 8.75<A u 0 , A u 0 >_{-1}^0.5 = 0.0001642018006803098
Newton iteration 1energy = 9.156866275697862<A u 1 , A u 1 >_{-1}^0.5 = 0.6277706466655382
Newton iteration 2energy = 9.251205359514927<A u 2 , A u 2 >_{-1}^0.5 = 0.8750251785365131
Newton iteration 3energy = 10.955507548351848<A u 3 , A u 3 >_{-1}^0.5 = 2.395912810707176
Newton iteration 4energy = 23.83351572024048<A u 4 , A u 4 >_{-1}^0.5 = 7.199410738437444
Newton iteration 5energy = -64.8797973992964<A u 5 , A u 5 >_{-1}^0.5 = 11.2623248051877
Newton iteration 6energy = 4.237327903418191<A u 6 , A u 6 >_{-1}^0.5 = 4.256140755927589
Newton iteration 7energy = 178.37626637980202<A u 7 , A u 7 >_{-1}^0.5 = 18.64681906266925
Newton iteration 8energy = 15.77863338872313<A u 8 , A u 8 >_{-1}^0.5 = 4.1651060295466245
Newton iteration 9energy = 9.550196724496905<A u 9 , A u 9 >_{-1}^0.5 = 4.326737556128749
Newton iteration 10energy = -64.77798929408894<A u 10 , A u 10 >_{-1}^0.5 = 11.605636769254128
Newton iteration 11energy = 17.13190846840228<A u 11 , A u 11 >_{-1}^0.5 = 3.640598274639897
Newton iteration 12energy = 121.52784084982778<A u 12 , A u 12 >_{-1}^0.5 = 15.313299440375175
Newton iteration 13energy = -12.662371812854389<A u 13 , A u 13 >_{-1}^0.5 = 8.06245346180259
Newton iteration 14energy = -631.071596582654<A u 14 , A u 14 >_{-1}^0.5 = 35.933903037700496
Newton iteration 15energy = -142.44196597610403<A u 15 , A u 15 >_{-1}^0.5 = 17.337300334580434
Newton iteration 16energy = 25.95442096038036<A u 16 , A u 16 >_{-1}^0.5 = 4.44123748900561
Newton iteration 17energy = -15.356509281883746<A u 17 , A u 17 >_{-1}^0.5 = 8.785883754803226
Newton iteration 18energy = 45.548819881554664<A u 18 , A u 18 >_{-1}^0.5 = 8.583076331355292
Newton iteration 19energy = 20.09330835163189<A u 19 , A u 19 >_{-1}^0.5 = 4.945433857526404
Newton iteration 20energy = -104.05841918738736<A u 20 , A u 20 >_{-1}^0.5 = 15.002666059845719
Newton iteration 21energy = -1.1632007370142503<A u 21 , A u 21 >_{-1}^0.5 = 1.9396458748867116
Newton iteration 22energy = -49.58562134418571<A u 22 , A u 22 >_{-1}^0.5 = 10.350558487882001
Newton iteration 23energy = -1.707584003728698<A u 23 , A u 23 >_{-1}^0.5 = 10.053874738531851
Newton iteration 24energy = -1671.978999010753<A u 24 , A u 24 >_{-1}^0.5 = 13.808052581330848
loadstep 1
Newton iteration 0energy = -3643134.375<A u 0 , A u 0 >_{-1}^0.5 = 2699.304839744679
Newton iteration 1energy = 4.75<A u 1 , A u 1 >_{-1}^0.5 = 2.5743298085364423
Newton iteration 2energy = -128.0<A u 2 , A u 2 >_{-1}^0.5 = 15.736270618966905
Newton iteration 3energy = 64.0<A u 3 , A u 3 >_{-1}^0.5 = 22.16969953640391
Newton iteration 4energy = -1024.0<A u 4 , A u 4 >_{-1}^0.5 = 32.0543455208382
Newton iteration 5energy = -4096.0<A u 5 , A u 5 >_{-1}^0.5 = 57.78833531492178
Newton iteration 6energy = -14336.0<A u 6 , A u 6 >_{-1}^0.5 = 105.61744752313625
Newton iteration 7energy = -25088.0<A u 7 , A u 7 >_{-1}^0.5 = 141.23036251503467
Newton iteration 8energy = 10848.0<A u 8 , A u 8 >_{-1}^0.5 = 158.68121691665937
Newton iteration 9energy = -294912.0<A u 9 , A u 9 >_{-1}^0.5 = 837.446695779949
Newton iteration 10energy = 1835008.0<A u 10 , A u 10 >_{-1}^0.5 = 1357.896840873672
Newton iteration 11energy = -131072.0<A u 11 , A u 11 >_{-1}^0.5 = 3592.090061192222
Newton iteration 12energy = 5242880.0<A u 12 , A u 12 >_{-1}^0.5 = 6123.539939898264
Newton iteration 13energy = -607125504.0<A u 13 , A u 13 >_{-1}^0.5 = 16272.62534992344
Newton iteration 14energy = -1342177280.0<A u 14 , A u 14 >_{-1}^0.5 = 63698.26131152963
Newton iteration 15energy = 0.0<A u 15 , A u 15 >_{-1}^0.5 = 186260.59674799797
Newton iteration 16energy = -103079215104.0<A u 16 , A u 16 >_{-1}^0.5 = 241320.10850585124
Newton iteration 17energy = 481036337152.0<A u 17 , A u 17 >_{-1}^0.5 = 2974595.389440306
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 = 0.41450282223244284<A u 0 , A u 0 >_{-1}^0.5 = 0.28092314371084526
Newton iteration 1energy = 0.3769004492603475<A u 1 , A u 1 >_{-1}^0.5 = 0.01917625885151113
Newton iteration 2energy = 0.3767159759702836<A u 2 , A u 2 >_{-1}^0.5 = 9.630978095564052e-05
Newton iteration 3energy = 0.3767159713324167<A u 3 , A u 3 >_{-1}^0.5 = 2.5088242424872063e-09
Newton iteration 4energy = 0.3767159713324165<A u 4 , A u 4 >_{-1}^0.5 = 3.006244530976361e-16
t = 0.1
Newton iteration 0energy = 0.34388864618592396<A u 0 , A u 0 >_{-1}^0.5 = 0.23599849082178062
Newton iteration 1energy = 0.31712741612159606<A u 1 , A u 1 >_{-1}^0.5 = 0.013451967104651866
Newton iteration 2energy = 0.3170367312150468<A u 2 , A u 2 >_{-1}^0.5 = 4.66463400725311e-05
Newton iteration 3energy = 0.3170367301270973<A u 3 , A u 3 >_{-1}^0.5 = 5.773857991310932e-10
Newton iteration 4energy = 0.31703673012709743<A u 4 , A u 4 >_{-1}^0.5 = 3.445193849894897e-16
t = 0.2
Newton iteration 0energy = 0.2955289399084664<A u 0 , A u 0 >_{-1}^0.5 = 0.18333789232253336
Newton iteration 1energy = 0.2792210737880486<A u 1 , A u 1 >_{-1}^0.5 = 0.00802042217248658
Newton iteration 2energy = 0.27918886698940865<A u 2 , A u 2 >_{-1}^0.5 = 1.6284517755706075e-05
Newton iteration 3energy = 0.2791888668568156<A u 3 , A u 3 >_{-1}^0.5 = 6.887915489446651e-11
Newton iteration 4energy = 0.27918886685681554<A u 4 , A u 4 >_{-1}^0.5 = 3.328682324100485e-16
t = 0.30000000000000004
Newton iteration 0energy = 0.26701067131949147<A u 0 , A u 0 >_{-1}^0.5 = 0.13329656299311263
Newton iteration 1energy = 0.2583141901965072<A u 1 , A u 1 >_{-1}^0.5 = 0.004187366441885108
Newton iteration 2energy = 0.25830541710911625<A u 2 , A u 2 >_{-1}^0.5 = 4.373249880717528e-06
Newton iteration 3energy = 0.2583054170995535<A u 3 , A u 3 >_{-1}^0.5 = 4.8761362288601394e-12
Newton iteration 4energy = 0.25830541709955357<A u 4 , A u 4 >_{-1}^0.5 = 3.4653965502424423e-16
t = 0.4
Newton iteration 0energy = 0.25217703918435985<A u 0 , A u 0 >_{-1}^0.5 = 0.09212611622514379
Newton iteration 1energy = 0.24799424785134225<A u 1 , A u 1 >_{-1}^0.5 = 0.001980092473528173
Newton iteration 2energy = 0.24799228683269614<A u 2 , A u 2 >_{-1}^0.5 = 9.674620131215291e-07
Newton iteration 3energy = 0.24799228683222818<A u 3 , A u 3 >_{-1}^0.5 = 2.352109141921178e-13
Newton iteration 4energy = 0.24799228683222818<A u 4 , A u 4 >_{-1}^0.5 = 3.4483226640983087e-16
t = 0.5
Newton iteration 0energy = 0.24516619622475483<A u 0 , A u 0 >_{-1}^0.5 = 0.061423152197536905
Newton iteration 1energy = 0.24329759376849047<A u 1 , A u 1 >_{-1}^0.5 = 0.000874061455503885
Newton iteration 2energy = 0.24329721172242824<A u 2 , A u 2 >_{-1}^0.5 = 1.873318935832003e-07
Newton iteration 3energy = 0.2432972117224108<A u 3 , A u 3 >_{-1}^0.5 = 8.784766454877497e-15
t = 0.6
Newton iteration 0energy = 0.24207025658945602<A u 0 , A u 0 >_{-1}^0.5 = 0.0399862829041845
Newton iteration 1energy = 0.24127567813807255<A u 1 , A u 1 >_{-1}^0.5 = 0.0003690821207900598
Newton iteration 2energy = 0.2412756100231785<A u 2 , A u 2 >_{-1}^0.5 = 3.3391323540808995e-08
Newton iteration 3energy = 0.2412756100231781<A u 3 , A u 3 >_{-1}^0.5 = 4.829365730784197e-16
t = 0.7
Newton iteration 0energy = 0.24076310861693295<A u 0 , A u 0 >_{-1}^0.5 = 0.025661409740772197
Newton iteration 1energy = 0.24043513963301003<A u 1 , A u 1 >_{-1}^0.5 = 0.00015217410938012284
Newton iteration 2energy = 0.24043512805424114<A u 2 , A u 2 >_{-1}^0.5 = 5.749334600194143e-09
Newton iteration 3energy = 0.24043512805424122<A u 3 , A u 3 >_{-1}^0.5 = 4.2237822038956273e-16
t = 0.7999999999999999
Newton iteration 0energy = 0.24022546483918172<A u 0 , A u 0 >_{-1}^0.5 = 0.016372127934406275
Newton iteration 1energy = 0.24009177741713594<A u 1 , A u 1 >_{-1}^0.5 = 6.263686267922135e-05
Newton iteration 2energy = 0.24009177545542662<A u 2 , A u 2 >_{-1}^0.5 = 1.0306859724281195e-09
Newton iteration 3energy = 0.24009177545542662<A u 3 , A u 3 >_{-1}^0.5 = 4.364677451505078e-16
t = 0.8999999999999999
Newton iteration 0energy = 0.2400062744077331<A u 0 , A u 0 >_{-1}^0.5 = 0.0104864645029816
Newton iteration 1energy = 0.23995138191747392<A u 1 , A u 1 >_{-1}^0.5 = 2.6656671604435883e-05
Newton iteration 2energy = 0.239951381562183<A u 2 , A u 2 >_{-1}^0.5 = 2.2716183145683502e-10
Newton iteration 3energy = 0.2399513815621831<A u 3 , A u 3 >_{-1}^0.5 = 4.264118775503098e-16
t = 0.9999999999999999
Newton iteration 0energy = 0.23991578829736723<A u 0 , A u 0 >_{-1}^0.5 = 0.0068447552968940225
Newton iteration 1energy = 0.23989238972747973<A u 1 , A u 1 >_{-1}^0.5 = 1.2489092164807365e-05
Newton iteration 2energy = 0.23989238964949075<A u 2 , A u 2 >_{-1}^0.5 = 7.427827828091099e-11
Newton iteration 3energy = 0.23989238964949083<A u 3 , A u 3 >_{-1}^0.5 = 4.899636566134229e-16
t = 1.0999999999999999
Newton iteration 0energy = 0.2398766683553299<A u 0 , A u 0 >_{-1}^0.5 = 0.004664493817844107
Newton iteration 1energy = 0.23986579907505196<A u 1 , A u 1 >_{-1}^0.5 = 6.960493363908454e-06
Newton iteration 2energy = 0.2398657990508278<A u 2 , A u 2 >_{-1}^0.5 = 3.382730982279769e-11
Newton iteration 3energy = 0.2398657990508277<A u 3 , A u 3 >_{-1}^0.5 = 4.703931716766879e-16
t = 1.2
Newton iteration 0energy = 0.23985799635280028<A u 0 , A u 0 >_{-1}^0.5 = 0.003421663848452149
Newton iteration 1energy = 0.2398521466966484<A u 1 , A u 1 >_{-1}^0.5 = 4.710134579841151e-06
Newton iteration 2energy = 0.2398521466855556<A u 2 , A u 2 >_{-1}^0.5 = 1.859712624118175e-11
Newton iteration 3energy = 0.23985214668555566<A u 3 , A u 3 >_{-1}^0.5 = 4.767545860664359e-16
t = 1.3
Newton iteration 0energy = 0.23984754756902163<A u 0 , A u 0 >_{-1}^0.5 = 0.002752600824904327
Newton iteration 1energy = 0.2398437614948929<A u 1 , A u 1 >_{-1}^0.5 = 3.665276433548626e-06
Newton iteration 2energy = 0.23984376148817577<A u 2 , A u 2 >_{-1}^0.5 = 1.1531469382810009e-11
Newton iteration 3energy = 0.23984376148817574<A u 3 , A u 3 >_{-1}^0.5 = 4.688623449697754e-16
t = 1.4000000000000001
Newton iteration 0energy = 0.2398405116330511<A u 0 , A u 0 >_{-1}^0.5 = 0.0024031293480801475
Newton iteration 1energy = 0.23983762559026803<A u 1 , A u 1 >_{-1}^0.5 = 3.087220334776422e-06
Newton iteration 2energy = 0.23983762558550253<A u 2 , A u 2 >_{-1}^0.5 = 7.812214584990433e-12
Newton iteration 3energy = 0.23983762558550256<A u 3 , A u 3 >_{-1}^0.5 = 4.745866868777359e-16
t = 1.5000000000000002
Newton iteration 0energy = 0.23983499320021817<A u 0 , A u 0 >_{-1}^0.5 = 0.0022129311097539264
Newton iteration 1energy = 0.2398325456786903<A u 1 , A u 1 >_{-1}^0.5 = 2.7132456843986436e-06
Newton iteration 2energy = 0.23983254567500933<A u 2 , A u 2 >_{-1}^0.5 = 5.665179123070813e-12
Newton iteration 3energy = 0.2398325456750095<A u 3 , A u 3 >_{-1}^0.5 = 4.412031143442219e-16
t = 1.6000000000000003
Newton iteration 0energy = 0.2398302381096157<A u 0 , A u 0 >_{-1}^0.5 = 0.0020960078622131225
Newton iteration 1energy = 0.2398280422176908<A u 1 , A u 1 >_{-1}^0.5 = 2.4404707967673256e-06
Newton iteration 2energy = 0.2398280422147128<A u 2 , A u 2 >_{-1}^0.5 = 4.324031272293299e-12
Newton iteration 3energy = 0.23982804221471277<A u 3 , A u 3 >_{-1}^0.5 = 4.367197053984415e-16
t = 1.7000000000000004
Newton iteration 0energy = 0.23982593874004046<A u 0 , A u 0 >_{-1}^0.5 = 0.002011953735829304
Newton iteration 1energy = 0.23982391531464325<A u 1 , A u 1 >_{-1}^0.5 = 2.223996485646607e-06
Newton iteration 2energy = 0.23982391531217015<A u 2 , A u 2 >_{-1}^0.5 = 3.42446126472881e-12
Newton iteration 3energy = 0.23982391531217015<A u 3 , A u 3 >_{-1}^0.5 = 4.488908088274565e-16
t = 1.8000000000000005
Newton iteration 0energy = 0.2398219628758993<A u 0 , A u 0 >_{-1}^0.5 = 0.0019431518886679963
Newton iteration 1energy = 0.23982007538882258<A u 1 , A u 1 >_{-1}^0.5 = 2.042505797811638e-06
Newton iteration 2energy = 0.23982007538673664<A u 2 , A u 2 >_{-1}^0.5 = 2.7826612448875328e-12
Newton iteration 3energy = 0.23982007538673675<A u 3 , A u 3 >_{-1}^0.5 = 4.2097970682770793e-16
t = 1.9000000000000006
Newton iteration 0energy = 0.23981824793588372<A u 0 , A u 0 >_{-1}^0.5 = 0.0018821371909694753
Newton iteration 1energy = 0.23981647706316292<A u 1 , A u 1 >_{-1}^0.5 = 1.8850840284062163e-06
Newton iteration 2energy = 0.2398164770613862<A u 2 , A u 2 >_{-1}^0.5 = 2.3016458392371904e-12
Newton iteration 3energy = 0.23981647706138626<A u 3 , A u 3 >_{-1}^0.5 = 4.214491705755986e-16
t = 2.0000000000000004
Newton iteration 0energy = 0.23981475961775717<A u 0 , A u 0 >_{-1}^0.5 = 0.0018257325375344146
Newton iteration 1energy = 0.2398130932535269<A u 1 , A u 1 >_{-1}^0.5 = 1.7456684941714879e-06
Newton iteration 2energy = 0.23981309325200315<A u 2 , A u 2 >_{-1}^0.5 = 1.9275569903888303e-12
Newton iteration 3energy = 0.23981309325200323<A u 3 , A u 3 >_{-1}^0.5 = 4.3002386453871115e-16
t = 2.1000000000000005
Newton iteration 0energy = 0.23981147563059332<A u 0 , A u 0 >_{-1}^0.5 = 0.0017725381181930738
Newton iteration 1energy = 0.23980990492349197<A u 1 , A u 1 >_{-1}^0.5 = 1.6205829233769044e-06
Newton iteration 2energy = 0.23980990492217882<A u 2 , A u 2 >_{-1}^0.5 = 1.6286631901268343e-12
Newton iteration 3energy = 0.2398099049221788<A u 3 , A u 3 >_{-1}^0.5 = 4.3904616979880167e-16
t = 2.2000000000000006
Newton iteration 0energy = 0.2398083792159378<A u 0 , A u 0 >_{-1}^0.5 = 0.0017218913657836724
Newton iteration 1energy = 0.23980689696317323<A u 1 , A u 1 >_{-1}^0.5 = 1.5073985459147084e-06
Newton iteration 2energy = 0.2398068969620372<A u 2 , A u 2 >_{-1}^0.5 = 1.3852117186986472e-12
Newton iteration 3energy = 0.23980689696203716<A u 3 , A u 3 >_{-1}^0.5 = 4.326899231247316e-16
t = 2.3000000000000007
Newton iteration 0energy = 0.23980545650590596<A u 0 , A u 0 >_{-1}^0.5 = 0.0016734406119765489
Newton iteration 1energy = 0.239804056477282<A u 1 , A u 1 >_{-1}^0.5 = 1.4043855040836676e-06
Newton iteration 2energy = 0.23980405647629577<A u 2 , A u 2 >_{-1}^0.5 = 1.184181222843902e-12
Newton iteration 3energy = 0.23980405647629577<A u 3 , A u 3 >_{-1}^0.5 = 4.345886890347035e-16
t = 2.400000000000001
Newton iteration 0energy = 0.23980269539820454<A u 0 , A u 0 >_{-1}^0.5 = 0.001626969986090236
Newton iteration 1energy = 0.23980137203197138<A u 1 , A u 1 >_{-1}^0.5 = 1.31023498306856e-06
Newton iteration 2energy = 0.239801372031113<A u 2 , A u 2 >_{-1}^0.5 = 1.0165320499198098e-12
Newton iteration 3energy = 0.23980137203111296<A u 3 , A u 3 >_{-1}^0.5 = 4.1099073931727053e-16
t = 2.500000000000001
Newton iteration 0energy = 0.23980008504012346<A u 0 , A u 0 >_{-1}^0.5 = 0.0015823266755118724
Newton iteration 1energy = 0.23979883329107032<A u 1 , A u 1 >_{-1}^0.5 = 1.2239101235926281e-06
Newton iteration 2energy = 0.2397988332903212<A u 2 , A u 2 >_{-1}^0.5 = 8.756362329898839e-13
Newton iteration 3energy = 0.23979883329032126<A u 3 , A u 3 >_{-1}^0.5 = 4.2623037284730425e-16
t = 2.600000000000001
Newton iteration 0energy = 0.23979761556362392<A u 0 , A u 0 >_{-1}^0.5 = 0.001539390031738811
Newton iteration 1energy = 0.23979643081609264<A u 1 , A u 1 >_{-1}^0.5 = 1.1445610184716526e-06
Newton iteration 2energy = 0.23979643081543767<A u 2 , A u 2 >_{-1}^0.5 = 7.565310732742972e-13
Newton iteration 3energy = 0.2397964308154377<A u 3 , A u 3 >_{-1}^0.5 = 4.660894012758241e-16
t = 2.700000000000001
Newton iteration 0energy = 0.23979527792877278<A u 0 , A u 0 >_{-1}^0.5 = 0.0014980579568581978
Newton iteration 1energy = 0.23979415593924858<A u 1 , A u 1 >_{-1}^0.5 = 1.0714732744954146e-06
Newton iteration 2energy = 0.23979415593867454<A u 2 , A u 2 >_{-1}^0.5 = 6.553770379939637e-13
Newton iteration 3energy = 0.23979415593867448<A u 3 , A u 3 >_{-1}^0.5 = 4.357400795100921e-16
t = 2.800000000000001
Newton iteration 0energy = 0.23979306381744556<A u 0 , A u 0 >_{-1}^0.5 = 0.001458240554129904
Newton iteration 1energy = 0.23979200067199583<A u 1 , A u 1 >_{-1}^0.5 = 1.0040351240519976e-06
Newton iteration 2energy = 0.23979200067149173<A u 2 , A u 2 >_{-1}^0.5 = 5.691378349981974e-13
Newton iteration 3energy = 0.2397920006714918<A u 3 , A u 3 >_{-1}^0.5 = 4.2945099131223385e-16
t = 2.9000000000000012
Newton iteration 0energy = 0.23979096555290583<A u 0 , A u 0 >_{-1}^0.5 = 0.0014198569182498702
Newton iteration 1energy = 0.2397899576330501<A u 1 , A u 1 >_{-1}^0.5 = 9.417153346805539e-07
Newton iteration 2energy = 0.23978995763260666<A u 2 , A u 2 >_{-1}^0.5 = 4.953690606456294e-13
Newton iteration 3energy = 0.2397899576326067<A u 3 , A u 3 >_{-1}^0.5 = 4.1485795479196675e-16
t = 3.0000000000000013
Newton iteration 0energy = 0.23978897603449173<A u 0 , A u 0 >_{-1}^0.5 = 0.0013828333391534794
Newton iteration 1energy = 0.2397880199885049<A u 1 , A u 1 >_{-1}^0.5 = 8.840477145468194e-07
Newton iteration 2energy = 0.2397880199881141<A u 2 , A u 2 >_{-1}^0.5 = 4.3208633739203237e-13
Newton iteration 3energy = 0.23978801998811416<A u 3 , A u 3 >_{-1}^0.5 = 4.1889039384022973e-16
t = 3.1000000000000014
Newton iteration 0energy = 0.23978708868229584<A u 0 , A u 0 >_{-1}^0.5 = 0.0013471021789486376
Newton iteration 1energy = 0.23978618140041508<A u 1 , A u 1 >_{-1}^0.5 = 8.306198222530883e-07
Newton iteration 2energy = 0.23978618140007013<A u 2 , A u 2 >_{-1}^0.5 = 3.7768445965447377e-13
Newton iteration 3energy = 0.2397861814000702<A u 3 , A u 3 >_{-1}^0.5 = 3.95762202557423e-16
t = 3.2000000000000015
Newton iteration 0energy = 0.23978529738916515<A u 0 , A u 0 >_{-1}^0.5 = 0.0013126010930648482
Newton iteration 1energy = 0.23978443598182622<A u 1 , A u 1 >_{-1}^0.5 = 7.810644621865449e-07
Newton iteration 2energy = 0.2397844359815212<A u 2 , A u 2 >_{-1}^0.5 = 3.3075993438410995e-13
Newton iteration 3energy = 0.2397844359815212<A u 3 , A u 3 >_{-1}^0.5 = 4.2814700263579335e-16
t = 3.3000000000000016
Newton iteration 0energy = 0.23978359647845926<A u 0 , A u 0 >_{-1}^0.5 = 0.0012792724425239811
Newton iteration 1energy = 0.23978277825700817<A u 1 , A u 1 >_{-1}^0.5 = 7.350530890736731e-07
Newton iteration 2energy = 0.2397827782567381<A u 2 , A u 2 >_{-1}^0.5 = 2.901885892642499e-13
Newton iteration 3energy = 0.23978277825673805<A u 3 , A u 3 >_{-1}^0.5 = 4.142414331682174e-16
t = 3.4000000000000017
Newton iteration 0energy = 0.23978198066655107<A u 0 , A u 0 >_{-1}^0.5 = 0.001247062821864828
Newton iteration 1energy = 0.23978120312604317<A u 1 , A u 1 >_{-1}^0.5 = 6.922905624823563e-07
Newton iteration 2energy = 0.23978120312580362<A u 2 , A u 2 >_{-1}^0.5 = 2.5506378441000593e-13
Newton iteration 3energy = 0.2397812031258036<A u 3 , A u 3 >_{-1}^0.5 = 4.0490307976335146e-16
t = 3.5000000000000018
Newton iteration 0energy = 0.23978044502934845<A u 0 , A u 0 >_{-1}^0.5 = 0.0012159226630448561
Newton iteration 1energy = 0.23977970583314206<A u 1 , A u 1 >_{-1}^0.5 = 6.525108847324154e-07
Newton iteration 2energy = 0.23977970583292924<A u 2 , A u 2 >_{-1}^0.5 = 2.2467289655797467e-13
Newton iteration 3energy = 0.23977970583292918<A u 3 , A u 3 >_{-1}^0.5 = 4.081509809497932e-16
t = 3.600000000000002
Newton iteration 0energy = 0.23977898497228517<A u 0 , A u 0 >_{-1}^0.5 = 0.0011858058930362873
Newton iteration 1energy = 0.23977828193819858<A u 1 , A u 1 >_{-1}^0.5 = 6.154736727297335e-07
Newton iteration 2energy = 0.2397782819380092<A u 2 , A u 2 >_{-1}^0.5 = 1.9819662278146196e-13
Newton iteration 3energy = 0.2397782819380092<A u 3 , A u 3 >_{-1}^0.5 = 4.2603397129405805e-16
t = 3.700000000000002
Newton iteration 0energy = 0.23977759620333955<A u 0 , A u 0 >_{-1}^0.5 = 0.0011566696317935063
Newton iteration 1energy = 0.23977692729118208<A u 1 , A u 1 >_{-1}^0.5 = 5.809611925812358e-07
Newton iteration 2energy = 0.2397769272910133<A u 2 , A u 2 >_{-1}^0.5 = 1.7522476903707717e-13
Newton iteration 3energy = 0.23977692729101338<A u 3 , A u 3 >_{-1}^0.5 = 4.047409254982618e-16
t = 3.800000000000002
Newton iteration 0energy = 0.23977627470871896<A u 0 , A u 0 >_{-1}^0.5 = 0.0011284739221579458
Newton iteration 1energy = 0.23977563800903529<A u 1 , A u 1 >_{-1}^0.5 = 5.487758354142881e-07
Newton iteration 2energy = 0.23977563800888466<A u 2 , A u 2 >_{-1}^0.5 = 1.5507997069304132e-13
Newton iteration 3energy = 0.23977563800888463<A u 3 , A u 3 >_{-1}^0.5 = 4.323166319223727e-16
t = 3.900000000000002
Newton iteration 0energy = 0.23977501673089996<A u 0 , A u 0 >_{-1}^0.5 = 0.001101181486078337
Newton iteration 1energy = 0.2397744104547916<A u 1 , A u 1 >_{-1}^0.5 = 5.18737945006562e-07
Newton iteration 2energy = 0.239774410454657<A u 2 , A u 2 >_{-1}^0.5 = 1.3758973718264115e-13
Newton iteration 3energy = 0.23977441045465703<A u 3 , A u 3 >_{-1}^0.5 = 4.0557757650951884e-16
t = 4.000000000000002
Newton iteration 0energy = 0.2397738187487627<A u 0 , A u 0 >_{-1}^0.5 = 0.0010747575032145978
Newton iteration 1energy = 0.23977324121866775<A u 1 , A u 1 >_{-1}^0.5 = 4.906839324936977e-07
Newton iteration 2energy = 0.23977324121854734<A u 2 , A u 2 >_{-1}^0.5 = 1.2219907041575643e-13
Newton iteration 3energy = 0.23977324121854734<A u 3 , A u 3 >_{-1}^0.5 = 3.840194540426063e-16
t = 4.100000000000001
Newton iteration 0energy = 0.23977267745958777<A u 0 , A u 0 >_{-1}^0.5 = 0.0010491694090460297
Newton iteration 1energy = 0.23977212710091578<A u 1 , A u 1 >_{-1}^0.5 = 4.644646276708653e-07
Newton iteration 2energy = 0.23977212710080797<A u 2 , A u 2 >_{-1}^0.5 = 1.0871003686839808e-13
Newton iteration 3energy = 0.23977212710080797<A u 3 , A u 3 >_{-1}^0.5 = 4.03574599200634e-16
t = 4.200000000000001
Newton iteration 0energy = 0.23977158976271712<A u 0 , A u 0 >_{-1}^0.5 = 0.001024386710281008
Newton iteration 1energy = 0.23977106509624788<A u 1 , A u 1 >_{-1}^0.5 = 4.399438299407741e-07
Newton iteration 2energy = 0.23977106509615115<A u 2 , A u 2 >_{-1}^0.5 = 9.690346949553966e-14
t = 4.300000000000001
Newton iteration 0energy = 0.23977055274470216<A u 0 , A u 0 >_{-1}^0.5 = 0.0010003808158116111
Newton iteration 1energy = 0.2397700523796682<A u 1 , A u 1 >_{-1}^0.5 = 4.1699702703592823e-07
Newton iteration 2energy = 0.23977005237958127<A u 2 , A u 2 >_{-1}^0.5 = 8.654960332599887e-14
t = 4.4
Newton iteration 0energy = 0.23976956366578206<A u 0 , A u 0 >_{-1}^0.5 = 0.0009771248817591054
Newton iteration 1energy = 0.2397690862935644<A u 1 , A u 1 >_{-1}^0.5 = 3.95510258344093e-07
Newton iteration 2energy = 0.2397690862934863<A u 2 , A u 2 >_{-1}^0.5 = 7.733972019754257e-14
t = 4.5
Newton iteration 0energy = 0.23976861994755688<A u 0 , A u 0 >_{-1}^0.5 = 0.0009545936693687135
Newton iteration 1energy = 0.23976816433593098<A u 1 , A u 1 >_{-1}^0.5 = 3.7537910338097784e-07
Newton iteration 2energy = 0.23976816433586054<A u 2 , A u 2 >_{-1}^0.5 = 6.931204261703242e-14
t = 4.6
Newton iteration 0energy = 0.23976771916173212<A u 0 , A u 0 >_{-1}^0.5 = 0.0009327634146649046
Newton iteration 1energy = 0.23976728414960677<A u 1 , A u 1 >_{-1}^0.5 = 3.565077784943188e-07
Newton iteration 2energy = 0.2397672841495433<A u 2 , A u 2 >_{-1}^0.5 = 6.220590664124038e-14
t = 4.699999999999999
Newton iteration 0energy = 0.23976685901982442<A u 0 , A u 0 >_{-1}^0.5 = 0.0009116117088952735
Newton iteration 1energy = 0.23976644351242662<A u 1 , A u 1 >_{-1}^0.5 = 3.3880832906193704e-07
Newton iteration 2energy = 0.23976644351236903<A u 2 , A u 2 >_{-1}^0.5 = 5.586233544733547e-14
t = 4.799999999999999
Newton iteration 0energy = 0.2397660373637348<A u 0 , A u 0 >_{-1}^0.5 = 0.0008911173888813228
Newton iteration 1energy = 0.23976564032819303<A u 1 , A u 1 >_{-1}^0.5 = 3.2219990500109006e-07
Newton iteration 2energy = 0.23976564032814107<A u 2 , A u 2 >_{-1}^0.5 = 5.0261538711424355e-14
t = 4.899999999999999
Newton iteration 0energy = 0.23976525215710134<A u 0 , A u 0 >_{-1}^0.5 = 0.0008712604364703288
Newton iteration 1energy = 0.23976487261839027<A u 1 , A u 1 >_{-1}^0.5 = 3.0660811024381e-07
Newton iteration 2energy = 0.23976487261834328<A u 2 , A u 2 >_{-1}^0.5 = 4.5328641659897583e-14
t = 4.999999999999998
Minimal energy extension (postscript in unit-2.1.3 )¶
\[u \in V^{ho,disc}, \quad u^{lo,cont} \in V^{lo,cont}, \quad \lambda \in V^{lo,disc},\]
[14]:
from netgen.geom2d import unit_square
mesh = Mesh(unit_square.GenerateMesh(maxh=0.1))
fes_ho = Discontinuous(H1(mesh, order=10))
fes_lo = H1(mesh, order=1, dirichlet=".*")
fes_lam = Discontinuous(H1(mesh, order=1))
fes = FESpace([fes_ho, fes_lo, fes_lam])
uho, ulo, lam = fes.TrialFunction()
\[\int_{\Omega} \frac12 \Vert \nabla u \Vert^2 - u + \sum_T \sum_{V \in V(T)} ((u-u^{lo})\cdot \lambda)|_{V} \longrightarrow \operatorname{min}!\]
[15]:
a = BilinearForm(fes)
a += Variation(0.5 * grad(uho)*grad(uho)*dx
- 1*uho*dx
+ (uho-ulo)*lam*dx(element_vb=BBND))
gfu = GridFunction(fes)
solvers.Newton(a=a, u=gfu)
Draw(gfu.components[0])
Newton iteration 0
err = 0.3929793997717295
Newton iteration 1
err = 6.621326119794413e-15
The minimization problem is solved by the solution of the PDE:
\[ \begin{align}\begin{aligned} \int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} 1 \cdot v \quad \forall ~ v \in V^{ho,disc}\\under the constraint\end{aligned}\end{align} \]
\[u(v) = u^{lo}(v) \quad \text{ for all vertices } v \in V(T) \text{ for all } T.\]